Problemar Statistica

Statistica Prof.univ.dr.Virgil VOINEAGU, Prof.univ.dr.Constatntin MITRUT, Conf.univ.dr.Emilia TITAN,

Lect.univ.dr.Simona GHITA, Lect.univ.drd.Mihaela VATUI

Cuprinsul cărţii:

LUCRARI PRACTICE Capitolul 1 CONCEPTE DE BAZA ALE STATISTICII Capitolul 2 PRELUCRAREA PRIMARA A DATELOR STATISTICE Capitolul 3 INDICATORI STATISTICI Capitolul 4 ANALIZA SERIILOR DE REPARTITIE UNIDIMENSIONALE Capitolul 5 SONDAJUL STATISTIC Capitolul 6 IPOTEZE STATISTICE. VERIFICAREA NORMALITATII REPARTITIILOR EMPIRICE

Capitolul 7 ANALIZA STATISTICA A DEPENDENTELOR DINTRE VARIABILE Capitolul 8 SERII CRONOLOGICE Capitolul 9 INDICII STATISTICI

Lucrari aplicative propuse spre rezolvare Anexe Bibliografie

Cuprinsul testelor:

Teste grila Solutii ale testelor grila Anexe Bibliografie

http://www.ase.ro/biblioteca/pagina.asp?id=C1









http://www.ase.ro/biblioteca/pagina.asp?id=Lucrari

http://www.ase.ro/biblioteca/pagina.asp?id=Anexe

http://www.ase.ro/biblioteca/pagina.asp?id=Bib

http://www.ase.ro/biblioteca/pagina.asp?id=Teste

http://www.ase.ro/biblioteca/pagina.asp?id=Solutii

http://www.ase.ro/biblioteca/pagina.asp?id=Anexe2

http://www.ase.ro/biblioteca/pagina.asp?id=bib2

11

ÍQWUHE UL

¾ Care este obiectul de studiu al statisticii? ¾ &DUH�VXQW�SDUWLFXODULW LOH�PHWRGHL�VWDWLVWLFLL" ¾ &DUH�VXQW�SULQFLSDOHOH�QR LXQL�úL�FRQFHSWH�IRORVLWH�vQ�OLPEDMXO�VWDWLVWLFLL" ¾ Clasificarea caracteristicilor statistice. ¾ &H�UHSUH]LQW �FROHFWLYLWDWHD�VWDWLVWLF " ¾ &DUH�VXQW�SULQFLSDOHOH�PHWRGH�GH�REVHUYDUH�VWDWLVWLF " ¾ &DUH�VXQW�HWDSHOH�SURLHFW ULL�XQHL�REVHUY UL�VWDWLVWLFH" ¾ &H�UHSUH]LQW �WLPSXO�úL�ORFXO�REVHUY ULL" ¾ Clasificarea erorilor de observare. ¾ Cum se poate realiza controlul datelor statistice?

&21&(37(�'(�%$= �$/(�67$7,67,&,,

([HUFL LL�üL�SUREOHPH

1. 3HQWUX�FDUDFWHUL]DUHD�XQHL�JUXSH�GH�VWXGHQ L�GLQ�DQXO�,�DO�)DFXOW LL�GH�&LEHUQHWLF ��6WDWLVWLF �úL�

,QIRUPDWLF �(FRQRPLF ��vQ�IXQF LH�GH�PHGLD�OD�DGPLWHUH��XQLWDWHD�GH�REVHrvare este: a) grupa; b) studentul; c) media la admitere.

5 VSXQV�� &ROHFWLYLWDWHD� VWDWLVWLF � HVWH� JUXSD� GH� VWXGHQ L�� XQLWDWHD� VWDWLVWLF este studentul, iar PHGLD�GH�DGPLWHUH�UHSUH]LQW �YDULDELOD�FHUFHWDW �

2. ,GHQWLILFD L�WUHL�WLSXUL�GLIHULWH�GH�GDWH�VWDWLVWLFH�FH pot fi colectate pentru a reflecta popularitatea

D�FLQFL�SXEOLFD LL�SHULRGLFH�VLPLODUH��3UHFL]D L�VFDOD�GH�P VXUDUH�D�ILHF UXL�WLS�GH�GDWH�

5 VSXQV� 1XP U�GH�DERQDPHQWH��VFDO �GH�UDSRUW� 1XP U�GH�H[HPSODUH�YkQGXWH��VFDO �GH�UDSRUW� 7LUDM��VFDO �GH�UDSRrt)

3. 3UHFL]D L�FDUH�GLQ�XUP WRDUHOH� DVRFLHUL� vQWUH�YDULDELOHOH�VWDWLVWLFH� úL� VFDOD�GH�P VXUDUH�QX�HVWH�

DGHY UDW � a) PDUFD�GH�FDOFXODWRU�FXPS UDW�FHO�PDL�GH�FXUkQG�GH�F WUH��GH�RDPHQL�GH�DIDFHUL�–�VFDO �QRPLQDO � b) VDODULXO�PHGLX�SH�UDPXUL�DOH�HFRQRPLHL�QD Lonale –�VFDO �GH�UDSRUW� c) OXQD�GLQ�DQXO��vQ�FDUH��GH�ILUPH�VHOHFWDWH�DOHDWRU�DX�UHDOL]DW�FHOH�PDL�PDUL�YkQ] UL�–

VFDO �QRPLQDO � d) SXQFWDMXO�RE LQXW�GH�F WUH��FRQFXUHQ L�OD�XQ�FRQFXUV�GH�FXOWXU �JHQHUDO �–�VFDO �QRPLQDO � e) afilierea la un partid politiF�D��GH�GLUHFWRUL�H[HFXWLYL�VHOHFWD L�DOHDWRU�–�VFDO �QRPLQDO �

5 VSXQV� c,d.

4. ,GHQWLILFD L� FDUH� GLQ� XUP WRDUHOH� GDWH� QX� VXQW� YDULDELOH� VWDWLVWLFH� SHQWUX� VWXGLXO� FROHFWLYLW LL�

VWXGHQ LORU�GH�OD�)DFXOWDWHD�GH�&LEHUQHWLF ��6WDWLVWLF �úL�,QIRUPDWLF �(FRQRPLF � a) grupa de studiu; b) PHGLD�RE LQXW �OD�FRQFXUVXO�GH�DGPLWHUH� c) QXP UXO�GH�DQL�GH�SUHJ WLUH�vQ�LQIRUPDWLF � d) facultatea de studiu; e) vârsta.

5 VSXQV� G��IDFXOWDWHD�GH�VWXGLX�UHSUH]LQW �R�FRQVWDQW �FH�GHILQHúWH�FROHFWLYLWDWHD�VWDWLVWLF �DQDOL]DW ��

5. Pentru�XUP WRDUHOH�FD]XUL��SUHFL]D L�XQLWDWHD�VWDWLVWLF ��LGHQWLILFD L�YDULDELOD�VWDWLVWLF �VWXGLDW �úL�

WLSXO�GH�YDULDELO ��3UHFL]D L�GDF �YDULDELOD�HVWH�FDQWLWDWLY �VDX�FDOLWDWLY ��GDF �HD�HVWH�FRQWLQX �

VDX�GLVFRQWLQX � a) 7LPSLL�GH�H[HFX LH��vQ�VHFXQGH��D��GH�programe BASIC; b) $EVHQWHLVPXO�DQJDMD LORU��]LOH�� c) 3URIHVLLOH�D��GH�VDODULD L� d) 1XP UXO�SHUVRQDOXOXL�GLQ��GH�vQWUHSULQGHUL� e) 1XP UXO�FRSLLORU�GLQ��GH�IDPLOLL�

5 VSXQV� a) XQLWDWHD�VWDWLVWLF : programul BASIC;

YDULDELOD�VWDWLVWLF ��WLPSXO�GH�H[HFX LH�– FDQWLWDWLY ��FRQWLQX � b) XQLWDWHD�VWDWLVWLF ��DQJDMDWXO�

YDULDELOD�VWDWLVWLF ��QXP UXO�GH�]LOH�DEVHQWDWH�–�QXPHULF ��GLVFUHW � c) XQLWDWHD�VWDWLVWLF ��VDODULDWXO�

YDULDELOD�VWDWLVWLF : profesia –�FDOLWDWLY � d) XQLWDWHD�VWDWLVWLF ��vQWUHSULQGHUHD�

variabila statisWLF ��QXP UXO�GH�SHUVRQDO�–�FDQWLWDWLY ��GLVFUHW � e) XQLWDWHD�VWDWLVWLF ��IDPLOLD�

�YDULDELOD�VWDWLVWLF ��QXP UXO�GH�FRSLL�–�FDQWLWDWLY ��GLVFUHW �

6. &ODVLILFD L� XUP WRDUHOH� H[HPSOH� GH� GDWH� VWDWLVWLFH� vQ� QXPHULFH� �FDQWLWDWLYH�� úL� QHQXPHULFH�

�FDOLWDWLYH��$U WD L�VFDOD�GH�P VXUDUH� a) UDWD�OXQDU �D�úRPDMXOXL�vQ�MXGH XO�$��vQ�DQXO�� b) FLIUD�GH�DIDFHUL�UHDOL]DW �GH�ILUPHOH�XQXL�RUDú��vQ�SULPXO�WULPHVWUX�DO�DQXOXL�� c) DILOLHUHD�OD�XQ�SDUWLG�SROLWLF�D��GH�GLUHFWRUL�H[HFXWLYL�VHOHFWD L�vQWkPSO WRU��ILHFDUH�GLUHFtor

H[HFXWLY�SRDWH�DYHD�R�VLQJXU �DILOLHUH�OD�XQ�SDUWLG�SROLWLF� d) cursul valutar zilnic oficial al leului în raport cu U.S.D; e) WLSXO�GH�XOHL� GH�PRWRU� FXPS UDW� FHO�PDL� UHFHQW�GH� ��GH�SRVHVRUL� GH�DXWRPRELOH� VHOHFWD L�

vQWkPSO WRU��ILHFDUH�SRVHVRU�SRDWH�SUHFL]D�R�VLQJXU �FDWHJRULH� f) QXP UXO�SHUVRDQHORU�GH�VH[�IHPLQLQ�FDUH�vQGHSOLQHVF�IXQF LD�GH�GLUHFWRU�HFRQRPLF�SH�MXGH H� g) VDODULXO�PHGLX�SH�UDPXUL�DOH�HFRQRPLHL�QD LRQDOH��PDUWLH�� h) MXGH HOH�vQ�FDUH�FLQFL�FRWLGLHQH�DX�UHDOL]DW�FHOH�PDL�PXOWH�DERQDPHQWH�vQ�DQXO�1997; i) QXP UXO�GH�RUH�GH�SURJUDP�VSRUWLY�WUDQVPLVH�GH��WHOHYL]LXQL�SULQ�FDEOX� j) procentul mediu dintr-R�]L�GH�OXFUX��FRQVXPDW�vQ�úHGLQ H�GH�F WUH��GH�GLUHFWRUL� k) PDUFD�GH�FDOFXODWRU�FXPS UDW�FHO�PDL�GH�FXUkQG�GH��GH�SHUVRDQH� l) SUH XO�FDOFXODWRUXOXL�FXPS rat cel mai de curând de 20 de oameni de afaceri; m) OXQD�GLQ�DQXO��vQ�FDUH��GH�DJHQ L�HFRQRPLFL�VHOHFWD L�vQWkPSO WRU�DX�UHDOL]DW�FHOH�PDL�

PDUL�YkQ] UL� n) FXORDUHD�GH�WDSHW��DOWD�GHFkW�DOE ��OD�FDUH�FLQFL�GLQ�FHL�PDL�LPSRUWDQ L�LPSRUWDWRUL�GH�WDSHWH�

au rHDOL]DW�FHOH�PDL�PDUL�YHQLWXUL�GLQ�YkQ] UL��SHQWUX�ILUPD� o) LQGLFHOH�SUH XULORU�GH�FRQVXP�vQ�XOWLPHOH��OXQL� p) FKHOWXLHOLOH�FX�UHFODPD�I FXWH�GH��GH�ILUPH�VHOHFWDWH�DOHDWRU� q) WLSXO�GH�UHFODP �SXEOLFLWDU �SUHIHUDW �GH��GH�ILUPH�VHOHFWDWH�DOHDWRU� r) marca de aXWRPRELOH�DPHULFDQH�SH�FDUH�R�LQGLF ��GH�PHFDQLFL�DXWR�FD�SURGXFkQG�FHOH�PDL�

FUHGLELOH�PDúLQL� 5 VSXQV�

a) FDQWLWDWLY ��VFDO �GH�UDSRUW� b) FDQWLWDWLY ��VFDO �GH�UDSRUW� c) FDOLWDWLY ��VFDO �QRPLQDO � d) FDQWLWDWLY ��VFDO �GH�UDSRUW� e) FDOLWDWLY ��VFDO �QRPLQDO � f) caQWLWDWLY ��VFDO �GH�UDSRUW� g) FDQWLWDWLY ��VFDO �GH�UDSRUW� h) FDOLWDWLY ��VFDO �QRPLQDO � i) FDQWLWDWLY ��VFDO �GH�UDSRUW� j) FDQWLWDWLY ��VFDO �GH�UDSRUW� k) FDOLWDWLY ��VFDO �QRPLQDO � l) FDQWLWDWLY ��VFDO �GH�UDSRUW�

m) FDOLWDWLY ��VFDO �RUGLQDO � n) FDOLWDWLY ��VFDOD�QRPLQDO ��HYHQWXDO�RUGLQDO ��GDF �VH�LQWURGXFH�FULWHULXO�LQWHQVLW LL�FXORULL�� o) FDQWLWDWLY ��VFDO �GH�UDSRUW� p) FDQWLWDWLY ��VFDO �GH�UDSRUW� q) FDOLWDWLY ��VFDO �QRPLQDO � r) FDOLWDWLY ��VFDO �QRPLQDO �

7. 3UHVXSXQH L�F �OXFUD L�SHQWUX�R�ILUP �GH�VRQGDUH�D�RSLQLHL�SXEOLFH�úL�GRUL L�V �HVWLPD L�SURSRU LD�

FHW HQLORU� FDUH�� vQ� HYHQWXDOLWDWHD� RUJDQL] ULL� GH� DOHJHUL� DVW ]L�� DU� YRWD� FX� SDUWLGXO� GH�

JXYHUQ PkQW�� 'HILQL L� SRSXOD LD� VWDWLVWLF � SH� FDUH� R� HúDQWLRQD L�� 'DU� GDF � Y-DU� LQWHUHVD� V �HVWLPD L�SURSRU LD�FHW HQLORU�FDUH��OD�YLLWRDUHOH�DOHJHUL��DU�YRWD�FX�SDUWLGXO�GH�JXYHUQ PkQW��FDUH�

DU�IL�SRSXOD LD�HúDQWLRQDW " 5 VSXQV�� în primul caz:�SRSXOD LD�FX�GUHSW�GH�YRW�DVW ]L� în al doilea caz:�SRSXOD LD�FX�GUHSW�GH�YRW�DVW ]L��SUHFXP�úL�FHL�FDUH�YRU�DYHD�GUHSW�GH�YRW�

la viitoareOH�DOHJHUL��GDU�FDUH�QX�DX�DVW ]L�

8. 2� FRPSDQLH� GH� DVLJXU UL� GRUHúWH� V � GHWHUPLQH� SURSRU LD� PHGLFLORU� FDUH� DX� IRVW� LPSOLFD L� vQ�

XOWLPXO� DQ� vQ� XQD� VDX� PDL� PXOWH� DF LXQL� MXGLFLDUH� GH� UHOH� SUDFWLFL�� &RPSDQLD� VHOHFWHD] �

vQWkPSO WRU� �� GH�PHGLFL� FDUH� DX� SUDFWLFDW� vQ� XOWLPXO� DQ� úL� GHWHUPLQ �SURSRU LD�� ,GHQWLILFD L�

SRSXOD LD�GH�LQWHUHV��HúDQWLRQXO�úL�LQIHUHQ D�VWDWLVWLF �GRULW � 5 VSXQV� SRSXOD LD – medicii care au practicat în ultimul an; HúDQWLRQXO – cei 500 de medici care

DX� SUDFWLFDW� vQ� XOWLPXO� DQ�� VHOHFWD i; LQIHUHQ D�� SURSRU LD� FHORU� LPSOLFD L� vQ� DF LXQL�

judiciare de rele practici.

9. 2� FRPSDQLH� GH� SURGXVH� DOLPHQWDUH� GRUHúWH� V � FRPHUFLDOL]H]H� XQ� QRX� SURGXV� GH� VQDFN-food.

3HQWUX�D�YHGHD�FXP�UHDF LRQHD] �FXPS U WRULL�OD�DFHVW�SURGXV��FRPSDQLD�RUJDQL]HD] �R�WHstare a JXVWXULORU�� SHQWUX� �� GH� FXPS U WRUL� VHOHFWD L� vQWkPSO WRU� OD� XQ� PDJD]LQ� VXEXUEDQ��

&XPS U WRULL�VXQW�UXJD L�V �JXVWH�SURGXVXO�úL�DSRL�V �FRPSOHWH]H�XQ�FKHVWLRQDU�FX�XUP WRDUHOH�

vQWUHE UL� a) &DUH�HVWH�YkUVWD�GXPQHDYRDVWU " b) 6XQWH L�SHUVRDQD�FDUH�IDFH�GH�RELFHL�FXPS U WXUL�SHQWUX�IDPLOLD�GY�" c) Câte persoane sunt în familia dv.? d) &XP�QRWD L��SH�R�VFDO �GH�OD��OD��JXVWXO�SURGXVXOXL��GDF ��HVWH�FHO�PDL�SX LQ�JXVWRV" e) 9H L�FXPS UD�DFHVW�SURGXV�GDF �YD�IL�GLVSRQLELO�vQ�PDJD]LQH" f) 'DF �U VSXQVXO�OD�H��HVWH�³'D´��FkW�GH�GHV�YH L�FXPS UD�SURGXVXO"

&ODVLILFD L�GDWHOH�RIHULWH�GH�U VSXQVXUL�vQ�FDQWLWDWLYH�úL�FDOLWDWLYH�úL�LQGLFD L�VFDOD�GH�P VXUDUH�

pentru fiecare dintre ele. 5 VSXQV� a) FDQWLWDWLY �–�VFDO �GH�UDSRUW� b) FDOLWDWLY �–�VFDO �QRPLQDO � c) cantitatiY �–�VFDO �GH�UDSRUW� d) FDOLWDWLY �–�VFDO �RUGLQDO � e) FDOLWDWLY �–�VFDO �QRPLQDO � f) FDOLWDWLY �–�VFDO �RUGLQDO ��GDF �U VSXQVXO�HVWH�GH�WLSXO��UDU��GHV��IRDUWH�GHV�HWF��

10. 8Q�FHUFHW WRU�HVWH�LQWHUHVDW�V �FRPSDUH�VDODULXO�GH�vQFDGUDUH�SHQWUX�E UED LL�úL�IHPHLOH�FDUH�LQWU �

vQ�VHUYLFLX�LPHGLDW�GXS �DEVROYLUHD�IDFXOW LL��6XQW�FHUFHWD L��GH�E UED L�úL��GH�IHPHL� a) 'HVFULH L�SRSXOD LD� b) 'HVFULH L�HúDQWLRQXO� c) 'HVFULH L�LQIHUHQ D�FDUH�LQWHUHVHD] �

5 VSXQV�

a) GRX � SRSXOD LL – cea a femeilor care s-au încDGUDW� SH� XQ� SRVW� LPHGLDW� GXS � DEVROYLUHD�

IDFXOW LL�úL�FHD�D�E UED LORU�FDUH�V-DX�vQFDGUDW�LPHGLDW�GXS �DEVROYLUHD�IDFXOW LL� d) HúDQWLRDQHOH –�FHL��GH�E UED L�úL��GH�IHPHL� e) LQIHUHQ HOH –�VDODULLOH�PHGLL�GH�vQFDGUDUH�SHQWUX�E UED L�úL�SHQWUX�IHPHL�

Probleme propuse

1. 5 VSXQGH L�OD�XUP WRDUHOH�vQWUHE UL�

a) (OHPHQWHOH�DVXSUD�F URUD�VH�HIHFWXHD] �XQ�VWXGLX�VWDWLVWLF�VXQW�GHQXPLWH�³YDULDELOH´" b) 3DUWLFXODULW LOH��vQVXúLULOH�SUH]HQWDWH�GH�R�XQLWDWH�VWDWLVWLF �VXQW�QXPLWH�FDUDFWHULVWLFL" c) 2�VLQJXU �XQLWDWH�VWDWLVWLF �SRDWH�DYHD�PDL�PXOWH�FDUDFWHULVWLFL" d) 2�FDUDFWHULVWLF �PDL�HVWH�FXQRVFXW �úL�VXE�GHQXPLUHD�GH�� e) &XP�VH�LGHQWLILF �IUHFYHQ D�XQHL�YDORUL�LQGLYLGXDOH" f) &DUH�HVWH�GHRVHELUHD�vQWUH�YDULDQW �úL�YDULDELO �VWDWLVWLF "

2. &ROHFWLYLWDWHD�VWDWLVWLF �UHSUH]LQW :

a) P ULPL�FRQFUHWH��RE LQXWH�vQ�HWDSH�GLIHULWH�GH�REVHUYDUH�VDX�GLQ�FDOFXOH� b) HOHPHQWXO�FRQVWLWXWLY�DO�SRSXOD LHL�VWDWLVWLFH�SHQWUX�FDUH�VH�FXOHJ�GDWH� c) WRWDOLWDWHD�HOHPHQWHORU�GH�DFHHDúL�QDWXU �VXSXVH�VWXGLXOXL�VWDWLVWLF� d) H[SUHVLD� QXPHULF � D� XQHL� GHWHUPLQDUL� FDOLWDWLYH� RELHFWLYH�� REWLQX � vQ� XUPD� XQHL� FHUFHW UL�

statistice; e) SURSULHWDWHD�FRPXQ � WXWXURU�XQLW LORU�XQHL�SRSXOD LL� VWDWLVWLFH�úL� FDUH� DUH� vQVXúLUHD�GH�D-úL�

modifica nivelul de la o unitate la alta.

3. 'HRVHELUHD�vQWUH�YDULDQW �úL�YDULDELO �VWDWLVWLFD�HVWH�XUP WRDUHD�

a) YDULDELOD�HVWH�HOHPHQWXO�DVXSUD�F UXLD�VH�HIHFWXHD] �XQ�VWXGLX�VWDWLVWLF�� vQ� WLPS�FH�YDULDQWD�

este o însusire, o proprietate a variabilei. b) YDULDQWD�HVWH�IRUPD�FRQFUHW �GH�PDQLIHVWDUH�D�YDUDLELOHL��QLYHOXO�HL�� c) QX�H[LVW �QLFL�R�GHRVHELUH� d) o�XQLWDWH�VWDWLVWLF �SRDWH�DYHD�PDL�PXOWH�YDULDQWH��GDU�QXPDL�R�VLQJXU �YDULDELO � e) WRWDOLWDWHD� YDULDQWHORU� úL� D� YDULDELOHORU� VWDWLVWLFH� IRUPHD] � FROHFWLYLWDWHD� VWDWLVWLF �� RELHFWXO�

demersului statistic.

4. ([SOLFD L�úL�MXVWLILFD L�GDF �YDULDELOD�³SRSXOD LD� ULORU�OD��LXQLH��´�HVWH�R�FDUDFWHULVWLF �GH�

tip continuu sau discret.

5. $U WD L� FH� WLS� GH� YDULDELO � úL� SH� FH� VFDO � GH� P VXUDUH� VH� P VRDU � FDUDFWHULVWLFD� ³SURILWXO�

întreprinderilor”.

6. /HJ WXULOH�GLQWUH�IHQRPHQHOH�HFRQRPLFR-VRFLDOH�GH�PDV �VXQW��

a) leg WXUL�IXQF LRQDOH�� b) OHJ WXUL�FDUH�VH�VXSXQ�DF LXQLORU�OHJLORU�VWRFKDVWLFH�� c) OHJ WXUL�FH�SRW�IL�SXVH�vQ�HYLGHQ ��OD�QLYHOXO�ILHF UHL�XQLW L�vQ�SDUWH�� d) OHJ WXUL�XQLYRF�GHWHUPLQDWH�� e) OHJ WXUL�FH�SRW�IL�SXVH�vQ�HYLGHQ �OD�QLYHOXO�DQVDPEOXOXL�

7. 5HFHQV PkQWXO��FD�PHWRG �GH�REVHUYDUH�VWDWLVWLF �

a) QX�SUHVXSXQH�FXOHJHUHD�GDWHORU�GH�OD�WRDWH�XQLW LOH�SRSXOD LHL�VWDWLVWLFH�ELQH�GHWHUPLQDWH� b) are exclusiv un caracter demografic; c) VH�vQFDGUHD] �vQ�VIHUD�REVHUY ULORU�FX�FDUDFWHU�SHUPDQHQW� d) VH�RUJDQL]HD] �FX�R�DQXPLW �Seriodicitate; e) HVWH�R�REVHUYDUH�WRWDO �

22

ÍQWUHE UL ¾ &H�UHSUH]LQW �FODVLILFDUHD�VWDWLVWLF " ¾ Care sunt etapele procesului de grupare a datelor pe intervale egale? ¾ &DUH�VXQW�HOHPHQWHOH�GH�FRQ LQXW�úL�GH�IRUP �DOH�XQXL�WDEHO�VWDWLVWLF" ¾ &H�HVWH�VHULD�VWDWLVWLF " ¾ Care sunt principalele rHJXOL�GH�UHSUH]HQWDUH�JUDILF �D�GDWHORU�VWDWLVWLFH" ¾ Care sunt principalele tipuri de grafice statistice? ¾ &XP�VH�UHSUH]LQW �JUDILF�R�VHULH�FURQRORJLF "�'DU�XQD�WHULWRULDO "

35(/8&5$5($�35,0$5 �$�'$7(/25�67$7,67,&(


1. *UXSDUHD�MXGH HORU�5RPkQLHL�GXS �QXP UXO�WRWDO�DO�DQJDMD ilor cu studii superioare din ultimii

10 ani este o: a) JUXSDUH�WHULWRULDO � b) JUXSDUH�FURQRORJLF � c) JUXSDUH�GXS �R�FDUDFWHULVWLF �QXPHULF � d) JUXSDUH�GXS �R�FDUDFWHULVWLF �GH�WLS�FDOLWDWLY� e) JUXSDUH�FRPELQDW �

5 VSXQV��F��&DUDFWHULVWLFD�HVWH�³QXP UXO�WRWDO�DO�DQJDMD LORU�FX�VWXGLL�VXSHULRDUH�GLQ�XOWLPLL��DQL´�

2. 3UHVXSXQHP�F �R�ILUP �FH�SURGXFH�WUHL�SURGXVH�VLPLODUH��GRUHúWH�V �FRPSDUH��SULQ�LQWHUPHGLXO�

UHSUH]HQW ULORU� JUDILFH�� SRQGHUHD� YkQ] ULORU� SH� SURGXVH� vQ� WRWDOXO� YkQ] ULORU�� vQ� GRL� DQL�

consecutivi. ReprezeQWDUHD�JUDILF �YD�IL� a) KLVWRJUDPD�IUHFYHQ HORU�UHODWLYH� b) GLDJUDPD�GH�VWUXFWXU � c) cronograma; d) corelograma; e) GLDJUDPD�SULQ�VXSUDIH H�

5 VSXQV��E��6H�FRQVWUXLHúWH�FkWH�R�GLDJUDP �GH�VWUXFWXU �SHQWUX�ILHFDUH�DQ�

3. ([SOLFD L�FDUH�GLQ�XUP WRDUHOH�FRPHQWDULL�DS UXWH�vQ�GRX �]LDUH�GHVSUH�JUDILFXO�DO WXUDW��HVWH�FRUHFW�

Fig. 2.1.

a) Anul 1996 este cel de-DO�GRLOHD�DQ�vQ�FDUH�SURGXF LD�D�FRQWLQXDW�V �VFDG ��6LWXD LD�HVWH�UHD�� b) Anul 1996 este cel de-DO�WUHLOHD�DQ�FkQG�SURGXF LD�VH�VLWXHD] �OD�FHOH�PDL�vQDOWH�FRWH��6LWXD LD�

HVWH�IRDUWH�EXQ �� 5 VSXQV��1LFL�XQ�FRPHQWDULX�QX�HVWH�SH�GHSOLQ�FRUHFW�� vQWUXFkW�HVWH�DGHY UDW�F �SURGXF LD�HVWH� vQ�

XúRDU �VF GHUH� vQ�DQXO��GDU� WUHEXLH� LQXW� VHDPD�úL�GH�QLYHOXO� ULGLFDW� OD�FDUH�VH�DIO ��SH�

ansamblul perioadei remarcându-VH� WRWXúL� R� WHQGLQ � FUHVF WRDUH� � D� SURGXF LHL�� ÌQ� DQDOL] �

trebuie luat întregul orizont de timp.

��

!� "� #� $% &� '� (� )� *� !� �

!�*�*� %+,!�*�*�!-+,!�*�*-"�+.!�*�*�#%+,!�*�*�$�+,!�*�*�&%+,!�*�*-'�+/1032

4 5678 96:

4. 8Q�PDJD]LQ�GH�DUWLFROH�GH�vPEU F PLQWH�SHQWUX�IHPHL��VLWXDW�vQ�FHQWUXO�XQXL�RUDú�PDUH��GRUHúWH�

V -úL� GHVFKLG � R� ILOLDO � vQWU-R� ]RQ � SHULIHULF �� 3HQWUX� D� RE LQH� LQIRUPD LL� GHVSUH� GLVWULEX LD�

JHRJUDILF � D� FOLHQ LORU�SUH]HQ L��PDQDJHUXO�PDJD]LQXOXL�RUJDQL]HD] �R�DQFKHW � vQ� FDUH�ILHFDUH�

client dintr-R�]L�HVWH� vQWUHEDW�XQGH� ORFXLHúWH�� UDSRUWDW� OD�FHOH�SDWUX�FDGUDQH�DOH�RUDúXOXL��QRUG-vest (NV), nord-est (NE), sud-YHVW��69��úL�VXG-HVW��6(��&OLHQ LL�GLQ�DIDUD�RUDúXOXL�VXQW�H[FOXúL�GLQ�DQFKHW ��5H]XOWDWXO�DQFKHWHL�VH�SUH]LQW �DVWIHO�

Tabelul 2.1.

Clientul =RQD�GH�UHúHGLQ Clientul =RQD�GH�UHúHGLQ

1 NV 16 NE

2 SE 17 NE

3 SE 18 NV

4 NV 19 NV

5 SV 20 SV

6 NV 21 NE

7 NE 22 NV

8 SV 23 SV

9 NV 24 SE

10 SE 25 SV

11 NV 26 NV

12 SE 27 NV

13 SV 28 SE

14 NV 29 NE

15 SV 30 SV

Se cere:

a) 6LVWHPDWL]D L�GDWHOH�GLVSRQLELOH� b) 3UH]HQWD L�JUDILF�LQIRUPD LLOH�VLVWHPDWL]DWH�

6ROX LH�

D��6H�HIHFWXHD] �FODVLILFDrea datelor: 9DULDELOD� GXS � FDUH� VH� HIHFWXHD] � FODVLILFDUHD�HVWH�]RQD� GH� UHúHGLQ ��$FHDVW �YDULDELO � DUH�SDWUX�

YDULDQWH��GHFL�VH�SRDWH�UHDOL]D�R�JUXSDUH�FKLDU�SH�DFHVWH�YDULDQWH��6H�RE LQH�WDEHOXO� Tabelul 2.2.

*UXSH�GH�FOLHQ L�GXS �]RQD�GH�UHúHGLQ 1XP U�GH�FOLHQ L��ni) NE 5 NV 11 SE 6 SV 8

Total 30

b)

Fig. 2.2 'HFL�HVWH�PDL�DYDQWDMRV�FD�PDQDJHUXO�PDJD]LQXOXL�V �GHVFKLG �QRXD�ILOLDO �vQ�]RQD�GH�QRUG-

YHVW�D�RUDúXOXL��GH�XQGH�SURYLQ�FHL�PDL�PXO L�GLQWUH�FOLHQ LL�V L�

5. Un analist fiQDQFLDU�HVWH� LQWHUHVDW� vQ�FXQRDúWHUHD�UHVXUVHORU�FKHOWXLWH�GH�FRPSDQLLOH�GH�KDUG�úL�

soft pentru cercetare-GH]YROWDUH�� 3HQWUX� �� GH� DVWIHO� GH� ILUPH� DOHVH� vQWkPSO WRU�� FDOFXOHD] �

procentul cheltuielilor pentru cercetare-dezvoltare din totalul veniturilor din ultimul an. Datele UH]XOWDWH�VXQW�XUP WRDUHOH��

Tabelul 2.3. Compania Procentul din venituri

cheltuit pentru C-D (%) Compania Procentul din venituri

cheltuit pentru C-D (%) 1 10,5 26 9,5 2 8,4 27 8,2 3 10,5 28 6,9 4 9,0 29 7,2 5 9,2 30 8,2 6 9,7 31 9.6 7 6,6 32 7,2 8 10,6 33 8,8 9 10,1 34 11,3 10 7,1 35 8,5 11 8,0 36 9,4 12 7,9 37 10,5 13 6,8 38 6,9 14 9,5 39 6,5 15 8,1 40 7,5 16 11,5 41 7,1 17 9,9 42 5,2 18 6,9 43 7,7 19 7,5 44 5,9 20 11,1 45 5,2 21 8,2 46 5,6 22 8,0 47 11,7 23 7,7 48 6,0 24 7,4 49 7,8 25 6,5 50 6,5

;=< >�?A@A< B�C < DFE�G < H�I < G J @LKLC�M N�JLI%DFK�H�@ H�O�H%KL< I

PQRSTUVPU�Q

WLX W�Y Z X Z�Y[�\�]_â`�bdc b�e_b�`�f ]

g h�ijkl mnl

Se cere: a) 6 �VH�VLVWHPDWL]H]H�GDWHOH��JUXSkQGX-VH�SH�LQWHUYDOH�HJDOH�GH�YDULD LH� b) 6 �VH�FDOFXOH]H�IUHFYHQ HOH�FXPXODWH� c) 6 � VH� UHSUH]LQWH� JUDILF� GLVWULEX LD� FRPSDQLLORU� GXS � SURFHQWXO� GLQ� YHQLWXUL� DORFDW� SHQWUX�

cercetare-GH]YROWDUH�� 6 � VH� ILJXUH]H� vQ� FDGUXO� XQXL� JUDIic curbele cumulative ale IUHFYHQ HORU�

6ROX LH�

a) 6H�QRWHD] �FX�;�– caracteristica de grupare (procentul din venituri alocat pentru cercetare). 6H�SDUFXUJ�XUP WRULL�SDúL�

� VH�FDOFXOHD] �DPSOLWXGLQHD�YDULD LHL�FDUDFWHULVWLFLL��Ax) Ax = xmax - xmin = 11,7 – 5,2 = 6,5 %

� VH�VWDELOHúWH�QXP UXO�GH�JUXSH��r) r = 1 + 3,322 lg n = 1 + 3,322 lg 50 = 6,64 ≈ 7. (formula lui Sturges)

� VH�VWDELOHúWH�P ULPHD�LQWHUYDOXOXL�GH�JUXSDUH��h) h = Ax / r = 6,5 / 7 ≈ 1 %

� VH�VWDELOHVF�LQWHUYDOHOH�GH�YDULD LH�úL�VH�HIHFWXHD] �JUXSarea.

&X�YDULD LH�FRQWLQX

Varianta I Varianta II Tabelul 2.4 Tabelul 2.5

,QWHUYDOH�GH�YDULD LH�D�

procentului din venituri cheltuit pentru C-D

Nr. companii (ni)


procentului din venituri cheltuit pentru C-D

Nr. companii (ni)

5,0 – 6,0 4 5,0 – 6,0 5 6,0 – 7,0 9 6,0 – 7,0 8 7,0 – 8,0 11 7,0 – 8,0 13 8,0 – 9,0 9 8,0 – 9,0 8

9,0 – 10,0 8 9,0 – 10,0 7 10,0 – 11,0 5 10,0 – 11,0 5 11,0 - 12,0 4 11,0 - 12,0 4

Total 50 Total 50 o1p-q r�s t uvt q wxt y�z|{_}At p�w_} t y�~�s �� y

interval. o�p-q r�s t u�t q wa��{_}�t p%w_} t y�~�s �� y�t y�q {_}|��w_s �

&X�YDULD LH�GLVFRQWLQX Tabelul 2.6

,QWHUYDOH�GH�YDULD LH�D�SURFHQWXOXL�GLQ�YHQLWXUL�FKHOWXLW�

pentru C-D Nr. companii

(ni) 5,0 – 5,9 4 6,0 – 6,9 9 7,0 – 7,9 11 8,0 – 8,9 9 9,0 – 9,9 8

10,0 – 10,9 5 11,0 – 11,9 4

Total 50

0 ULPHD�LQWHUYDOXOXL��vQ�FD]XO�YDULD LHL�GLVFUHWH��VH�FDOFXOHD] �DVWIHO� � hi = limsup i – liminf i + ∆, unde ∆�UHSUH]LQW �YDORDUHD�FX�FDUH�VH�GLVFUHWL]HD] �LQWHUYDOHOH� � ³K´�VH�PDL�SRDWH�FDOFXOD�FD�GLIHUHQ �vQWUH�GRX �OLPLWH�VXSHULRDUH�DO WXUDWH��VDX�GRX �OLPLWH�

LQIHULRDUH�DO WXUDWH� hi=limsup i+1 – lim sup i

hi=liminf i+1 – lim inf i

b) Pentru Varianta I:

Tabelul 2.7. FrecvHQ H�FXPXODWH ,QWHUYDOH�GH�YDULD LH�D�SURFHQWXOXL�GLQ

venituri cheltuit pentru C-D Nr. companii

(ni) &UHVF WRU 'HVFUHVF WRU 5,0 – 6,0 4 4 50 6,0 – 7,0 9 13 46 7,0 – 8,0 11 24 37 8,0 – 9,0 9 33 26

9,0 – 10,0 8 41 17 10,0 – 11,0 5 46 9 11,0 – 12,0 4 50 4

Total 50 - -

c)

Fig. 2.3.

Fig. 2.4.

�� 3�� a� � ��3�x��a� � � �d�d�d�a� �a�V�a��3��a��d� �� a� ��a��=�� a� ��=��3��V��

�� ¡¢£-�£��

¤�¥ ��¦¡�¥ �

¡�¥ ��¦§�¥ �

§�¥ �%¦¢�¥ �

¢�¥ ��¦¨�¥ �

¨�¥ ��¦£-��¥ �

£-��¥ ��¦£%£�¥ �

£%£�¥ ��¦£��¥ �

©«ª-¬ -®�¯%° ®�¯-¬ ±%° ¯

² ³�´µ¶·¸¹º»»

¼¾½x¿�À=Á=Â ÁÄÃ�¿�Á3Å Æ=Á=Ç Á=Â ÈÉ¿dÅ=½�Ê�½xÂ Ë�Ì�Á

ÍÎ�ÍÏ�ÍÐ�ÍÑ%ÍÒ�ÍÓ�Í

Ò%Ô ÒÕÓ�Ô Ò×Ö�Ô ÒÙØ�Ô ÒÙÚ%Ô ÒÛÎ�Í%Ô ÒÜÎ�Î-Ô ÒÝ-Þ ß�à�á|â�ã�ä_å â�æ�á|â�å ã ç�Þ å�è�ã é�êLë ì

í îïð ñ òð óô óõ ö÷ ï

ø�ù ú�û�üLý û�þ�ÿ þ��ý û�ù ú��û�ý� ù ú�û-üLý û�þ�ÿ þ�� ý� ú��û�ù ú��%û�ý

6. Un fabrLFDQW� GH� UR L� LQGXVWULDOH�E QXLHúWH� F � XQHOH� FRPHQ]L� SURILWDELOH� VXQW�SLHUGXWH�

GLQ�FDX]D�WLPSXOXL�SUHD�PDUH��QHFHVDU�ILUPHL�SHQWUX�D�VWDELOL�SUH XO�FHUXW�SRWHQ LDOLORU�

FOLHQ L�� 3HQWUX� D� LQYHVWLJD� DFHDVW � SRVLELOLWDWH�� GH� FHUHUL� GH� IL[DUH� D� SUH XOXL� DX�

fRVW� DOHVH� DOHDWRU� GLQ� WRWDOXO� XOWLPXOXL� DQ� úL� DX� IRVW� FDOFXOD L� WLPSLL� QHFHVDUL� SHQWUX�IL[DUHD�SUH XOXL��7LPSLL��vQ�]LOH��vPSUHXQ �FX�FODVLILFDUHD�FRPHQ]LL�vQ�³SLHUGXW ´�VDX�

nu, sunt: Tabelul 2.8

Nr. comenzii

Timpul pt. calculul SUH XOXL��]LOH�

&RPDQG �

pLHUGXW Nr.

comenzii Timpul pt. calculul SUH XOXL��]LOH�

&RPDQG �

SLHUGXW 1 6,4 Nu 21 3,3 Nu 2 5,7 Nu 22 6,0 Nu 3 6,6 Nu 23 5,9 Nu 4 10,0 Da 24 7,3 Da 5 4,9 Nu 25 4,0 Nu 6 6,5 Nu 26 11,4 Da 7 6,0 Nu 27 3,2 Nu 8 4,7 Nu 28 6,4 Nu 9 3,9 Nu 29 5,1 Nu

10 6,7 Da 30 7,8 Da 11 3,9 Nu 31 4,4 Nu 12 6,5 Nu 32 5,5 Nu 13 8,6 Nu 33 7,5 Nu 14 6,4 Nu 34 6,2 Nu 15 9,5 Da 35 8,2 Da 16 4,9 Nu 36 4,4 Nu 17 3,9 Nu 37 5,9 Nu 18 7,7 Da 38 9,9 Da 19 10,3 Da 39 12,2 Da 20 5,5 Nu 40 9,0 Nu

6 �VH�VLVWHPDWL]H]H�GDWHOH�SUH]HQWDWH�úL�E��V �VH�UHSUH]LQWH�JUDILF�GLVWULEX LLOH� 6ROX LH�

1RW P�;�–�WLPSXO�QHFHVDU�VWDELOLULL�SUH XOXL��6H�SDUFXUJ�XUP WRDUHOH�HWDSH�vQ�YHGHUHD�HIHFWX ULL�

JUXS ULL�SH�LQWHUYDOH�GH�YDULD LH� � VH�FDOFXOHD] �DPSOLWXGLQHD�YDULD LHL�FDUDFWHULVWLFLL��Ax)

Ax = xmax - xmin = 12,2 – 3,2 = 9 zile. � VH�VWDELOHúWH�QXP UXO�GH�JUXSH��r)

r = 1 + 3,322 lg n = 1 + 3,322 lg 40 = 1 + 3,322 × 1,60206 = 6,32 ≈ 6 (formula lui Sturges)

� VH�VWDELOHúWH�P ULPHD�LQWHUYDOXOXL�GH�JUXSDUH��h) h = Ax / r = 9 / 6 ≈ 1,5 zile.

� VH�VWDELOHVF�LQWHUYDOHOH�GH�YDULD LH�úL�VH�HIHFWXHD] �JUXSDUHD�

Tabelul 2.9. ,QWHUYDOH�GH�YDULD LH�D�WLPSXOXL�GH�FDOFXO�DO�

SUH XOXL 1XP U�FRPHQ]L

3,2 – 4,7 8 4,7 – 6,2 11 6,2 – 7,7 10 7,7 – 9,2 5

9,2 – 10,7 4 10,7 – 12,2 2

Total 40 1RW :�OLPLWD�LQIHULRDU �LQFOXV �vQ�LQWHUYDO�

5HSUH]HQWDUHD�JUDILF �D�GLVWULEX LHL�VH�J VHúWH�vQ�ILJXUD��

Fig. 2.5.

7. 8UP WRDUHOH�GDWH�UHSUH]LQW �FKHOWXLHOLOH�SHQWUX�FXPS UDUHD�XQXL�SURGXV��PLL�OHL��HIHFWXDWH�GH��

de familii într-R�V SW PkQ ��435; 430; 435; 439; 423; 431; 436; 435; 428; 436; 438; 437; 430; 435; 430; 434; 433; 429; 435; 427; 427; 428; 431; 432; 433; 435; 422; 434; 435; 433; 438; 436.

Se cere:

a) 6 �VH�FRQVWUXLDVF �R�GLVWULEX LH�GH� IUHFYHQ H�� IRORVLQG� LQWHUYDOH�GH�P ULPH�� vQFHSkQG�FX�

intervalul 421-��&DOFXOD L�FHQWUHOH�GH�LQWHUYDO� b) 6 �VH�VLVWHPDWL]H]H�GDWHOH�SH��LQWHUYDOH�HJDOH� c) 6 �VH�UHSUH]LQWH�JUDILF�GLVWULEX LLOH�RE LQXWH�

6ROX LH�

a)

∆+−= infsupiii xxh ,

unde hi� �P ULPHD�LQWHUYDOXOXL�i ∆� �YDORDUHD�FX�FDUH�VH�GLVFUHWL]HD] �LQWHUYDOHOH�

Cum pentru intervalul 421 – 422, 1infsup =− ii xx iar hi� ��UH]XOW �F �∆ = 1, iar centrele de

LQWHUYDO�VH�YRU�FDOFXOD�FX�UHOD LD�

2inf h

xCI ii +=

� �� "!#��$ % �&$�� '�� $��(� ��

)*+,-./)./*

021 *#3+�1 4

+21 4#3,51 *

,51 *#3451 4

451 4#3651 *

651 *#3./)51 4

./)21 4#3./*21 *

7�8 9#:<;>=@? A/8 = B/C

D E FGHIJKLM

Intervalele pe care se va efectua gruparea sunt: 421 – 422; 423 – 424; 425 – 426; 427 – 428; 429 – 430; 431 – 432; 433 – 434; 435 – 436; 437 – 438; 439 – 440

6H�HIHFWXHD] �JUXSDUHD� Tabelul 2.10 ,QWHUYDOH�GH�YDULD LH�D�FKHOWXLHOLORU��PLL�OHL� 1XP U�IDPLOLL Centre de interval

421 – 422 1 422 423 – 424 2 424 425 – 426 1 426 427 – 428 4 428 429 – 430 4 430 431 – 432 5 432 433 – 434 8 434 435 – 436 11 436 437 – 438 7 438 439 – 440 2 440

Total 45 -

b) Ax = 440 – 422 = 18 mii lei r = 6 h = Ax / r = 18 / 6 = 3 mii lei.

Tabelul 2.11 ,QWHUYDOH�GH�YDULD LH�D�FKHOWXLHOLORU��PLL�OHL� 1XP U�IDPLOLL

422 – 425 3 425 – 428 3 428 – 431 6 431 – 434 9 434 – 437 15 437 – 440 9

Total 45

c) 5HSUH]HQW ULOH�JUDILFH�VH�IDF�SULQ�KLVWRJUDP ��ILJXULOH��úL��

Fig. 2.6.

' LV WULE X LD �ID P LOLLOR U �G X S �F K H OWX LH OL

NOPQRS2NS5O

P�O�STP�O�O

P�O�UTP�O�P

P�O�VTP�O�Q

P�O�WTP�O�R

P�O�XTP�U�N

P�U�STP�U�O

P�U�UTP�U�P

P�U�VTP�U�Q

P�U�WTP�U�R

P�U�XTP�P�N

Y[Z�\�] ^ _�` \�] `5acbd` `5] \�` e

f g(hi jklmll

Fig. 2.7.

8. 6H� FRQVLGHU � XUP WRDUHD� UHSUH]HQWDUH� JUDILF � SULYLQG� GLVWULEX LD� VDODULD LORU� GXS � WLPSXO�

nelucrat:

Fig. 2.8.

d) &H�VH�UHSUH]LQW �SULQ�DFHVW�JUDILF" e) ([SOLFD L�GH�FH�YDORDUHD�GH�SH�D[D�2\�DIHUHQW �XOWLPXOXL�LQWHUYDO�HVWH��"� f) 7UDQVIRUPD L� GLVWULEX LD� UHSUH]HQWDW � JUDILF� vQWU-R� GLVWULEX LH� GH� IUHFYHQ H� UHODWLYH� úL�

UHSUH]HQWD L-o grafic. g) &DUH�HVWH�SRQGHUHD�VDODULD LORU�FDUH�DX�PDL�SX LQ�GH��PLQ��WLPS�QHOXFUDW"�'DU�D�FHORU�FDUe

au peste 20 de min. timp nelucrat? 6ROX LH�

a) )UHFYHQ H�UHODWLYH�FXPXODWH�FUHVF WRU��úL�H[SULPDWH�SULQ�FRHILFLHQ L� b) 'HRDUHFH� vQWRWGHDXQD� IUHFYHQ D�FXPXODW � FUHVF WRU� DIHUHQW �XOWLPXOXL� LQWHUYDO� FRLQFLGH�FX�

VXPD� IUHFYHQ HORU� GLQ� FDUH� V-DX� FDOFXODW� IUHFYHQ HOH� FXPXODWH�� )LLQG� YRUED� GH� IUHFYHQ H�UHODWLYH�H[SULPDWH�SULQ�FRHILFLHQ L��VXPD�ORU�HVWH��

' LVWULEX LD�IDP LOLLORU�GXS �FKHOWX LHOL

nopqrs�ns�os ps�q

p o�outp o v

p o#v&tp o r

p o rwtp"x s

p x sytp"x'p

p x�p tp"x�z

p"x�z tp#p n

{�|�}�~ ��"� }�~ �'�� ~ }��

� �>��

�2� ��2� �

�� 2� �

�

��2� ��2� ��2� ��2� ��

�>� �

��c�<� �<�� 5� �5�5� �� 5� �5�� '� ��¡ >¢�£ ¤>¥�¦ §<¨2© �¡� 5ª «

c) Tabelul 2.12

)UHFYHQ H�UHODWLYH�FXPXODWH ,QWHUYDOH�GH�YDULD LH�D�WLPSXOXL�QHOXFUDW�

(min.) )UHFYHQ H relative FUHVF WRU GHVFUHVF WRU

0 – 10 0,1 0,1 1,0 10 – 20 0,2 0,3 0,9 20 – 30 0,45 0,75 0,7 30 – 40 0,15 0,9 0,25 40 – 50 0,1 1,0 0,1 Total 1,00 - -

Fig. 2.9.

d) 3RQGHUHD�VDODULD LORU�FDUH�DX�PDL�SX LQ�GH��PLQ�� WLPS�QHOXFUDW�HVWH�GH�� LDU�D�FHORU�

care au peste 20 min. timp nelucrat de 70%.

Probleme propuse

1. &kQG� GRX � ILUPH� DQXQ � GRULQ D� GH� D� IX]LRQD�� VH� vQWkPSO � GHVHRUL� FD� vQ� GHFXUV� GH� FkWHYD�

V SW PkQL��XQD�GLQ�ILUPH�V �GHYLQ �QHPXO XPLW �GH�FRQVHFLQ HOH�IX]LRQ ULL�úL�IX]LRQDUHD�V �ILH�

UH]LOLDW �� 'RGG� �� D� UDSRUWDW� F � GLQ� �� de firme pe care le-D� LGHQWLILFDW� F � DX� DQXQ DW�

fuzionarea, 80 au anulat-R��'H�DFHHD��GLQ�PRPHQWXO� vQ�FDUH�R�ILUP �DQXQ �IX]LXQHD��H[LVW �R�

PDUH� GR] � GH� LQFHUWLWXGLQH� GDF � IX]LXQHD� YD� DYHD� ORF� �SRW� WUHFH� OXQL� GXS � DQXQ � SkQ � FkQG�

fuziunea are loc). Edger��D�SXEOLFDW�QXP UXO�GH�]LOH�GH�WUDGLQJ��vQWUH�]LXD�vQ�FDUH�V-D�I FXW�DQXQ XO�SRWHQ LDOHL�IX]LXQL�vQ�:DOO�6WUHHW�-RXUQDO�úL�]LXD�IX]LXQLL�HIHFWLYH��3HQWUX��GH�ILUPH��VH�

G �QXP UXO�GH�]LOH�VFXUV�vQWUH�GDWD�DQXQ XOXL�SRWHQ LDOHL�IX]LXQL�úL�GDWD�IX]LXnii efective: 74, 42, 155, 80, 214, 97, 225, 182, 92, 116, 140, 162, 192, 178, 40, 193, 94, 157, 123, 128, 92, 173, 173, 216, 135, 120, 64, 184, 255, 27, 123, 80, 143, 212, 176, 200, 164, 86.

6 � VH� JUXSH]H� GDWHOH� SH� LQWHUYDOH� GH� P ULPH� HJDO � úL� V � VH� UHSrezinte grafic rezultatele

JUXS ULL�

2. 3HQWUX� SUREOHPD� �� GLQ� DFHVW� FDSLWRO�� V � VH� HIHFWXH]H� JUXSDUHD� FRPELQDW � GXS � FHOH� GRX �

caracteristici considerate concomitent.

¬¬2 ®¬2 ¯¬2 °¬2 ±¬2 ²

¬�³(®<¬ ®<¬�³ ¯2¬´¯2¬�³ °5¬´°5¬>³ ±2¬µ±�¬�³ ²5¬¶'· ¸¡¹�º¼»5½ ¾>¿�À Á<Â�Ã ¸�· º>Ä

Å ÆÇÈ ÉÇÊÇÆÇË ÌÍÎ ÉÇ

33

ÍQWUHE UL ¾ &DUH�VXQW�IXQF LLOH�LQGLFDWRULORU�VWDWLVWLFL" ¾ Ce sunt indicatorii primari? ¾ 0 ULPLOH�UHODWLYH�GH�VWUXFWXU ��GHILQL LH��PRG�GH�FDOFXO��XWLOL]DUH��UHSUH]HQWDUH�JUDILF � ¾ 0 ULPLOH�UHODWLYH�GH�GLQDPLF ��GHILQL LH��PRG�GH�FDOFXO��XWLOL]DUH��UHSUH]HQWDUH�JUDILF � ¾ 0 ULPLOH�UHODWLYH�GH�LQWHQVLWDWH��GHILQL LH��PRG�GH�FDOFXO��XWLOL]DUH��UHSUH]HQWDUH�JUDILF � ¾ 0HGLD�DULWPHWLF ��GHILQL LH��XWLOL]DUH��SURSULHW L� ¾ Care sunt principalele medii de calcul? ¾ &DOFXOXO�úL�XWLOL]DUHD�SULnFLSDOLORU�LQGLFDWRUL�PHGLL�GH�SR]L LH�

INDICATORI STATISTICI


1. 3HQWUX�FHL��GH�VDODULD L�DL�XQXL�DWHOLer din cadrul unui agent economic cu profil industrial s-a

vQUHJLVWUDW�SURGXF LD��vQ�EXF L��UHDOL]DW �vQ�OXQD�PDL��SUHFXP�úL�HFKLSHOH�GLQ�FDUH�IDF�SDUWH�

DFHúWL�VDODULD L��WDEHOXO�� Se cere:

a) 6 �VH�HIHFWXH]H�FkWH�R�JUXSDUH�VLPSO �GXS �YDULD LD�ILHF UHL�FDUDFWHULVWLFL��3HQWUX�YDULDELOD�

QXPHULF �VH�YRU�IRUPD��JUXSH�GH�P ULPH�HJDO �úL�VH�YRU�UHSUH]HQWD�JUDILF�GDWHOH�JUXSDWH� b) )RORVLQG�JUXSDUHD�VLPSO �GXS �FDUDFWHULVWLFD�QHQXPHULF �� V � VH�FHQWUDOL]H]H�GDWHOH�SULYLQG�

YDORDUHD�SURGXF LHL� Tabelul 3.1 Salariat 3URGXF LD�UHDOL]DW �

vQ�OXQD�PDL��EXF L� Echipa Salariat 3URGXF LD�UHDOL]DW �vQ�OXQD�

PDL��EXF L� Echipa

1 18 A 11 15 C 2 11 A 12 15 A 3 14 B 13 15 C 4 17 A 14 15 C 5 12 A 15 14 B 6 14 B 16 13 D 7 16 A 17 15 C 8 14 D 18 13 D 9 16 B 19 16 A

10 13 B 20 12 B

c) 8WLOL]D L�UH]XOWDWHOH�GH�OD�SXQFWXO�SUHFHGHQW��6H�PDL�FXQRVF� Tabelul 3.2

Echipa 0 ULPHD�UHODWLY �GH�GLQDPLF �D�SURGXF LHL�

UHDOL]DWH�vQ�PDL�ID �GH�DSULOLH�� 6WUXFWXUD�SURGXF LHL�SODQLILFDWH�

în luna mai A 105 0,333 B 110,6 0,296 C 120 0,204 D 160 0,167

ùWLLQG�F �SURGXF LD�SODQLILFDW �D� OXQLL�PDL�OD�QLYHOXO� vQWUHJXOXL�DWHOLHU�HVWH�FX��PDL�PDUH�GHFkW�

SURGXF LD�WRWDO �UHDOL]DW �D�OXQLL�DSULOLH��V �VH�FDOFXOH]H: c1�� SURGXF LD� UHDOL]DW � vQ� OXQD� DSULOLH�� SUHFXP� úL� FHD� SODQLILFDW � vn luna mai, pe fiecare

HFKLS �úL�SH�WRWDO�DWHOLHU� c2��VWUXFWXUD�SURGXF LHL�UHDOL]DWH�vQ�DSULOLH��SUHFXP�úL�FHD�D�SURGXF LHL�UHDOL]DWH�vQ�OXQD�PDL��

6 �VH�UHSUH]LQWH�JUDILF�XQD�GLQWUH�HOH� c3�� P ULPLOH� UHODWLYH� GH� FRRUGRQDUH� SHQWUX� SURGXF LD� HIHFWLY� UHDOL]DW � vQ� DSULOLH� úL� V � VH�

reprezinte grafic; c4��P ULPLOH�UHODWLYH�DOH�SODQXOXL�SH�ILHFDUH�HFKLS �úL�SH�WRWDO�DWHOLHU� c5��P ULPHD�UHODWLY �GH�GLQDPLF �OD�QLYHOXO�vQWUHJXOXL�DWHOLHU� c6��R�P ULPH�UHODWLY �GH�LQWHQVLWDWH�SH�ILHFDUH�HFKLS �úL�SH�WRWDO�DWHOLHU�

6ROX LH� a)

a1) *UXSDUHD��FODVLILFDUHD��VDODULD LORU�GXS �FDUDFWHULVWLFD�QHQXPHULF ��HFKLSD� 6H� HIHFWXHD] � R� FODVLILFDUH� SH� YDULDQWH�� vQWUXFkW� FDUDFWHULVWLFD� DUH� XQ� QXP U� PLF� GH�

variante. Tabelul 3.3

Echipa 1XP U�VDODULD L��ni)

A 7

B 6

C 4

D 3

Total 20

Fig. 3.1.

a2) *UXSDUHD�VDODULD LORU�GXS �FDUDFWHULVWLFD�QXPHULF ��SURGXF LD�–�vQ�EXF L� 6H�QRWHD] �FX�;�–�FDUDFWHULVWLFD�GH�JUXSDUH��SURGXF LD�–�vQ�EXF L��6H�SDUFXUJ�XUP WRULL�SDúL�

� VH�FDOFXOHD] �DPSOLWXGLQHD�YDULD LHL�FDUDFWHULVWLFLL��Ax) Ax = xmax - xmin = 18 –�� EXF L

� VH�VWDELOHúWH�QXP UXO�GH�JUXSH��r) r = 4

� VH�VWDELOHúWH�P ULPHD�LQWHUYDOXOXL�GH�JUXSDUH��h) h = Ax / r = 7/4 ≈ 1,75 ≈ 2

� VH�VWDELOHVF�LQWHUYDOHOH�GH�YDULD LH�úL�VH�HIHFWXHD] �JUXSDUHD� Tabelul 3.4

,QWHUYDOH�GH�YDULD LH�D�SURGXF LHL��EXF L� 1U��VDODULD L

11-13 3

13-15 7

15-17 8

17-19 2

Total 20

1RW ��OLPLWD�LQIHULRDU �LQFOXV �vQ�LQWHUYDO�

��

�� !"#

$ % & '(�)�*�+ ,.-

/ 0�1234 305 35

Fig. 3.2.

b) Tabelul 3.5

Echipa 3URGXF LD�FHQWUDOL]DW ��EXF�� A 105 B 83 C 60 D 40

Total 288

c) 6H�QRWHD] � ix0 � �SURGXF LD�UHDOL]DW �GH�HFKLSD�³i” în luna aprilie;

ix1 �SURGXF LD�UHDOL]DW �GH�HFKLSD�³i” în luna mai; iplx �SURGXF LD�SODQLILFDW �GH�HFKLSD�³i” în luna mai;

X0 �SURGXF LD�WRWDO �D�DWHOLHUXOXL�UHDOL]DW �vQ�OXQD�DSULOLH� X1 �SURGXF LD�WRWDO �D�DWHOLHUXOXL�UHDOL]DW �vQ�OXQD�PDL� Xpl �SURGXF LD�WRWDO �D�DWHOLHUXOXL�SODQLILFDW �vQ�OXQD�PDL� 'H�OD�SXQFWXO�SUHFHGHQW�VH�FXQRDúWH�SURGXF LD�UHDOL]DW �GH�FHOH��HFKLSH�vQ�OXQD�PDL��DGLF ��

ix1 �úL�X1� ��EXF L� Tabelul 3.6

Echipa ik 0/1 (%)

iplg ix1

(buc.) A 105 0,333 105 B 110,6 0,296 83 C 120 0,204 60 D 160 0,167 40

Total 115,2 1,0 288

c1) i

ii

i

ii

k

xx

x

xk

0/1

10

0

10/1

100100

⋅=⇒⋅= (vezi coloana 1 din tabelul 3.7)

buc. 25000 =∑=i

ixX

687 9;:�< 7 =�> 7 ?@9A?CB ?�< 7 ? 7 B DE<�FE>�G G�< D�FE>CH 7 ?< I;?�B 7 J.?;: K LMB >�LA?ON�?�7

PQRSTU P

U�U V U�W U�W�V U X U X�V U Y U Y�V U Z[A\ ]_^;` a b�c d�` e

f g hijk jgl jl

buc. 27025008,108,1 00/0

0/ =⋅=⋅=⇒== XKXX

XK plpl

plpl

plipl

ipl

pl

ipl

i

ipl

ipli

pl XgxX

x

x

xg ⋅=⇒=

∑= (vezi coloana 2 din tabelul 3.7)

c2) 1001000

0

0

00 ⋅=⋅

∑=

X

x

x

xg

i

i

i

ii (vezi coloana 3 din tabelul 3.7)

1001001

1

1

11 ⋅=⋅

∑=

X

x

x

xg

i

i

ii (vezi coloana 4 din tabelul 3.7).

Fig. 3.3.

c3) AlegHP� FD� ED] � GH� UDSRUWDUH� HFKLSD� '� úL� FDOFXO P� UDSRDUWHOH� GH� FRRUGRQDUH� vPS U LQG�

SURGXF LD�UHDOL]DW �vQ�LDQXDULH�GH�ILHFDUH�HFKLS �OD�SURGXF LD�UHDOL]DW �vQ�LDQXDULH�GH�HFKLSD�'�

D

i

Dix

xk

0

0/ = (vezi coloana 5 din tabelul 3.7)

c4)

1000

0/ ⋅=i

ipli

plx

xk (vezi coloana 6 din tabelul 3.7)

1001/1 ⋅=

ipl

ii

plx


Pe total:

%)7,106( 067,1270

2881/1 ===

plpl X

XK

3URGXF LD�UHDOL]DW �SH�vQWUHJ�DWHOLHUXO�vQ�PDL�HVWH�GH��RUL�PDL�PDUH�GHFkW�FHD�SODQLILFDW �

vQ�OXQD�PDL��DGLF �FX��,7% mai mare.

6WUXFWXUD�SURGXF LHL�UHDOL]DWH�vQ�OXQD�DSULOLH�

SH�HFKLSH

��

��

��

��

$

%

&

'

c5) %)2,115( 152,1250

288

0

10/1 ===

X

XK

3URGXF LD� UHDOL]DW � vQ� OXQD� PDL� GH� vQWUHJ� DWHOLHUXO� HVWH� GH� �� RUL� PDL� PDUH� GHFkW� FHD�

UHDOL]DW �vQ�OXQD�DSULOLH��VDX�FX��PDL�PDUH� c6��6H�SRDWH�GHWHUPLQD�SURGXFWLYLWDWD�PXQFLL�vQ�OXQD�PDL��SH�ILHFDUH�HFKLS �úL�SH�WRWDO�

i

ii

n

xw 1

1 = (vezi coloana 8 din tabelul 3.7)

La nivel total:

iatbuc./salar 4,1420

28811

111 ===== ∑

∑∑

∑∑ n

ii

i

ii

i

igw

n

nw

n

xW

Tabelul 3.7

Echipa ix0 (buc.)

iplx

(buc.)

ig0 (%)

ig1 (%)

0/ Dik i

plk 0/

(%)

iplk /1

(%)

iw1 buc/ pers

0 1 2 3 4 5 6 7 8

A 100 90 40 36 4 90 116,67 15 B 75 80 30 29 3 106,67 103,75 13,83 C 50 55 20 21 2 110 109,09 15 D 25 45 10 14 1 180 88,89 13,33

Total 250 270 100 100 - 108 106,67 14,4

2. Despre activitatea unei socLHW L�FRPHUFLDOH�VH�FXQRVF�XUP WRDUHOH�GDWH��SULYLQG�OXQLOH�LDQXDULH-

februarie 1999: Tabelul 3.8 6HF LD 6WUXFWXUD�SURGXF LHL�vQ�

SHULRDGD�GH�ED] ��

(ianuarie) ig0

Procentul sarcinii de SODQ�vQ�IHEUXDULH�ID �

de ianuarie (%) iplk 0/

0 ULPHD�UHODWLY �D�

UHDOL] ULL�SODQXOXL�vQ�

febr. i

plk /1

Structura personalului în

ianuarie tig0

0 1 2 3 4 I 20 125 1,20 0,18 II 40 175 0,8571 0,41 III 25 100 1,60 0,30 IV 15 200 1,333 0,11

ùWLLQG�F vDORDUHD�WRWDO �D�SURGXF LHL�vQ�OXQD�LDQXDULH�D�IRVW�GH��PLOLRDQH�OHL��LDU�QXP UXO�WRWDO�

al personalului a fost de 700 persoane, se cere:

a) 6 � VH� GHWHUPLQH� QLYHOXO� SURGXF LHL� GLQ� OXQD� LDQXDULH�� FD� úL� DO� FHOHL� SODQLILFDWH� úL� HIHFWLY�

realizate în februarie��SH�ILHFDUH�VHF LH�úL�SH�WRWDO�VRFLHWDWH�FRPHUFLDO � b) 6 �VH�GHWHUPLQH�VWUXFWXUD�SURGXF LHL�SODQLILFDWH�úL�D�FHOHL�HIHFWLY�UHDOL]DWH�vQ�OXQD�IHEUXDULH��

reprezentându-VH�JUDILF�P ULPLOH�UHODWLYH�GH�VWUXFWXU � c) 6 �VH�FDOFXOH]H�P ULPLOH�UHODWLYH�GH�GLQDPLF �SH�ILHFDUH�VHF LH� d) 6 � VH� FDOFXOH]H� P ULPLOH� UHODWLYH� GH� GLQDPLF � úL� DOH� SODQXOXL� SH� vQWUHDJD� VRFLHWDWH�

FRPHUFLDO ��FX�VSHFLILFDUHD�WLSXOXL�GH�PHGLH�XWLOL]DW �GH�ILHFDUH�GDW � e) 6 �VH�UHSUH]LQWH�JUDILF�P ULPLOH�UHODWLYH�GH�GLQDPLF �úL�DOH�SODQXOXL� f) 6 �VH�FDOFXOH]H�R�P ULPH�UHODWLY �GH�LQWHQVLWDWH�úL�V �VH�UHSUH]LQWH�JUDILF�

6ROX LH�

a) 6H�QRWHD] � ix0 �SURGXF LD�UHDOL]DW �GH�HFKLSD�³i” în luna ianuarie;

ix1 �SURGXF LD�UHDOL]DW �GH�HFKLSD�³i” în luna februarie; iplx �SURGXF LD�SODQLILFDW �GH�HFKLSD�³i” în luna februarie;

X0 �SURGXF LD�WRWDO �D�DWHOLHUXOXL�UHDOL]DW �vQ�OXQD�LDQXDULH� X1 �SURGXF LD�WRWDO �D�DWHOLHUXOXL�UHDOL]DW �vQ�OXQD�IHEUXDULH� Xpl �SURGXF LD�WRWDO �D�DWHOLHUXOXL�SODQLILFDW �vQ�OXQD februarie;

100100100 00

00

0

0

00

Xgx

X

x

x

xg

ii

i

i

i

ii ⋅=⇒⋅=⋅


Tabelul 3.9 6HF LD ix0

(mil.lei)

iplx

(mil. lei)

ix1

(mil. lei)

iplg

(%)

ig1

(%)

ik 0/1 it0

(pers)

iw0

(mii lei/ pers)

0 1 2 3 4 5 6 7 8

I 1000 1250 1500 16,67 17,65 1,5 126 7930,65

II 2000 3500 3000 46,67 35,29 1,5 287 6960,864

III 1250 1250 2000 16,67 23,53 1,6 210 5950,238

IV 750 1500 2000 20,00 23,53 2,66 77 9740,026

Total 5000 7500 8500 100,0 100,0 700

100100

00/

00/

iipli

pli

ipli

plxk

xx

xk


mil.lei 7500=∑=i

iplpl xX

ipl

ipl

iipl

ii

pl xkxx

xk ⋅=⇒= /11

1/1 (vezi coloana 3 din tabelul 3.9)

mil.lei 850011 =∑=i

ixX

b) 100100 ⋅=⋅=∑ pl

ipl

i

ipl

ipli

pl X

x

x

xg (vezi col. 4 din tabelul 3.9) (Fig. 3.4.)

1001001

1

1

11 ⋅=⋅

∑=

X

x

x

xg

i

i

i

ii (vezi coloana 5 din tabelul 3.9) (Fig. 3.5.)

6WUXFWXUD�SURGXF LHL�SODQLILFDWH�vQ�IHEUXDULH

m�n�o nqpsrt n�o nqpqr

m�n�o nqpqruqv@o vsvqr

wMxsy z {}|wMxsy z {}| |wMxsy z {}| | |wMxsy z {}| ~

Fig. 3.4.

Fig. 3.5.

c) ipl

ipli

ipl

ipl

i

i

ii kk

x

x

x

x

x

xk 0//1

0

1

0

10/1 ⋅=⋅== (vezi coloana 6 din tabelul 3.9)

d) %)150( 5,15000

7500

0

0

000/ ==

∑

∑ ⋅=

∑

∑==

i

iipl

i

iplpl

plx

xk

x

x

X

XK

$FHDVWD� vQVHDPQ � F � vQ� OXQD� IHEUXDULH�� V-D� SODQLILFDW� R� SURGXF LH� WRWDO � OD� QLYHOXO� vQWUHJLL�

VRFLHW L�FRPHUFLDOH�GH��RUL�PDL�PDUH�GHFkW�SURGXF LD�UHDOL]DW �vQ�OXQD�LDQXDULH��VDX�FX��PDL�

mare. 3HQWUX�FDOFXOXO�P ULPLL�UHODWLYH�D�VDUFLQLL�GH�SODQ�OD�Qivel total se poate utiliza, deci, media DULWPHWLF �SRQGHUDW � ùL�SHQWUX�FDOFXOXO�P ULPLL�UHODWLYH�D�UHDOL] ULL�SODQXOXL�OD�QLYHO�WRWDO�VH�SRDWH�XWLOL]D�R�PHGLH�

DULWPHWLF �SRQGHUDW �

%)113( 13,17500

8500111/1 ==

∑

∑ ⋅=

∑∑==

ipl

ipl

i

ipl

i

plpl

x

xk

x

x

X

XK

6WUXFWXUD�SURGXF LHL�UHDOL]DWH�vQ�IHEUXDULH

�� s�s�

�8�� q�s��q�@� �8�q�

�q�@� �8�s��O�s� � ��O�s� � �� O�s� � �� O�s� � ��

În luna februarie s-D�UHDOL]DW�R�SURGXF LH�WRWDO �GH��RUL�PDL�PDUH�GHFkW�FHD�SODQLILFDW ��VDX�FX��PDL�PDUH�GHFkW�FHD�SODQLILFDW �vQ�IHEUXDULH�

%)170( 7,113,15,15000

85000//1

0

1

0

10/1 =⋅==⋅=⋅== plpl

pl

plKK

X

X

X

X

X

XK e)

e) 5HSUH]HQWDUHD�JUDILF �D�P ULPLORU� UHODWLYH�DOH�SODQXOXL� úL�DOH�GLQDPLFLL� OD�QLYHOXO� VHF LLORU�

HVWH�UHDOL]DW �vQ�ILJXUD��

Fig. 3.6.

f) &XQRVFkQG�VWUXFWXUD�SHUVRQDOXOXL�vQ�LDQXDULH��YH]L�FRO��GLQ�WDEHOXO��úL�QXP UXO�WRWDO�GH�

VDODULD L� �70� � �� SHUVRDQH�� VH� GHWHUPLQ � QXP UXO� GH� VDODULD L� SHQUX� ILHFDUH� VHF LH� �W0) astfel:

000000

00 Tgtgt

t

tg i

tii

ti

i

iit ⋅=∑⋅=⇒


i

ii

t

xw

0

00 = (vezi coloana 8 din tabelul 3.9), unde iw0 � UHSUH]LQW � SURGXFWLYLWDWHD�PXQFLL� vQ�

SHULRDGD�GH�ED] �–�LDQXDULH��vQ�VHF LD�³i”. /D�QLYHOXO�VRFLHW LL�FRPHUFLDOH�

lei/pes. mii 86,7142700

5000

0

00

0

0

0

00 ==

∑∑ ⋅=

∑∑==

i

ii

i

i

t

tw

t

x

T

XW

3. Fondul� GH� VDODUL]DUH� SHQWUX� DQJDMD LL� D� GRX � ILOLDOH� DOH� XQHL� E QFL� D� IRVW� GH� ��PLO�� OHL�� LDU�

raportul de coordonare pentru acest fond k1/2 = 1,63��ùWLLQG�F �VDODULXO�PHGLX�DO�DQJDMD LORU�GLQ�

ILOLDOD� �� D� IRVW�GH� ��PLL� OHL� úL� F � vQ� ILOLDOD� �� OXFUHD] � ��GLQ� DQJDMD LL� E QFLL�� V � VH� DIOH�

VDODULXO�PHGLX�vQ�ILOLDOD��úL�SH�vQWUHDJD�EDQF � 6ROX LH� 1RW P� FS1, FS2 –�)RQGXO�GH�VDODULL�vQ�ILOLDOD��UHVSHFWLY��D�E QFLL�

21 , SS –�VDODULXO�PHGLX�vQ�ILOLDOD��UHVSHFWLY��D�E QFLL

0DULPLOH�UHODWLYH�DOH�SODQXOXL�úL�GH�GLQDPLF �SH�

FHOH��VHF LL�DOH�VRFLHW LL�FRPHUFLDOH

�� A�� E��

�s�� s�E� � �� s�E� � �� s�E� � �� ¡�s�E� � �

¢

� £A¤¦¥ ¤�� £C�E§¨� ��E© ��¥ ª�« ¥¬ �E�E« � E¥ ªE« ¥

T1, T2 –�QXP UXO�SHUVRQDOXOXL�vQ�ILOLDOD��UHVSHFWLY��D�E QFLL gT1, gT2 –�*UHXWDWHD�VSHFLILF �D�SHUVRQDOXOXL�vQ�ILOLDOD��UHVSHFWLY��D�E QFLL� FS –�IRQGXO�GH�VDODULL�SH�vQWUHDJD�EDQF

S -�VDODULXO�PHGLX�SH�vQWUHDJD�EDQF T – personalul întregii�E QFL FS = 184 mil. lei

63,12

12/1 ==

FS

FSk 222/11 63,1 FSFSkFS ⋅=⋅=⇒

lei mii 35002 =S %)60( 6,0

1=Tg

FS = FS1 + FS2 = 184 = 1,63 FS2 + FS2 =184

lei mil. 7063,2

1842 ==⇒ FS

FS1 = 1,63 x 70 = 114 mil. lei

persoane 203500

70000

2

22

2

22 ===⇒=

S

FST

T

FSS

%)60( 6,01

=Tg persoane 504,0

204,0 2

2==⇒==⇒ T

T

TgT

T1 = 50 – 20 = 30 persoane

lei mii 380030

114

1

11 ===

T

FSS

lei mii 368050

1000184

21

21 =⋅==++=

T

FS

TT

FSFSS

4. 6H�FXQRVF�XUP WRDUHOH�GDWH��SULYLQG�DFWLYLWDWHD�XQRU�ILUPH�vQ�OXQLOH�LDQXDULH�úL�IHEUXDULH� Tabelul 3.10 Firma 6WUXFWXUD�SURGXF LHL�

realizate în ian. (%) Raportul de coordonare a

SURGXF LHL�SODQLILFDWH�vQ�IHEUXDULH Procentul de realizare a planului în februarie (%)

A 1 2 3 1 10 1,4 120 2 20 2,4 80 3 25 2,8 100 4 40 4,0 75 5 5 1,0 150

Total 100 - -

ùWLLQG�F �SURGXF LD�UHDOL]DW �GH�ILUPD�QU. 5 în luna februarie a fost de 750 mil. lei, ceea ce a vQVHPQDW�R�GHYDQVDUH�D�SURGXF LHL�GLQ�OXQD�LDQXDULH�FX��se cere:

a) 1LYHOXULOH�SURGXF LHL�UHDOL]DWH�vQ�OXQD�LDQXDULH�úL�IHEUXDULH�úL�D�FHOHL�SODQLILFDWH�SH�IHEUXDULH��

SHQWUX�ILHFDUH�ILUP �úL�SH�WRWal. b) 0 ULPLOH�UHODWLYH�DOH�SODQXOXL�SH�ILHFDUH�ILUP �úL�SH�WRWDO��FX�VSHFLILFDUHD�WLSXOXL�GH�PHGLH�

XWLOL]DW �SHQWUX�FDOFXOXO�LQGLFDWRULORU�SH�WRWDO� c) 6 �VH�FRQVWUXLDVF �XQ�JUDILF�SHQWUX�UHSUH]HQWDUHD�P ULPLORU�UHODWLYH�DOH�SODQXOXL�

6ROX LH:

a) 1RW P�

ix0 �SURGXF LD�UHDOL]DW �GH�ILUPD�³i” în luna ianuarie;

ix1 �SURGXF LD�UHDOL]DW �GH�ILUPD�³i” în luna februarie; iplx �SURGXF LD�SODQLILFDW �GH�ILUPD�³i” în luna februarie;

X0 �SURGXF LD�WRWDO ��SH�DQVDPEOXO�FHORU��ILUPH��UHDOL]DW �vQ�OXQD�LDQXDULH� X1 �SURGXF LD�WRWDO ��SH�DQVDPEOXO�FHORU��ILUPH��UHDOL]DW �vQ�OXQD�IHEUXDULH� Xpl �SURGXF LD�WRWDO ��SH�DQVDPEOXO�FHORU��ILUPH��SODQLILFDW �vQ�OXQD�IHEUXDULH�

lei mil. 75051 =x

lei mil. 2503

750

3%300

515

050

515

0/1 ===⇒== xx

x

xk

1001000

0

0

00 ⋅=⋅

∑=

X

x

x

xg

i

i

i

ii , deci pentru firma nr. 5 vom avea:

lei mil. 50005

100250100

50

50 100

00

505

0 =⋅==⇒⋅=⋅

g

xX

X

xg

Pentru celelalte patru firme avem:

10000

00

00100igXi

ii x

X

xg


Pentru firma 5 avem:

lei mil. 500150

100750100100

5/1

515

5

515

/1 =⋅=⋅=⇒=pl

plpl

plk

xx

x

xk

Raportul de coordonarH�DO�SURGXF LHL�SODQLILFDWH��OXkQG�FD�ED] �GH�UDSRUWDUH�SURGXF LD�SODQLILFDW �D�

firmei 5, va fi:

55/55/ pl

pli

ipl

pl

iplpl

i xkxx

xk ⋅=⇒= (vezi coloana 5 din tabelul 3.11)

lei mil. 5800=∑=i

iplpl xX

100100 /1

11

/1

ipl

ipli

ipl

ii

plxk

xx

xk


lei mil. 545011 =∑=i

ixX

b) 1RW P�

0/plK P ULPHD�UHODWLY �D�VDUFLQLL�GH�SODQ�SH�DQVDPEOXO�FHORU��ILUPH

plK /1 �P ULPHD�UHODWLY �D�UHDOL] ULL�SODQXOXL�SH�DQVDPEOXO�FHORU��ILUPH

0/1K P ULPHD�UHODWLY ��GH�GLQDPLF �SH�DQVDPElul celor 5 firme.

%)94( 94,05800

545011/1 ==

∑∑==

ipl

i

plpl

x

x

X

XK

3URGXF LD� WRWDO � D� FHORU� FLQFL� ILUPH� UHDOL]DW � vQ� IHEUXDULH� D� UHSUH]HQWDW� GRDU� �� GLQ� FHD�

SODQLILFDW �vQ�DFHDVW �OXQ ��UHVSHFWLY��FX��PDL�PLF �GHFkW�FHD�SURGXF LD�WRWDO �SURSXV �D�IL�

UHDOL]DW �vQ�IHEUXDULH�

1000

0/ ⋅=i

ipli

plx


%)116( 16,15000

5800

000/ ==

∑

∑==

i

iplpl

plx

x

X

XK

3URGXF LD�WRWDO �SURSXV �SH�DQVDPEOXO�FHORU�FLQFL�ILUPH��vQ�OXQD�IHEUXDULH�D�IRVW�GH��RUL�

PDL�PDUH�GHFkW�SURGXF LD�WRWDO �UHDOL]DW �vQ�LDQXDULH�DGLF �FX��PDL�PDUH�GHFkW�DFHDVWD�GLQ�XUP � 0 ULPLOH�UHODWLYH�DOH�SODQXOXL�SH�WRWDO�VH�PDL�SRW�FDOFXOD�FX�UHOD LD�

∑

∑==

i

iplpl

plx

x

X

XK

000/

Dar iipl

ipli

ipli

pl xkxx

xk 00/

00/ ⋅=⇒= .

ÌQORFXLQG�vQ�UHOD LD�VDUFLQLL�GH�SODQ�SH�WRWDO��J VLP�

( )∑ ⋅=∑

∑ ⋅=

∑

∑== ii

pli

iipl

i

iplpl

pl gkx

xk

x

x

X

XK 00/

0

00/

000/ ,

unde ig0 este� VWUXFWXUD�SURGXF LHL� UHDOL]DWH� vQ� OXQD�LDQXDULH��$úDGDU�� V-D�IRORVLW�R�PHGLH�DULWPHWLF �

SRQGHUDW � $VHP Q WRU�VH�SURFHGHD] �úL�vQ�FD]XO�UHDOL] ULL�SODQXOXL�

∑∑==

ipl

i

plpl

x

x

X

XK 11

/1

Dar ipl

ipl

iipl

ii

pl xkxx

xk ⋅=⇒= /11

1/1 .

ÌQORFXLQG�vQ�UHOD LD�P ULPLL�UHODWLYH�D�UHDOL] ULL�SODQXOXL�SH�WRWDO��J VLP�

( )∑ ⋅=∑

∑ ⋅=

∑∑== i

pli

plipl

ipl

ipl

ipl

i

plpl gk

x

xk

x

x

X

XK /1

/111/1 ,

unde iplg HVWH�VWUXFWXUD�SURGXF LHL�SODQLILFDWH�vQ�OXQD�IHEUXDULH��$úDGDU��úL�vQ�DFHVW�FD]�V-a folosit o

PHGLH�DULWPHWLF �SRQGHUDW �

®

¯ ®

°�®�®

° ¯ ®

±�®�®

�

° ± ² ³ ¯

)LUPD

0 ULPLOH�UHODWLYH�DOH�SODQXOXL�OD�QLYHO�WRWDO

5HDOL]�SODQ�

6DUF�SODQ

Tabelul 3.11 Firma L

�J

(%)

plik 5/ i

plk /1

(%)

ix0 (mil.lei)

iplx

(mil.lei)

ix1 (mil.lei)

iplk 0/

(%) 0 1 2 3 4 5 6 7 1 10 1,4 120 500 700 840 140 2 20 2,4 80 1000 1200 960 120 3 25 2,8 100 1250 1400 1400 112 4 40 4,0 75 2000 2000 1500 100 5 5 1,0 150 250 500 750 200

Total 100 5000 5800 5450 c)

Fig.3.7.

5. 6H�FXQRVF�XUP WRDUHOH�GDWH� Tabelul 3.12 Departamentul S.C. Procentul sarcinii de plan al

YDORULL�YkQ] ULORU�� Procentul îndeplinirii SODQXOXL�YkQ] ULORU��

Structura valorii YkQ] ULORU�UHDOL]DWH�vQ�

SHULRDGD�GH�ED] �� A 100 90 15 B 120 100 35 C 105 115 40 D 100 75 10

Se cere:

a) 6 �VH�UHSUH]LQWH�JUDILF�JUDGXO�vQGHSOLQLULL�SODQXOXL�OD�YDORDUHD�YkQ] ULORU�SH�GHSDUWDPHQWH�úL�

VWUXFWXUD�YDORULL�YkQ] ULORU�UHDOL]DWH��SHQWUX�SHULRDGD�GH�ED] �� b) 6 �VH�FDOFXOH]H�SURFHQWXO�PHGLX�DO�VDUFLQLL�GH�SODQ��DO� vQGHSOLQLULL�SODQXOXL� úL�DO�GLQDPLFLL�

YDORULL�YkQ] ULORU��SH�vQWUHDJD�VRFLHWDWH�FRPHUFLDO �

´µ�´¶ ´·�´¸E´¹CÉ´¹Aµ�´

º

» ¼ ½ ¾¾�¿ÁÀ�ÂÁÃ Ä ÂÁÅ¨¿¦Æ�Ä ¿

Ç¨È É�ÊqË8ÌÎÍ�ÏÁÊ¦Ð¦Ñ8Ì Ò ÏsÒ È�Ò Ò�Ñ8Ì É¦ÏsË8Ì Ë8Ò�Ì ÉÓÏsÒ ÔÁÐ¦Ì Ë8Ì�Ê8Ð¦Ñ�É¦È Õ ÉÁÖÓÐ¦ÏEÕ Ð¦Ì ×qÈØ ×8ÙÁÒ Ð�Õ Ò Ò�ÙE×sÖÓÐ¦È�ÙÁÒ É¦Ì Ð

6ROX LH� a)

Fig. 3.8.

Fig. 3.9.

b) 6H�IRORVHVF�DFHOHDúL�QRWD LL�FD�úL�OD�SUREOHPHOH�DQWHULRDUH�

∑

∑==

i

iplpl

plx

x

X

XK

000/

Dar iipl

ipli

ipli

pl xkxx

xk 00/

00/ ⋅=⇒= .

ÌQORFXLQG�vQ�UHOD LD�P ULPLL�UHODWLYH�D�VDUFLQLL�GH�SODQ�SH�WRWDO��RE LQHP�

( ) %10900/0

00/

000/ =∑ ⋅=

∑

∑ ⋅=

∑

∑== ii

pli

iipl

i

iplpl

pl gkx

xk

x

x

X

XK

unde ig0 HVWH�VWUXFWXUD�YDORULL�YkQ] ULORU�UHDOL]DWH�vQ�SHULRDGD�GH�ED] ��YH]L�FRORDQD��D�WDEHOXOXL��

6WUXFWXUD�YDORULL�YkQ] ULORU�UHDOL]DWH�vQ�SHULRDGD�GH�

ED]

��

��

��

��

$

%

&

'

'HFL��OD�QLYHOXO�VRFLHW LL�FRPHUFLDOH�V-DX�SODQLILFDW�R�YDORDUH�D�YkQ] ULORU�vQ�SHULRDGD�FXUHQW �FX�

��PDL�PDUH�GHFkW�YDORDUHD�UHDOL]DW �D�YkQ] ULORU�WRWDOH�vQ�SHULRDGD�GH�ED] ��VDX�GH��RUL��

∑∑==

i

i

x

x

X

XK

0

1

0

10/1

Dar iiii

ii xkx

x

xk 00/11

0

10/1 ⋅=⇒= .

ÌQORFXLQG�vQ�UHOD LD�UHDOL] ULL�SODQXOXL�SH�WRWDO��J VLP�

( ) %3,11100/10

00/1

0

1

0

10/1 =∑ ⋅=

∑∑ ⋅=

∑∑== ii

i

ii

i

igk

x

xk

x

x

X

XK

unde ig0 HVWH�VWUXFWXUD�YDORULL�YkQ] ULORU�UHDOL]DWH�vQ�SHULRDGD�GH�ED] ��

ipl

ipli

ipl

ipl

i

i

ii kk

x

x

x

x

x

xk 0//1

0

1

0

10/1 ⋅=⋅== (vezi coloana 5 din tabelul 3.13).

'HFL��YDORDUHD�YkQ] ULORU�UHDOL]DWH�OD�QLYHOXO�VRFLHW LL�FRPHUFLDOH�vQ�SHULRDGD�FXUHQW �D�IRVW�

GH��RUL�PDL�PDUH�GHFkW�FHD�HIHFWLY�UHDOL]DW �vQ�SHULDRGD�GH�ED] ��DGLF �FX��PDL�PDUH�

%)1,102( 021,1109

3,111

0/

0/1/1 ===

plpl K

KK

9DORDUHD� YkQ] ULORU� UHDOL]DW � vQ� SHULRDGD� FXUHQW � SH� WRWDO� VRFLHWDWH� FRPHUFLDO � D fost de

��RUL�PDL�PDUH�GHFkW�FHD�SODQLILFDW �vQ�DFHHDúL�SHULRDG ��DGLF �FX��PDL�PDUH� Tabelul 3.13

Dep. iplk 0/

(%)

iplk /1

(%)

ig0

(%)

iipl gk 00/ ⋅

(%)

ik 0/1

(%)

ii gk 00/1 ⋅

(%) 0 1 2 3 4 5 6 A 100 90 15 15 90 13,5 B 120 100 35 42 120 42 C 105 115 40 42 120,75 48,3 D 100 75 10 10 75 7,5

Total - - 100 109 - 111,3

6. 6H�FXQRVF�XUP WRDUHOH�GDWH� Tabelul 3.14

Firma 6WUXFWXUD�FLIUHL�GH�DIDFHUL�UHDOL]DW �vn SHULRDGD�FXUHQW ��

0 ULPHD�UHODWLY �D�vQGHSOLQLULL�SODQXOXL�OD�FLIUD�

GH�DIDFHUL�vQ�SHULRDGD�FXUHQW A 20 115 B 50 120 C 30 125

6 �VH�GHWHUPLQH�SURFHQWXO�PHGLX�DO�vQGHSOLQLULL�SODQXOXL�OD�FLIUD�GH�DIDFHUL�vQ�SHULRDGD�FXUHQW �

6ROX LH:

=++

====∑∑

∑∑∑

125

30

120

50

115

2011

1

/1

11

/1

11/1

plpl

plpl

k

gx

k

x

x

xK

(120%) 2,18306,0

1

24,04167,01733,0

1 ==++

=

7. 'RX �JUXSH�GH�VWXGHQ L�� FX�HIHFWLYH�GH��úL� UHVSHFWLY��GH�SHUVRDQH��DX�VXV LQXW�XQ� WHVW�GH�

FXOWXU �JHQHUDO ��3ULPD�JUXS �D�RE LQXW�PHGLD��LDU�D�GRXD��&HUFHW WRUXO�HVWH�LQWHUHVDW�vQ�D�

determina nota medie pe DQVDPEOXO�FHORU�GRX �JUXSH�

6ROX LH: 1RWD�PHGLH�SH�DQVDPEOXO�FHORU�GRX �JUXSH�VH�SRDWH�GHWHUPLQD�FD�R�PHGLH�GH�PHGLL�SDU LDOH��

6 �QRW P�

n1 = 25 úL� 8,71 =x

n2 = 32 úL� 4,82 =x

Atunci: ∑

∑=i

ii

n

nxx

14,857

8,268195

3225

324,8258,7

21

2211 =+=+

⋅+⋅=+

⋅+⋅=

nn

nxnxx puncte

8. &RQVLGHUD L�XUP WRDUHOH�YDORUL��

a) &DOFXOD L�PHGLD�DULWPHWLF �úL�PHGLDQD�DFHVWXL�VHW�GH�GDWH��FH�YDORDUH�YL�VH�SDUH�PDL�SRWULYLW �

SHQWUX�D�FDUDFWHUL]D�WHQGLQ D�FHQWUDO ��PHGLD�VDX�PHGLDQD" b) ÌQORFXL L� YDORDUHD� �� FX� YDORDUHD� �� úL� UHFDOFXOD L� FHL� GRL� LQGLFDWRUL� DL� WHQGLQ HL� FHQWUDOH��

([SOLFD L�PRGLILF ULOH�VXUYHQLWH� c) $G XJD L��ILHF UHL�YDORUL�LQL LDOH��&XP�VH�YD�PRGLILFD�PHGLD"

6ROX LH:

a) 28,77

51 ==∑=n

xx i

Pentru determinarea medianei, datele� VH�RUGRQHD] ��6H�GHWHUPLQ � ORFXO�

medianei în serie:

42

17

2

1 =+=∑ +

= inMeLoc

6HULD�DYkQG�QXP U�LPSDU�GH�WHUPHQL��PHGLDQD�HVWH�HJDO �FX�YDORDUHD�WHUPHQXOXL�FHQWUDO��DO�

patrulea), deci Me = 5. Mediana este un indicator potrivit� SHQWUX� D� VWXGLD� WHQGLQ D� FHQWUDO � vQ� DFHVW� VHW� GH� GDWH��GHRDUHFH�YDORDUHD��HVWH�R�YDORDUH�H[WUHP ��FH�DIHFWHD] ��SULQ�PDJQLWXGLQH��QLYHOXO�WRWDO�DO�

YDULDELOHL�úL�GHFL�YDORDUHD�PHGLHL��ùDVH�GLQ�FHOH�úDSWH�YDORUL�VH�VLWXHD] �VXE�YDORDUHD�PHGLHL�

aritmetice��FHHD�FH�IDFH�FD�LQGLFDWRUXO�PHGLDQ �V �H[SULPH�PDL�FRUHFW�WHQGLQ D�FHQWUDO �

b) Noile valori sunt: 4, 4, 5, 5, 6, 7, 8.

57,5=x , Me = 5. 9DORDUHD�PHGLDQHL�U PkQH�QHVFKLPEDW ��HD� LQkQG�FRQW�QXPDL�GH�QXP UXO�YDORULORU�PDUL��QX�

úL�GH�YDORDUHD�ORU�HIHFWLY � c) Valorile sunt: 55, 57, 54, 55, 70, 56, 54.

5028,577

401’

’+===∑= x

n

xx i .

6H�YHULILF �SURSULHWDWHD�PHGLHL�DULWPHWLFH�GH�D�VH�PRGLILFD��±) cu “ a ´�XQLW L��GDF �YDORULOH�

VH�PRGLILF ��±) fiecare cu câte “ a ´�XQLW L��

9. Un analist financiar al uneL�ILUPH�HVWH�LQWHUHVDW�vQ�D�GHWHUPLQD�VDODULXO�PHGLX�DFRUGDW�DQJDMD LORU�

FHORU��ILOLDOH�DOH�ILUPHL��3HQWUX�DFHDVWD��HO�FXOHJH�GDWH�SULYLQG�VDODULXO�PHGLX�SH�ILHFDUH�ILOLDO �úL�

fondurile de salarizare (Tabelul 3.14’) Tabelul 3.14’

Filiala Salariul mediu (mii UM) Fond de salarizare (mil. UM) 1 540 45,90 2 620 33,48 3 480 16,80 4 700 19,60

6ROX LH: Cum fondul de salarizare la nivelul unei filiale este:

iii NSFS ⋅= iar la nivelul întregii firme este:

∑⋅=∑= ii NSFSFS

vom putea calcula salariul mediu S FD�R�PHGLH�DUPRQLF �SRQGHUDW �

∑ ⋅

∑=i

i

ih

nx

nx

1

UMmii 465,573202

115840

19600700

116800

480

133480

620

145900

540

119600168003348045900

1

==

=+++

+++=⋅

=∑∑

∑∑

ii

i

i

ih

FSS

FS

N

FSx

10. ,QGLFLL�DQXDOL�DL�SUH XULORU�GH�FRQVXP�OD�P UIXULOH�DOLPHQWDUH�vQ�SHULRDGD��-1994 au fost: Tabelul 3.14’’

Anul Indice (anul precedent = 100) 1991 286,2 1992 336,6 1993 348,9 1994 236,2

Sursa: Anuarul Statistic al României, 1995, pg. 411.

6 �VH�GHWHUPLQH� LQGLFHOH�PHGLX� OD�SUH XULORU�GH�FRQVXP�OD�P UIXULOH�DOLPHQWDUH�� vQ�DFHDVW �

SHULRDG �

6ROX LH:

Pentru calculul indicelui PHGLX� YRP� XWLOL]D� PHGLD� JHRPHWULF �� vQWUXFkW� vQWUH� QLYHOXULOH�

YDULDELOHL�VWXGLDWH��LQGLFH�VH�DIO �R�UHOD LH�GH�SURGXV�

nig xx ∏=

%5,2989850,2362,2489,3366,3862,24

4 93/9492/9391/9290/914

==⋅⋅⋅

=⋅⋅⋅== ∏ pppppp IIIIII

Probleme propuse

1. 3HQWUX� FHOH� WUHL� DWHOLHUH� GH� SURGXF LH� GLQ� FDGUXO� XQXL� DJHnt economic cu profil industrial, se

FXQRVF�XUP WRDUHOH�GDWH��SULYLQG�OXQLOH�DSULOLH�úL�PDL�� Tabelul 3.15

Atelierul 3URGXF LD�UHDOL]DW �

în mai (buc.) 3URFHQWXO�GH�GLQDPLF �DO�SURGXF LHL�

UHDOL]DWH�vQ�PDL�ID �GH�DSULOLH 6WUXFWXUD�SURGXF LHL�

planificate în mai (%) A 200 125 25 B 100 120 35 C 300 150 40

ùWLLQG�F �SURGXF LD�WRWDO �SODQLILFDW �D�OXQLL�PDL�OD�QLYHOXO�vQWUHJXOXL�DJHQW�HFRQRPLF�HVWH�FX�

��PDL�PDUH�GHFkW�SURGXF LD�WRWDO �UHDOL]DW �D�OXQLL�DSULOLH��se cere: a) 6 � VH� FDOFXOH]H� QLYHOXULOH� SURGXF LLORU� UHDOL]DWH� vQ� OXQD� DSULOLH�� SUHFXP� úL� DOH� SURGXF LLORU�

SODQLILFDWH�vQ�OXQD�PDL��SHQWUX�ILHFDUH�DWHOLHU�úL�SH�WRWDO�DJHQW�HFRQRPLF� b) 6 �VH�GHWHUPLQH�VWUXFWXUD�SURGXF LHL�UHDOL]DWH�vQ�DSULOLH�úL�V �VH�UHSUH]LQWH�JUDILF� c) 6 �VH�FDOFXOH]H�R�P ULPH�UHODWLY �GH�FRRUGRQDUH�SHQWUX�SURGXF LD�UHDOL]DW �vQ�DSULOLH� d) 6 �VH�FDOFXOH]H�P ULPLOH�UHODWLYH�DOH�SODQXOXL�SH�ILHFDUH�DWHOLHU�úL�SH�WRWDO�DJHQW�HFRQRPLF�

2. Se cunosc datele: Tabelul 3.16 Departament Procentul sarcinii de

plan al cifrei de afaceri (%)

Procentul îndeplinirii planului la cifra de afaceri

(%)

Structura cifrei de afaceri UHDOL]DW �vQ�SHULRDGD�GH�ED] �

(%) A 110 114 30 B 125 120 40 C 115 125 30

Se cere:

a) 6 �VH�FDOFXOH]H�SURFHQWXO�PHGLX�DO�VDUFLQLL�GH�SODQ�úL�SURFHQWXO�PHGLX�DO�vQGHSOLQLULL�SODQXOui la cifra de afaceri.

b) 6 � VH� GHWHUPLQH� P ULPHD� UHODWLY � GH� GLQDPLF � D� FLIUHL� GH� DIDFHUL� OD� QLYHOXO� ILHF UXL�

GHSDUWDPHQW�úL�SH�DQVDPEOXO�VRFLHW LL�FRPHUFLDOH� c) 6 � VH� UHSUH]LQWH� JUDILF� VWUXFWXUD�FLIUHL�GH�DIDFHUL� UHDOL]DW � vQ� SHULRDGD�GH�ED] �� SUHFXP� úL�

P ULPLOH�UHODWLYH�DOH�SODQXOXL�SH�ILHFDUH�GHSDUWDPHQW�DO�VRFLHW LL�FRPHUFLDOH�

3. 6H�FXQRVF�GDWHOH��SULYLQG�YkQ] ULOH�GLQ�SDWUX�XQLW L�FRPHUFLDOH� Tabelul 3.17

Unitatea FRPHUFLDO

6WUXFWXUD�YDORULL�YkQ] ULORU�

vQ�SHULRDGD�GH�ED] � (%)

Procentul sarcinii de plan al valorii YkQ] ULORU��

Procentul îndeplinirii planului la valoarea YkQ] ULORU��

1 20 110 115 2 30 150 160 3 10 120 130 4 40 140 110

ùWLLQG�F �YDORDUHD�WRWDO �D�Yk] ULORU�SODQLILFDWH�D�IRVW�GH��PLO��OHL��V �VH�FDOFXOH]H:

a) 0 ULPLOH�UHODWLYH�DOH�SODQXOXL�úL�GH�GLQDPLF �SH�DQVDPEOXO�FHORU�WUHL�XQLW L�FRPHUFLDOH� b) 9DORDUHD�YkQ] ULORU�UHDOL]DWH�vQ�SHULRDGD�GH�ED] �SH�ILHFDUH�XQLWDWH�FRPHUFLDO �úL�SH�WRWDO� c) 6 �VH�UHSUH]LQWH�JUDILF�VWUXFWXUD�YDORULL�YkQ] ULORU�UHDOL]DWH�vQ�SHULRDGD�FXUHQW �

4. Se cuQRVF�XUP WRDUHOH�GDWH��SULYLQG�DFWLYLWDWHD�XQHL�VRFLHW L�FRPHUFLDOH�vQ�LDQXDULH�úL�IHEUXDULH�

1999: Tabelul 3.18 Filiala 6WUXFWXUD�SURGXF LHL�

realizate în feb. 1999 3URFHQWXO�VDUFLQLL�GH�SODQ�DO�SURGXF LHL�

feb/ian. (%) 3URFHQWXO�UHDOL] ULL�

planului în feb. (%) A 20 120 120 B 30 110 200 C 40 150 120 D 10 140 ?

ùWLLQG�F �SH�WRWDOXO�6�&��YDORDUHD�SODQLILFDW �D�SURGXF LHL�D�IRVW�GH��PLO��OHL�úL�F �LQGLFHOH�

UHDOL] ULL�SODQXOXL�SURGXF LHL�SH�WRWDO�6�&��D�IRVW�GH��se cere: a) 6 �VH�GHWHUPLQH�QLYHOXULOH�SURGXF LHL�UHDOL]DWH�vQ�LDQXDULH��QLYHOXULOH�SURGXF LHL�SODQLILFDWH�úL�

HIHFWLY�UHDOL]DWH�vQ�IHEUXDULH��SH�ILHFDUH�ILOLDO �úL�SH�WRWDO�6�&� b) 6 �VH�FDOFXOH]H�úL�V �VH�UHSUH]LQWH�JUDILF�VWUXFWXUD�SURGXF LHL�UHDOL]DWH�vQ�LDQXDULH� c) 6 �VH�GHWHUPLQH�P ULPHD�UHODWLY �D�UHDOL] ULL�SODQXOXL�SHQWUX�GHSDUWDPHQWXO�'� d) 6 �VH�FDOFXOH]H�SURFHQWXO�PHGLX�DO�VDUFLQLL�GH�SODQ�úL�DO�GLQDPLFLL�SURGXF LHL�vQ�IHEUXDULH�ID �

de ianuarie.

5. 6H� FXQRVF� XUP WRDUHOH� GDWH�� SULYLQG� DFWLYLWDWHD� D� WUHL� ILOLDOH� DOH� XQHL� VRFLHW L� FRPHUFLDOH� vn

OXQLOH�RFWRPEULH�úL�QRLHPEULH��

Tabelul 3.19 Filiala 5DSRUWXO�GH�FRRUGRQDUH�DO�YkQ] ULORU�UHDOL]DWH�vQ�

noiembrie 6WUXFWXUD�YkQ] ULORU�vQ�

octombrie A 2,0 0,5 B 1,0 0,3 C 1,5 0,2

ùWLLQG�F �YDORDUHD�WRWDO �D�YkQ] ULORU�VRFLHW LL�FRPHUFLDOH�SH�OXna octombrie a fost de 200 POG��OHL�úL�F �SHQWUX�ILOLDOD�%�P ULPHD�UHODWLY �GH�GLQDPLF �D�YkQ] ULORU�UHDOL]DWH�vQ�QRLHPEULH�ID �

de octombrie a fost de 150%, se cere:

a) 6 � VH� GHWHUPLQH� YDORDUHD� YkQ] ULORU� UHDOL]DWH� vQ� OXQD� RFWRPEULH� úL� D� FHOHL� UHDOL]DWH� vQ�

nRLHPEULH��SH�ILHFDUH�ILOLDO �úL�SH�WRWDO�6�&� b) 6 �VH�FDOFXOH]H�VWUXFWXUD�YDORULL�YkQ] ULORU�vQ�QRLHPEULH�úL�V �VH�UHSUH]LQWH�JUDILF� c) 6 � VH� GHWHUPLQH�P ULPLOH� UHODWLYH� GH� GLQDPLF �D� YDORULL� YkQ] ULORU� SH� ILHFDUH� ILOLDO � úL� SH�

total.

6. Despre activitatea a 3 HFKLSH�DOH�XQXL�DWHOLHU��SHQWUX�OXQLOH�DSULOLH�úL�PDL��VH�FXQRVF�GDWHOH�

Tabelul 3.20 Echipa 6WUXFWXUD�SURGXF LHL�

în aprilie (%) ,QGLFHOH�GH�GLQDPLF �DO�YDORULL�

SURGXF LHL�PDL�DSU� 0 ULPHD�UHODWLY �D�UHDOL] ULL�

SODQXOXL�SURGXF LHL�vQ�PDL 1 25 1,6 1,33 2 30 2,0 2,5 3 35 1,8 1,5 4 10 0,7 0,5

ùWLLQG�F �YDORDUHD�SURGXF LHL�UHDOL]DWH�vQ�DSULOLH�GH�HFKLSD��D�IRVW�GH��PLO��OHL��V �VH�GHWHUPLQH� a) 1LYHOXO�SURGXF LHL�UHDOL]DWH�vQ�DSULOLH��DO�SURGXF LHL�SODQLILFDWH�úL�HIHFWLY�UHDOL]DWH�vQ�PDL��SH�

fiecare HFKLS �úL�SH�WRWDO� b) 6WUXFWXUD�SURGXF LHL�UHDOL]DWH�vQ�PDL�úL�D�FHOHL�SODQLILFDWH�vQ�PDL� c) 0 ULPHD�UHODWLY �D�VDUFLQLL�GH�SODQ�SH�ILHFDUH�HFKLS � d) 0DULPLOH�UHODWLYH�GH�GLQDPLF �úL�DOH�SODQXOXL�SH�WRWDO�DWHOLHU�

7. Se cunosc datele: Tabelul 3.21

6HF LD StructXUD�SURGXF LHL�vQ�SHULRDGD�GH�ED] ��

Procentul sarcinii de plan (%)

Procentul îndeplinirii planului (%)

I 25 105,2 100,0 II 35 110,0 98,0 III 40 107,0 105,0

1) Se cere: a) ùWLLQG� F �YROXPXO�DFWLYLW LL� GLQ�SHULRDGD�GH�ED] � D� IRVW� GH� ��PLO�� OHL�� V � VH� Ueprezinte

JUDILF�VWUXFWXUD�SURGXF LHL�vQ�SHULRDGD�FXUHQW �úL�vQ�SHULRDGD�GH�ED] � b) 6 �VH�FDOFXOH]H�SURFHQWXO�PHGLX�DO�VDUFLQLL�GH�SODQ�úL�DO�vQGHSOLQLULL�SODQXOXL� c) ùWLLQG� F � vQ� vQWUHSULQGHUH� DX� OXFUDW� vQ� SHULRDGD�GH� ED] � �� GH�PXQFLWRUL�� LDU� vQ� SHULRDGD�

cuUHQW � OXFUHD] � �� GH�PXQFLWRUL�� V � VH� FDOFXOH]H� LQGLFHOH� GH� GLQDPLF � DO� SURGXFWLYLW LL�

PXQFLL�vQ�SHULRDGD�FXUHQW �ID �GH�SHULRDGD�GH�ED] � d) 6 � VH� DUDWH� SHQWUX� FDUH� GLQWUH� SHULRDGH� �GH� ED] � VDX� FXUHQW �� FROHFWLYLWDWHD� HVWH� PDL�

RPRJHQ �GLQ�SXQFWXO�GH�YHGHUH�DO�SURGXF LHL�

8. 3HQWUX�XUP WRDUHOH�GDWH��SUHFL]D L�FDUH�DILUPD LH�HVWH�DGHY UDW �

a) 14== Mex ; b) Me = Mo = 20; c) Me = 13; d) Me = 11,5; e) Me = 16.

9. &RQVLGHUkQG�XUP WRDUHOH�YDORUL��H[SOLFD L�ce indicator vi se

SDUH�PDL�SRWULYLW�SHQWUX�D�H[SULPD�WHQGLQ D�FHQWUDO �

10. /D�VIkUúLWXO�DQXOXL��0F�'RQDOG¶V�DYHD��GH�UHVWDXUDQWH�úL�XQ�WRWDO�GH�YkQ] UL�DQXDOH�GH�

��PLOLDUGH��%XUJHU�.LQJ�DYHD��GH�UHVWDXUDQWH�úL��PLOLDUGH��0F�'RQDOG¶V�GHschide 500 de noi restaurante pe an, iar Burger King 200 pe an. Se cere: a) &DOFXOD L�PHGLD�YkQ] ULORU�SHQWUX�0F�'RQDOG¶V�vQ��úL�FRPSDUD L-o cu cea a lui Burger

King. b) &kWH�UHVWDXUDQWH�VH�GHVFKLG��vQ�PHGLH��SH�OXQ ��SH�V SW PkQ ��SH�]L��SHQWUX�ILHFDUH�ILUP � c) &DUH�HVWH�VXUSOXVXO�GH�YkQ] UL�UDOL]DWH�SH�V SW PkQ �SULQ�GHVFKLGHUHD�D�QRL�UHVWDXUDQWH�

11. * VL L�PHGLD��PHGLDQD�úL�PRGXO�GLQ�XUP WRDUHOH�YDORUL��'HPRQVWUD L�FXP�

ILHFDUH�GLQ�LQGLFDWRULL�WHQGLQ HL�FHQWUDOH�VH�VFKLPE ��GDF �XQD�GLQ�YDORULOH��VH�vQORFXLHúWH�FX��. &XP�YD�DIHFWD�LQGLFDWRULL�WHQGLQ HL�FHQWUDOH�vQORFXLUHD�XQHL�YDORUL��FX�YDORDUHD��"

'HPRQVWUD L�SH�ED]D�YDORULORU�LQL LDOH�SURSULHWDWHD�PHGLHL�DULWPHWLFH� ( ) 0=∑ − xxi .

12. În anul 1994, indicele mediu lunar al� SUH XULORU� GH� FRQVXP� OD� VHUYLFLL� D� IRVW� SHQWUX� SHULRDGD�

ianuarie –�DXJXVW��úL�SHQWUX�SHULRDGD�VHSWHPEULH-decembrie 1,0319. Care a fost indicele mediu lunar pe întregul an 1994?

13. 8Q�FRQGXF WRU� DXWR�FXPS U � XOHL� GH�PRWRU�� OD� SUH XO� GH� �� OHL�OLWUX�� vQ�YDORDUH� WRWDO � GH�

100.000 lei dintr-XQ�PDJD]LQ�úL� vQ�YDORDUH�GH��OHL�GLQ�DOW�PDJD]LQ��OD�SUH XO�GH��OHL�OLWUX��&DUH�D�IRVW�SUH XO�PHGLX�SH�OLWUX�SH�FDUH�O-D�SO WLW"

44

ÍQWUHE UL ¾ &XP�VH�UHSUH]LQW �JUDILF�R�VHULH�GH�GLVWULEX LH�GH�IUHFYHQ H" ¾ ,QGLFDWRULL�IUHFYHQ HORU��PRG�GH�FDOFXO��VHPQLILFD LH��UHSUH]HQWDUH�JUDILF � ¾ 'HWHUPLQDUHD�LQGLFDWRULORU�WHQGLQ HL�FHQWUDOH�vQWU-R�VHULH�GH�GLVWULEX LH�GH�IUHFYHQ H� ¾ &H�FRQFOX]LL�SXWH L�IRUPXOD�SH�ED]D�UHOD LHL�GH�RUGLQH�vQWUH�PHGLH��PHGLDQ �úL�PRG" ¾ Ce sunt cuartilele? Dar decilele? ¾ &DUH�VXQW�SULQFLSDOHOH�YDORUL�FRJQLWLYH�DOH�LQGLFDWRULORU�VLPSOL�úL�VLQWHWLFL�DL�YDULD LHL" ¾ &RPSDUDUHD�RPRJHQLW LL�D�GRX �VHULL� ¾ &DUH�HVWH�XQLWDWHD�GH�P VXU �D�DEDWHULL�VWDQGDUG"�'DU�D�GLVSHUVLHL"�'DU�D�FRHILFLHQWXOXL�GH�YDULD LH" ¾ &XP�VH�GHWHUPLQ �PHGLD�úL�GLVSHUVLD�XQHL�FDUDFWHULVWLFL�DOWHUQDWLYH" ¾ &DUH�VXQW�SULQFLSDOLL�LQGLFDWRUL�GH�DQDOL] �D�IRUPHL�GLVWULEX LHL" ¾ 5HJXOD�GH�DGXQDUH�D�GLVSHUVLLORU��PRG�GH�FDOFXO��XWLOL]DUH��UHOD LL�vQWUH�GLIHULWHOH�WLSXUL�GH�dispersii. ¾ &DOFXOXO�úL�LQWHUSUHWDUHD�FRHILFLHQWXOXL�GH�GHWHUPLQD LH�

$1$/,=$�6(5,,/25�'(�5(3$57, ,(�

UNIDIMENSIONALE


1. 'LUHFWRUXO�XQXL�SRVW�GH�UDGLR�FRPDQG �R�FHUFHWDUH�VWDWLVWLF �vQ�VFRSXO�GHWHUPLQ ULL�DXGLHQ HL�GH�

FDUH� VH� EXFXU � SRVWXO� V X�� &HUFHW WRUXO� FXOHJH� GDWH� SULYLQG� QXP UXO� GH� RUH� DXGLDWH� SH� XQ�

HúDQWLRQ�úL�DIO �F ��GLQ�SHUVRDQH�DVFXOW �SRVWXO�GH�UDGLR�PDL�SX LQ�GH��RUH��vQWUH��úL��

RUH��vQWUH��úL��RUH��vQWUH��úL��RUH��LDU�UHVWXO�SHVWH��RUH��(O�XUP UHúWH�GHWHUPLQDUHD�

QXP UXOXL�PHGLX�GH�RUH�GH�DXGL LH�SH�R�SHUVRDQ � 6ROX LH: Datele pot fi sistematizate ca în tabelul 4.1: Tabelul 4.1.

,QWHUYDOH�H�YDULD LH�D�

QXP UXOXL�GH�RUH�GH�DXGL LH 1XP U�GH�SHUVRDQH�� xi *

%ii nx

sub 2 10 1 10 2 – 4 25 3 75 4 – 6 40 5 200 6 – 8 20 7 140

��úL peste 5 9 45 Total 100 - 470

6H�YD�XWLOL]D�PHGLD�DULWPHWLF �SRQGHUDW ��FDOFXODW �SH�ED]D�IUHFYHQ HORU�UHODWLYH�

ore 7,4100

470

100

*% ==∑= ii nx

x

$úDGDU��vQ�PHGLH��R�SHUVRDQ �GLQ�HúDQWLRQXO�VWXGLDW�DXGLD] ��RUH�GH�SURJUDP�OD�SRVWXO�GH�

radio respectiv.

2. CânG� GRX � ILUPH� DQXQ � GRULQ D� GH� D� IX]LRQD�� VH� vQWkPSO � GHVHRUL� FD� vQ� GHFXUV� GH� FkWHYD�

V SW PkQL�� XQD� GLQ� ILUPH� V � GHYLQ � QHPXO XPLW � GH� FRQVHFLQ HOH� IX]LRQ ULL� úL� DFHDVWD� V � ILH�

UH]LOLDW �� 'H� DFHHD�� GLQ� PRPHQWXO� vQ� FDUH� VH� DQXQ � IX]LRQDUHD�� H[LVW � R� PDUH� GR] � GH�

LQFHUWLWXGLQH� GDF � YD� DYHD� ORF� HIHFWLY� úL� SRW� WUHFH� FKLDU� OXQL� GH� ]LOH� GXS � DQXQ � SkQ � FkQG�

IX]LRQDUHD�YD�DYHD�ORF��8Q�QXP U�GH��GH�ILUPH�DX�IRVW�JUXSDWH�GXS �QXP UXO�GH�]LOH�GH�OD�

GDWD�DQXQ XOXL�SRWHQ LDOHL�IX]LRQ UL�SkQ �OD�GDWD�IX]LRQ ULL�HIHFtive (Tabelul 4.2.) Tabelul 4.2.

,QWHUYDOH�GH�YDULD LH�D�QXP UXOXL�GH�]LOH 1XP U�GH�ILUPH 40 - 80 50

80 - 120 70 120 – 160 90 160 – 200 100 200 – 240 50 240 – 280 20

Total 380

$QDOL]D�DFHVWHL�VHULL�GH�UHSDUWL LH�GH�IUHFYHQ H�XUP UHúWH�GHWHUPLQDUHD�QXP UXOXL�PHGLX�GH�

]LOH�³GH�DúWHSWDUH´��SUHFXP�úL�D�LQGLFDWRULORU�PHGLL�SR]L LH�

6ROX LH: 5HSUH]HQWDUHD� JUDILF � D� GLVWULEX LHL� GH� IUHFYHQ H� FX� DMXWRUXO� KLVWRJUDPHL� úL� SROLJRQXOXL�

IUHFYHQ HORU��DUH�FD�VFRS�HYLGHQ LHUHD�IRUPHL�UHSDUWL LHL�úL�WHQGLQ HL�GH�QRUPDOLWDWH��)LJXULOH��úL��

,QGLFDWRUXO� SRWULYLW� SHQWUX� D� GHVFULH� WHQGLQ D� FHQWUDO � HVWH�PHGLD� DULWPHWLF �� x �� GHWHUPLQDW � SH�baza datelor din tabelul 4.3.

∑∑=

i

ii

n

nxx

��

!" # $" %� &' ( &)!�

+*#

# ,*%�

%� ,*&)!"

&)!� -&)$�

&)$" -!" �

!" � -! #

! # -!"%�

.0/ 1'2�3+4(5 6 3

7 8:9; <= > 8?<

Fig. 4.1.

3ROLJRQXO�IUHFYHQ HORU

�

��

��

��

��

��

��

��

��

��

��

��±

��

��±

��

��±

��

��±

��

��

��

1U��GH�]LOH

1U�

�GH

�ILU

PH

Fig.4.2. Tabelul 4.3. ,QWHUYDOH�GH�YDULD LH�D�QU��GH�]LOH ni xi xi ni Fi

0 1 2 3 4 40 – 80 50 60 3000 50 80 – 120 70 100 7000 120

120 – 160 90 140 12600 210 160 – 200 100 180 18000 310 200 – 240 50 220 11000 360 240 – 280 20 260 5200 380

Total 380 - 56800 -

unde: ni �IUHFYHQ HOH�GH�DSDUL LH��QXP UXO�GH�ILUPH�FH�VH�vQFDGUHD] �vQ�ILHFDUH�LQWHUYDO�GH�YDULD LH�D�

³SHULRDGHL�GH�DúWHSWDUH´� Fi� �IUHFYHQ HOH�FXPXODWH� xi = centrele de interval:

rixxh

xx iiiii ,1 ,

22

supinfinf =

+=+= (r = num UXO�LQWHUYDOHORU�

zile 47,149380

56800 ==∑

∑=i

ii

n

nxx

ÌQ�PHGLH��³SHULRDGD�GH�DúWHSWDUH´�vQWUH�GDWD�DQXQ XOXL�SRWHQ LDOHL�IX]LXQL�úL�GDWD�IX]LRQ ULL�HIHFWLYH�

este de 149,47 zile. ,QGLFDWRULL�PHGLL�GH�SR]L LH�

Mediana (Me)�VH�GHWHUPLQ �SULQ�LQWHUSRODUH�SH�LQWHUYDlul median:

5,1902

381

2

1

2

1 ==∑ +

=+= innMeloc

1XP UXO�PHGLDQ�GH�]LOH�VH�J VHúWH�vQWUH�QXP UXO�GH�]LOH�DO�FHOHL�GH-a 190-D�ILUP �úL�DO�FHOHL�

de-a 191-a. Intervalul median: [120-��]LOH�HVWH�SULPXO�LQWHUYDO�vQ�FDUH�IUHFYHQ D�FXPXODW �Fi este mai

mare decât locul�PHGLDQHL��GHRDUHFH�SkQ �OD�OLPLWD�VXSHULRDU �D�DFHVWXL�LQWHUYDO�VH�J VHVF��ILUPH�

�GHFL�vQ�DFHVW�LQWHUYDO�VH�YRU�J VL�ILUPHOH�FX�QXP UXO��úL��

Me

Me

n

FMelockxMe 1

0 −−

+=

unde: x0� �OLPLWD�LQIHULRDU �D�LQWHUYDOXOXL�PHGLDQ�� k� �P ULPea intervalului median = 40; FMe-1� �IUHFYHQ D�FXPXODW �D�LQWHUYDOXOXL�DQWHULRU�FHOXL�PHGLDQ� �� nMe� �IUHFYHQ D�LQWHUYDOXOXL�PHGLDQ� ��

zile 33,15190

1205,19040120 =−+=Me

$úDGDU��GLQ�ILUPH�DX�R�SHULRDG �GH�DúWHSWDUH�PDL�PLF �GHFkW�YDORDUHD�PHGLDQHL��51,33

]LOH��LDU��GLQ�ILUPH�DX�R�SHULRDG �GH�DúWHSWDUH�PDL�PDUH�GHFkW�YDORDUHD�PHGLDQHL� 6H�SRDWH�SXQH�SUREOHPD�GHWHUPLQ ULL�QXP UXOXL�GH�]LOH�GH�DúWHSWDUH�SkQ �OD�FDUH�VH�VLWXHD] �

cele mai rapide 25% din firme, quartila 1 (Q1):

25,954

381

4

1

4

1 1 ==∑ +

=+= innQloc

Intervalul quartilei 1: [80 –��HVWH�SULPXO�LQWHUYDO�FDU�DUH�IUHFYHQ D�FXPXODW �Fi mai mare GHFkW�ORFXO�TXDUWLOHL��3kQ �OD�OLPLWD�VXSHULRDU �D�DFHVWXL�LQWHUYDO�VH�VLWXHD] ��GH�ILUPH�

1

1 1101

Q

Q

n

FQlockxQ

−−+=

unde: x0 = limita inferioDU �D�LQWHUYDOXOXL�TXDUWLOHL�� k� �P ULPHD�LQWHUYDOXOXL�TXDUWLOHL�� FQ1-1� �IUHFYHQ D�FXPXODW �SkQ �OD�LQWHUYDOXO�TXDUWLOHL�� nQ1 �IUHFYHQ D�LQWHUYDOXOXL�SULPHL�TXDUWLOH�

zile 86,10570

5025,9540801 =−+=Q

$úDGDU��FHOH�PDL�UDSLGH��GLQ�ILUPH�vQ�SULYLQ D�IX]LRQ ULL�DX�XQ�QXP U�GH�]LOH�GH�DúWHSWDUH�

GH�SkQ �OD��

Quartila 2 (Q2)�HVWH�HJDO �vQWRWGHDXQD�FX�0H��Q2 = 151,33 zile

Quartila 3 (Q3)� LGHQWLILF � QXP UXO� GH� ]LOH� SHVWH� FDUH� VH� VLWXHD] � FHOH�PDL� OHQWH� �� GLQ�

firme, din punctul de vedere al deciziei de fuzionare:

75,2854

3813

4

)1(3

4

)1(3 3 =⋅=∑ +

=+= innQloc

Intervalul quartilei 3: [160 - 200).

3

3 1303

Q

Q

n

FQlockxQ

−−+=

unde: x0� �OLPLWD�LQIHULRDU �D�LQWHUYDOXOXL�TXDUWLOHL�� k� �P ULPHD�LQWHUYDOXOXL�TXDUWLOHL�� FQ3-1� �IUHFYHQ D�FXPXODW �SkQ �OD�LQWHUYalul quartilei 3; nQ3 �IUHFYHQ D�LQWHUYDOXOXL�TXDUWLOHL��

zile 3,190100

21075,285401603 =−+=Q

&HOH�PDL�OHQWH��GLQ�ILUPH�vQ�SULYLQ D�IX]LRQ ULL�DX�XQ�WLPS�GH�DúWHSWDUH�PDL�PDUH de 190,3 zile. Modul (Mo)�LGHQWLILF �QXP UXO�GH�]LOH�GH�DúWHSWDUH�SH�FDUH�vO�vQUHJLVWUHD] �FHOH�PDL�PXOWH�ILUPH��

vQ�SULYLQ D�IX]LRQ ULL��(O�VH�GHWHUPLQ �SULQ�LQWHUSRODUH�SH�LQWHUYDOXO�FX�IUHFYHQ �PD[LP ��– 200).

21

10 ∆+∆

∆+= kxMo

unde: x0� �OLPLWD�LQIHULRDU �D�LQWHUYDOXOXL�PRGDO� k� �P ULPHD�LQWHUYDOXOXL�PRGDO� ∆1 = nMo – nMo-1� �IUHFYHQ D� LQWHUYDOXOXL�PRGDO�PLQXV�IUHFYHQ D�LQWHUYDOXOXL�DQWHULRU�FHOXL�

modal; ∆2 = nMo - nMo+1� �IUHFYHQ D�LQWHUYDOXOXL�PRGDO�PLQXV�IUHFYHQ D�LQWHUYDOXOXL�XUP WRU�FHOXL�

modal;

zile 67,166)50100()90100(

9010040160 =

−+−−+=Mo

$úDGDU�� FHOH� PDL� PXOWH� ILUPH� vQUHJLVWUHD] � R� SHULRDG � GH� DúWHSWDUH� GH� �� ]LOH�� 'LQ�

YDORULOH� FHORU� WUHL� LQGLFDWRUL� DL� WHQGLQ HL� FHQWUDOH� � MoMex << �� SUHFXP� úL� GLQ� KLVWRJUDP �� VH�

REVHUY �F � UHSDUWL LD� GH� IUHFYHQ H�QX�HVWH� SHUIHFW� VLPHWULF �� vQ� HD� SUHGRPLQkQG� � YDORULOH�PLci ale YDULDELOHL�³QXP U�GH�]LOH�GH�DúWHSWDUH´�

3. 'LVWULEX LD� UDWHL�SURILWXOXL�SH� OXFUDUH�� vQUHJLVWUDW�SHQWUX� ��GH� OXFU UL�GH� FRQVWUXF LL�� OD� GRX �

PDUL�FRPSDQLL�GH�FRQVWUXF LL�VH�SUH]LQW �DVWIHO��GDWH�FRQYHQ LRQDOH��

)LJ��'LVWULEX LD�OXFU ULORU�GXS �UDWD�SURILWXOXL�

6 �VH�DQDOL]H]H� WHQGLQ D�FHQWUDO �úL�YDULD LD�GLQ�FHOH�GRX �FROHFWLYLW L�SHQWUX�FDUDFWHULVWLFD�

SUH]HQWDW � 6ROX LH� 'HRDUHFH� GLVWULEX LLOH� GH� IUHFYHQ H� VXQW� SHUIHFW� VLPHWULFH�� UDWD� PHGLH� D� SURILWXOXL� YD� IL��

evident, centrul de inWHUYDO�FX�IUHFYHQ D�PD[LP �

%15=Ax %15=Bx ,QGLFDWRULL�VLPSOL�DL�YDULD LHL��FDOFXOD L�SHQWUX�FHOH�GRX �FROHFWLYLW L�VXQW� Pentru compania A Tabelul 4.4. ,QWHUYDOH�GH�YDULD LH�

a ratei profitului (%) 1U��GH�OXFU UL

ni xi di

(%) inxxi − ( ) inxxi2

−

0 1 2 3 4 5 -10 – 0 15 -5 -20 300 6000 0 – 10 20 5 -10 200 2000

10 – 20 30 15 0 0 0 20 – 30 20 25 10 200 2000 30 – 40 15 35 20 300 6000 Total 100 - - 1000 16.000

$PSOLWXGLQHD�YDULD LHL� A = xmax - xmin AA = 40 – (-10) = 50 % $EDWHUHD�LQGLYLGXDO ��FRO��WDEHOXO��

xxd ii −= $EDWHUHD�PD[LP �SR]LWLY �

xxd poz −= maxmax

%251540 max =−=Apozd

D��&RPSDQLD�$

@BA C"DE D

C�D @)A

FG�FH)FIBFJ'FKBFL)FM)FNBF

O G�F O FPF O G�FQG�F O H)FRH)F O IBFSIBF O J'F5DWD�SURILWXOXL��

1U�

�OX

FU

UL

T�UWVYX�Z\[�]W^�_ ]a`

b c:d

e d

c�d bfghfi�fj�fk fl fm�fn fo�f

p g'f p f f p ghf ghf p i�fqi�f p j�frj�f p k fsut"v txw�y z�{ | v }�~ }�|h� ��

� ��

$EDWHUHD�PD[LP �QHJDWLY �

xxd neg −= min max

%251510 max −=−−=Anegd

Pentru compania B AB = 40 – (-10) = 50 %

di (calculat în col. 3, tabelul 4.5): Tabelul 4.5. ,QWHUYDOH�GH�YDULD LH�D�

ratei profitului (%) 1U��GH�OXFU UL

ni xi di

(%) inxxi − h

axi −

i

i nh

ax 2

−

0 1 2 3 4 5 6 -10 – 0 5 -5 -20 100 -2 20 0 – 10 10 5 -10 100 -1 10 10 – 20 70 15 0 0 0 0 20 – 30 10 25 10 100 1 10 30 – 40 5 35 20 100 2 20 Total 100 - - 400 - 60

%251540 max =−=Bpozd

%251510 max −=−−=Bnegd

'XS �FXP�VH�REVHUY ��LQGLFDWRULL�VLPSOL�DL�YDULD LHL�VXQW�HJDOL�SHQWUX�FHOH�GRX �FRPSDQLL�úL�

QX� RIHU � LQIRUPD LL� VXILFLHQWH� SHQWUX� FDUDFWHUL]DUHD� VLQWHWLF � úL� FRPSDUDUHD� YDULDELOLW LL� vQ�

FROHFWLYLW L��'HúL�DPSOLWXGLQLOH�VXQW�HJDOH�úL�LQGLFDWRULL�WHQGLQ HL�FHQWUDOH�VXQW�DFHLDúL��SHQWUX�DFHVWH�

GRX �VHWXUL�GH�GDWH�VLPHWULFH��H[LVW �R�GLIHUHQ �HYLGHQW �vQWUH�FHOH�GRX �FRPSDQLL�SULYLQG�GLVWULEX LD�

UDWHL�SURILWXOXL��'LIHUHQ D�HVWH�DFHHD�F �OD�FRPSDQLD�%�UDWD�SURILWXOXL�WLQGH�V �VH�IRFDOL]H]H�vQ�MXUXO�

valorii centrale. Prin contrast, rata profitului la coPSDQLD�$�HVWH�PXOW�PDL� vPSU úWLDW �GH-a lungul DPSOLWXGLQLL��LQGLFkQG�R�LQFLGHQ �FUHVFXW �D�OXFU ULORU�FX�UDW �PDUH�D�SURILWXOXL��GDU�GH�DVHPHQHD�úL�

D� FHORU� FX� ULVF� FUHVFXW� GH� SLHUGHUH�� ,QGLFDWRULL� VLQWHWLFL� DL� YDULD LHL� YRU� OXD� vQ� FRQVLGHUDUH� WRDWH�

valorile înregistrate: Pentru compania A

$EDWHUHD�PHGLH�OLQLDU �

∑

∑ −=

i

ii

n

nxxd

'LQ�FRO��WDEHOXO��UH]XOW �F � %10100

1000 ==Ad

'LVSHUVLD�VH�RE LQH�DSOLFkQG�IRUPXOD�

( )

∑∑ −=

i

ii

n

nxx2

2σ

'LQ�FRORDQD��D�WDEHOXOXL��UH]XOW �F � 160100

160002 ==Aσ

6H�SRDWH�DSOLFD�FRUHF LD�OXL�6KHSSDUG��SHQWUX�D�HOLPLQD�HIHFWXO�GH�JUXSDUH�

12

222 h

cor −= σσ

67,15112

1001602

=−=corAσ

$EDWHUHD�PHGLH�S WUDWLF �VH�RE LQH�FX�IRUPXOD�

( ) 2

2

σσ =∑

∑ −=i

ii

n

nxx

%32,1267,151 ==Aσ &RHILFLHQWXO�GH�YDULD LH�

100x

vσ=

%1,8210015

32,12 ==Av

Pentru compania B vom utiliza datele din tabelul 4.6.:

%4=Bd

( )22

2

2 axhn

nh

ax

i

ii

−−∑

∑

−

=σ

a = 15, h = 10 (vezi col. 6 din tabelul 4.6.)

( ) 601515100100

60 22 =−−⋅=Bσ

67,5112

100602

=−=corBσ

%188,767,51 ==Bσ

%92,4710015

188,7 ==Bv

$úDGDU��GDF �SHQWUX�FRPSDQLD�$�UDWD�SURILWXOXL�SH�OXFUDUH�VH�DEDWH�vQ�PHGLH�FX��ID �

de rata medie de 15%, în schimb pentru compania B, rata profitului se abate în medie FX��ID �

GH�UDWD�PHGLH�D�SURILWXOXL�GH��DU WkQG�R�VWDELOLWDWH�FUHVFXW �D�YDORULORU��YA < vB).

4. Într-R�FROHFWLYLWDWH�VWDWLVWLF �V-DX�FXOHV�GDWH�SULYLWRDUH�OD�GRX �YDULDELOH�QXPHULFH��RE LQkQGX-se:

{ } { }1,2,3,6,5,1,1,4=ix úL�{ } { }70,100,50,70,80,40,90,100=iy

6 �VH�DUDWH�GXS �FDUH�GLQ�YDULDELOH�FROHFWLYLWDWHD�HVWH�PDL�RPRJHQ � 6ROX LH� Pentru variabila X vom calcula:

875,28

23 ==∑=n

xx i

( )

839,37

875,26

1

22 ==

−∑ −=

n

xxs i

x

sau prin metoda momentelor:

( ) ( )839,3

7

125,6693

78

2393

1

222

2 =−=−

=−

∑ ∑−=

nn

xx

s

ii

x

959,12 == xx ss

%1,68100875,2

959,1100 ===

x

sv x

x

iar pentru variabila Y:

758

600 ==∑=n

yy i

( )

71,4857

3400

1

22 ==

−∑ −=

n

yys i

y

039,222 == yy ss

%38,2910075

039,22100 ===

y

sv

yy

6HULD� IRUPDW � GXS � FDUDFWHULVWLFD� <� HVWH�� GHFL�� PDL� RPRJHQ �� GHFkW� FHD� IRUPDW � GXS �

caracteristica X.

5. Un fDEULFDQW� GH� EDWHULL� GH� DXWRPRELOH� GHWHUPLQ �GXUDWD�PHGLH� GH� IXQF LRQDUH� D� EDWHULLORU�� FD�

ILLQG��GH�OXQL��FX�R�DEDWHUH�PHGLH�S WUDWLF �GH��OXQL��3UHVXSXQkQG�F �GLVWULEX LD�GH�IUHFYHQ H�

D�GXUDWHL�GH�IXQF LRQDUH�HVWH�FXQRVFXW �FD�ILLQG�vQ�IRUPD�FORSRWXOXL lui Gauss-Laplace, se cere: a) $SUR[LPDWLY� FH�SURFHQW�GLQ� EDWHULLOH� IDEULFDWH� YRU� DYHD�PDL�PXOW� GH� ��GH� OXQL� GXUDW � GH�

IXQF LRQDUH"�'DU�R�R�GXUDW �PDL�PLF �GH��GH�OXQL" b) &DUH�HVWH�P ULPHD�DSUR[LPDWLY �D�DPSOLWXGLQLL�YDULD LHL�SHQWUX�GXUDWD�GH�IXQF LRQDUH" c) &H�SXWH L�VSXQH�GHVSUH�DEDWHUHD�PHGLH�OLQLDU "�'DU�GHVSUH�DEDWHUHD�LQWHUTXDUWLOLF "

6ROX LH�

a) &RQIRUP� UHJXOLL� HPSLULFH� FH� GHFXUJH� GLQ� WHRULD� OXL� &HEkúHY�� SXWHP� FRQVWUXL� JUDILFXO�

GLVWULEX LHL�GXUDWHL�GH�IXQF LRQDUH��FD�vQ�ILJXUD�� 6H�REVHUY � LPHGLDW�F � vQ�FD]XO�GLVWULEX LHL� VLPHWULFH�VXE�IRUP �GH�FORSRW�SURFHQWXO�

EDWHULLORU�FX�GXUDW �GH�IXQF LRQDUH�PDL�PDUH�GH��GH�OXQL�YD�IL�DSUR[LPDWLY��vQWUH��úL�

60 de luni) + 50% (mai mare de 60 de luni) = 84%. 'H�DVHPHQHD��SURFHQWXO�EDWHULLORU�GXS �GXUDWD�GH�IXQF LRQDUH�GH��GH�OXQL�YD�IL�DSUR[LPDWLY��

b) 'XUDWD�GH�IXQF LRQDUH�D�PDMRULW LL�FRYkUúLWRDUH�D�EDWHULLORU�YD�IL�FXSULQV �DSUR[LPDWLY�vQWUH�

σ3−x �úL� σ3+x , deci între 60-�[��úL��[��OXQL��$WXQFL� luni 603090minmax ≈−≈−= xxA

$úDGDU��QH�DúWHSW P�FD� vQWUH�R�EDWHULH�FX�GXUDW �GH�IXQF LRQDUH�PD[LP �úL�FHD�FX�R�GXUDW �

PLQLP �GH�IXQF LRQDUH�V �VH�vQUHJLVWUH]H�R�YDULD LH�GH��GH�OXQL�

c) &XP�UHSDUWL LD�VH�SUHVXSXQH�F �HVWH�VLPHWULF ��DWXQFL�

σ5

4≈d

Y FrecYHQ H�UHODWLYH

≈34% ≈34% 68% 95% ≈2,5% ≈2,5%

30 40 50 =x 60 70 80 90 X Durata GH�IXQF LRQDUH��OXQL�

Fig. 4.4.

luni 8105

4 ≈⋅≈d

σ3

2≈Q

luni 67,6103

2 ≈⋅≈Q

$úDGDU�� vQWUH�FHOH��GLQ�EDWHULL� VLWXDWH� vQ�PLMORFXO�GLVWULEX LHL�H[LVW �R� DEDWHUH� HJDO �FX�

DPSOLWXGLQHD�LQWHUTXDUWLOLF � luni 34,132 13 =−= QQQ iar cele mai performante 25% din baterii, din punctul de vedere al variabilei studiate, vor avea GXUDWD�GH�IXQF LRQDUH�PDL�PDUH�GH��OXQL� 'LVWULEX LD�ILLQG�SUHVXSXV �VLPHWULF ��UH]XOW �F

luni

QMeQMeQQQMeMeQ

67,6667,6603313

=+==+=⇒−=⇒−=−

6. 2� PDUH� DJHQ LH� GH� vQFKLULDW� DXWRPRELOH� KRW U úWH� V -úL� YkQG � DXWRPRELOHOH� GXS � XWLOL]DUHD�

acestora timp de un an. Managerul firmei preVXSXQH�F �GLVWDQ D�SDUFXUV �GH�PDúLQL�LQIOXHQ HD] �

FRVWXO� GH� vQWUH LQHUH� DO� DXWRYHKLFXOHORU� úL� GHFL� SUH XO� GH� YkQ]DUH� DO� DFHVWRUD�� 3HQWUX� D� YHULILFD�

DFHDVW � SUHVXSXQHUH�� VH� vQUHJLVWUHD] �� SHQWUX� XQ� QXP U� GH� �� GH�PDúLQL�� GLVWDQ D� SDUFXUV � vQ�

ultimul an (îQ�PLL� NP�� úL� FRVWXO� GH� vQWUH LQHUH�DO� DFHVWRUD� �vQ� VXWH�PLL� OHL�� 6H� DOF WXLHVF� SDWUX�

JUXSH�GXS �GLVWDQ D�SDUFXUV ��VXE��PLL�NP��-60 mii km, 60-��PLL�NP��úL�SHVWH��PLL�NP.

Tabelul 4.7.

&RVW�GH�vQWUH LQHUH��VXWH�PLL�Oei) *UXSH�GXS �

GLVWDQ D�SDUFXUV 5 – 7 7 – 9 9 – 11 11 – 13 Total

20 – 40 18 28 14 - 60 40 – 60 10% 25% 35% 30% 100%

ùWLLQG�F � � pentru grupa a 3-a (60 – 80 mii km) s-au înregistrat date pentru 40 de autovehicule, costul

PHGLX�GH�vQWUH LQHUH�HVWH�GH��OHL��FX�XQ�FRHILFLHQW�GH�YDULD LH�GH��LDU � pentru grupa a 4-D��FH�FXSULQGH��GLQ�DXWRYHKLFXOH��FKHOWXLHOLOH�WRWDOH�GH�vQWUH LQHUH�DX�IRVW�

de 264 sute mii lei, iar abaterea standard de 0,98 sute mii lei, se cere: a) 6 � VH� P VRDUH� LQIOXHQ D� GLVWDQ HL� SDUFXUVH� DVXSUD� YDULD LHL� FKHOWXLHOLORU� GH� vQWUH LQHUH��

XWLOL]kQG�FRHILFLHQWXO�GH�GHWHUPLQD LH� b) 6 � VH� DUDWH� GDF � YDORDUHD� PHGLH� D� FKHOWXLHOLORU� SHQWUX� FHOH� �� GH� DXWRYHKLFXOH� HVWH� R�

YDORDUH�UHSUH]HQWDWLY � c) 6 �VH�GHWHUPLQH�PHGLD�úL�GLVSHUVLD�DXWRYHKLFXOHORU�FDUH�DX�SDUFXUV�R�GLVWDQ �PDL�PLF �VDX�

HJDO �FX��PLL�NP� d) 6 � VH� DSOLFH� WHVWXO� )� GH� DQDOL] � GLVSHUVLRQDO �� SHQWUX� D� YHULILFD� VHPQLILFD LD� LQIOXHQ HL�

GLVWDQ HL�SDUFXUVH�DVXSUD�FRVWXOXL�GH�vQWUH LQHUH�

6ROX LH� a) 1RW P�FX�;�–�FDUDFWHULVWLFD�³GLVWDQ D�SDUFXUV ´�– factorul de gUXSDUH�úL�FX�<�- caracteristica

³FRVWXO�GH�vQWUH LQHUH´� i = 1 (grupa sub 40 mii km).

Tabelul 4.8. &RVW�GH�vQWUH LQHUH��VXWH�

mii km) n1j yj yjn1j

1yy j − ( ) jj nyy 12

1−

5 – 7 18 6 108 -1,87 62,9442 7 – 9 28 8 224 0,13 0,4732

9 – 11 14 10 140 2,13 63,5166 Total 60 - 472 - 126,934

lei mii sute 87,760

472

1

11 ==

∑∑

=j

jj

n

nyy

( )

1156,260

934,126

1

12

121 ==

∑∑ −

=j

jj

n

nyyσ

i = 21 (grupa 40 - 60 mii km). Tabelul 4.9.

&RVW�GH�vQWUH LQHUH (sute mii km)

*(%)2 jn yj *

(%)2 jjny ( ) *(%)2

22 jj nyy −

5 - 7 10 6 60 136,90 7 – 9 25 8 200 72,25 9 – 11 35 10 350 3,15 11 - 13 30 12 360 158,70 Total 100 - 970 371,00

lei mii sute 7,9100

970

100

*(%)2

*(%)2

*(%)2

2 ==∑

=∑

∑= jj

j

jj ny

n

nyy

( ) ( )71,3

100

371

100

*(%)2

22

*(%)2

*(%)2

222

2 ==∑ −

=∑

∑ −= jj

j

jj nyy

n

nyyσ

i = 3 (grupa 60 – 80 mii km.)

lei mii sute 4,103 =y

n3 = 40

lei mii sute 2,1100

4,10538,11

100100 33

33

33 =⋅=

⋅=⇒=

yvyv σ

σ

44,123 =σ

L� ��JUXSD��úL�SHVWH��PLL�NP��

OHDXWRYHKLFX��20200100

10n 4 ==

lei mii sute 2,1320

2644 ==y

9604,0)98,0( 224 ==σ

Atunci media dispersiilor grupelor va fi:

6467,2200

344,529

200

209604,06044,18071,3601156,222

==

=⋅+⋅+⋅+⋅==∑

∑i

ii

n

nσσ

n2 = n - ( n1 + n3 + n4 ) = 200 - ( 60 + 40 + 20 ) =80 autovehicule Media întrHJLL�FROHFWLYLW L�GH��GH�PDúLQL�SRDWH�IL�FDOFXODW �FD�PHGLH�D�PHGLLORU�SDU LDOH�

OHL��PLL��VXWH��64,9200

1928

200

264404,10807,9472 ==+⋅+⋅+==∑

∑i

ii

n

nyy

'LVSHUVLD�GLQWUH�JUXSH�FDUH�VLQWHWL]HD] �LQIOXHQ D�IDFWRUXOXL�GH�JUXSDUH�;�DVXSUD�YDULD LHL�YDULDELOHL�<�HVWH�

( )=

−=δ

∑∑

i

ii

n

nyy2

2

Din reJXOD�GH�DGXQDUH�D�GLVSHUVLLORU�UH]XOW �

9709,43242,26467,2222 =+=+= δσσ

( ) ( ) ( ) ( )

3242,2200

838,464200

2064,92,134064,94,108064,97,96064,987,7 2222

==

=⋅−+⋅−+⋅−+⋅−=

LDU�FRHILFLHQWXO�GH�GHWHUPLQD LH�HVWH�

%76,461009709,4

3242,2100

2

22 ===

σδ

R

$úDGDU�� GLQ� YDULD LD� FDUDFWHULVWLFLL� <� �FRVWXO� GH� vQWUH LQHUH�� HVWH� LQIOXHQ DW � GH�

GLVWDQ D�SDUFXUV ��LDU�LQIOXHQ D�DOWRU�IDFWRUL��FRQVLGHUD L�DOHDWRUL��HVWH�

%24,5310012

222 ==−=

σσ

RK

b) 5HSUH]HQWDWLYLWDWHD�PHGLHL�HVWH�LQIOXHQ DW �GH�RPRJHQLWDWHD�VHULHL�GH�GDWH�úL�VH�P VRDU �FX�

DMXWRUXO�FRHILFLHQWXOXL�GH�YDULD LH�

%13,2310064,9

9709,4100 ===

yv

σ

&XP� Y� �� UH]XOW � F � LQGLFDWRUXO� FRVWXO� PHGLX� GH� vQWUH LQHUH� HVWH� R� YDORDUH�

UHSUH]HQWDWLY �

c) 9DULDELOD� ³GLVWDQ D� SDUFXUV � PDL� PLF � VDX� HJDO � FX� �� PLL� NP�´� � HVWH� R� FDUDFWHULVWLF �

DOWHUQDWLY ��$WXQFL� nivelul mediu al caracteristicii alternative este:

N

Mp = ,

unde: M� �QXP UXO�GH�U VSXQVXUL�IDYRUDELOH��QXP UXO�XQLW LORU�FDUH�SRVHG �FDUDFWHULVWLFD� N = QXP UXO�WRWDO�GH�U VSXQVXUL��YROXPXO�WRWDO�DO�FROHFWLYLW LL�� ÌQ�LSRWH]D�UHSDUWL LHL�QRUPDOH�D�XQLW LORU�SH�LQWHUYDOXO��-��PLL�NP��SDUFXUúL��QXP UXO�

XQLW LORU�FH�U VSXQG�FRQGL LHL�GDWH�HVWH�GH��GH�DXWRPRELOH�SH�DFHVW�LQWHUYDO��úL�GHFL M = 60 + 40 = 100 automobile p=�� GLQ�XQLW L�SRVHG �FDUDFWHULVWLFD�VWXGLDW �

dispersia variabilei alternative este:

( ) 25,05,05,012 =⋅=−= pppσ

dispersie care araW �R�vPSU úWLHUH�PD[LP �D�XQHL�FDUDFWHULVWLFL�GH�WLS�DOWHUQDWLY��R�MXP WDWH�

GLQ�XQLW L�U VSXQG�FULWHULXOXL��LDU�R�MXP WDWH�QX�U VSXQG�FDUDFWHULVWLFLL��

d) 9DULDQ D�WRWDO �HVWH�� ( )∑ ∑ −= 2yyS ij

9DULDQ D�GLQWUH�JUXSH�� ( ) ii nyyS2

1 ∑ −=

9DULDQ D�Gin interiorul grupelor: ( )∑ ∑ −= 22 iij yyS

S = S1 + S2 S = 4,9709 ⋅ 200 = σ2 ⋅ n = 994,18 S1 = 2,3242 ⋅ 200 = δ2 ⋅ n = 464,84

34,5292006467,22

2 =⋅=⋅= nS σ Dispersiile corectate vor fi:

95,15414

84,464

112

1 =−

=−

=r

Ss

7,24200

34,529222 =

−=

−=

rn

Ss

4,577,2

95,15422

21 ===⇒

s

sFcalc

68,2196;31;05,0 == =−=−= rnrtab FF α

Cum Fcalc > Ftab�� UH]XOW � F � IDFWRUXO� GH� JUXSDUH� �GLVWDQ D� SDUFXUV �� LQIOXHQ HD] � VHPQLILFDWLY�

YDULD LD�FRVWXOXL�GH�vQWUH LQHUH��DGLF �VH�UHVSLQJH�LSRWH]D�QXO �SULYLQG�HJDOLWDWHD�GLVSHUVLLORU�

7. Se cunosc datele: Tabelul 4.10. *UXSH�GH�VDODULD L�GXS �GXUDWD�

medie a zilei de lucru (ore)

1U��GH�VDODULD L Salariul mediu (sute

mii lei)

Coeficientul de

YDULD LH�SH�JUXS ��

0 1 2 3

7 – 7,2 100 9,2 6,52

7,2 – 7,4 120 10,4 10,58

7,4 – 7,6 180 11,6 12,07

7,6 – 7,8 140 11,8 10,17

��úL�SHVWH 130 12,0 7,50

Total 670 - -

Se cere:

a) 6 �VH�P VRDUH�LQIOXHQ D�GXUDWHL�PHGLL�D�]LOHL�GH�OXFUX�DVXSUD�YDULD LHL�VDODULXOXL��IRORVLQG� a1��FRHILFLHQWXO�GH�GHWHUPLQD LH� a2��WHVWXO�³)´�GH�DQDOL] �GLVSHUVLRQDO �

6ROX LH�

a1) 1002

22

σδ=R

222 δσσ +=

Din 100i

ii

yv

σ= ��FRO��WDE��VH�GHWHUPLQ �DEDWHULOH�PHGLL�S WUDWLFH�SH�JUXSH�

100

iii

yv ⋅=σ (col. 1 tab. 4.11)

Tabelul 4.11. *UXSH�GH�VDODULD L�GXS �GXUDWD�PHGLH�D�

zilei de lucru (ore) σi 2

iσ ii n2σ ii ny ( ) ii nyy2

−

0 1 2 3 4 5 7 – 7,2 0,6 0,36 36 920 380,25

7,2 – 7,4 1,1 1,21 145,2 1248 67,50 7,4 – 7,6 1,4 1,96 352,8 2088 36,45 7,6 – 7,8 1,2 1,44 201,6 1652 59,15 ��úL�SHVWH 0,9 0,81 105,3 1560 93,92

Total - - 840,9 7468 637,27

0HGLD�GLVSHUVLLORU�GH�JUXS �HVWH�

255,1670

9,84022

==∑

∑i

ii

n

nσσ

3HQWUX�GHWHUPLQDUHD�GLVSHUVLHL�GLQWUH�JUXSH��FDOFXO P�

15,11670

7468 ===∑

∑i

ii

n

nyy sute mii lei.

Dispersia dintre grupe:

( )

951,0670

27,6372

2 ==−

=∑

∑i

ii

n

nyyδ

'LVSHUVLD�WRWDO �

206,2951,0255,1222 =+=+= δσσ

%11,43100206,2

951,0100

2

22 ===

σδ

R

'XUDWD�PHGLH�D�]LOHL�GH�OXFUX�LQIOXHQ HD] �YDULD LD�VDODULXOXL�vQ�SURSRU LH�GH��

a2)�9DULDQ D�WRWDO �HVWH��

( ) 1,1478670206,222 =⋅=⋅=−= ∑ nyyS j σ

9DULDQ D�IDFWRULDO ��

( ) 2,637221 =⋅=−= ∑ nnyyS ii δ

9DULDQ D�UH]LGXDO �

( ) 9,84022

2 =⋅=−= ∑ ∑ nyyS ij σ

Dispersiile corectate vor fi:

3,1594

2,637

112

1 ==−

=r

Ss

265,1665

9,840222 ==

−=

rn

Ss

45,29,125265,1

3,159665;4;05,02

2

21 ==>===⇒ =αFF

s

sF tabcalc

Cum Fcalc > Ftab�� UH]XOW � F � LQIOXHQ D�GXUDWHL�PHGLL� D� ]LOHL� GH� OXFUX� DVXSUD� VDODULXOXL� HVWH�

VHPQLILFDWLY ��

8. 3HQWUX�R�YDULDELO ��DOHDWRDUH�;��GLVWULEXLW �QRUPDO�úL�FX�REOLFLWDWH�PRGHUDW ��VH�úWLH�F ��GLQ�

YDORUL��VLWXDWH�vQ�FHQWUXO�GLVWULEX LHL��VXQW�FXSULQVH�vQWUH��úL��FRHILFLHQWXO�GH�YDULD LH�HVWH�

GH��LDU�YDORDUHD�PRGDO ��6 �VH�P VRDUH�úL�LQWHUSUHWH]H�DVLPHWULD�GLVWULEX LHL� 6ROX LH� 6H� úWLH� F � SHQWUX� R� GLVWULEX LH� QRUPDO � VLPHWULF � VDX� PRGHUDW� REOLF �� DPSOLWXGLQHD�

VHPLLQWHUTXDUWLOLF �UHSUH]LQW �DSUR[LPDWLY��GLQ�DEDWHUHD�PHGLH�S WUDWLF ��$WXQFL��FXP�Q1 = 35,2 úL�Q3� ��UH]XOW �F

7,62

2,356,48

213 =−=

−=

QQQ �úL� 10

67,0≅≅ Qσ

'LQ�UHOD LD�FRHILFLHQWXOXL�GH�YDULD LH��

4025

10010100%25100 =⋅=⋅=⇒==

vx

xv

σσ�úL� 18,0

10

2,3840 =−=−=σMox

Cas

6HULD�DQDOL]DW �SUH]LQW �R�DVLPHWULH�PRGHUDW ��SR]LWLY ��vQ�VHULH�SUHGRPLQ �YDORULOH�PDUL��

9. Într-R� úFRDO � JHQHUDO � VH� RUJDQL]HD] � XQ� WHVW� GH� Yerificare a aptitudinilor elevilor pentru

LQIRUPDWLF ��'LVWULEX LD�HOHYLORU�GXS �SXQFWDMXO�RE LQXW�HVWH�DSUR[LPDWLY�QRUPDO ��3ULPLL��

GLQ� FHL� PDL� SX LQ� vQFOLQD L� VSUH� DFHDVW � GLVFLSOLQ � RE LQ� SkQ � OD� �� SXQFWH�� LDU� FHL� PDL�

SHUIRUPDQ L��RE LQ�SHVWH��GH�SXQFWH��'DF �YDORDUHD�PHGLDQ �HVWH�GH��GH�SXQFWH��V �VH�

VWXGLH]H�REOLFLWDWHD�GLVWULEX LHL� 6ROX LH: $VLPHWULD��REOLFLWDWHD��GLVWULEX LHL�SRDWH�IL�VWXGLDW �FX�LQGLFDWRUXO��FRHILFLHQWXO�OXL�%RZOH\��

( ) ( )

Q

MeQQ

Q

QMeMeQAs

2

2

23113 −+

=−−−

=

sau

( ) ( )

13

13

QQ

QMeMeQAs

−−−−

=

$úDGDU��

( ) ( )

4,030

921

1848

18272748 =−=−

−−−=As

9DORDUHD�LQGLFDWRUXOXL�GH�REOLFLWDWH�DUDW �F ��vQ�FROHFWLYLWDWHD�DQDOL]DW ��SUHGRPLQ �HOHYLL�FX�

SXQFWDM�PDUH�OD�WHVW��$VLPHWULD�HVWH�SR]LWLY ��PRGHUDW �

10. 8WLOL]kQG�GLVWULEX LD�GH�OD�SUREOHPD��FDS��V �VH�VWXGLH]H�REOLFLWDWHD�úL�H[FHVXO�UHSDUWL LHL�

6ROX LH: 'LVWULEX LD�FHORU��GH�ILUPH�GXS �QXP UXO�GH�]LOH�GH�DúWHSWDUH�SkQ �OD�UHDOL]DUHD�IX]LRQ ULL�HVWH�

Tabelul 4.12. ,QWHUYDOH�GH�YDULD LH�D�QXP UXOXL�GH�]LOH 1XP U�GH�ILUPH

40 – 80 50 80 – 120 70

120 – 160 90 160 – 200 100 200 – 240 50 240 – 280 20

Total 380

În problema 2 s-DX�RE LQXW�

zilex 47,149= Me = 151,33 zile; Q1 = 105,86 zile; Q3 = 190,3 zile; Mo = 166,67 zile; σ2=3068,14; σ = 55,39 zile.

6H�REVHUY �F �DVLPHWULD�HVWH�QHJDWLY ��SHQWUX�F � x < Me < Mo. Într-DGHY U��GLQ�IRUPXOD��

31,039,55

67,16647,149 −=−=−=σMox

Cas

úL� ( ) ( )

25,039,55

33,15147,14933 −=−=−=σ

MexCas � �DVLPHWULH� QHJDWLY � PRGHUDW �� FRDGD� PDL�

OXQJ �D�GLVWULEX LHL�HVWH�VSUH�VWkQJD��

2EOLFLWDWHD�SRDWH�IL�VWXGLDW �úL�SH�ED]D�FRHILFLHQWXOXi lui Bowley:

( ) ( )08,0

44,84

5,6

86,1053,190

33,151286,1053,190

13

13 −=−=−

⋅−+=−

−−−=

QQ

QMeMeQQ

2�P VXU �D�H[FHVXOXL�XQHL�GLVWULEX LL�XQLPRGDOH�HVWH�GDW �GH�FRHILFLHQWXO�OXL�3HDUVRQ�

22

42

µµβ =

Cum µ2 = σ2�UH]XOW �F � 483.413.922 =µ

( )

491.876.20380

1093306,7 94

4 =⋅=∑

∑ −=

i

ii

n

nxxµ

218,2483.413.9

491.876.202 ==β

sau γ2 = β2 – 3 = 2,218 – 3 = -0,782 &XUED�HVWH�VODE�SODWLFXUWLF ��DSODWL]DW ��GHRDUHFH�β2 < 3, iar γ2 < 0.

Probleme propuse

1. 8WLOL]kQG�LQIRUPD LLOH�XUP WRDUH��LGHQWLILFD L�IRUPD�ILHF UHL�GLVWULEX LL�

42=x ; Me = 42; Mo = 42;

30=x ; Me = 27; Mo = 25;

3,10=x ; Me = 10,5; Mo = 10,6;

25=x ; Me = 25; Mo =��úL��

2. 8UP WRDUHOH�GDWH�UHSUH]LQW �YkUVWD�PHGLH�SHQWUX�ILHFDUH�HFKLS �GLQ�1%$�úL�QXP UXO�GH�MXF WRUL�

la începutul sezonului 1992 – 199��&DUH�HVWH�YkUVWD�PHGLH�D�vQWUHJLL�DVRFLD LL�GH�EDVFKHW"

Tabelul 4.13. Nr. crt. Echipa 1U��MXF WRUL Vârsta medie

1 Indiana 12 24,72 2 Portland 12 24,72 3 Golden State 15 24,96 4 Dallas 13 25,17 5 New Jersey 13 25,26 6 Chicago 12 25,27 7 Kansas City 12 25,32 8 Detroit 13 25,65 9 Philadelphia 12 25,96 10 Phoenix 13 26,10 11 Washington 13 26,12 12 Utah 12 26,13 13 San Antonio 13 26,25 14 Los Angeles 11 26,42 15 New York 14 26,45 16 Cleveland 14 26,48 17 San Diego 13 26,71 18 Boston 12 26,94 19 Seattle 11 27,09 20 Atlanta 12 27,12 21 Denver 11 27,59 22 Houston 12 29,21 23 Milwaukee 13 29,72

Sursa: Basketball Weekly, vol. 16, nr. 5, 1993.

3. 9HULILFD L�SURSULHW LOH�PHGLHL�DULWPHWLFH�SH�XUP WRDUHOH�VHULL�GH�UHSDUWL LH�

a)

35 50 65 30 20 n

201 195 189 182 170 x

i

i

b)

40 60 100 60 40 n

2500 2300 2100 1900 1700 x

i

i

4. 'LVWULEX LD�F V WRULLORU�vQ�DQXO��GXS �YkUVWD�VR XOXL�OD�F V WRULH�VH�SUH]LQW �DVWIHO� Tabelul 4.14.

*UXSD�GH�YkUVW �D�VR XOXL��DQL� 1XP U�F V WRULL sub 24 113.808 25 – 29 40.250 30 – 34 17.028 35 – 39 8.395 40 – 44 4.185 45 – 49 2.459 50 – 54 2.241 55 – 59 1.574

��úL�SHVWH 2.712 Total 192.652

Sursa: Anuarul Statistic al României, 1995, p. 116.

6 �VH�GHWHUPLQH�YkUVWD�PHGLH�D�VR XOXL�OD�F V WRULH��XWLOL]kQG�WLSXO�GH�PHGLH�GH�FDOFXO�FHO�PDL�SRWULYLW�

5. $WXQFL�FkQG�R�ILUP �LPSRUWDQW �KRW U úWH�V �YkQG �R�VHF LH��R�OLQLH�GH�SURGXF LH�VDX�DOWH�EXQXUL�XQHL�DOWH�

ILUPH��XQ�DQDOLVW�ILQDQFLDU�vúL�SRDWH�SXQH�vQWUHEDUHD��FXP�YRU�LQWHUSUHWD�LQYHVWLWRULL�DFHDVW �DF LXQH"�&D�R�

LQIRUPD LH�SR]LWLY �VDX�QHJDWLY �SHQWUX�ILUP "�3UHVXSXQHP�F �DQDOLVWXO�D�FXOHV�GDWH�SULYLWRDUH�OD�YDULD LD�

UHODWLY �D�VWRFXULORU�SHQWUX��GH�]LOH� vQDLQWH�úL��GH�]LOH�GXS �HIHFWXDUHD�DQXQ XOXL�GH�YkQ]Dre. Data HIHFWX ULL�DQXQ XOXL�D�IRVW��,��&XP�YD�SXWHD�FHUFHW WRUXO�DQDOL]D�VWDWLVWLF�LQIRUPD LLOH�FXOHVH"

Tabelul 4.15. Data 9DULD LD�UHODWLY �D�VWRFXULORU Data 9DULD LD�UHODWLY �D�VWRFXULORU

3.I.1999 -0,0078 1.II. -0,0254 4.I. -0,0329 2.II. -0,0062 5.I. 0,204 5.II. -0,0025 8.I. -0,0067 6.II. 0,0025 9.I. 0,0255 7.II. 0,0249 10.I. 0,0131 8.II. -0,0121 11.I. 0,0259 9.II. 0,0074 12.I. -0,0126 12.II. 0,0110 15.I. -0,0038 13.II. -0,0084 16.I. -0,0077 14.II. 0,0097 17.I. -0,0052 15.II. -0,0169 18.I. 0,0026 16.II. -0,0024 19.I. 0,0000 19.II. 0,0000 22.I. -0,0091 20.II. 0,0197 23.I. -0,0065 21.II. 0,0012 24.I. -0,0118 22.II. 0,0301 25.I. -0,0200 23.II. 0,0164 26.I. -0,0163 26.II. -0,0008 29.I. -0,0080 27.II. -0,0011 30.I. 0,0270 28.II. -0,0006 31.I. 0,389

,QGLFD LH��9DULDELOD�UHODWLY �D�VWRFXULORU�D�IRVW�FDOFXODW �FD�(St - St-1) / St-1, unde St este stocul valoric OD�VIkUúLWXO�]LOHL��LDU�St-1�VWRFXO�YDORULF�OD�VIkUúLWXO�]LOHL�SUHFHGHQWH�

6. Procentul din profit cheltuit de 70 de firme SHQWUX� UHFODP �úL�SXEOLFLWDWH� vQ�DQXO��D�DYXW�

XUP WRDUHD�GLVWULEX LH�

Tabelul 4.16. ,QWHUYDOH�GH�YDULD LH�D�SURFHQWXOXL 1XP U�GH�ILUPH

0,6 – 1,0 14 1,0 – 1,4 18 1,4 – 1,8 24 1,8 – 2,2 8 2,2 – 2,6 4 2,6 – 3,0 2

Total 70

Se cere: a) &DUDFWHUL]D L� WHQGLQ D� FHQWUDO � SULYLQG� SURFHQWXO� GLQ� SURILW� FKHOWXLW� SHQWUX� UHFODP � úL�

publicitate în aceaVW �FROHFWLYLWDWH� b) 3HVWH�FH�OLPLW �VH�VLWXHD] �SURFHQWXO�FKHOWXLW�GH�SULPHOH��GLQ�ILUPH��FX�SURFHQWHOH�FHOH�

PDL�PDUL�FKHOWXLWH�SHQWUX�UHFODP �úL�SXEOLFLWDWH"�'ar primele 10% din firmele care au avut FKHOWXLHOLOH�UHODWLYH�FHOH�PDL�VF ]XWH"

c) &DUH�HVWH�SURFHQWXO�FKHOWXLW�SHQWUX�UHFODP �úL�SXEOLFLWDWH�FHO�PDL�GHV�vQWkOQLW�OD�ILUPHOH�GLQ�

FROHFWLYLWDWHD�DQDOL]DW "

7. 'LVWULEX LD�vQWUHSULQGHULORU�GH�FRQVWUXF LL�vQ�DQWUHSUL] �GXS �QXP UXO�VDODULD LORU��vQ�DQXO��D�

IRVW�FHD�SUH]HQWDW �vQ�WDEHOXO�� Se cere:

a) &DUH�HVWH�QXP UXO�PHGLX�DO�VDODULD LORU�SH�vQWUHSULQGHUH" b) &DUH� HVWH� QXP UXO� GH� DQJDMD L� FHO� PDL� GHV� vQWkOQLW� vQ� vQWUHSULQGHULOH� GH� FRQVWUXF LL� vQ�

DQWUHSUL] �vQ�DQXl 1994? c) ÌQWUH�FH�OLPLWH�VH�VLWXHD] �QXP UXO�GH�DQJDMD L�DO�FHORU��GLQ�ILUPH�FDUH�VH�DIO �vQ�PLMORFXO�

GLVWULEX LHL"

Tabelul 4.17. *UXSH�GXS �QXP UXO�VDODULD LORU Structura întreprinderilor (%)

SkQ �OD�� 54,4 6 – 19 17,1

20 – 49 10,5 50 – 99 5,5

100 – 199 4,4 200 – 499 4,8 500 – 999 2,1

��VDODULD L�úL�SHVWH 1,2 Total 100

Sursa: Anuarul Statistic al României, 1995, p.546

8. 3HQWUX��GH�VDODULD L�DL�XQHL�VRFLHW L�FRPHUFLDOH��IRQGXO�GH�VDODUL]DUH�D�IRVW�� vQWU-R�OXQ ��GH�

��PLOLRDQH� OHL��ùWLLQG�F �FHL�PDL�PXO L�GLQWUH�VDODULD L�DX�DYXW�XQ�VDODULX�GH��PLL�OHL��LDU�

FRHILFLHQWXO� GH� DVLPHWULH� DO� UHSDUWL LHL� GXS � VDODULX� D� IRVW� GH� –�� V � VH� DUDWH� GDF � DFHDVW �

FROHFWLYLWDWH�HVWH�RPRJHQ �GLQ�SXQFWXO�GH�YHGHUH�DO�VDODULXOXL�

9. 'LVWULEX LD�HOHYLlor în ciclul primar dintr-R�úFRDO �JHQHUDO ��GXS �QXP UXO�PDWHULLORU�OD�FDUH�DX�

RE LQXW�PHGLD��HVWH�XUP WRDUHD�

Tabelul 4.18. Nr. materii 0 1 2 3 4 5 6 7 Nr. elevi 15 70 85 160 230 129 78 33

6 �VH�VWXGLH]H�DVLPHWULD�úL�RPRJHQLWDWHD�DFHVWHL�GLVWULEX LL�

10. 'LVWULEX LD�XQHL�YDULDELOH�DOHDWRDUH�X��HVWH�SHUIHFW�VLPHWULF ��GH�PHGLH� 20=x ��'DF �VH�DGDXJ �

GRX � XQLW L� VWDWLVWLFH�� FX� YDORULOH� x1 �� úL� x2� � �� FH� SXWH L� VSXQH� GHVSUH� REOLFLWDWHD� QRLL�

GLVWULEX LL"

11. Pentru o colectivitate sWDWLVWLF �� UHSDUWL LLOH� GXS � GRX � YDULDELOH� DOHDWRDUH�X� úL�Y, considerate

independente, sunt simetrice (Cas� � �� &H� SXWH L� DILUPD� GHVSUH� DVLPHWULD� GLVWULEX LHL� GXS �

variabila Z = X + Y"�([SOLFD L�

12. 'LVWULEX LD�XQXL�ORW�GH�FXWLL�GH�ODSWH�GXS �FRQ LQXWXO�Qet este:

Tabelul 4.19. ,QWHUYDOH�GH�YDULD LH�D�JUHXW LL��PO�� Nr. cutii lapte

950 – 960 12 960 – 970 43 970 – 980 59 980 – 990 74 990 – 1000 135

1000 – 1010 92 1010 – 1020 55 1020 – 1030 24 1030 – 1040 6

Total 500

6 �VH�VWXGLH]H�YDULD LD��REOLFLWDWHD�úL�H[FHVXO�GLVWULEX LHL�

13. 3HQWUX� D� vQWRFPL� R� VLWXD LH� DVXSUD� FKHOWXLHOLORU� HIHFWXDWH� GH� FHW HQL� FX� vQWUH LQHUHD�

DSDUWDPHQWHORU� vQ� OXQD� RFWRPEULH� D� DQXOXL� �� VH� vQUHJLVWUHD] � QXP UXO� GH� FDPHUH� úL�

FKHOWXLHOLOH�FX�vQWUH LQHUHD�SHQWUX��GH�DSDUWDPHQWH��'DWHOH�JUXSDWH�VH�SUH]LQW �DVWIHO�

Tabelul 4.20. &KHOWXLHOL�GH�vQWUH LQHUH��PLL�OHL� *UXSH�GH�DSDUWDPHQWH�GXS �

QXP UXO�FDPHUHORU 200-300 300-400 400-500 500-600 Total

Garsoniere 35 10 5 - 50 2 camere 10 50 30 10 100 3 camere 10 10 40 15 75

��úL�SHVWH��FDPHUH - 5 5 15 25 Total 55 75 80 40 250

Se cere: a) 6 � VH� SUHFL]H]H� FDUH� JUXS � GH� DSDUWDPHQWH� HVWH� PDL� RPRJHQ � GLQ� SXQFWXO� GH� YHGHUH� DO�

FKHOWXLHOLORU�GH�vQWUH LQHUH� b) 6 �VH�DQDOL]H]H�úL�V �VH�UHSUH]LQWH�JUDILF�VWUXFWXUD�FKHOWXLHOLORU�WRWDOH�GH�vQWUH LQHUH�SH�WLSXUL�

GH�DSDUWDPHQWH�GXS �QXP UXO�FDPHUHORU� c) 6 � VH�P VRDUH� LQIOXHQ D� QXP UXOXL� GH� FDPHUH� DVXSUD� YDULD LHL� FKHOWXLHOLORU� GH� vQWUH LQHUH��

folosind: � FRHILFLHQWXO�GH�GHWHUPLQD LH� � WHVWXO�)�GH�DQDOL] �GLVSHUVLRQDO �

55

ÍQWUHE UL ¾ &DUH�VXQW�SULQFLSDOHOH�QR LXQL�VSHFLILFH�DOH�VRQGDMXOXL�VWDWLVWLF" ¾ &DUH�VXQW�SULQFLSDOHOH�SURFHGHH�GH�HúDQWLRQDUH" ¾ (VWLPDUHD�PHGLHL�úL�GLVSHUVLHL�SRSXOD LHL�vQ�VRQGDMXO�DOHDWRU�VLPSOX�UHSHWDW�úL�QHUHSHWDW� ¾ Determinarea indicatorilor GH�VHOHF LH�vQ�FD]XO�VRQGDMXOXL�DOHDWRU�VWUDWLILFDW� ¾ 'HWHUPLQDUHD�YROXPXOXL�GH�VRQGDM�vQ�FD]XO�VHOHF LHL�DOHDWRDUH�VLPSOH�úL�VWUDWLILFDWH� ¾ ([WLQGHUHD�UH]XOWDWHORU�VHOHF LHL�SH�WLSXUL�GH�VRQGDMH�

SONDAJUL STATISTIC


1. ÌQ� VFRSXO� DQDOL]HL� FDSDFLW LL� GH� FD]DUH� WXULVWLF � GLQWU-R� ]RQ � PRQWDQ �� R� ILUP � LQWHUHVDW �

HIHFWXHD] � XQ� VRQGDM� VLPSOX� DOHDWRU� GH� YROXP� �� GLQ� FROHFWLYLWDWHD� JHQHUDO �� RE LQkQG�

XUP WRDUHOH�GDWH�VLVWHPDWL]DWH�

Tabelul 5.1. ,QWHUYDOH�GH�YDULD LH�D�FDSDFLW LL�GH�FD]DUH�

(locuri/unitate) NXP U�GH�XQLW L�WXULVWLFH

20 – 40 8 40 – 60 10 60 – 80 32 80 – 100 7

��úL�SHVWH 3 Total 60

Se cere: a) 6 � VH� GHWHUPLQH� FDSDFLWDWHD�PHGLH� GH� FD]DUH� vQ� HúDQWLRQ� úL� V � VH� SUHFL]H]H� GDF � YDORDUHD�

PHGLH�RE LQXW �HVWH�UHSUH]HQWDWLY �SHQWUX�FROHFWLYLWDWHD�GH�VHOHF LH�DQDOL]DW � b) 6 � VH� FDOFXOH]H� HURDUHD� PHGLH� SUREDELO � GH� VHOHF LH� úL� HURDUHD� PD[LP � DGPLVLELO �� GDF �

UH]XOWDWHOH� VH�JDUDQWHD] �FX�R�SUREDELOLWDWH�GH��SHQWUX�FDUH� ]� �� vQ� LSRWH]D�XQHL�

VHOHF LL�DOHDWRDUH�UHSHWDWH�úL�vQ�LSRWH]D�XQHL�VHOHF LL aleatoare nerepetate); c) 6 �VH�HVWLPH]H�OLPLWHOH�vQWUH�FDUH�VH�YD�VLWXD�FDSDFLWDWHD�PHGLH�GH�FD]DUH�SH�XQLWDWH�WXULVWLF �

vQ�FROHFWLYLWDWHD�JHQHUDO �úL�OLPLWHOH�vQWUH�FDUH�VH�YD�VLWXD�FDSDFLWDWHD�WRWDO �GH�FD]DUH� d) 6 �VH�GHWHUPLQH�QRXO�YROXP�GH�VHOHF LH�FDUH�YD�IL�QHFHVDU��GDF �ILUPD�FH�HIHFWXHD] �DQDOL]D�

GRUHúWH�FD�HURDUHD�OLPLW ��PD[LP �DGPLVLELO ��V �VH�UHGXF �OD�MXP WDWH��LDU�SUREDELOLWDWHD�FX�

FDUH�VH�JDUDQWHD] �UH]XOWDWHOH�V �ILH�GH��]� �� e) 6 � VH� HVWLPH]H� vQWUH� FH� OLPLWH� VH� YD� vQFDGUD� SRQGHUHD� XQLW LORU� WXULVWLFH� FX� FDSDFLWDWH� GH�

FD]DUH�GH��úL�SHVWH��GH�ORFXUL�úL�OLPLWHOH�vQWUH�FDUH�VH�YD�VLWXD�QXP UXO�DFHVWRU�XQLW L��

GDF �UH]XOWDWHOH�VH�JDUDQWHD] �FX�R�SUREDELOLWDWH�GH��]� �� f) &DUH�YD�IL�QRXO�YROXP�DO�HúDQWLRQXOXL�QHFHVDU�SHQWUX�D�HVWLPD�DFHDVW �SRQGHUH�FX�R�HURDUH�

OLPLW �PDL�PLF �FX��

6ROX LH:

a) 3HQWUX�GHWHUPLQDUHD�FDSDFLW LL�PHGLL�GLQ�FROHFWLYLWDWHD�GH�VHOHF LH��YRP�FDOFXOD�

7,6560

3940 ===∑

∑i

ii

n

nxx locuri/unitate. (tabelul 5.2.)

ÌQ� VFRSXO� YHULILF ULL� RPRJHQLW LL� FROHFWLYLW LL� GH� VHOHF LH� VWXGLDWH� úL� D� UHSUH]HQWDWLYLW LL�

�VHPQLILFD LHL��PHGLHL�RE LQXWH��YRP�GHWHUPLQD� Tabelul 5.2.

xi ni xini ( ) ii nxx2−

30 8 240 10195,92 50 10 500 2464,90 70 32 2240 591,68 90 7 630 4133,43 110 3 330 5887,47

Total 60 3940 23273,4

100x

vσ= , unde

( )∑

∑ −==

i

ii

n

nxx2

2σσ

( )

89,38760

4,232732

2 ==−

=∑

∑i

ii

n

nxxσ

7,192 == σσ locuri/unitate

&DSDFLWDWHD�GH�FD]DUH�WXULVWLF �VH�DEDWH��DúDGDU��vQ�PHGLH��vQ�SOXV�VDX�vQ�PLQXV��FX��ORFXUL�

de la capacitatea medie de cazare.

%301007,65

7,19100 ===

xv

σ

&DSDFLWDWHD�PHGLH�GH�FD]DUH��ORFXUL�XQLWDWH��HVWH�R�YDORDUH�VHPQLILFDWLY ��FH�FDUDFWHUL]HD] �

WHQGLQ D�FHQWUDO �GLQ�FROHFWLYLWDWHD�GH�VHOHF LH� 3HQWUX�XUP WRDUHOH�SXQFWH�VH�YD�HIHFWXD�DQDOL]D�VHSDUDW� vQ� LSRWH]D�XQHL� VHOHF LL� UHSHWDWH�úL�

DSRL�vQ�LSRWH]D�XQHL�VHOHF LL�DOHDWRDUH�I U �UHYHQLUH�

,SRWH]D�VHOHF LHL�DOHDWRDUH�FX�UHYHQLUH

b) (URDUHD�PHGLH�SUREDELO �GH�VHOHF LH�

54,260

89,3872===

nxσσ ��ORFXUL�XQLWDWH�WXULVWLF

(URDUHD�PD[LP �DGPLVLELO ��OLPLW ��

98,454,296,1 =⋅=σ⋅=∆ xx z ��ORFXUL�XQLWDWH�WXULVWLF

c) (VWLPDUHD�SH�LQWHUYDO�GH�vQFUHGHUH�D�PHGLHL�GLQ�FROHFWLYLWDWHD�JHQHUDO �

xx xxx ∆+≤≤∆− 0

98,47,6598,47,65 0 +≤≤− x �ORFXUL�XQLWDWH�WXULVWLF

68,7072,60 0 ≤≤ x �ORFXUL�XQLWDWH�WXULVWLF

Estimarea pe interval dH�vQFUHGHUH�D�FDSDFLW LL�WRWDOH�GH�FD]DUH�

( ) ( )xix xNxxN ∆+≤≤∆− ∑

&XP�VRQGDMXO�HVWH�GH�YROXP��GLQ�FROHFWLYLWDWHD�JHQHUDO

%10100 =N

n��UH]XOW �F �N� ��XQLW L�WXULVWLFH

68,7060072,60600 ⋅≤≤⋅ ∑ ix locuri

4240836432 ≤≤ ∑ ix locuri GH� FD]DUH� vQ� XQLW LOH� WXULVWLFH� GLQ� ]RQD� PRQWDQ �

(garantat cu o probabilitate de 95%).

d) 'DF �IDFWRULL�LQWHUHVD L�GRUHVF�V �RE LQ �LQIRUPD LLOH�FX�R�SUHFL]LH�FUHVFXW ��HURDUH�OLPLW �PDL�

PLF ��úL�V �JDUDQWH]H�UH]XOWDWHOH�FX�R�SUREDELOLWDWH�PDL�PDUH��HVWH�FODU�F �YD�IL�QHFHVDU�V �VH�

VHOHFWH]H�XQ�HúDQWLRQ�GH�YROXP�PDL�PDUH�GLQ�FROHFWLYLWDWHD�JHQHUDO �GH��GH�XQLW L�

Într-DGHY U�

( )

25049,2

89,3874’’

22’

22≈⋅=

∆=

x

zn

σ��XQLW L�WXULVWLFH

unde 49,22

98,4

2’ ==

∆=∆ x

x locuri/unitate

DúDGDU��VH�YRU�H[WUDJH��GH�XQLW L�vQ�HúDQWLRQ��SHQWUX�FDUH�VH�YRU�VLVWHPDWL]D�GDWHOH��VH�YRU�

SUHOXFUD�úL�VH�YD�HIHFWXD�LQIHUHQ D�VWDWLVWLF �

e) 9DULDELOD� ³FDSDFLWDWH� GH� FD]DUH� GH� �� úL� SHVWH� �� GH� ORFXUL� SH� R� XQLWDWH´� HVWH� R� YDULDELO �

DOWHUQDWLY ��FDUH�DFFHSW �GRX �YDULDQWH�GH�U VSXQV��XQLW Lle statistice pot avea capacitatea de FD]DUH� GH� �� úL� SHVWH� �� GH� ORFXUL� VDX� FDSDFLWDWHD�PDL�PLF � GH� �� GH� ORFXUL�� $WXQFL�� vQ�

HúDQWLRQ�

� media: 167,060

10 ===n

mw (16,7%)

XQGH�P� �QXP UXO�FD]XULORU�FX�U VSXQV�IDYRUDELO�� dispersia: ( ) 139,0833,0167,01 =⋅=− ww � HURDUHD�PHGLH�SUREDELO �GH�UHSUH]HQWDWLYLWDWH�

( )048,0

60

139,01 ==−=n

wwwσ (4,8%)

� HURDUHD�OLPLW �DGPLVLELO �SHQWUX�R�SUREDELOLWDWH�GH�JDUDQWDUH�D�UH]XOWDWHORU�GH�� 144,0048,03 =⋅==∆ ww zσ (14,4%)

/LPLWHOH�vQWUH�FDUH�VH�YD�VLWXD�SRQGHUHD�XQLW LORU�FX�FDSDFLWDWH�GH�FD]DUH�GH��úL�SHVWH��GH�ORFXUL�

%1,31%%3,2

311,0023,0

144,0167,0144,0167,0

≤≤≤≤

+≤≤−∆+≤≤∆−

p

p

p

wpw ww

(VWLPDUHD�SH�LQWHUYDO�GH�vQFUHGHUH�D�QXP UXOXL�GH�XQLW L�FX�FDSDFLWDWH�GH��úL�SHVWH��GH�

locuri: ( ) ( )

187 14 ≤≤∆+≤≤∆−

M

wNMwN ww ��XQLW L�WXULVWLFH

f) 1RXO�YROXP�DO�HúDQWLRQXOXL��GDF �VH�GRUHúWH�FD�HURDUHD�OLPLW �V �ILH�PDL�PLF �FX��

108,0144,075,075,0’ =⋅=∆⋅=∆ ww

( )( ) ( )

107108,0

139,031’

2

2

2’

2≈⋅=

∆

−=w

wwzn XQLW L�WXULVWLFH�VHOHFWDWH�SHQWUX�FDUH�VH�YRU�HIHFWXD�

HWDSHOH�GHVFULSWLY �úL�GH�LQIHUHQ �VWDWLVWLF �

,SRWH]D�VHOHF LHL�DOHDWRDUH�VLPSOH�I U �UHYHQLUH

b) 41,29,060

89,3871

2=⋅=

−=

N

n

nxσσ �ORFXUL��XQLWDWH�WXULVWLF

72,441,296,1 =⋅=⋅=∆ xx z σ �ORFXUL��XQLWDWH�WXULVWLF

c) xx xxx ∆+≤≤∆− 0

72,47,6572,47,65 0 +≤≤− x �ORFXUL�XQLWDWH�WXULVWLF

42,7098,60 0 ≤≤ x �ORFXUL�XQLWDWH�WXULVWLF

( ) ( )xix xNxxN ∆+≤≤∆− ∑

42,7060098,60600 ⋅≤≤⋅ ∑ ix

4225236588 ≤≤ ∑ ix locuri d) locuri/unitate

( )190

600

89,387436,2

89,3874

’

’’

2222’

22≈

⋅+

⋅=

+∆

=

N

z

zn

x

σσ

�XQLW L�WXULVWLFH

e) ( )

045,09,060

139,01

1 =⋅=

−−=

N

n

n

wwwσ (4,5%)

135,0045,03 =⋅==∆ ww zσ (13,5%)

%2,30%%2,3

302,0032,0

135,0167,0135,0167,0

≤≤≤≤

+≤≤−∆+≤≤∆−

p

p

p

wpw ww

( ) ( )

turisticeunitati 181 19

302,0600032,0600

≤≤⋅≤≤⋅

∆+≤≤∆−

M

M

wNMwN ww

��XQLW L�WXULVWLFH

f) 10,0135,075,075,0’ =⋅=∆⋅=∆ ww

( )

( ) ( ) ( )104

600

139,031,0

139,03

1

1’

22

2

22’

2≈

⋅+

⋅=−+∆

−=

N

wwz

wwzn

w

XQLW L�WXULVWLFH

2. $IOD L�DUJXPHQWXO�IXQF LHL�GH�SUREDELOLWDWH�FX�FDUH�VH�JDUDQWHD] �UH]XOWDWHOH��zα/2��úL�QLYHOXO�GH�

FRQILGHQ ��vQFUHGHUH��-α)100, pentru α = 0,10.

6ROX LH� α /2 = 0,05, deci zα/2 = 1,645; (1-α)100 = 95%.

3. &DUH�HVWH�QLYHOXO�GH�vQFUHGHUH��SUREDELOLWDWHD�FX�FDUH�VH�JDUDQWHD] �UH]XOWDWHOH��SHQWUX�LQWHUYDOXO�

de încredere nx /96,1 σ⋅± �DO�PHGLHL�GLQ�FROHFWLYLWDWHD�JHQHUDO �� 0x )? 6ROX LH: zα/2 = 1,96, deci 0,5 - α /2 = 0,4750 (α /2 = 0,025) 1-α = 0,4750 Â�� 3UREDELOLWDWHD�FX�FDUH�VH�JDUDQWHD] �UH]XOWDWHOH��-α)100 = 95%.

4. 8Q�HúDQWLRQ�DOHDWRU�GH��GH�REVHUYD LL�D� IRVW�VHOHFWDW�GLQWU-R�SRSXOD LH�QRUPDO�GLVWULEXLW �� ÌQ�

urma calculelor a rezultDW�YDORDUHD�PHGLH�vQ�HúDQWLRQ� 1,14=x �úL�DEDWHUHD�PHGLH�S WUDWLF ��

6 � VH� GHWHUPLQH� LQWHUYDOXO� GH� vQFUHGHUH�� JDUDQWDW� FX� R� SUREDELOLWDWH� GH� �� SHQWUX� PHGLD�

FROHFWLYLW LL�JHQHUDOH�� 0x ). 6ROX LH:

29,080

6,2 ===nx

σσ

5684,029,096,129,0025,02/ =⋅=⋅=⋅=∆ zz xx σα

xx xxx ∆+≤≤∆− 0

5684,01,145684,01,14 0 +≤≤− x

67,1453,13 0 ≤≤ x

5. 3UHVXSXQHP�F �GRULP�V �GHWHUPLQ P�SURSRU LD�DEVROYHQ LORU�XQXL�OLFHX��FDUH�VH�YRU� vQVFULH�vQ�

vQY PkQWXO�VXSHULRU��1XP UXO�HOHYLORU�GH�FODVD�D�;,I –a din liceul supus analizei este 500. Un HúDQWLRQ�GH��GH�HOHYL�HVWH�VHOHFWDW�DOHDWRU�I U �UHYHQLUH��LDU�UH]XOWDWHOH�FHUFHW ULL��DUDW �F ��

HOHYL�YRU�V �VH�vQVFULH�OD�R�LQVWLWX LH�GLQ�vQY PkQWXO�VXSHULRU��8WLOL]kQG�DFHVWH�LQIRUPD LL��V �VH�

estimeze SURSRU LD�vQ�vQWUHDJD�FROHFWLYLWDWH��S��SHQWUX�XQ�QLYHO�GH�vQFUHGHUH�GH��

6ROX LH� Avem: N = 500 n = 30 w = 19 / 30 = 0,6333

3HQWUX� D� HVWLPD� LQWHUYDOXO� GH� vQFUHGHUH� YRP� FDOFXOD� XQ� HVWLPDWRU� DO� GLVSHUVLHL� SURSRU LHL�

SRSXOD LHL� ( ) ( )

00729,01500

30500

30

6333,016333,0

1

12 =−

−⋅−=−−⋅−=

N

nN

n

wwwσ úL� XQ� HVWLPDWRU� DO� DEDWHULL�

PHGLL�S WUDWLFH��HURDUHD�PHGLH�SUREDELO �GH�UHSUH]HQWDWLYLWDWH��

( )

0854,01

1 =

−−−=

N

nN

n

wwwσ

Intervalul de încredere pentru o probabilitate 100(1-α) este: ww zwpzw σσ αα ⋅+≤≤⋅− 2/2/

Pentru a construi un interval de încredere, pentru un nivel de încredere de 90%, vom folosi z0,05 = 1,645:

7738,04928,0

0854,0645,16333,00854,0645,16333,0

�� ≤≤⋅+≤≤⋅−

p

p

&HHD�FH�vQVHDPQ �F �QH�SXWHP�DúWHSWD��FX�R�SUREDELOLWDWH�GH��FD�SURSRU LD�DEVROYHQ LORU�

GH�OLFHX�FDUH�VH�YRU�vQVFULH�OD�IDFXOWDWH�V �ILH�FXSULQV �vQWUH��úL��%.

6. 3UHVXSXQHP� F � XQ� FHUFHW WRU� HVWH� LQWHUHVDW� vQ� GHWHUPLQDUHD� VDODULXOXL�PHGLX� SHQWUX� DQJDMD LL�

XQHL�ILUPH��ÌQ�ILUP �OXFUHD] ��GH�SHUVRDQH��GLQ�FDUH��GH�DQJDMD L�SHUPDQHQ L�úL��GH�

FRODERUDWRUL��(O�GHFLGH�V �VHOHFWH]H�DOHDWRU�VWUDWLILFDW��GH�SHUVRDQH��GH�DQDJDMD L�SHUPDQHQW�

úL� �� GH� FRODERUDWRUL� úL� GRUHúWH� V � JDUDQWH]H�HVWLPD LLOH� I FXWH� FX� R� SUREDELOLWDWH� GH� �� ÌQ�

XUPD�SUHOXFU ULL�GDWHORU��RE LQH�XUP WRDUHOH�UH]XOWDWH�

$QJDMD L�SHUPDQHQW Colaboratori

16201 =x mii lei 21002 =x mii lei σ1 = 235 mii lei σ2 = 410 mii lei n1 = 50 n2 = 35 N1 = 500 N2 = 350

6ROX LH: 6DODULXO�PHGLX�vQ�HúDQWLRQ�HVWH�

647,181785

7350081000

85

352100501620 =+=⋅+⋅==∑

∑i

ii

n

nxx mii lei

0HGLD�GLVSHUVLLORU�GH�JUXS �vQ�HúDQWLRQ�HVWH�

94,702.10185

3541050235 2222

=⋅+⋅==∑

∑i

ii

n

nσσ

Eroarea medie�SUREDELO �GH�UHSUH]HQWDWLYLWDWH�HVWH�

8,32350500

8519,0851

2

=

+−⋅=

−=

N

n

nstrxσσ mii lei

(URDUHD�OLPLW �DGPLVLELO �SHQWUX�R�SUREDELOLWDWH�GH�JDUDQWDUH�D�UH]XOWDWHORU�GH��HVWH��zα/2 = z0,025 =1,96): 288,648,3296,1 str 025,0str =⋅=⋅=∆ xx z σ mii lei

Intervalul de încredere pentru salaULXO�PHGLX�DO�DQJDMD LORU�YD�IL�

strxstrx xxx 0 ∆+≤≤∆−

288,64647,1817288,64647,1817 0 +≤≤− x mii lei

9,188136,1753 0 ≤≤ x mii lei &DUH�YD�IL�QRXO�YROXP�DO�HúDQWLRQXOXL�VHOHFWDW�DOHDWRU�VWUDWLILFDW��GDF �FHUFHW WRUXO�GRUHúWH�V �

HVWLPH]H�VDODULXO�PHGLX�DO�DQJDMD LORU�FX�R�HURDUH�OLPLW �DGPLVLELO �GH��PLL�OHL�GH�ILHFDUH�SDUWH�D�

mediei ( 30±x PLL�OHL��LDU�UH]XOWDWHOH�V �OH�JDUDQWH]H�FX�R�SUREDELOLWDWH�GH��"�6 �VH�UHSDUWL]H]H�SURSRU LRQDO�úL�RSWLP��SH�FHOH�GRX �FDWHJRULL�GH�DQJDMD L�

( )399

850

94,702.101575,230

94,702.101575,2

’

’’

22

2

222’

22≈

⋅+

⋅=

+∆

=

N

z

zn

x

σ

σpersoane

30’ =∆ x mii lei

z’ = zα/2 = z0,005 =2,575 1RXO�YROXP�DO�HúDQWLRQXOXL�HVWH�GH��GH�SHUVRDQH��DQJDMD L��LDU�UHSDUWL LD�SH�JUXSH�HVWH�

� SHQWUX�VHOHF LH�WLSLF �SURSRU LRQDO ��∑

⋅=i

ii N

Nnn

2357,234850

5003991 ≈=⋅=n �SHUVRDQH��DQJDMD L�SHUPDQHQW�

1643,164850

3503992 ≈=⋅=n persoane (colaboratori)

� SHQWUX�VHOHF LH�WLSLF �RSWLP ��∑

⋅=ii

iii N

Nnn

σσ

1806,179410350235500

2355003991 ≈=

⋅+⋅⋅⋅=n �SHUVRDQH��DQJDMD L�SHUPDQHQW�

2194,219410350235500

4103503992 ≈=

⋅+⋅⋅⋅=n persoane (colaboratori)

7. Un analist conWDELO� GRUHúWH� V � HVWLPH]H� YDORDUHD� PHGLH� D� IDFWXULORU� HPLVH� GH� ILUPD� OD� FDUH�

OXFUHD] ��(O� KRW U úWH� V � H[WUDJ � XQ� HúDQWLRQ� VLPSOX� DOHDWRU� GLQ� WRWDOXO� FHORU� �� GH� IDFWXUL�

HPLVH� vQ� FDGUXO� XQHL� SHULRDGH�� 'LQ� FHUFHW UL� DQWHULRDUH� FXQRDúWH� GLVSHUVLD� vQ� FROHFWivitate

425.22 =σ . Se cere: a) &H�YROXP�GH�VRQGDM�DU�IL�QHFHVDU�GDF �GRUHúWH�V �HVWLPH]H�YDORDUHD�PHGLH�D�IDFWXULORU�HPLVH�

FX�XQ�QLYHO�GH�SUHFL]LH��P VXUDW�GH�HURDUHD�PHGLH�GH�UHSUH]HQWDWLYLWDWH� 7,8=xσ mii lei;

b) Ce volum de sondaM�HVWH�QHFHVDU�GDF �VH�GRUHúWH�FD�HURDUHD�PHGLH�GH�UHSUH]HQWDWLYLWDWH�V �

fie 7=xσ mii lei.

c) 3UHVXSXQHP�F �DQDOLVWXO�QX�HVWH� VLJXU� FDUH�HVWH�YDORDUHD�HURULL�PHGLL�GH� UHSUH]HQWDWLYLWDWH�

QHFHVDU ��7RWXúL��HO�GRUHúWH�V �HVWLPH]H�UH]XOWDWHOH�SH un interval de încredere de 10 mii lei GH�ILHFDUH�SDUWH�D�PHGLHL�HúDQWLRQXOXL��FX�R�SUREDELOLWDWH�GH��&DUH�YD�IL�DWXQFL�YROXPXO�

HúDQWLRQXOXL�QHFHVDU" 6ROX LH:

a) nNNnN

n

n xx22

21 σσσσσ −=⋅⋅⇒

−=

$úDGDU��YROXPXO�HúDQWLRQXOXL�QHFHVDU�HVWH�

3206,31

1025

24257,75

24252

2

2

22

2≈=

+=

+

=+

=

N

N

Nn

xx

σσ

σσσ

σfacturi

3HQWUX�D�HVWLPD�UH]XOWDWHOH�FX�R�HURDUH�PHGLH�GH�UHSUH]HQWDWLYLWDWH�GH��PLL�OHL��YD�WUHEXL�V �

selecteze aleator nerepetat 32 de facturi.

b) 4821,47

1025

242549

24252

2

2≈=

+=

+

=

N

n

x

σσ

σ facturi

'XS � FXP� VH� REVHUY �� GDF � HURDUHD�PHGLH� SUREDELO � VFDGH� �DGLF � SUHFL]LD� GRULW � FUHúWH��

DWXQFL�YROXPXO�HúDQWLRQXOXL�YD�IL�PDL�PDUH�

c) (URDUHD�OLPLW �GH�UHSUH]HQWDWLYLWDWH�HVWH�

xx z σα ⋅=∆2/

�úL�z’ = zα/2 = z0,025 =1,96

facturi 864,851025

242596,110

242596,11

22

2

22’

222

≈=

=⋅+

⋅=

+∆

=⇒

−=∆

N

z

zn

N

n

nz

x

x σσσ

$úDGDU��SHQWUX�D�HVWLPD�UH]XOWDWHOH�SH�XQ�LQWHUYDO�GH�vQFUHGHUH�GH��PLL�OHL ( 100 ±x ), cu o

SUREDELOLWDWH�GH��YD�WUHEXL�V �HúDQWLRQH]H�DOHDWRU�QHUHSHWDW��GH�IDFWXUL�

8. 2�WHOHYL]LXQH�SDUWLFXODU �GRUHúWH�V �RUJDQL]H]H�R�HúDQWLRQDUH�DOHDWRDUH�VLPSO �QHUHSHWDW �SHQWUX�

D� HVWLPD� SURFHQWXO� ORFXLQ HORU� XQXL� MXGH �� FDUH� XUP UHVF� úWLULOH� GH� OD� RUD� �� &DUH� HVWH�

YROXPXO� HúDQWLRQXOXL� QHFHVDU�� SHQWUX� R� HURDUH� OLPLW � GH� �� úL� R� SUREDELOLWDWH� GH� JDUDQWDUH� D�

rezultatelor de 95%? 6ROX LH: 'HúL�QX�VH�FXQRDúWH�GLVSHUVLD�vQ�SRSXOD LH��úWLP�WRWXúL�F �vPSU úWLHUHD�PD[LP �HVWH w(1-w) = 0,5 x 0,5 = 0,25. Atunci:

−−=⋅=∆

N

n

n

wwzz ww 1

)1(σ ; z0,025 = 1,96

&XP� QXP UXO� ORFXLQ HORU� MXGH XOXL� �1�� HVWH� IRDUWH� PDUH�� VH� SRDWH� FRQVLGHUD� FRHILFLHQWXO�

FRUHF LHL�ILQLWH�vQ�SRSXOD LH� 11 ≈−N

n

( )

60004,0

5,05,096,1)1(2

2

2

2=⋅⋅=

∆

−=w

wwzn persoane

9. 5HIHULWRU� OD� SUREOHPD� �� FDS�� SUHVXSXQHP� F � GDWHOH� SURYLQ� GLQWU-XQ� VRQGDM� WLSLF� DOHDWRU� I U �

UHYHQLUH��GH�YROXP��GLQ�FROHFWLYLWDWHD�JHQHUDO ��6H�FHUH�V �VH�HVWLPH]H�LQWHUYDOXO�GH� vQFUHGHUH�

SHQWUX�VDODULXO�PHGLX�úL�IRQGXO�GH�VDODUL]DUH��SHQWUX�XQ�QLYHO�de încredere de 1-α = 0,95 (z = 1,96). 6ROX LH� Eroarea medie de probabilitate:

042,095,0670

255,11

2

=⋅=

−=

N

n

nyσσ sute mii lei.

(URDUHD�OLPLW �PD[LP �DGPLVLELO � 082,0042,096,1 =⋅=⋅=∆ yy z σ sute mii lei

Intervalul de încredere al salariului mediu:

yy yyy ∆+≤≤∆− 0

082,015,11082,015,11 0 +≤≤− y

232,11068,11 0 ≤≤ y sute mii lei

)()( yiy yNyyN ∆+≤≤∆− ∑

232,1113400068,1113400 ⋅≤≤⋅ ∑ iy

88,1505012,14831 ≤≤ ∑ iy mil. lei

Probleme propuse

1. $IOD L�DUJXPHQWXO�IXQF LHL�GH�SUREDELOLWDWH�FX�FDUH�VH�JDUDQWHD] �UH]XOWDWHOH��zα/2��úL�QLYHOXO�GH�

FRQILGHQ � �vQFUHGHUH�� - α�� SHQWUX� ILHFDUH� GLQ� XUP WRDUHOH� YDORUL� DOH� OXL� α (prag de VHPQLILFD LH�� a) α = 0,01; b) α = 0,05; c) α = 0,20; d) α = 0,0024.

2. &DUH�HVWH�QLYHOXO�GH� vQFUHGHUH� �SUREDELOLWDWHD�FX�FDUH�VH�JDUDQWHD] � UH]XOWDWHOH��SHQWUu fiecare

GLQWUH�XUP WRDUHOH�LQWHUYDOH�GH�vQFUHGHUH�DOH�PHGLHL�GLQ�FROHFWLYLWDWHD�JHQHUDO �� 0x )?

a) nx /645,1 σ⋅± ;

b) nx /575,2 σ⋅± ;

c) nx /282,1 σ⋅± ;

d) nx /99,0 σ⋅± ;

e) nx /35 σ⋅± ;

f) nx /25 σ⋅± .

3. 8Q�HúDQWLRQ�DOHDWRU�GH��GH�REVHUYD LL�D� IRVW�VHOHFWDW�GLQWU-R�SRSXOD LH�QRUPDO�GLVWULEXLW �� ÌQ�

XUPD�FDOFXOHORU�D�UH]XOWDW�YDORDUHD�PHGLH�vQ�HúDQWLRQ� 1,14=x �úL�DEDWHUHD�PHGLH�S WUDWLF �� Se cere: a) * VL L�LQWHUYDOXO�GH�vQFUHGHUH��JDUDQWDW�FX�R�SUREDELOLWDWH�GH��SHQWUX�PHGLD�FROHFWLYLW LL�

generale ( 0x ). b) &H� VH� vQWkPSO � FX� P ULPHD� LQWHUYDOXOXL� GH� vQFUHGHUH�� GDF � YROXPXO� HúDQWLRQXOXL� U PkQH�

DFHODúL�� LDU� YDORDUHD� FRHILFLHQWXOXL� GH� vQFUHGHUH� �SUREDELOLWDWHD� FX� FDUH� VH� JDUDQWHD] �

rezultatele) scade? c) 3UHVXSXQHP�F �HúDQWLRQXO�VHOHFWDW�FRQ LQH�GRDU��GH�REVHUYD LL��5HFDOFXOD L� � LQWHUYDOXO�GH�

vQFUHGHUH� SHQWUX� PHGLD� FROHFWLYLW LL� JHQHUDOH�� GDF � UH]XOWDWHOH� VH� JDUDQWHD] � FX� R�

probabilitate de 95%. Ce se întâmSO �FX�P ULPHD� LQWHUYDOXOXL�GH� vQFUHGHUH��GDF �YROXPXO�

HúDQWLRQXOXL�VFDGH"�'DU�GDF �YROXPXO�HúDQWLRQXOXL�FUHúWH"

4. ÌQ�VFRSXO�GH�D�VWXGLD�PRGLILF ULOH�vQUHJLVWUDWH�vQ�QLYHOXO�FKLULHL�ORFXLQ HORU��XQ�DQDOLVW�VHOHFWHD] �

úL� LQWHUYLHYHD] � � �� GH� SRVHVRUL� GH� DSDUWDPHQWH� GH� vQFKLULDW�� HO� RE LQH� XUP WRDUHOH� GDWH�

UHIHULWRDUH�OD�PRGLILFDUHD�UHODWLY ��D�FKLULHL�vQWUH��OXQL�

1,13 3,32 0,62 2,03 1,82 2,61 1,79 3,59 2,27 1,80 -1,22 2,06 -0,45 0,05 0,34 7,33 5,20 1,25 1,42 0,68 2,03 0,68 0,00 1,41 1,39 2,42 1,00 0,90 0,61 0,00 1,00 -1,07

8WLOL]D L�R�SUREDELOLWDWH�GH��SHQWUX�D�HVWLPD�LQWHUYDOXO�GH�vQFUHGHUH�DO�PRGLILF ULL�PHGLL�

UHODWLYH�D�FKLULHL��vQ�FROHFWLYLWDWHD�JHQHUDO �

5. �� GH� DQDOLúWL� HFRQRPLFL�� VHOHFWD L� vQWkPSO WRU�� DX� IRVW� UXJD L� V � HVWLPH]H� YDORDUHD� LQGLFHOXL�

SUH XULORU�GH�FRQVXP�SHQWUX�XUP WRDUHOH��OXQL��9DORULOH�HVWLPDWH��vQ�PRGLILF UL� UHODWLYH��V-au VLWXDW�vQWUH��FHO�PDL�VF ]XW�QLYHO��úL��FHD�PDL�FUHVFXW �YDORDUH�� Se cere: a) 6 � VH� GHWHUPLQH� YDORDUHD� DSUR[LPDWLY � D� DEDWHULL� PHGLL� S WUDWLFH� �GHYLD LD� VWDQGDUG�� vQ�

HúDQWLRQXO�GH��GH�REVHUYD LL� b) �,QGLFD LH�� GLVWULEX LD� VH� SUHVXSXQH� D� IL� DSUR[LPDWLY� QRUPDO �� 6H� YD� XWLOL]D� IDSWXO� F �

DPSOLWXGLQHD�YDULD LHL�HVWH�DSUR[LPDWLY�HJDO �FX��σ��GHRDUHFH�úWLP�F ��GLQ�REVHUYD LL�

sunt cuprinse în intervalul σ3±x ). c) 3UHVXSXQHP�F �FHOH��GH�HVWLPD LL�DX�IRVW�VHOHFWDWH�DOHDWRU�GLQWU-o colectivitate de volum

PDUH��'DF �PHGLD�HúDQWLRQXOXL�D�IRVW�GH��J VL L�XQ�LQWHUYDO�GH�vQFUHGHUH��JDUDQWDW�FX�R�

probabilitate de 9��SHQWUX�PRGLILFDUHD�PHGLH�UHODWLY �D�LQGLFHOXL�SUH XULORU�GH�FRQVXP�vQ�

XUP WRDUHOH�GRX �OXQL�

6. 8Q�QRX�SURFHV�GH�SURGXF LH�D�IRVW�FDSDELO�V �WUDQVIRUPH�ILHUXO�RELúQXLW�vQWU-un material superior,

QXPLW�³VWLFO �PHWDOLF ´��6WLFOD�PHWDOLF �HVWH�GH�WUHL�SkQ �OD�SDWUX�RUL�PDL�WDUH�GHFkW�R HOXO�úL�GH�

SHVWH��GH�RUL�PDL�UH]LVWHQW �OD�FRUR]LXQL�GHFkW�R HOXO��8QD�GLQWUH�SUREOHPHOH�VWLFOHL�PHWDOLFH�

HVWH�� WRWXúL�� WHQGLQ D� GH� D� GHYHQL� FDVDELO � OD� R� DQXPLW � WHPSHUDWXU �� ÌQ� VFRSXO� GH� D� HVWLPD�

temperatura medie ( 0x ��OD�FDUH�DFHVW�WLS�GH�VWLFO �PHWDOLF �GHYLQH�FDVDELO��GH�EXF L�GH�VWLFO �

PHWDOLF �DX�IRVW�VHOHFWDWH�DOHDWRU�GLQWU-XQ�ORW�RE LQXW� FLHFDUH�EXFDW �D�IRVW�LQGHSHQGHQW�VXSXV �la temperaturi din ce în ce mai ridicate. Temperaturile la care�EXF LOH�DX�GHYHQLW�FDVDELOH�DX�

IRVW�QRWDWH�úL�VXSXVH�SUHOXFU ULL�VWDWLVWLFH��5H]XOWDWHOH�RE LQXWH�DX�IRVW�� 450=x ° F, σ = 10° F. Se cere: a) 6 � VH� HVWLPH]H�� FX� R� SUREDELOLWDWH� GH� �� LQWHUYDOXO� GH� vQFUHGHUH� SHQWUX� WHPSHUDWXUD�

medie la caUH�VWLFOD�PHWDOLF �GHYLQH�FDVDELO � b) 6 � VH� GHWHUPLQH� FH� YROXP� DO� HúDQWLRQXOXL� YD� IL� QHFHVDU�� GDF � GRULP� FD� UH]XOWDWHOH� V � VH�

garanteze cu o probabilitate de 99,73%.

7. 3UHVXSXQHP� F � XQ� DQDOLVW� ILQDQFLDU� GRUHúWH� V � DIOH� GDF � FHL� �� GH� DQJDMD L� DL� XQHL� X]LQH�

UHSUH]LQW �R�VXUV �GH�SRWHQ LDOL�FOLHQ L�SHQWUX�XQ�PDJD]LQ�FH�VH�YD�DPSODVD�OkQJ �SRDUWD�X]LQHL��

(O�GHFLGH�V �DQDOL]H]H�P ULPHD�YHQLWXULORU�FHORU��GH�DQJDMD L��8Q�HúDQWLRQ�VHOHFWDW�DOHDWRU�

I U � UHYHQLUH�� GH� �� GH� SHUVRDQH�� RIHU � LQIRUPD LL� FH� GXF� OD� FDlculul veniturilor medii pe HúDQWLRQ�� 1280=x PLL�OHL�úL�DEDWHULL�PHGLL�S WUDWLFH�vQ�HúDQWLRQ�σ = 175 mii lei. Utilizând aceste LQIRUPD LL��V �VH�GHWHUPLQH�LQWHUYDOXO�GH�vQFUHGHUH�SHQWUX�YHQLWXULOH�PHGLL�DOH�DQJDMD LORU��OD�XQ�

nivel de încredere de 95%.

8. 3UHVXSXQHP� F � VXQWHP� LQWHUHVD L� vQ� D� GHWHUPLQD� SURSRU LD� FHORU� �� GH� HOHYL� GLQ� FLFOXO�

JLPQD]LDO� DO� XQHL� úFROL�� FDUH� GRUHVF� V � IDF � RUH� VXSOLPHQWDUH� GH� LQL LHUH� vQ� LQIRUPDWLF �� 2�

DQFKHW �UHDOL]DW �SH�XQ�HúDQWLRQ�DOHDWRU�VLPSOX�I U �UHYHQLUH��GH��GH�HOHYL��DUDW �F ��GLQWUH�HL�

GRUHVF�DVWIHO� GH� RUH� VXSOLPHQWDUH��6 � VH� HVWLPH]H�SURSRU LD� �p�� GLQ� FROHFWLYLWDWHD� JHQHUDO � GH��GH�HOHYL��XWLOL]kQG�DFHVWH�LQIRUPD LL�

9. 'DF �GRUL L�V �HVWLPD L�PHGLD�XQHL�FROHFWLYLW L�� 0x ) cu R�HURDUH�OLPLW �GH��úL�R�SUREDELOLWDWH�

GH� JDUDQWDUH� D� UH]XOWDWHORU� GH� �� FH� YROXP� DO� HúDQWLRQXOXL� YD� IL� QHFHVDU�� úWLLQG� GLQWU-o FHUFHWDUH� DQWHULRDU � F � GLVSHUVLD�σ2� HVWH� DSUR[LPDWLY� HJDO � FX� ��"�'DU� GDF � GRUL L� FD� vQWUHJ�

LQWHUYDOXO�GH�vQFUHGHUH�V �ILe 0,2 ( 1,0± )?

10. 6 �VH�GHWHUPLQH�YROXPXO�HúDQWLRQXOXL�QHFHVDU�D�IL�H[WUDV�vQ�XUP WRDUHOH�LSRWH]H�

a) N = 4500, σ2 = 2,1, (1- α) 100 = 95%, 5,0=∆ x ��HúDQWLRQDUH�QHUHSHWDW ��I U �UHYHQLUH��

b) σ = 0,5, (1- α�� VHOHF ie cu revenire; c) Ax� ��GLVWULEX LH�DSUR[LPDWLY�QRUPDO ��α = 0,05, 1,2=xσ ��VHOHF LH�FX�UHYHQLUH�

11. 3UHVXSXQHP�F �vQWU-R�FHUFHWDUH�SULQ�VRQGDM��FRVWXO�FHUFHW ULL�XQHL�XQLW L�HVWH�GH��PLL�OHL�úL�

bugetul total este de 12.000 mii lei. EsWH�EXJHWXO�VXILFLHQW�GDF �GRULP�V �HVWLP P�PHGLD�vQWUHJLL�

FROHFWLYLW L�FX�R�SUREDELOLWDWH�GH��úL�SH�XQ�LQWHUYDO�GH�vQFUHGHUH�GH��XQLW L�� 2± XQLW L�"�

(Presupunem σ � ��'DU�GDF �GRULP�V �JDUDQW P�UH]XOWDWHOH�FX�R�SUREDELOLWDWH�de 90%? Care HVWH�SUREDELOLWDWHD�PLQLP �FX�FDUH�SXWHP�JDUDQWD�UH]XOWDWHOH�vQ�OLPLWD�EXJHWXOXL�GLVSRQLELO"�'H�

ce buget avem nevoie pentru a garanta rezultatele cu o probabilitate de 99,73%?

12. În urma unui sondaj nerepetat de 15%, efectuat asupra muncitorilor unei fabrici, s-DX�RE LQXW�

XUP WRDUHOH�GDWH�QHFHVDUH�JUXS ULL�PXQFLWRULORU�GXS �FDOLILFDUH��QHFDOLILFD L��FDOLILFD L��úL�GXS �

SURGXF LD�]LOQLF �UHDOL]DW � � YROXPXO�HúDQWLRQXOXL� ��GH�PXQFLWRUL� � s-DX�vQUHJLVWUDW�GDWH�SHQWUX��GH�PXQFLWRUL�QHFDOLILFD L� � OD�JUXSD�PXQFLWRULORU�QHFDOLILFD L��SURGXF LD�PHGLH�D�XQXL�PXQFLWRU�HVWH�GH��EXF��FX�XQ�

FRHILFLHQW�GH�YDULD LH�GH�� SURGXF LD�WRWDO �UHDOL]DW �GH�PXQFLWRULL�FDOLILFD L�D�IRVW�GH��EXF��YDORUHD�PRGDO �D�DFHVWHL�

grupe a fost de 19,33 buc., cu un coeficient de asimetrie de –0,42456.

Se cere: a) 3HQWUX�R�SUREDELOLWDWH�GH��SHQWUX�FDUH�]� �� V � VH�HVWLPH]H�LQWHUYDOXO�GH� vQFUHGHUH�

SHQWUX�SURGXF LD�PHGLH�úL�SURGXF LD�WRWDO � b) ÌQ�FRQGL LLOH�vQ�FDUH�DP�RUJDQL]D�XQ�QRX�VRQGDM�úL�DP�IRORVL�R�SUREDEilitate de 0,9545, pentru

FDUH�] ��V �VH�FDOFXOH]H�YROXPXO�QRXOXL�HúDQWLRQ�úL�V �VH�UHSDUWL]H]H�SURSRU LRQDO�úL�RSWLP�

SH�VXEHúDQWLRDQH� c) 6 � VH� YHULILFH� VHPQLILFD LD� IDFWRUXOXL� GH� JUXSDUH� IRORVLQG� UHJXOD� DGXQ ULL� GLVSHUVLLORU�� SH�

baza datelor din sondaj.

13. $QJDMD LL�GLQ�WUHL�VXFXUVDOH�DOH�XQHL�VRFLHW L�FRPHUFLDOH�SUH]LQW �GLVWULEX LD�GXS �WLPSXO�OXFUDW�

într-R�OXQ �GLQ�WDEHOXO��

Tabelul 5.3. Sucursala Nr.

DQJDMD L Timp mediu lucrat (ore)

&RHILFLHQW�GH�YDULD LH�D�WLPSXOXL�

OXFUDW�SH�VXFXUVDO �� 1XP U�GH�DQJDMD L�FDUH�DX�OXFUDW�

peste timpul mediu A 150 200 10 50 B 220 220 12 100 C 80 210 15 20

a) $QDOL]D L�GDF �H[LVW �R�GLIHUHQ �VHPQLILFDWLY �VWDWLVWLF� vQWUH�VXFXUVDOH� vQ�SULYLQ D� WLPSXOXL�

lucrat. b) 5HSUH]HQWD L�JUDILF�VWUXFWXUD�FROHFWLYLW LL�GH�VDODULD L�pe sucursale. c) 3UHVXSXQkQG� F � GDWHOH� SURYLQ� GLQWU-XQ� VRQGDM� WLSLF� SURSRU LRQDO� GH� YROXP� �� GLQ�

FROHFWLYLWDWHD� JHQHUDO �� V � VH� HVWLPH]H� FX� R� SUREDELOLWDWH� GH� �� vQWUH� FH� OLPLWH� VH� YD�

vQFDGUD�� vQ� FROHFWLYLWDWHD� JHQHUDO �� SRQGHUHD� VDODULD LORU� FX� WLPS� OXFrat situat peste timpul mediu lucrat.

d) &DUH�HVWH�QRXO�YROXP�DO�HúDQWLRQXOXL�GDF �GRULP�FD�HURDUHD�OLPLW �D�WLPSXOXL�OXFUDW�V �VFDG �

FX��SHQWUX�R�SUREDELOLWDWH�GH��]� ��6 �VH�UHSDUWL]H]H�RSWLP�YROXPXO�QRXOXL�

HúDQWLRDQH�JUXSH"

14. 6H�FXQRVF�XUP WRDUHOH�GDWH�GHVSUH�XQ�HúDQWLRQ� VHOHFWDW� vQWkPSO WRU�QHUHSHWDW�� GH�YROXP��

GLQ�FROHFWLYLWDWHD�JHQHUDO �

Tabelul 5.4. 9DORDUHD�vQFDV ULORU��PLO��OHL� 0 ULPHD�HFKLSHORU

de servire (pers.) 15 – 25 25 – 35 35 – 45 ��úL�SHVWH SkQ �OD�� 80 20 - -

10 – 20 40 90 70 - ��úL�SHVWH - 30 60 10

Se cere: a) 6 �VH�YHULILFH�UHJXOD�GH�DGXQDUH�D�GLVSHUVLLORU�úL�V �VH�FDOFXOH]H�FRHILFLHQWXO�GH�GHWHUPLQD LH� b) 6 �VH�UHSUH]LQWH�JUDILF�GLVWULEX LD�XQLW LORU�GXS �YDORDUHD�vQFDV ULORU� c) 6 �VH�VWXGLH]H�LQGLFDWRULL�PHGLL�GH�SR]L LH�SHQWUX�GLVWULEX LD�GXS �YDORDUHD�vQFDV ULORU� d) 6 �VH�GHWHUPLQH�OLPLWHOH� vQWUH�FDUH�VH�YRU� vQFDGUD�YDORULOH�PHGLL�úL� WRWDOH�DOH� vQFDV ULORU� vQ�

FROHFWLYLWDWHD�JHQHUDO ��SHQWUX�Φ(z) = 0,9545, pentru care z = 2.

15. 'LVWULEX LD�XQXL�HúDQWLRQ�GH�XQLW L�FRPHUFLDOH��VHOHFWDW�vQWkPSO WRU�úL�QHUHSHWDW��GH�YROXP��

GLQ�FROHFWLYLWDWHD�JHQHUDO ��GXS �YDORDUHD�GHVIDFHULORU�OXQDUH�VH�SUH]LQW �DVWIHO� Tabelul 5.5. ,QWHUYDOH�GH�YDULD LH�D�GHVIDFHULORU��PLO��OHL� 1U��XQLW L�FRPHUFLDOH

÷ 10 5 10 – 20 10 20 – 30 20 30 - 40 30 40 – 50 20

��úL�SHVWH 15 Total 100

Se cere: a) 6 �VH�UHSUH]LQWH�JUDILF�GLVWULEX LD�XQLW LORU�FRPHUFLDOH�GXS �YDORDUHD�GHVIDFHULORU�OXQDUH� b) 6 �VH�DQDOL]H]H�LQGLFDWRULL�WHQGLQ HL�FHQWUDOH�úL�DL�DVLPHWULHL� c) 6 �VH�DUDWH�GDF �HúDQWLRQXO�HVWH�RPRJHQ�GLQ�SXQFWXO�DO�GHVIDFHULORU�GH�P UIXUL� d) 6 �VH�FDOFXOH]H�PHGLD�úL�GLVSHUVLD�XQLW LORU�FRPHUFLDOH�FDUH�DX�R�YDORDUH�D�GHVIDFHULORU�GH�

P UIXUL�PDL�PDUH�VDX�HJDO �FX��PLO��OHL� e) 6 �VH�HVWLPH]H�OLPLWHOH�vQWUH�FDUH�VH�YD�vQFDGUD�YDORDUHD�WRWDO �D�GHVIDFHULORU�GH�P UIXUL�vQ�

vQWUHDJD�UH HD�GH�XQLW L�FRPHUFLDOH��GDF �UH]XOWDWHOH�VXQW�JDUDQWDWH�FX�R�SUREDELOLWDWH�Φ(z) = 0,9545, pentru care z = 2.

66

ÍQWUHE UL ¾ &H�UHSUH]LQW �LSRWH]D�VWDWLVWLF " ¾ &H�UHSUH]LQW �LSRWH]D�QXO "�'DU�LSRWH]D�DOWHUQDWLY " ¾ &H�UHSUH]LQW �HURDUHD�GH�JHQXO�vQWkL�vQWU-un proces de testare a ipotezelor statistice? ¾ &H�UHSUH]LQW �HURDUHD�GH�JHQXO�DO�GRLOHD�vQWU-un proces de testare a ipotezelor statistice? ¾ &H�UHSUH]LQW �UHJLXQHD�FULWLF " ¾ &H�UHSUH]LQW �ULVFXO�GH�JHQXO�vQWkL"�'DU riscul de genul al doilea? ¾ &H�UHSUH]LQW �SXWHUHD�XQXL�WHVW�VWDWLVWLF"

IPOTEZE STATISTICE. VERIFICAREA 1250$/,7 ,,�5(3$57, ,,/25�(03,5,&(


1. 5H]XOWDWHOH�D�QRX �P VXU WRUL�DVXSUD�FRQ LQXWXOXL�GH�ILEUH��OD�XQ�DQXPLW�VRL�GH�FkQHS �VXQW��

22,5; 26; 24,5; 23,5; 29,5; 32,5; 25,5; 26; 28. Din ceUFHW ULOH�DQWHULRDUH�VH�úWLH�F �YDULDELOD�

aleatoare X –�FRQ LQXWXO�GH�ILEUH��XUPHD] �R�OHJH�QRUPDO �1�P��3HQWUX�XQ�QLYHO�GH�

VHPQLILFD LH�α� ��V �VH�YHULILFH�LSRWH]D�� H0: m = 27.

6ROX LH� 6H�FXQRDúWH�F ��σ = 2,20; α = 0,05; n = 9. Se aplLF �testul z bilateral.

ασ αα −=

≤

−≤−

−−1

/2

1

0

21

zn

mxzP

96,1975,0

2

05,01

==−

zz

Limitele intervalului de încredere sunt:

nzmx

nzm

σσαα ⋅+≤≤⋅−

−−2

10

21

0

437,196,19

20,2

21

=⋅=⋅−α

σz

n

437,127437,127 +≤≤− x DGLF

437,28536,25 ≤≤ x

'LQ�VRQGDM�VH�RE LQH�

11,27=∑=n

xx i %

'HRDUHFH�PHGLD� GH� VRQGDM� H� FXSULQV � vQ� LQWHUYDOXO� GH� vQFUHGHUH� ��

��VH�DFFHSW �LSRWH]D�H0.�'HFL�VH�SRDWH�VSXQH�F �m = 27%.

2. &RQ LQXWXO�GH�JU VLPH�DO�ODSWHOXL�GH�YDF �LQWHJUDO�HVWH�VWDELOLW�OD��3HQWUX�D�YHULILFD�GDF �DFHDVW �

FRQGL LH�HVWH�VDWLVI FXW ��vQ�FD]XO�XQHL�IHUPH�GH�YDFL��V-DX�OXDW�QRX �SUREH��RE LQkQGX-VH�XUP WRDUHOH�

SURFHQWH�GH�JU VLPH��6 �VH�YHULILFH�LSRWH]D�F �P� �

��SURFHQWH�JU VLPL��FXQRVFkQGu-se σ� ��úL�QLYHOXO�GH�VHPQLILFD LH�α = 0,04. 6ROX LH�

ÌQWUXFkW� FDOLWDWHD� ODSWHOXL� HVWH� DVLJXUDW � GDF � SURFHQWXO� GH� JU VLPH� HVWH� HJDO� VDX�

GHS úHúWH�QLYHOXO�GH��VXQWHP�vQ�FD]XO�XQXL�test z unilateral. ÌQ�XUPD�GDWHORU�GH�VRQGDM��VH�RE LQH�

%4,39

6,30 ==∑=n

xx i

75,196,004,011 === −− zzz α

Intervalul de încredere este dat de:

ασα −=

−≤− − 1

/0

1n

mxzP

DGLF �

n

zmxσ

α ⋅−≥ −10

%495,375,19

18,06,310 =⋅−=⋅− −

nzm

σα

Cum %495,34,3 <=x �UH]XOW �F �QH�DIO P�vQ�UHJLXQHD�FULWLF ��GHFL�LSRWH]D�H0 se respinge.

3. 3HQWUX�D�YHULILFD�GDF �vQWUH�JUHXW LOH�PHGLL�DOH�úWLXOH LORU�GLQ�GRX �VRLXUL�GH�SRUXPE��H[LVW �R�

GLIHUHQ �VHPQLILFDWLY �VDX�QX��V-DX�OXDW�FkWH��SUREH�GLQ�ILHFDUH�VRL�úL�V-DX�RE LQXW�XUP WRDUHOH�

date (grame):

Soiul 1: 260; 250; 290; 300; 270; 310; 260; 280; 340; 300. Soiul 2: 240; 270; 230; 250; 220; 280; 210; 260; 290; 240.

'LQ�FHUFHW ULOH�DQWHULRDUH�VH�FXQRDúWH�F �� 27521 =σ úL� 2602

2 =σ .

6ROX LH� 3HQWUX�D�YHULILFD�GDF �H[LVW �R�GLIHUHQ �VHPQLILFDWLY �vQWUH�JUHXW LOH�PHGLL�DOH�úWLXOH LORU�GLQ�FHOH�

GRX �VRLXUL�GH�SRUXPE��VH�IRORVHúWH�testul z bilateral: Din datele de sondaj:

gr. 28610

28601 ==∑=

n

xx i

gr. 24910

24902 ==∑=

n

xx i

3HQWUX�XQ�QLYHO�GH�VHPQLILFD LH��α� ��VH�RE LQH�F �� 96,1975,0

2

05,01

21

===−−

zzz α

Intervalul de încredere este dat de:

ασσσσ

αα −=

+≤−≤+−

−−1

22

2

21

1

21

21

122

21

1

21

21 nn

zxxnn

zP

3472,1432,796,110

26027596,1

2

2

21

1

21

21

=⋅=+⋅=+− nn

zσσ

α

Cum 3472,143712 >=− xx �� QH� DIO P� vQ� UHJLXQHD� FULWLF �� VH� UHVSLQJH� +0�� GHFL� vQWUH� JUHXW LOH�

PHGLL�DOH�úWLXOH LORU�GH�SRUXPE�GLQ�FHOH�GRX �VRLXUL�H[LVW �GLIHUHQ H�VHPQLILFDtive.

4. 3UHVXSXQHP�F � SHQWUX� ��GH� REVHUYD LL� DVXSUD�XQHL� YDULDELOH� DOHDWRDUH�;�� V-D�RE LQXW�PHGLD�

110=x úL�DEDWHUHD�VWDQGDUG�� Se cere: a) 7HVWD L�LSRWH]D�QXO �F �m = 100, cu alternativa m > 100, utilizând α = 0,05. b) 7HVWD L� LSRWH]D� QXO � F �m = 100, cu alternativa m ≠ 100, utilizând α� � �� &RPSDUD L�

UH]XOWDWHOH�FHORU�GRX �WHVWH� 6ROX LH�

a) n = 100

110=x σ = 60 α = 0,05 &RQVLGHU P�� H0: m = 100 Ha: m > 100 6H�DSOLF �testul z unilateral dreapta:

ασ α −=

≤

−− 1

/1

0 zn

mxP

RegLXQHD�FULWLF �YD�IL�GDW �GH�

n

zmxσ

α ⋅+≥ −10

645,195,005,01 ==− zz

870,109645,1100

6010010 =⋅+=⋅+ −

nzm

σα

Cum 110=x !��VXQWHP�vQ�UHJLXQHD�FULWLF �GHFL�VH�UHVSLQJH�+0.

c) H0: m = 100 Ha: m ≠ 100 6H�DSOLF �testul z bilateral.

ασ αα −=

≤

−≤−

−−1

/2

1

0

21

zn

mxzP

96,1975,0

2

05,01

==−

zz

Limitele intervalului de încredere sunt:

nzmx

nzm

σσαα ⋅+≤≤⋅−

−−2

10

21

0

76,1196,1100

60

21

=⋅=⋅−α

σz

n

76,11124,88

76,1110076,11100

<<

+<<−

x

x

ÌQWUXFkW�QH�DIO P�vQ�LQWHUYDOXO�GH�vQFUHGHUH��VH�DFFHSW �LSRWH]D�+0.

5. 'LVWULEX LD� XQXL� ORW� GH� QRL� DXWRPRELOH�� GXS � FRQVXPXO� GH� FDUEXUDQW� OD� �� NP� SDUFXUúL�� VH�

SUH]LQW �DVWIHO�

Tabelul 6.1. ,QWHUYDOH�GH�YDULD LH��FRQVXPXOXL��O� Nr. automobile

6,2 – 6,6, 4 6,6 – 7,0 12 7,0 – 7,4 44 7,4 – 7,8 90 7,8 – 8,2 107 8,2 – 8,6 86 8,6 – 9,0 36 9,0 – 9,4 15 9,4 – 9,8 6

Total 400

6 �VH�YHULILFH�QRUPDOLWDWHD�GLVWULEX LHL�XWLOL]kQG�WHVWXO�χ2.

6ROX LH� 3HQWUX�YHULILFDUHD�QRUPDOLW LL�UHSDUWL LHL�HPSLULFH�VH�SDUFXUJ�XUP WRDUHOH�HWDSH�

1) 6H�FDOFXOHD] � x úL�σ��SHQWUX�GLVWULEX LD�SUH]HQWDW �vQ�WDEHOXO��

litri 8997,7400

8,3198 ≈==∑

∑=i

ii

n

nxx

( )3596,0

400

84,1432

2 ==∑

∑ −=

i

ii

n

nxxσ

σ = 0,5997 litri

2) 6H�FDOFXOHD] �YDORULOH�YDULDELOHL�QRUPDOH�QRUPDWH�SHQWUX�OLPLWHOH�VXSHULRDUH�DOH�LQWHUYDOHORU�

σxx

zi

i−

= sup (col. 5, tabelul 6.2.)

/LPLWD�VXSHULRDU �D�XOWLPHL�JUXSH�VH�FRQVLGHU ��∞�� LDU�OLPLWD�LQIHULRDU �D�SULPXOXL�LQWHUYDO�VH�FRQVLGHU �-∞�� LQkQG�FRQW�GH�IDSWXO�F �GLVWULEX LD�QRUPDO �WLQGH�F WUH��DVLPSWRWLF�

3) 6H�FDOFXOHD] �SUREDELOLW LOH�φ(zi)��FH�UHSUH]LQW �SUREDELOLW LOH�FD�R�YDORDUH�z�V �ILH�FXSULQV �vQWUH��úL�zi �FRO��WDEHOXO�� LQkQG�FRQW�F

)()( ii zz φφ −=−

4) 3UREDELOLW LOH�WHRUHWLFH�FD�R�YDORDUH�V �VH�VLWXH]H�vQ�LQWHUYDOXO�L�VXQW�

)()( sup1sup −−= iii zFzFp

sau )()( sup1sup −−= iii zzp φφ

��([FHS LH�IDFH�SULPXO�LQWHUYDO��Sentru care: )(5,0)5,0()( sup1sup11 zzp φφ +=−−=

��úL�XOWLPXO�LQWHUYDO��SHQWUX�FDUH� )(5,0 sup1−−= rr zp φ

��6XPD�SUREDELOLW LORU�GH�DSDUL LH�D�HYHQLPHQWHORU�HVWH��

5) )UHFYHQ HOH�WHRUHWLFH�VH�GHWHUPLQ ��SH�ED]D�SUREDELOLW LORU�WHRUHWLFH��DVWIHO��FRO��WDEHlul 6.2.):

iiteor pnn ⋅=

6) 6H�FDOFXOHD] �VWDWLVWLFD�( )

∑=

−=

k

i i

ii

np

npn

1

22χ

7) 'LQ�FRO��WDE��UH]XOW �� 44,32 =calculatχ .

��'LQ�WDEHOHOH�DQH[H�VH�FLWHúWH�YDORDUHD�OXL� 5916,1226,05,0, === ltab αχ

unde l �QXP UXO�JUDGHORU�GH�OLEHUWDWe, l = r – k – 1 = 9 - 2 - 1 = 6

Cum 22tabcalculat χχ < VH� DFFHSW � LSRWH]D� F � vQWUH� GLVWULEX LD� HPSLULF � úL� FHD� WHRUHWLF � H[LVW �

FRQFRUGDQ � Tabelul 6.2.

Limitele intervalelor

ni xi xini ii nxx 2)( −

0 1 2 3 4 6,2-6,6 4 6,4 25,6 10,24 6,6-7,0 12 6,8 81,6 17,28 7,0-7,4 44 7,2 316,8 28,16 7,4-7,8 90 7,6 684,0 14,40 7,8-8,2 107 8,0 856,0 0 8,2-8,6 86 8,4 722,4 13,76 8,6-9,0 36 8,8 316,8 23,04 9,0-9,4 15 9,2 138,0 21,60 9,4-9,8 6 9,6 57,6 15,36 Total 400 - 3198,8 143,84

σ

−=

xxz

supii

φ(z) pi npi ( )i

ii

np

npn 2−

5 6 7 8 9 -2,3340 -0,4901 0,0099 4 0 -1,6675 -0,4525 0,0376 15 0,6 -1,0005 -0,3413 0,1112 44 0 -0,3335 -0,1293 0,2120 85 0,29 0,3335 0,1293 0,2586 104 0,09 1,0005 0,3413 0,2120 85 0,01 1,6675 0,4525 0,1112 44 1,45 2,3340 0,4901 0,0376 15 0

∞ 0,5000 0,0099 4 1 - - 1,00 400 3,44

6. 8Q� SURGXF WRU� GH� GHWHUJHQ L� LQGXVWULDOL� SUHDPEDOHD] � SURGXVXO� OD� FXWLL� FH� WUHEXLH� V � DLE �� vQ�

PHGLH��NJ��(O�GRUHúWH�V �YHULILFH�FRUHFWLWXGLQHD�DPEDO ULL�úL�KRW U úWH�V �Rrganizeze un sondaj

GH�Q� ��REVHUYD LL��FXWLL��SHQWUX�FDUH�RE LQH�JUHXWDWHD�PHGLH� 85,11=x kg, cu o abatere medie S WUDWLF �σ� � �� NJ�� 6 � VH� WHVWH]H� LSRWH]D� FRQIRUP� F UHLD� vQ�PHGLH� FXWLLOH� DX� FkWH� �� NJ�� FX�

alternativa greutatea este difHULW �GH��NJ��NJ�VDX�!��NJ��SHQWUX�R�SUREDELOLWDWH�GH��. 6ROX LH�

H0: 120 =x kg

Ha: 120 ≠x kg ( 120 <x kg sau 120 >x kg).

Testul statistic: 0,310/5,0

1285,11

/

00 −=−=−=−=n

xxxxz

x σσ

Nivelul de încredere: 1 - α = 0,95 3UDJXO�GH�VHPQLILFD LH��α� ��GLQ�FDUH�UH]XOW �α/2 = 0,025 (deoarece avem test bilateral).

96,1025,02/ == zzα

Regiunea de respingere: 96,12/ −<αz sau 96,12/ >αz

Cum 2/αzzcalc > ��UH]XOW �F �LSRWH]D�QXO �HVWH�UHVSLQV �úL�VH�DFFHSW �LSRWH]D�DOWHUQDWLY ��DFHHD�F �

vQ�FXWLL�VH�DIO �R�FDQWLWDWH�GH�GHWHUJHQ L�VHPQLILFDWLY�GLIHULW �GH��NJ�

7. Într-o cercetare prin sondaj aleator privitoare la transportul în comun, au fost selectate 100 de

persoane pentru care s-D� FDOFXODW� YDORDUHD� PHGLH� D� ELOHWHORU� FXPS UDWH� vQWU-R� OXQ � SHQWUX�

WUDQVSRUW�vQ�FRPXQ�XUEDQ��PLL�OHL��FX�R�DEDWHUH�PHGLH�S WUDWLF �GH��PLL�OHL� Se cere: a) 7HVWD L�LSRWH]D�QXO ��DFHHD�FRQIRUP�F UHLD�YDORDUHD�PHGLH�D�ELOHWHORU�FXPS UDWH�vQWU-o lXQ ��

vQ� FROHFWLYLWDWHD� JHQHUDO �� HVWH� 1000 =x PLL� OHL�� FX� LSRWH]D� DOWHUQDWLY � 1000 >x mii lei, utilizând un nivel de încredere 1 - α = 0,95 (probabilitatea (1-α)⋅ ⋅100 = 95%).

b) 7HVWD L� LSRWH]D� QXO � 1000 =x mii leL�� FX� LSRWH]D� DOWHUQDWLY � 1000 ≠x PLL� OHL�� FX� DFHHDúL�

SUREDELOLWDWH��,QWHUSUHWD L�UH]XOWDWHOH� 6ROX LH�

a) H0: 1000 =x mii lei

Ha: 1000 >x mii lei

Testul statistic: 67,110/60

100110

/

00 =−=−=−=n

xxxxz

x σσ

Nivelul de încredere: 1 - α = 0,95 3UDJXO�GH�VHPQLILFD LH��α = 0,05 (test unilateral dreapta)

645,105,0 == zzα

Cum αzzcalc > �� UH]XOW � F � LSRWH]D� QXO � HVWH� UHVSLQV � � 1000 =x � PLL� OHL�� úL� VH� DFFHSW � LSRWH]D�

DOWHUQDWLY �� 1000 >x mii lei).

c) H0: 1000 =x mii lei

Ha: 1000 ≠x mii lei

Test statistic: 67,110/60

100110

/

00 =−=−=−=n

xxxxz

x σσ

Nivelul de încredere 1 - α = 0,95 3UDJXO�GH�VHPQLILFD LH�α� ��GLQ�FDUH�UH]XOW �α/2 = 0,025 (deoarece avem test bilateral).

96,1025,02/ == zzα

Cum 2/αzzcalc < ��UH]XOW �F �VH�DFFHSW �LSRWH]D�QXO ��YDORDUHD�PHGLH�D�ELOHWHORU�

FXPS UDWH�QX�HVWH�VHPQLILFDWLY�GLIHULW �GH��PLL�OHL��

8. 8Q�SURGXF WRU�GH�EDWHULL�GRUHúWH�V �YHULILFH�GDF �SURFHQWXO�EDWHULLORU�GHIecte este mai mic de

��3UHVXSXQHP�F � VXQW� VHOHFWDWH�DOHDWRU��GH�EDWHULL�� ILHFDUH�HVWH� WHVWDW � úL� VXQW�J VLWH��

EDWHULL�GHIHFWH��2IHU �DFHVWH�LQIRUPD LL�VXILFLHQWH�GRYH]L�F �SURFHQWXO�EDWHULLORU�GHIHFWH�HVWH�PDL�

PLF�GH��"�8WLOL]D L�R�SUREDELOLWDWH�GH 99% de garantare a rezultatelor 6ROX LH�

H0: 05,0=p Ha: p < 0,05

Testul statistic: ( )

n

ww

pwpwz

w −−=−=1σ

w = 10 / 300 = 0,03

( )01,00098,0

300

97,003,01 ≅=⋅=−=n

wwwσ

201,0

05,003,0 −=−=z

Nivelul de încredere 1 - α = 0,99 3UDJXO�GH�VHPQLILFD LH�α = 0,01 (test unilateral stânga)

33,201,0 == zzα

Cum αzzcalc < �� UH]XOW � F � LSRWH]D� QXO � QX� VH� UHVSLQJH�� DúDGDU� QX� DYHP� VXILFLHQWH�

GRYH]L�SHQWUX�D�DILUPD�F �SURFHQWXO�EDWHULLORU�GHIHFWH�HVWH�PDL�PLF�GH��

9. ��H[SHU L�vúL�H[SULP �RSLQLLOH�SULYLQG�SUH XO�XQXL�SURGXV��vQ�DQXO�YLLWRU��5H]XOWDWHOH�DQFKHWHL�VXQW�

exprimate de indicatorii sintetici: 63,20 =x mil. lei, s = 0,72 mil. lei. (( )

1

2

−−

= ∑n

xxs i ,

GHRDUHFH�HVWH�XQ�HúDQWLRQ�GH�YROXP�UHGXV��'DF �VH�úWLH�F �DQXO�DFHVWD�SUH XO�PHGLX�D�IRVW�GH�

��PLO�� OHL�� VXQW�PRWLYH�VXILFLHQWH�SHQWUX�D�VXV LQH� LSRWH]D�F �DQXO�YLLWRU�SUH XO�PHGLX�YD�IL�

VHPQLILFDWLY� PDL� PDUH� ID � GH� FHO� GH� DQXO� DFHVWD"� 6H� YD� XWLOL]D� R� SUREDELOLWDWH� GH� �� GH�

garantare a rezultatelor. 6ROX LH:

H0: 01,20 =x

Ha: 01,20 >x

Test statistic: 96,15/072

01,263,2

/

0 =−=−=ns

xxt

�3UHVXSXQHUH��GLVWULEX LD�vQ�SRSXOD LD�JHQHUDO �HVWH�DSUR[LPDWLY�QRUPDO �� Nivelul de încredere 1 - α = 0,95 3UDJXO�GH�VHPQLILFD LH�α = 0,05

132,24;05,01, ==− tt nα

Regiunea de respingere: 1, −> ntt α

Cum 1; −< ncalc tt α ��UH]XOW �F �QX�DYHP�PRWLYH�V �UHVSLQJHP�LSRWH]D�QXO �úL�V �DFFHSW P�

LSRWH]D�FRQIRUP�F UHLD�SUH XO�PHGLX�GH�DQXO�YLLWRU�YD�IL�VHPQLILFDWLY�PDL�PDUH�ID �GH�FHl de anul acesta.

10. 0DQDJHUXO� XQXL� UHVWDXUDQW� GRUHúWH� V � GHWHUPLQH� GDF � R� FRPSDQLH� SXEOLFLWDU � D� P ULW�

VHPQLILFDWLY�PHGLD�vQFDV ULORU�]LOQLFH��(O�FXOHJH�GDWH�SULYLWRDUH�OD��GH�]LOH�vQDLQWHD�FDPSDQLHL�

úL� OD��GH� ]LOH� GXS � vQFKHLHUHD� FDPSDQLHL� SXEOLFLWDUH��5H]XOWDWHOH�SHQWUX� FHOH� GRX �SHULRDGH�

sunt: Înaintea campaniei 'XS �FDPSDQLH

n1 = 50 n2 = 50

55,121 =x mil. lei 30,132 =x mil. lei

15,21 =σ mil. lei 38,22 =σ mil. lei

,QIRUPD LLOH� RE LQXWH� VXQW� VXILFLHQWH� SHQWUX� D� VXV LQH� LSRWH]D� FRQIRUP� F UHLD� PHGLLOH�

vQFDV ULORU�GLIHU �VHPQLILFDWLY��DGLF �VXQW�VHPQLILFDWLY�PDL�PDUL��GXS �FDPSDQLD�SXEOLFLWDU "�6 �VH�

utilizeze o probabilitate de 95%.

6ROX LH� H0: 00201 =− xx kg

Ha: 00201 <− xx NJ��DGLF 0201 xx < ) .

Test statistic: ( )

)(

21

21

0

xx

xxz

−

−−=σ

3UHVXSXQkQG�F �FHOH�GRX �HúDQWLRDQH�VXQW�LQGHSHQGHQWH�

( )2

22

1

21

21 nnxx

σσσ +=−

( ) ( )42,1

53,0

30,1355,12

30

38,2

50

15,2

30,1355,12

22−=−=

+

−=z

Regiunea de respingere: 645,105,0 => zz .

Cum 05,0=< αzzcalc �� UH]XOW � F � QX� DYHP� PRWLYH� V � UHVSLQJHP� LSRWH]D� QXO �� FRQIRUP� F UHLD�

GLIHUHQ D�GLQWUH�FHOH�GRX �PHGLL�QX�HVWH�VHPQLILFDWLY�GLIHULW �GH��QX�VXQW�VXILFLHQWH�GRYH]L�SHQWUX�D�

VXV LQH�FUHúWHUHD�vQFDV ULORU�GXS �FDPSDQLH��

Probleme propuse

1. 5 VSXQGH L�OD�XUP WRDUHOH�vQWUHE UL�

a) &DUH�HVWH�GLIHUHQ D�GLQWUH�+0�úL�+a? b) 'HILQL L�ILHFDUH�GLQ�XUP WRDUHOH�

� Eroare de tip I; � Eroare de tip II; � α; � β.

c) &kQG�ULVFD L�V �IDFH L�R�HURDUH�GH�WLS�,"�'DU�GH�WLS�,," d) ÌQ�WHVWDUHD�LSRWH]HORU��FH�GHWHUPLQ �P ULPHD�UHJLXQLi critice?

2. 3HQWUX�ILHFDUH�GLQ�XUP WRDUHOH�UHJLXQL�FULWLFH��VFKL D L�GLVWULEX LD�OXL�]�úL�LQGLFD L�ORFXO�UHJLXQLL�

critice Rc: a) z > 1,96; b) z > 1,645 c) z > 2,575 d) z < -1,29 e) z < -1,645 sau z > 1,645 f) z < -2,575 sau z > 2,575.

3HQWUX�ILHFDUH�UHJLXQH�FULWLF �VSHFLILFD L�FDUH�HVWH�SUREDELOLWDWHD�FD�R�HURDUH�GH�WLS�,�V �VH�IDF �DWXQFL�

când H0�HVWH�DGHY UDW �

3. 3UHVXSXQHP�F �VXQWH L�LQWHUHVD L�vQ�UH]ROYDUHD�WHVWXOXL�

H0: m = 200 Ha: m > 200

úL�GHFLGH L�V �XWLOL]D L�XUP WRDUHD�UHJXO �GH�GHFL]LH��VH�UHVSLQJH�+0�GDF �PHGLD�D��GH�REVHUYD LL�

este mai mare de 212. Se cere:

a) ([SULPD L�GHFL]LD�vQ�WHUPHQL�]� b) 'HWHUPLQD L�SUREDELOLWDWHD�HPLWHULL�HURULL�GH�JHQXO�,�

4. În 1930 s-a propus utilizarea poligrafului pentru detectarea minciunilor. Pentru a se

YHGHD� FkW� GH� ELQH� OXFUHD] �� s-D� HIHFWXDW� XUP WRDUHD� FHUFHWDUH�� GDF � �� GH� SHUVRDQH�

VXQW� WHVWDWH� úL� �� GH� SHUVRDQH� VSXQ� DGHY UXO�� LDU� �� PLQW�� SROLJUDIXO� D� LQGLFDW� F �

DSUR[LPDWLY��GH�SHUVRDQH�FDUH�DX�IRVW� LGHQWLILFDWH�FD�VSXQkQG�DGHY UXO��DX�PLQ LW� vQ�

realitate, iar 120 de persRDQH� FDUH� DX� IRVW� LQGLFDWH� F � DX� PLQ LW�� vQ� UHDOLWDWH� DX� VSXV�

DGHY UXO��'DF � H0��VSXQH�DGHY UXO Ha: minte.

a) Care este eroarea de tip I? Dar de tip II? b) Care este probabilitatea comiterii unei erori de tipul I? Dar probabilitatea comiterii unei

erori de tipul II?

5. 'DF � VH� HIHFWXHD] � WHVWDUHD� XQHL� LSRWH]H� VWDWLVWLFH� XWLOL]kQG� α = 0,05, pentru care din

XUP WRDUHOH�YDORUL�DOH�OXL�φ(z)�YD�IL�UHVSLQV �LSRWH]D�QXO " a) 0,44; b) 0,499; c) 0,40; d) 0,249; e) 0,49; f) 0,458.

6. Pentru fiecare din valorile φ(z)�úL�α observate, specificD L�GDF �LSRWH]D�QXO �HVWH�UHVSLQV �

a) α = 0,05; φ(z) = 0,40 b) α = 0,10; φ(z) = 0,45 c) α = 0,01; φ(z) = 0,499 d) α = 0,025; φ(z) = 0,45 e) α = 0,10; φ(z) = 0,05.

7. Într-XQ� WHVW� vQ� FDUH� VH� WHVWHD] � LSRWH]D� QXO � 100:0 ≤mH cu 100: >mH a VH� RE LQH valoarea

WHVWXOXL�]� ��$IOD L�SUDJXO�GH�VHPQLILFD LH�REVHUYDW�DO�WHVWXOXL�

8. 'DF � VH� WHVWHD] � 100:0 =mH cu 100: ≠mH a úL� VH� RE LQH� ]� � �� DIOD L� SUDJXO� GH�

VHPQLILFD LH�REVHUYDW�DO�WHVWXOXL�

9. Un fabricant de automobile a HVWLPDW�F �XQ�QRX�WLS�GH�PDúLQL�RE LQH�XQ�FRQVXP�PHGLX�GH��O��

�� NP�� GDU� FRPSDQLD� FDUH� GRUHD� V � FRPHUFLDOL]H]H� DFHVWH� PDúLQL� SUHWLQGHD� F � FRQVXPXO�

GHS úHúWH�HVWLPD LD�SURGXF WRUXOXL��3HQWUX�D�VXV LQH�DILUPD LD��FRPSDQLD�D�VHOHFWDW��GH�PDúLQL�

úL�D�vQUegistrat consumul. au rezultat datele:

8,11=x l / 100 km; σ = 3,0 l / 100 km. Se cere: a) 'DF �FHL�LQWHUHVD L�GRUHVF�V �DUDWH�F �PHGLD�FRQVXPXOXL�HVWH�PDL�PDUH�GH��O��NP��FDUH�

HVWH�LSRWH]D�QXO "�'DU�LSRWH]D�DOWHUQDWLY " b) (IHFWXD L�WHVWDrea pentru α = 0,05.

10. 2�FRPSDQLH�GRUHúWH�V �LQWURGXF �R�QRX �PHWRG �GH�SURGXFHUH�D�XQXL�SURGXV��6H�VHOHFWHD] ��

GH�SURGXVH�SHQWUX�FDUH�VH�vQUHJLVWUHD] �WLPSLL�GH�SURGXFHUH�FX�YHFKHD�PHWRG �úL��GH�SURGXVH�

SHQWUX�QRXD�PHWRG ��5H]XOWDWHOH�VXQW� Metoda actXDO � 1RXD�PHWRG �

n1 = 50 n2 = 50

3,271 =x minute 4,252 =x minute

7,31 =s minute 1,32 =s minute 6 �VH�DUDWH��XWLOL]kQG�R�SUREDELOLWDWH�GH��GDF � UH]XOWDWHOH�RIHU � VXILFLHQWH� LQIRUPD LL�pentru a LQGLFD�IDSWXO�F �QRXD�PHWRG �GXFH�OD�XQ�FRQVXP�GH�WLPS�VHPQLILFDWLY�PDL�PLF�

11. $QDOL]D�XQXL�HúDQWLRQ�DOHDWRU�GH��GH�REVHUYD LL�D�GXV�OD�XUP WRDUHOH�UH]XOWDWH�

7,200 =x ; ( ) 155,22 =−∑ xxi

Se cere: c) 7HVWD L�LSRWH]D�QXO � 47,00 =x ��FX�LSRWH]D�DOWHUQDWLY �� 47,00 <x utilizând o probabilitate de 90%.

d) 7HVWD L�LSRWH]D�QXO � 47,00 =x FX�LSRWH]D�DOWHUQDWLY � 47,00 ≠x utilizând o probabilitate de 90%.

12. Un analist financiar presupunH�F � YDORDUHD� IDFWXULORU� HPLVH� GH� ILUPD� OD�FDUH� OXFUHD] �HVWH�� vQ�

PHGLH��GH�SHVWH��PLO��OHL�IDFWXU ��3HQWUX�D�VXV LQH�DFHDVW �LSRWH] ��HO�VHOHFWHD] �DOHDWRU��GH�

IDFWXUL�HPLVH�vQ�XOWLPD�OXQ ��SHQWUX�FDUH�vQUHJLVWUHD] �YDORULOH��PLO��OHL��12, 8, 9, 10, 11, 9, 9, 10, 10, 9, 10, 10, 11, 9, 8, 9, 6, 10, 7, 10, 9, 13, 9, 10, 11, 10, 9, 9 , 9, 17, 12, 6, 10, ��6 � VH� WHVWH]H� LSRWH]D�DQDOLVWXOXL�SHQWUX�R�SUREDELOLWDWH�GH�

garantare a rezultatelor de 99%.

13. PresupXQHP� F � vQ� XUPD� XQXL� VRQGDM� DOHDWRU� GH� �� GH� REVHUYD LL�� SHQWUX� R� FDUDFWHULVWLF �

DOWHUQDWLY � V-D� RE LQXW� PHGLD� Z� � �� úL� GRULP� V � WHVW P� LSRWH]D� QXO � F � vQ� FROHFWLYLWDWHD�

JHQHUDO �PHGLD�p�HVWH��FX�LSRWH]D�DOWHUQDWLY �S��6 �VH�XWLOL]H]H�XQ�SUDJ�GH�VHPQLILFD LH�

α� � �� 6 � VH� WHVWH]H� SHQWUX� DFHODúL� SUDJ� GH� VHPQLILFD LH� LSRWH]D� QXO � p = 0,7 cu ipoteza DOWHUQDWLY ��p ≠ 0,7.

14. 8Q�HúDQWLRQ�DOHDWRU�GH��GH�SHUVRDQH�DX�IRVW� WHVWDWH�FX�SULYLUH� OD�XQ�QRX�SURGXV�GH�VQDFN�–

IRRG��5 VSXQVXULOH�DX�IRVW�FRGLficate (0: nu-PL�SODFH��vPL�SODFH��úL�OLVWDWH��1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1. Se cere: a) 8WLOL]D L�R�SUREDELOLWDWH�GH��SHQWUX�D�HVWLPD� LQWHUYDOXO� vQ�FDUH� VH�YD� vQFDGUD�SURSRU LD�

FHORU�F URUD�OH�SODFH�SURGXVXO��vQ�FROHFWLYLWDWHD�JHQHUDO � b) 7HVWD L�LSRWH]D�QXO �+0: p =��FX�LSRWH]D�DOWHUQDWLY �+D��p > 0,05, unde p�HVWH�SURSRU LD�

FHORU�F URUD�OH�SODFH�SURGXVXO��XWLOL]kQG�R�SUREDELOLWDWH�de 95%.

15. 3UHVXSXQHP�F �D L�VHOHFWDW�XQ�HúDQWLRQ�DOHDWRU�GH��GH�SHUVRDQH�GLQWU-R�SRSXOD LH�JHQHUDO �

FH� FRQ LQH� �� p = 0,2) persoane favorabile unei propuneri. Care este probabilitatea ca SURSRU LD�GLQ�HúDQWLRQ��Z��V �GLIHUH�GH�FHD�GLQ�FROHFWLYLWDWHD�JHQHUDO ��S��FX�PDL�PXOW�GH��"

16. 8Q� SURGXF WRU� GH� LPSULPDQWH� SHQWUX� FDOFXODWRDUH� SHUVRQDOH� GRUHúWH� V � HVWLPH]H� PHGLD�

QXP UXOXL�GH�FDUDFWHUH�WLS ULWH�SkQ �FkQG�VH�FRQVXP �FHUQHDOD��&RVWXO�FUHVFXW�DO�XQHL�DVWIHO�GH�

DQFKHWH�LPSXQH�XWLOL]DUHD�XQXL�HúDQWLRQ�GH�YROXP�UHGXV��3UHVXSXQHP�F �DX�IRVW�WHVWDWH�Q� ��

LPSULPDQWH�úL�V-au calculat: 13,10 =x milioane caractere, s = 0,27 milioane caractere. Se cere:

a) 6 � VH� WHVWH]H�FX�R� SUREDELOLWDWH� GH� �� LQWHUYDOXO� GH� vQFUHGHUH� SHQWUX� QXP UXO�PHGLX� GH�

caractere.

b) 6 � VH� WHVWH]H� LSRWH]D� QXO � 00,10 =x PLOLRDQH�� FX� LSRWH]D� DOWHUQDWLY � 00,10 >x milioane FDUDFWHUH��XWLOL]kQG�R�SUREDELOLWDWH�GH�� vQ� LSRWH]D�GLVWULEX LHL�QRUPDOH�D�QXP UXOXL�GH�

FDUDFWHUH�vQ�FROHFWLYLWDWHD�JHQHUDO �

77

ÍQWUHE UL� ¾ ([SOLFD L�FRQFHSWXO�GH�OHJ WXU �VWDWLVWLF � ¾ &ODVLILFDUHD�SULQFLSDOHORU�WLSXUL�GH�OHJ WXUL�vQWUH�YDULDELOHOH�HFRQRPLFR-sociale. ¾ &DUH�VXQW�PHWRGHOH�HOHPHQWDUH�GH�FDUDFWHUL]DUH�D�OHJ WXULORU�GLQWUH�YDULDELOH" ¾ ÌQ�FH�FRQVW �PHWRGD�WDEHOXOXL�GH�FRUHOD LH�úL�OD�FH�VH�IRORVHúWH" ¾ 5HJUHVLD�VLPSO �OLQLDU ��FRQVWUXF LD�PRGHOXOXL��LQWHUSUHWDUHD�FRHILFLHQ LORU�GH�UHJUHVLH� ¾ 3ULQFLSDOHOH�PRGHOH�GH�UHJUHVLH�XQLIDFWRULDO � ¾ 0RGHOH�GH�UHJUHVLH�PXOWLIDFWRULDO � ¾ Care sunt principalii indicatori sintetici aL�FRUHOD LHL" ¾ 7HVWH�GH�YHULILFDUH�D�VHPQLILFD LHL�HFXD LHL�GH�UHJUHVLH�úL�D�FRHILFLHQ LORU�GH�FRUHOD LH� ¾ &DUH� VXQW� SULQFLSDOHOH� PHWRGH� QHSDUDPHWULFH� GH� P VXUDUH� D� LQWHQVLW LL� OHJ WXULORU� GLQWUH�

fenomene? ¾ &RHILFLHQWXO�GH�DVRFLHUH�úL�GH�FRQWLQJHQ ��0RG�GH�calcul, utilizare, interpretare. ¾ &RHILFLHQWXO�GH�HODVWLFLWDWH��&RQFHSW��UHOD LL�GH�FDOFXO� ¾ (ODVWLFLWDWHD�FHUHULL�XQXL�SURGXV�vQ�UDSRUW�FX�SUH XO�DFHVWXLD�

$1$/,=$�67$7,67,& �$�'(3(1'(1 (/25 DINTRE VARIABILE


1. 8Q�DQDOLVW�GH�PDUNHWLQJ�XUP UHúWH�vQ��OXQL�HIHFWXO�UHFODPHORU�DVXSUD�YHQLWXULORU�GLQ�YkQ] UL��(O�

vQUHJLVWUHD] � FKHOWXLHOLOH� FX� UHFODPD� �;�� PLOLRDQH� OHL�� úL� YHQLWXULOH� GLQ� YkQ] UL�� <� �VXWH�PLOLRDQH�OHL��ÌQ�XUPD�SUHOXFU ULL�UH]XOWDWHORU��RE LQH�

∑ =15ix ; ∑ = 10iy ; ∑ = 552ix ; ∑ = 37ii yx .

ÌQ�LSRWH]D�XQHL�GHSHQGHQ H�OLQLDUH��V �VH�GHWHUPLQH�OLQLD�GH�UHJUHVLH�

a) xy 1,07,0ˆ −= ; b) xy 7,01,0ˆ +−= ; c) xy 7,02ˆ −= ; d) xy 5,57,3ˆ += ; e) xy 7,0ˆ =

6ROX LH�

( )

∑ ∑

∑ ∑ ∑

−

−==

n

xx

n

yxxy

s

yxb

x2

22

),cov(

n

xb

n

yxbya ∑∑ ⋅−=⋅−=

7,0

5

)15(55

5

101537

2=

−

⋅−=b

1,05

157,0

5

10 −=⋅−=a

⇒ xy 7,01,0ˆ +−= (b)

2. 3HQWUX�GRX �YDULDELOH�VWDWLVWLFH�vQWUH�FDUH�H[LVW �R�GHSHQGHQ �OLQLDU �V-au înregistrat datele, care

vQ�XUPD�SUHOXFU ULL��VH�SUH]LQW �DVWIHO�

∑ = 91,183ix ; ∑ = 968iy ; ∑ = 8,40992ix ; ∑ = 9,19485ii yx ; ∑ = 6,962992

iy , 11,1=i . 'HVSUH�GHSHQGHQ D�GLQWUH�FHOH�GRX �YDULDELOH�VH�SRDWH�DILUPD�

a) GHSHQGHQ D�HVWH�LQYHUV � b) coHILFLHQWXO�GH�FRUHOD LH�r = 0,978; c) HFXD LD�GH�UHJUHVLH�HVWH� xy 2213,314,34ˆ += ; d) HVWLPDUHD�FRYDULDQ HL�HVWH�cov (x,y) = 130,82; e) HFXD LD�GH�UHJUHVLH�HVWH� xy 2213,388ˆ −=

6ROX LH� (VWLP ULOH�GLVSHUVLLORU�YDULDELOHORU�;�úL�<�VXQW�

( )499,102

1011

91,1838,4099

1

222

2 =−

=−

−=

∑ ∑

nn

xx

si

x

( )56,1111

1011

9686,96299

1

222

2 =−

=−

−=

∑ ∑

nn

yy

si

y

LDU�DEDWHULOH�PHGLL�S WUDWLFH�

34,33

124,10

2

2

==

==

yy

xx

ss

ss

(VWLPDUHD�FRYDULDQ HL�GLQWUH�YDULDELOHOH�;�úL�<�HVWH�

182,33010

11

96891,1839,19485

1),cov( =

⋅−=

−

−=

∑ ∑ ∑

nn

yxxy

yx

Atunci

2213,3499,102

182,330),cov(2

===xs

yxb

úL�

14,3472,162213,388 =⋅−=⋅−=⋅−= ∑∑n

xb

n

yxbya

$úDGDU��HFXD LD�GH�regresie este:

xy 2213,314,34ˆ += &RHILFLHQWXO�GH�FRUHOD LH�ry/x este:

( )\

Uss

x,y

x/xy

cov= sau y

xxy s

sbr ⋅=/

978,034,33124,10

182,330/ =

⋅=xyr sau 978,0

34,33

124,102213,3/ =⋅=xyr

'HFL�DILUPD LLOH�E��úL�F��VXQW�DGHY UDWH�

3. 6H�FXQRVF�XUP WRDUHOH�GDWH��SULYLQG�FLIUD�GH�DIDFHUL�úL�FKHOWXLHOLOH�SHQWUX�FHUFHWDUH�HIHFWXDWH�GH�

500 de firme: Tabelul 7.1. *UXSH�GH�ILUPH�GXS �FLIUD�GH�DIDFHUL�

(U.M.) 1XP U�ILUPH Cheltuieli medii pentru cercetare

(U.M.) sub 50 70 8 50 – 70 100 7 70 – 90 200 12

90 – 110 80 15 peste 110 50 20

Total 500 - 3UHVXSXQkQG�F �vQWUH�FHOH�GRX �YDULDELOH�H[LVW �R�OHJ WXU �OLQLDU ��VH�FHUH�

a) 6 �VH�DMXVWH]H�FKHOWXLHOLOH�SHQWUX�FHUFHWDUH�vQ�IXQF LH�GH�FLIUD�GH�DIDFHUL��IRORVLQG�IXQF LD�GH�UHJUHVLH�DGHFYDW �

b) 6 �VH�WHVWH]H�VHPQLILFD LD�SDUDPHWULORU�IXQF LHL�GH�UHJUHVLH� c) 6 � VH� P VRDUH� LQWHQVLWDWHD� OHJ WXULL� GLQWUH� FHOH� GRX � YDULDELOH�� IRORVLQG� FRHILFLHQWXO� GH�

FRUHOD LH�OLQLDU �VLPSO �úL�UDSRUWXO�GH�FRUHOD LH��7HVWD L�VHPQLILFD LD�LQGLFDWRUXOXL�IRORVLW� 6ROX LH�

a) 1RW P�FX�;�– cifra de afDFHUL� �YDULDELOD�IDFWRULDO �� úL�FX�<�– cheltuielile pentru cercetare �YDULDELOD�UH]XOWDWLY ��

ε++= iix bxaY , unde Yxi – valorile teoretice, ajustate ale lui Y.

=+

=+

∑ ∑∑

∑ ∑

iiiiiii

iiii

nyxnxbnxa

nynxbna

2

( )=

−

⋅−⋅==

∑∑∑∑∑∑

∑∑∑∑∑∑∑

22

2

2

2

iiii

iiiiiiiii

iiii

ii

iiiii

iiii

nxnxn

nxnyxnxny

nxnx

nxn

nxnyx

nxny

a

66,0388003272000500

49640038800327200058602

−=−⋅

⋅−⋅=a

( )=

−

⋅−==

∑∑∑∑∑

∑∑∑

∑∑∑

22

2iiii

iiiiiii

iiii

ii

iiiii

ii

nxnxn

nynxnyxn

nxnx

nxn

nyxnx

nyn

b

16,0159,0130560

20832 ≈==b ��7DEHOXO��úL�WDEHOXO��

iix xY ⋅+−= 16,066,0 �úL�VXQW�FDOFXODWH�vQ�WDEHOXO��FRO��

Tabelul 7.2. xi yi ni xini yini 2

ix ii nx2 0 1 2 3 4 5 6

40 8 70 2800 560 1600 112.000 60 7 100 6000 700 3600 360.000 80 12 200 16.000 2400 6400 1.280.000

100 15 80 8000 1200 10000 800.000 120 20 50 6000 1000 14400 720.000

- - 500 38.800 5860 3.272.000

Tabelul 7.3. xi yi xiyini

ii ny 2 Yxi ( ) ixii nYy 2− 0 1 2 3 4 5

40 8 22.400 4480 5,74 357,532 60 7 42.000 4900 8,94 376,36 80 12 192.000 28800 12,14 3,92

100 15 120.000 18000 15,34 9,248 120 20 120.000 20000 18,54 106,58

- - 496.400 76.180 853,64

Interpretarea lui “b” (coeficientul de regresie)��GDF �FLIUD�GH�DIDFHUL�FUHúWH�FX�R8�0��DWXQFL�FKHOWXLHOLOH�PHGLL�SW��FHUFHWDUH�YRU�FUHúWH�FX��8�0�

b) 7HVWDUHD�VHPQLILFD LHL�SDUDPHWUXOXL�³a”: 6H�FRQVWUXLHúWH�VWDWLVWLFD��WHVWXO��t, astfel:

352,1136,223,1

66,0 =⋅−== ns

at

( )3,1

498

64,853

2

2==

−−

=∑

∑i

ixii

n

nYys

6H�FRPSDU �YDORDUHD�FDOFXODW �D�VWDWLVWLFLL�t�FX�YDORDUHD�WDEHODU ��WHRUHWLF �W0,01;498 = 2,568. Cum tcalc > t0,01;498� ��UH]XOW �F ��³D´�HVWH�VHPQLILFDWLY� 7HVWDUHD�VHPQLILFD LHL�SDUDPHWUXOXL�³b”:

( ) 568,227,1792611203,1

16,0498;01,0

2 =>=⋅=−= ∑ tnxxs

bt iicalc

deci “b” este semnificativ.

6,77500

800.38 ===∑

∑i

ii

n

nxx

Tabelul 7.4. xi yi ( ) ii nxx

2− ( ) ii nyy2

−

0 1 2 3 40 8 98.963,2 968,688 60 7 30.976 2227,84 80 12 1.152 15,68 100 15 40.140,8 860,672 120 20 89.888 3427,92

- - 261.120 7500,8

c) &RHILFLHQWXO�GH�FRUHOD LH�OLQLDU �VLPSO �

( )( )=

σσ−−

=σσ

= ∑yx

iii

yxxy n

nyyxxyxr

),cov(/

( )[ ] ( )[ ]∑ ∑∑ ∑

∑ ∑∑

−−

−=

2222iiiiiiii

iiiiiii

nynynnxnxn

nynxnyxn

( )( ) 94,0586076180500388003272000500

586038800496400500

22=

−⋅−⋅

⋅−⋅=

5DSRUWXO�GH�FRUHOD LH�VH�FDOFXOHD] �FX�GDWHOH�GLQ�WDE��FRO��úL�WDE��FRO��

( )( )

94,08,7500

64,85311

2

2

/ =−=−

−−=

∑

∑

ii

iiixy

nyy

nYxyR

72,11500

5860 ===∑

∑i

ii

n

nyy

Ry/x = ry/x� ��GHFL�H[LVW �R�OHJ WXU �OLQLDU ��SXWHUQLF ��GLUHFW ��vQWUH�FHOH�GRX �YDULDELOH�

7HVWDUHD�VHPQLILFD LHL�FRHILFLHQWXOXL�GH�FRUHOD LH�³r” se face cu ajutorul testului t:

5,6131,22341,0

94,0498

94,01

94,02

1 22=⋅=

−=−

−= n

r

rt

568,2498;01,0 => ttcalc , deci “r” este semnificativ.

4. Cheltuielile cu reclamD�úL�FDQWLW LOH�GLQ�SURGXVXO�³;´�YkQGXWH��SHQWUX��GH�PDJD]LQH�DOH�XQXL�

JUXS�GH�ILUPH�FX�SURILO�FRPHUFLDO��VH�SUH]LQW �DVWIHO� Tabelul 7.5.

&DQWLW L�YkQGXWH�GLQ�SURGXVXO�;��]HFL�PLL�EXF L� Cheltuieli cu reclama (mil. lei) 0,6 – 1,4 1,4 – 2,2 2,2 – 3,0 3,0 – 3,8 3,8 – 4,6 50 – 150 6 1 - - - 150 – 250 - 8 5 - - 250 – 350 - 1 8 - - 350 – 450 - - - 6 2 450 - 550 - - - - 4

Se cere:

a) $MXVWD L� FDQWLW LOH� YkQGXWH� GLQ� SURGXVXO� ³;´�� IXQF LH� GH� FKHOWXLHOLOH� FX� UHFODPD�� IRORVLQG�IXQF LD�GH�UHJUHVLH�DGHFYDW ��7HVWD L�VHPQLILFD LD�SDUDPHWULORU�HFXD LHL�GH�UHJUHVLH�

b) 0 VXUD L� LQWHQVLWDWHD� OHJ WXULL�GLQWUH�FHOH�GRX �YDULDELOH�� IRORVLQG�XQ� LQGLFDWRU�GH�FRUHOD LH�DGHFYDW��7HVWD L�VHPQLILFD LD�LQGLFDWRUXOXL�FDOFXODW�

c) 9HULILFD L�YDOLGLWDWHD�PRGHOXOXL�GH�UHJUHVLH�Iolosit.

6ROX LH� a) $GPLWHP�F � vQWUH� FHOH� GRX �YDULDELOH� H[LVW �R� OHJ WXU � GH� IRUP � OLQLDU �SULQ� LQWHUSUHWDUHD�

FRUHORJUDPHL��$WXQFL�YRP�XWLOL]D�IXQF LD�GH�UHJUHVLH�OLQLDU �

ε++= iix bxaY , unde Yxi – valorile teoretice, ajustate ale lui Y.

6H�RE LQH sistemul:

=+

=+

∑ ∑∑∑

∑∑∑ ∑

i jijji

iii

iii

jjj

iii

i jij

nyxnxbnxa

nynxbna

2

4465,0

000.680.3200.11

200.1141

000.680.332960

200.114,103

2

2

===

∑∑

∑

∑∑ ∑

∑∑

iii

iii

iii

iii

i jijji

iii

jjj

nxnx

nxn

nxnyx

nxny

a

0076,0

000.680.3200.11

200.1141

32960200.11

4,10341

2

===

∑∑

∑

∑ ∑∑

∑

iii

iii

iii

i jijji

iii

jjj

nxnx

nxn

nyxnx

nyn

b

³E´�HVWH�FRHILFLHQWXO�GH�UHJUHVLH��SDQWD�GUHSWHL��&XP�E�!��DYHP�R�OHJ WXU �GLUHFW �vQWUH�YDULDELOH� 'DF �FKHOWXLHOLOH�FX�UHFODPD�FUHVF�FX�XQ�PLOLRQ�OHL��YkQ] ULOH�FUHVF în medie cu 0,0076 zeci mii buc.

ixi xY ⋅+= 0076,04465,0 �&DOFXOHOH� LQWHUPHGLDUH� QHFHVDUH� GHWHUPLQ ULL� SDUDPHWULORU� ³D´� úL� ³E´��

SUHFXP�úL�YDORULOH�DMXVWDWH�VH�J VHVF�vQ�WDEHOXO��

7HVWDUHD�VHPQLILFD LHL�SDUDPHWUXOXL�³a”:

563,741378,0

4465,0 =⋅== ns

at

Tabelul 7.6. &DQWLW L�YkQGXWH��]HFL�PLL�EXF�� Chelt. cu

reclama (mil. lei)

1 1,8 2,6 3,4 4,2 in iinx

ii nx2 ∑j

ijji nyx

Yxi

100 6 1 - - - 7 700 70.000 780 1,196 200 - 8 5 - - 13 2600 520.000 5480 1,946 300 - 1 8 - - 9 2700 810.000 6780 2,696 400 - - - 6 2 8 3200 1.280.000 11.520 3,446 500 - - - - 4 4 2000 1.000.000 8400 4.196

jn 6 10 13 6 6 41 11200 3.680.000 32.960

jj ny 6 18 33,8 20,4 25,2 103,4

jj ny 2 6 32,4 87,88 69,36 105,84 301,48

( )378,0

39

565,5

2

2

==−

−

=∑ ∑

∑ ∑

i jij

i jijxij

n

nYy

s

&DOFXOHOH�LQWHUPHGLDUH�VH�J VHVF�vQ�WDEHOXO�� Tabelul 7.7.

yj Yxi 1 1,8 2,6 3,4 4,2

( )∑ −j

ijjxi nyY 2

1,196 0,23 0,365 - - - 0,595 1,946 - 0,1705 2,138 - - 2,3085 2,696 - 0,803 0,074 - - 0,877 3,446 - - - 0,276 1,508 1,784 4,196 - - - - 0,000064 0,000064 Total 5,565

021,239;05,02, ==− tt nα . Cum tcalc > ttab��UH]XOW �F �³a” este semnificativ.

7HVWDUHD�VHPQLILFD LHL�SDUDPHWUXOXL�³b”

( )∑ −= iicalc nxxs

bt

2

17,273==∑

∑i

ii

n

nxx mil. lei

( ) ( ) ( )( ) ( ) ( ) =⋅−+⋅−+⋅−+

+⋅−+⋅−=−∑417,273500817,273400917,273300

1317,273200717,273100222

222ii nxx

= 620.487,76

021,269,11176,487.620378,0

0076,0 =>=⋅= tabcalc tt , deci “b´�GLIHU �VHPQLILFDWLY�GH�]HUR�

'DF �α�úL�β�VXQW�SDUDPHWULL�WHRUHWLFL�FRUHVSXQ] WRUL�OXL�a�úL�b, intervalul de încredere va fi:

n

sta

n

sta nn ⋅+<<⋅− −− 2;2; αα α

41

378,0021,24465,0

41

378,0021,24465,0 ⋅+<<⋅− α

0,3271 < α < 0,5658

( ) ( )∑∑ −⋅+<<

−⋅− −−

ii

n

ii

n

nxx

stb

nxx

stb

22,

22, αα β

76,487.620

378,0021,20076,0

76,487.620

378,0021,20076,0 ⋅+<<⋅− β

0,0066 < β < 0,0086

b)

( ) ( )

−

−

−=

σσ=

∑ ∑∑ ∑∑∑ ∑ ∑

2222/

),cov(

jjjjiiii

jjiiijji

yxxy

nynynnxnxn

nynxnyxnyxr

ryx� ��GHFL�H[LVW �R�OHJ WXU �SXWHUQLF �GLUHFW �vQWUH�FHOH�GRX �YDULDELOH�

( )( )∑

∑ ∑

−

−−==−=

jj

ijxij

y

yY

y

Yyxy

nyy

nYyR

2

2

2

2/

2

2/

/ 11σ

σ

σ

σ

( )9927,0

41

7,402

2 ==−

=∑

∑

j

jjy n

nyyσ

52,241

4,103 ===∑

∑

j

jj

n

nyy zeci mii buc.

93,07,40

565,51/ =−=xyR ��GHFL�R�OHJ WXU �SXWHUQLF ��GLUHFW �

c) 22

21

s

sF =

( )135,35

12

565,57,40

11

212

1 =−

−=−−

=−

=∑

r

nyY

r

Ss

iix

( )1427,0

241

565,52

222 =

−=

−

−=

−=

∑ ∑rn

nYy

rn

Ss

ijixj

Fcalc = 246,2

Fα,r-1,n-r = F0,05, 1,39 = 4,08

Cum Fcalc > Ftab �vQVHDPQ �F �;�LQIOXHQ HD] �VHPQLILFDWLY�SH�<�

5. 6H� FXQRVF� XUP WRDUHOH� GDWH�� SULYLQG� YkQ] ULOH� OD� SURGXVXO� DOLPHQWDU� Ä$´�� vQUHJLVWUDWH� vQ� ��

XQLW L�FRPHUFLDOH� Tabelul 7.8. 1XP UXO�FXUHQW�DO�XQLW��

comerciale

9kQ] UL��EXF L� 1XP U�YkQ] WRUL�

(persoane)

6XSUDID D�FRPHUFLDO �

(mp)

1 22 7 98

2 20 5 90

3 23 8 110

4 26 9 130

5 30 12 140

6 32 15 145

7 45 22 156

8 50 25 160

9 52 32 164

10 60 40 175

ÌQ�LSRWH]D�XQHL�GHSHQGHQ H�OLQLDUH�vQWUH�FHOH�WUHL�YDULDELOH��se cere:

a) 6 �VH�HVWLPH]H�SDUDPHWULL�PRGHOXOXL�GH�UHJUHVLH� b) 6 �VH�GHWHUPLQH�HURULOH�UH]LGXDOH� c) 6 �VH�VWDELOHDVF �LQWHQVLWDWHD�GHSHQGHQ HL�YkQ] ULORU�GH�FHL�GRL�IDFWRUL�GH�LQIOXHQ ��XWLOL]kQG�

XQ�LQGLFDWRU�GH�FRUHOD LH�DGHFYDW� d) 6 �VH�WHVWH]H�YDOLGLWDWHD�PRGHOXOXL�GH�UHJUHVLH�úL�D�LQGLFDWRUXOXL�GH�FRUHOD LH�XWLOL]DW��úWLLQG�

F �F2;7� ��úL�F �t0,05;8 = 2,306. 6ROX LH�

a) 221102,1 xaxaaY xx ++= , unde X1� �QXP UXO�YkQ] WRULORU��X2� �VXSUDID D�FRPHUFLDO ��LDU�

Yx1,x2� �YkQ] ULOH�

( ) PLQLP→−∑ 22,1 xxi Yy

( ) PLQLP→−−−∑ 222110 xaxaayi

6H�RE LQH�VLVWHPXO�GH�HFXD LL�

=++

=++

=++

∑ ∑∑∑∑∑∑∑

∑ ∑ ∑

i

i

i

yxxaxxaxa

yxxxaxaxa

yxaxana

222221120

121221110

22110

&DOFXOHOH�LQWHUPHGLDUH�VH�J VHVF�vQ�WDEHOXO��

Tabelul 7.9 yi x1 x2 2

1x 22x x1x2 x1yi x2yi

0 1 2 3 4 5 6 7 22 7 98 49 9604 686 154 2156 20 5 90 25 8100 450 100 1800 23 8 110 64 12100 880 184 2530 26 9 130 81 16900 1170 234 3380 30 12 140 144 19600 1680 360 4200 32 15 145 225 21025 2175 480 4640 45 22 156 484 24336 3432 990 7020 50 25 160 625 25600 4000 1250 8000 52 32 164 1024 26896 5248 1664 8528 60 40 175 1600 30625 7000 2400 10500

360 175 1368 4321 194786 26721 7816 52754 Tabelul 7.9 continuare

Yxi,x2 ( )22,1 xxi Yy − ( )2yyi − 2iy ( )22,1 yY xx −

8 9 10 11 12 22 0 196 484 196 19 1 256 400 289 24 1 169 529 144 27 1 100 676 81 31 1 36 900 25 34 4 16 1024 4 42 9 81 2025 36 46 16 196 2500 100 53 1 256 2704 289 62 4 576 3600 676 38 1882 14.842 1840

=++=++

=++

754.52786.194267211368

7816267214321175

360136817510

210

210

210

aaa

aaa

aaa

6H�UH]ROY �VLVWHPXO�úL�VH�J VHúWH� a0� ��EXF L a1� ��EXF L a2� ��EXF L

212,1 104,097,0798,4 xxY xx ⋅+⋅+=

Parametrul “a0”�DUH�FDUDFWHU�GH�P ULPH�PHGLH��DUDW �F �V-a înregistrat o vânzare medie de

��EXF L�OD�XQ�PDJD]LQ��GDF �HOH�DU�IL�IRVW�LQIOXHQ DWH�GH�DO L�IDFWRUL��H[FHSWkQG�YDULDELOHOH�;1 úL�X2, incluse în model.

'DF �QXP UXO�GH�YkQ] WRUL�DU�FUHúWH�FX�R�SHUVRDQ ��DWXQFL�YkQ] ULOH�DU�FUHúWH�FX��EXF L��LDU�GDF �VXSUDID D�FRPHUFLDO �FUHúWH�FX��PS��DWXQFL�YkQ] ULOH�FUHVF�FX��EXF L�

b) 2,1 xxi Yy −=ε

Tabelul 7.10 yi 22 20 23 26 30 32 45 50 52 60

Yx1,x2 22 19 24 27 31 34 42 46 53 62 ε 0 +1 -1 -1 -1 -2 +3 +4 -1 -2 ε2 0 1 1 1 1 4 9 16 1 4

c) 3HQWUX�P VXUDUHD�LQWHQVLW LL�OHJ WXULL�GLQWUH�YDULDELOH�IRORVLP�UDSRUWXO�GH�FRUHOD LH�PXOWLSO �

( )( )

99,01882

3811

2

22,1

2,1/ =−=−

−−=

∑∑

yy

YyR

i

xxixxy

Datele sunt preluDWH�GLQ�WDE��FRO��úL��

36=∑=n

yy i �EXF L�PDJD]LQ

Valoarea lui Ry/x1,x2�DUDW �R�GHSHQGHQ �SXWHUQLF ��vQWUH�FHOH�WUHL�YDULDELOH� 'HRDUHFH�OHJ WXUD�HVWH�OLQLDU ��Ry/x1,x2�VH�PDL�SRDWH�FDOFXOD�úL�DVWIHO�

22/1

2,12/1/2

2/2

1/2,1/

1

2

xx

xxxyxyxyxyxxy

r

rrrrrR

−

⋅⋅⋅−+=

( ) ( )=

−

−

−=

∑ ∑∑ ∑∑ ∑ ∑

2221

21

11/ 1

ii

iixy

yynxxn

yxyxnr

( )( )22 3601484210175432110

360175781610

−⋅−⋅

⋅−⋅=

985,0

1/ =xyr

( ) ( )=

−

−

−=

∑ ∑∑ ∑∑ ∑ ∑

2222

22

22/ 2

ii

iixy

yynxxn

yxyxnr

( )( )22 3601484210136819478610

36013685275410

−⋅−⋅

⋅−⋅=

924,0

2/ =xyr

( ) ( )=

−

−

−=

∑ ∑∑ ∑∑ ∑ ∑

22

22

21

21

2121/ 21

xxnxxn

xxxxnr xx

( )( )22 136819478610175432110

13681752672110

−⋅−⋅

⋅−⋅=

897,02/1 =xxr

987,08046,01

6328,18537,097,02,1/ =

−−+=xxyR

ry/x1, ry/x2�úL�rx1/x2�VXQW�FRHILFLHQ L�GH�FRUHOD LH�VLPSO �

G��3HQWUX�WHVWDUHD�YDOLGLW LL�PRGHOXOXL�GH�UHJUHVLH�VH�IRORVHúWH�WHVWXO�)�GH�DQDOL] �GLVSHUVLRQDO �

22

21

s

sFcalc =

( )920

13

1840

11

22112

1 =−

=−

∑ −=

−=

r

yY

r

Ss xx

( )43,5

7

3822122

2 ==−

∑ −=

−=

rn

Yy

rn

Ss xxi

55,943,16943,5

920 =>== tabcalc FF

$úDGDU��PRGHOXO�GH�UHJUHVLH�PXOWLIDFWRULDO �HVWH�ELQH�DOHV��3HQWUX�WHVWDUHD�VHPQLILFD LHL�UDSRUWXOXL�GH�FRUHOD LH�PXOWLSO �IRORVLP�WHVWXO�³W´�

306,286,1921099,01

99,02

18;05,0

22=>=−

−=−

−= tn

r

rtcalc

'HFL�FRHILFLHQWXO�GH�FRUHOD LH�HVWH�VHPQLILFDWLY�

6. 3HQWUX�RSW�MXGH H�DOH�5RPkQLHL��VH�FXQRVF��VXSUDID D�FXOWLYDW �vQ��KD��úL�SURGXF LD�DJULFRO �

WRWDO �vQ�DFHODúL�DQ��PLO��OHL�SUH XUL�FXUHQWH�� Tabelul 7.11

-XGH XO 6XSUDID D�FXOWLYDW �vQ�� (ha)

3URGXF LD�DJULFRO (mil. lei)

Alba 127.296 751.173 Arad 342.158 1.123.600

%LVWUL D�1 V XG 97.925 644.173 %RWRúDQL 286.469 894.013 %UDúRY 111.445 803.597 %X] X 252.483 842.665

&DUDú�6HYHULQ 124.274 620.277 & O UDúL 404.147 964.966

6 � VH� VWXGLH]H� H[LVWHQ D�� GLUHF LD� úL� LQWHQVLWDWHD� OHJ WXULL� GLQWUH� VXSUDID D� FXOWLYDW � úL�

SURGXF LD�DJULFRO ��FX�DMXWRUXO�FRHILFLHQ LORU�GH�FRUHOD LH�D�UDQJXULORU�DL�OXL�6SHDUPDQ�úL�.HQGDOO�

6ROX LH��

Coeficientul lui Spearman�VH�GHWHUPLQ �FX�UHOD LD�

( )1

61

2

2

−−= ∑

nn

dr is ,

unde di� � UHSUH]LQW � GLIHUHQ D� vQWUH� UDQJXULOH� FHORU� GRX � YDULDELOH�� DIHUHQWH� DFHOHLDúL� XQLW L�� 6H�RUGRQHD] �SHUHFKLOH�GH�YDORUL�vQ�RUGLQHD�FUHVF WRDUH�D�YDORULORU�xi, apoi se atribuie ranguri valorilor FHORU�GRX �YDULDELOH��7DE�� Tabelul 7.12

-XGH Supr. (xi)

Prod. (yi)

Rx Ry di 2id

%LVWUL D�1 V XG 97.925 644.173 1 2 -1 1 %UDúRY 111.445 803.597 2 4 -2 4

&DUDú�6HYHULQ 124.274 620.277 3 1 2 4 Alba 127.296 751.173 4 3 1 1 %X] X 252.483 842.665 5 5 0 0 %RWRúDQL 286.469 894.013 6 6 0 0

Arad 342.158 1.123.600 7 8 -1 1 & O UDúL 404.147 964.966 8 7 1 1

Total 12

86,0638

1261 =

⋅⋅−=sr

YDORDUH�FH�LQGLF �R�OHJ WXU �GLUHFW ��SXWHUQLF ��vQWUH�FHOH�GRX �YDULDELOH� Coeficientul lui Kendall�VH�FDOFXOHD] �FX�UHOD LD�

)1(

2

−=

nn

Srk ,

XQGH�6�UHSUH]LQW �GLIHUHQ D�vQWUH�VXPD�UDQJXULORU�VXSHULRDUH��3��úL�VXPa rangurilor inferioare (Q), XUP WRDUH�ILHF UXL�UDQJ�FXUHQW�GXS �\��7DEHOXO��

71,078

202 =⋅⋅=kr

YDORDUH�D�F UHL�LQWHUSUHWDUH�HVWH�DFHHDúL�FD�vQ�FD]XO�FRHILFLHQWXOXL�OXL�6SHDUPDQ� Tabelul 7.13

-XGH Rx Ry P Q S=P-Q %LVWUL D�1 V XG 1 2 6 1 5

%UDúRY 2 4 4 2 2 &DUDú�6HYHULQ 3 1 5 0 5

Alba 4 3 4 0 4 %X] X 5 5 3 0 3 %RWRúDQL 6 6 2 0 2

Arad 7 8 0 1 -1 & O UDúL 8 7 0 0 0

Total - - 24 4 20

7. 'DWHOH�VLVWHPDWL]DWH��RE LQXWH�vQ�XUPD�XQXL�VWXGLX�VWDWLVWLF�SULYLQG�YHFKLPHD�vQ�PXQF �úL�WLPSXO�

zilnLF�QHOXFUDW��HIHFWXDW�SH��GH�VDODULD L�DL�XQHL�VRFLHW L�FRPHUFLDOH��VXQW� Tabelul 7.14

Vechimea Timp nelucrat (u.m.) sub 60 u.m. peste 60 u.m.

peste 10 ani 300 150 sub 10 ani 100 250

6 �VH�SUHFL]H]H�GDF �vQWUH�FHOH�GRX �YDULDELOH�H[LVW �R�OHJ WXU �

6ROX LH� 6H�IRORVHúWH�FRHILFLHQWXO�GH�DVRFLHUH��FDOFXODW�FX�UHOD LD�

bcad

bcadQa +

−= , unde a = 300, b = 150, c = 100, d = 250.

67,0000.90

000.60

150100250300

150100250300 ==⋅+⋅⋅−⋅=Qa

Cum ad > bc�UH]XOW �F �vQWUH�FHOH�GRX �YDULDELOH�H[LVW �R�OHJ WXU �GLUHFW �GH�LQWHQVLWDWH�Pedie.

8. S-DX�vQUHJLVWUDW�YDORULOH�YDULDELOHORU�;�úL�<�OD�QLYHOXO�D��XQLW L�

Tabelul 7.15 xi 5 6 10 12 15 20 22 30 35 40 yi 70 53 44 38 35 28 26 17 11 5

Se cere:

a) 6 �VH�DQDOL]H]H�H[LVWHQ D��GLUHF LD�úL�IRUPD�OHJ WXULL�GLQWUH�YDULDELOHOH�;�úL�<� b) 6 �VH�GHWHUPLQH�SDUDPHWULL�IXQF LHL�GH�UHJUHVLH� c) 6 �VH�SUHFL]H]H�LQWHQVLWDWHD�OHJ WXULL�GLQWUH�;�úL�<�

6ROX LH� a) 6H�DOF WXLHúWH�FRUHORJUDPD��ILJXUD�� úL� VH�REVHUY �F � vQWUH�FHOH�GRX �YDULDELOH�H[LVW �R�

OHJ WXU ��LQYHUV ��QHOLQLDU � b) 3HQWUX�PRGHODUHD�WHQGLQ HL�OHJ WXULL��IRORVLP�PRGHOXO�GH�UHJUHVLH�KLSHUEROLF�

i

xi xbaY

1+=

Fig. 7.1

�

��

��

��

� �

� �

��

�

��

� ��

��

$SOLFkQG�PHWRGD�FHORU�PDL�PLFL�S WUDWH��RE LQHP�VLVWHPXO�

=+

=+

∑∑∑

∑∑

iiii

ii

yxx

bx

a

yx

bna

111

1

2

=

−

⋅−⋅==

∑∑

∑∑∑∑

∑∑

∑

∑∑

∑∑

2

2

2

2

2

11

111

11

1

11

1

ii

ii

iii

ii

i

ii

i

ii

xxn

xy

xxy

xx

xn

xy

x

xy

a

18,76257,00938,010

943,35791,00938,0327 =−⋅

⋅−⋅=

=

−

⋅−⋅==

∑∑

∑∑∑

∑∑

∑

∑∑

∑

2

2

2

11

11

11

1

11

ii

ii

ii

ii

i

iii

i

xxn

yx

yx

n

xx

xn

yxx

yn

b

68,322791,00938,010

327791,0943,35102

=−⋅

⋅−⋅=

ixi x

Y1

68,32218,7 +=

9DORULOH�DMXVWDWH�DOH�OXL�<�úL�FDOFXOHOH�LQWHUPHGLDUH�VH�J VHVF�vQ�WDEHOXO��

Tabelul 7.16. xi yi 1/xi yi(1/xi) (1/xi)

2 Yxi (yi-Yxi)2 ( )2i yy −

5 70 0,200 14,000 0,0400 71,58 2,49 1391,29 6 53 0,160 8,480 0,0256 58,70 32,49 412,09 10 44 0,100 4,400 0,0100 39,38 21,34 127,69 12 38 0,083 3,154 0,0069 33,90 16,81 28,09 15 35 0,067 2,345 0,0044 28,75 39,01 5,29 20 28 0,050 1,400 0,0025 23,28 22,28 22,09 22 26 0,045 1,170 0,0020 21,67 18,75 44,89 30 17 0,033 0,561 0,0010 17,80 0,64 246,49 35 11 0,028 0,308 0,0008 16,20 27,04 470,89 40 5 0,025 0,125 0,0006 15,23 104,65 767,29 - 327 0,791 35,943 0,0938 285,50 3516,10

c) ,QWHQVLWDWHD�OHJ WXULL�VH�VWDELOHúWH�FX�UDSRUWXO�GH�FRUHOD LH�

( )( )

96,01,3516

5,28511

2

2=−=

−

−−=

∑

∑

yy

YyR

i

xiiyx

7,32== ∑n

yy i

5DSRUWXO�GH�FRUHOD LH�DUDW �R�OHJ WXU �SXWHUQLF �vQWUH�FHOH�GRX �YDULDELOH�

&RHILFLHQWXO�GH�GHWHUPLQD LH 92,02/ =xyR DUDW �F ��GLQ�YDULD LD�YDULDELOHL�<�HVWH�H[SOLFDW �SULQ�

LQIOXHQ D�OXL�;�DVXpra lui Y.

Probleme propuse

1. 3DWURQXO� XQHL� ILUPH�FH� YLQGH�DXWRWXULVPH�SUHVXSXQH� F � UHOD LD� GLQWUH� QXP UXO�GH�DXWRWXULVPH�

vândute într-R�]L��<��úL�QXP UXO�GH�YkQ] WRUL�FDUH�OXFUHD] �vQ�VDORQXO�GH�SUH]HQWDUH��;��SRDWH�IL�PRGHODW �SULQWU-R�OLQLH�GUHDSW ��Datele culese pentru 5 zile sunt:

Tabelul 7.17.

xi 6 6 4 2 3 yi 20 18 10 6 11

&DUH�GLQ�DILUPD LLOH�XUP WRDUH�VXQW�DGHY UDWH�úL�FDUH�VXQW�IDOVH�

a) /HJ WXUD� GLQWUH� FHOH� GRX � YDULDELOH� SRDWH� IL� PRGHODW � SULQ� HFXD LD� GH� UHJUHVLH��xy 125,3125,0ˆ += ;

b) PDQWD�OLQLHL�GUHSWH�GH�UHJUHVLH�HVWH��LDU�SXQFWXO�GH�LQWHUFHS LH�FX�2<�HVWH�–0,125; c) &RYDULDQ D�GLQWUH�;�úL�<��cov (x,y)� ��DUDW �R�OHJ WXU �SXWHUQLF �vQWUH�FHOH�GRX �YDULDELOH� d) &RHILFLHQWXO�GH�FRUHOD LH�ry/x este semnificativ statistic, pentru o probabilitate de 95%, pentru

care ttab;0,025;3 = 3,182; e) $SUR[LPDWLY� �� GLQ� YDULD LD� YDORULORU� FDUDFWHULVWLFLL� HIHFW� <� vQ� MXUXO� PHGLHL� y este

H[SOLFDW �SULQ�OHJ WXUD�OLQLDU �GLQWUH�;�úL�<� ,QGLFD LH��3HQWUX�YHULILFDUHD�VHPQLILFD LHL�FRHILFLHQWXOXL�GH�FRUHOD LH�VH�XWLOL]HD] �WHVWXO�W�

21 2

−−

= nr

rtcalc

&RHILFLHQWXO�GH�GHWHUPLQD LH�VH�FDOFXOHD] �FX�UHOD LD�

( )( )2

22

/

ˆ1

∑∑

−

−−=

yy

yyR xy

2. 3UHVXSXQHP� F �� vQ� XUPD� DSOLF ULL� PHWRGHL� GH� UHJUHVLH� OLQLDU �� V-a calculat, pentru 12 valori

perechL�� VXPD� S WUDWHORU� DEDWHULORU� YDORULORU� HPSLULFH� GH� OD� YDORULOH� DMXVWDWH�

( )∑=

=−12

1

2 507,0î

ii yy . Atunci 2es ��HVWLPDWRUXO�GLVSHUVLHL�DEDWHULORU�vQWkPSO WRDUH�ε, este:

a) 0,04225; b) 0,04609; c) 0,0507; d) 0,2252; e) 0,2055.

3. 'RX �YDULDELOH�VWDWLVWLFH�DX�IRVW�PRGHODWH�FX�DMXWRUXO�HFXD LHL�GH�UHJUHVLH�OLQLDU �� xy 645ˆ += .

ùWLLQG� F � ( )∑ =− 302

ii xx , ( )∑ =− 1960ˆ 2ii yy �� SHQWUX� Q� � �� XQLW L� VWDWLVWLFH�� DWXQFL�

intervalul de încredere pentru panta liniei de regresie la o probabilitate de 95%, pentru care

12;025,0=αtabt :

a) (63,53; 64,47); b) (4,53; 5,47); c) (58,92; 69,08); d) (61,35; 66,65); e) (58,92; 66,65);

4. &RQVLGHU P�XUP WRDUHOH�SHUHFKL�GH�REVHUYD LL�SHQWUX�YDULDELOHOH�;�úL�<� Tabelul 7.18.

xi 1 4 3 2 5 6 0 yi 1 3 3 1 4 7 2

&DUH�GLQ�XUP WRDUHOH�DILUPD LL�VXQW�DGHY UDWH�úL�FDUH�VXQW�IDOVH�

a) (FXD LD�GH�UHJUHVLH�OLQLDU �HVWH�� xy 821,0536,0ˆ −= ;

b) Pentru 571,25;025,0 =t , coeficientul de regresie este semnificativ statistic;

c) (FXD LD�GH�UHJUHVLH�OLQLDU �HVWH�� xy 821,0536,0ˆ += ; d) Intervalul de încredere pentru coeficientul de regresie este (0,243; 1,399); e) &RHILFLHQWXO�GH�FRUHOD LH�ry/x= -0,92.

5. 6H�FXQRDúWH�QXP UXO�GH�FRSLL�vQVFULúL�vQ��XQLW L�SUHúFRODUH��SUHFXP�úL�QXP UXO�SHUVRQDOXOXL�

didacWLF�GLQ�DFHVWH�XQLW L� Tabelul 7.19

Nr. crt. 1 2 3 4 5 6 7 8 9 10 Nr. copii 20 323 156 180 98 73 334 20 52 203 Pers. did. 2 21 18 14 11 6 21 1 2 17

Se cere: a) 6 �VH�SUHFL]H]H��SH�ED]D�PHWRGHL�JUDILFH��H[LVWHQ D��GLUHF LD�úL�IRUPD�OHJ WXULL�GLQWUH�QXP Uul

GH�FRSLL�vQVFULúL�vQ�XQLW LOH�SUHúFRODUH�úL�QXP UXO�SHUVRQDOXOXL�GLGDFWLF� b) 6 �VH�DMXVWH]H�QXP UXO�SHUVRQDOXOXL�GLGDFWLF�vQ�IXQF LH�GH�QXP UXO�FRSLLORU�vQVFULúL��XWLOL]kQG�

IXQF LD�GH�UHJUHVLH�DGHFYDW � c) 6 �VH�VWDELOHDVF �LQWHQVLWDWHD�OHJ WXULL�GLQWUH�FHOH��GRX �YDULDELOH��IRORVLQG�XQ�LQGLFDWRU�GH�

FRUHOD LH�DGHFYDW��V �VH�WHVWH]H�VHPQLILFD LD�LQGLFDWRUXOXL�IRORVLW� d) 6 � VH� VWDELOHDVF � LQWHQVLWDWHD� OHJ WXULL� GLQWUH� FHOH� GRX � YDULDELOH�� IRORVLQG� FRHILFLHQ LL� OXL�

6SHDUPDQ�úL�.HQGDOO� e) 6 �VH�FRPSOHWH]H�XUP WRUXO�WDEHO�GH�FRQWLQJHQ ��DVRFLHUH��úL�V �VH�FDOFXOH]H�FRHILFLHQWXO�GH�

FRQWLQJHQ ��DVRFLHUH��V �VH�LQWHUSUHWH]H�UH]XOWDWHOH�RE LQXWH� Tabelul 7.20

Y X

y ≤ y y > y

x ≤ x a = ? b = ?

x > x c = ? d = ?

6. 3HQWUX��OXFU WRUL�GLQ�VHF LD�GH�SURGXF LH�D�XQXL�DJHQW�HFRQRPLF�FX�SURILO�LQGXVWULDO��VH�FXQRVF��

WLPSXO�OXFUDW��X�P��úL�QXP UXO�GH�SLHVH�UHDOL]DWH�]LOQLF��EXF�� Tabelul 7.21.

Nr. crt. 1 2 3 4 5 6 7 8 9 10 Timp lucrat (u.m.) 4 4,5 6 4,2 3,5 7 7 7,5 6,3 8

Nr. piese (buc.) 6 8 10 7 4 12 13 15 11 19 Se cere:

a) 6 � VH� VWDELOHDVF � IXQF LD� GH� UHJUHVLH� FDUH� H[SULP � WHQGLQ D� OHJ WXULL� GLQWUH� FHOH� GRX �variabile pe baza corelogramei;

b) 6 �VH�SUHFL]H]H�LQWHQVLWDWHD�OHJ WXULL�GLQWUH�WLPSLL�OXFUD L�úL�QXP UXO�GH�SLHVH�UHDOL]DWH�]LOQLF�GH�FHL��OXFU WRUL��IRORVLQG� b1��FRHILFLHQWXO�GH�FRUHOD LH� b2��UDSRUWXO�GH�FRUHOD LH� b3) coeficientul lui Spearman; b4) coeficientul lui Kendall.

c) &DUH� YD� IL� QXP UXO� SUREDELO� GH� SLHVH� SH� FDUH� OH� YD� UHDOL]D� XQ� OXFU WRU�� FDUH� DUH� XQ� WLPS�mediu lucrat zilnic de 5 u.m.?

d) 6 � VH� FRQVWUXLDVF � XQ� WDEHO� GH� FRQWLQJHQ � úL� V � VH� DQDOL]H]H� OHJ WXUD� GLQWUH� FHOH� GRX �YDULDELOH�SULQ�SULVPD�FRHILFLHQWXOXL�GH�FRQWLQJHQ �úL�GH�DVRFLHUH�

e) 6 �VH�LQWHUSUHWH]H�UH]XOWDWHOH�RE LQXWH�

88

ÍQWUHE UL ¾ &XP�VH�GHILQHúWH�R�VHULH�FURQRORJLF " ¾ &DUH�VXQW�SDUWLFXODULW LOH�úL�WLSRORJLD�VHULLORU�FURQRORJLFH" ¾ &XP�VH�UHSUH]LQW �JUDILF�R�VHULH�FURQRORJLF " ¾ &XP�VH�FDOFXOHD] �PHGLD�FURQRORJLF " ¾ Care sunt componentele unei serii crolonogice? ¾ 0RGHOXO�DGLWLY�úL�PRGHOXO�PXOWLSOLFDWLY�GH�FRPELQDUH�D�FRPSRQHQWHORU�XQHL�VHULL�FURQRORJLFH� ¾ &H�UHSUH]LQW �DMXVWDUHD�XQHL�VHULL�FURQRORJLFH" ¾ &XP�VH�GHWHUPLQ �WUHQGXO�XQHL�VHULL�FURQRORJLFH�IRORVLQG�PHWRGH�PHFDQLFH" ¾ Utilizarea metodelor analitice de determinare a trendului. ¾ &DUH�VXQW�FULWHULLOH�GH�DSUHFLHUH�D�FDOLW LL�DMXVW ULL" ¾ 6LVWHPXO�GH�LQGLFDWRUL�FH�FDUDFWHUL]HD] �R�VHULH�FURORQRJLF � ¾ 'HVFULH L�PHWRGD�PHGLLORU�PRELOH��FD�PHWRG �PHFDQLF �GH�DMXVWDUH�D�XQHL�VHULL�FURQRORJLFH� ¾ Cum se� VWXGLD] � VWDWLVWLF� VH]RQDOLWDWHD�XQHL� VHULL� FURQORJLFH"�3DUWLFXODUL]D L� SHQWUX�XQ�PRGHO�

DGLWLY�úL�SHQWUX�XQXO�PXOWLSOLFDWLY� ¾ $XWRFRUHOD LD�WHUPHQLORU�XQHL�VHULL�FURQRORJLFH� ¾ Metodele de extrapolare pe baza datelor unei serii cronologice.

SERII CRONOLOGICE

([HUFL LL�üi probleme

1. 3URGXF LD�GH�SODFDMH�GH�OHPQ��PLL�P

3) a avut în perioada 1990-��XUP WRDUHD�HYROX LH� Tabelul 8.1

Anii 3URGXF LD��PLL�P3) (yt)

1990 110 1991 104 1992 100 1993 88 1994 84 1995 90 1996 93 1997 97 Total 766

Se cere: a) 6 �VH�FDOFXOH]H�SULQFLSDOLL�LQGLFDWRUL�VWDWLVWLFL�FH�FDUDFWHUL]HD] �DFHDVW �VHULH�FURQRORJLF � b) 6 �VH�GHWHUPLQH�WUHQGXO��V �VH�DMXVWH]H�VHULD�FURQRORJLF ��IRORVLQG�R�PHWRG �DGHFYDW � c) 6 � VH� SUHYL]LRQH]H� YROXPXO� SURGXF LHL� vQ� �� úL� �� vQ� LSRWH]D� F � VHULD� vúL� YD� S VWUD�

DFHHDúL�WHQGLQ �GH�HYROX LH�FD�FHD�GHWHUPLQDW �SULQ�DMXVWDUH�

6ROX LH� a) Principalii indicatori sunt:

� PRGLILFDUHD�DEVROXW �FX�ED] �IL[

11/ yytt −=∆ (col. 2 tab. 8.2) � PRGLILFDUHD�DEVROXW �FX�ED] �vQ�ODQ

11/ −− −=∆ tttt yy (col. 3 tab. 8.2) � LQGLFHOH�GH�PRGLILFDUH�FX�ED] �IL[

1001

%1/ ⋅=

y

yI tt (col. 4 tab. 8.2)

� LQGLFHOH�GH�PRGLILFDUH�FX�ED] �vQ�ODQ

1001

%1/ ⋅=

−−

t

ttt y

yI (col. 5 tab. 8.2)

� ULWPXO�GH�PRGLILFDUH��PRGLILFDUHD�UHODWLY ��FX�ED] �IL[

( ) 1001001 %1/1/1/ −=−= ttt IIR (col. 2 tab. 8.3)

� ULWPXO�GH�PRGLILFDUH��PRGLILFDUHD�UHODWLY ��FX�ED] �vQ�ODQ

( ) 1001001 %1/1/1/ −=−= −−− tttttt IIR (col. 3 tab. 8.3)

� YDORDUHD�DEVROXW �D�XQXL�SURFHQW�GLQ�ULWPXO�GH�PRGLILFDUH�FX�ED] �IL[

1001

1/

1/1/

y

RA

t

tt =

∆= (col. 4 tab. 8.3)

� YDORDUHD�DEVROXW �D�XQXL�SURFHQW�GLQ�ULWPXO�GH�PRGLILFDUH�FX�ED] �vQ�ODQ

1001

1/

1/1/

−

−

−− =

∆== t

tt

tttt

y

RA (col. 5 tab. 8.3)

� nivelul mediu al seriei cronologice:

75,958

766 === ∑n

yy t mii m3/an

ÌQ�PHGLH��vQ�SHULRDGD�DQDOL]DW ��SURGXF LD�DQXDO �GH�SODFDMH�GH�OHPQ�D�IRVW�GH��PLL�P3.

Tabelul 8.2

Anii (yt) (mii m3) 1/t∆ 1/ −∆ tt %1/tI %

1/ −ttI

0 1 2 3 4 5 1990 110 0 - 100 - 1991 104 -6 -6 94,5 94,5 1992 100 -10 -4 90,9 96,2 1993 88 -22 -12 80,0 88,0 1994 84 -26 -4 76,4 95,5 1995 90 -20 6 81,8 107,1 1996 93 -17 3 84,5 103,3 1997 97 -13 4 88,2 104,3 Total 766 - -13 - -

Tabelul 8.3

Anii (yt) (mii m3) 1/tR 1/ −ttR 1/tA

(mii m3) 1/−ttA

(mii m3) 0 1 2 3 4 5

1990 110 0 - - - 1991 104 -5,5 -5,5 1,1 1,10 1992 100 -9,1 -3,8 1,1 1,04 1993 88 -2,0 -12 1,1 1,00 1994 84 -23,6 -4,5 1,1 0,88 1995 90 -18,2 7,1 1,1 0,84 1996 93 -15,5 3,3 1,1 0,90 1997 97 -11,8 4,3 1,1 0,93 Total 766 - - - -

� modificarea medie abVROXW �

857,17

13

11/ −=−=

−∆

=∆n

n mii m3/an

ÌQ�PHGLH��vQ�SHULRDGD�DQDOL]DW ��SURGXF LD�GH�SODFDMH�D�VF ]XW�FX��PLL�P3 pe an. � indicele mediu de modificare:

982,0882,0711/ === −n

nII (98,2%)

� ritmul mediu de modificare:

( ) ( ) %8,11001982,01001 −=−=⋅−= IR În medie, în pHULRDGD�DQDOL]DW ��SURGXF LD�D�VF ]XW�FX��GH�OD�XQ�DQ�OD�DOWXO� 6H� REVHUY � F �PRGLILFDUHD� PHGLH� DEVROXW � úL� LQGLFHOH�PHGLX� GH� PRGLILFDUH� QX� VXQW� YDORUL�

UHSUH]HQWDWLYH�SHQWUX�VHULD�FURQRORJLF �DQDOL]DW � 'H�DOWIHO�� GLQ�DQDOL]D� GDWHORU� VH� REVHUY � F � vQ�Serioada 1990-��SURGXF LD�D� VF ]XW� úL� vQ�

SHULRDGD�XUP WRDUH�QLYHOXULOH�SURGXF LHL�DX�FXQRVFXW�R�HYROX LH�DVFHQGHQW ��&RHILFLHQWXO�GH�YDULD LH�

DO�PRGLILF ULL� DEVROXWH�HVWH�R�YDORDUHD�PDL�PDUH�GHFkW��FDUH�DUDW �R� HWHURJHQLWDWH� FUHVFXW �D�

acestor valori:

( )1001100

2

∆−

∆−∆

=∆

=

∑∆ nv

σ

b) $úD� FXP� VH� REVHUY � GLQ� UHSUH]HQWDUHD� JUDILF �� ILJ�� PHWRGD� DQDOLWLF �

DGHFYDW � HVWH� DMXVWDUHD� SH� ED]D� SDUDEROHL� GH� JUDGXO� GRL�� 6H� YD� IDFH� DMXVWDUHD� úL�

SULQ� PHWRGD� OLQLDU �� GXS � FDUH� VH� YD� DOHJH� DFHD� PHWRG � GH� DMXVWDUH� FDUH�

GHWHUPLQ �FHOH�PDL�PLFL�S WUDWH� DOH� DEDWHULORU� WHUPHQLORU� HPSLULFL� GH� OD�YDORULOH�

teoretice.

Ajustarea prin parabola de gradul II

=++

=++

=++

∑ ∑ ∑∑

∑∑∑∑

∑∑∑

yttctbta

tytctbta

ytctbna

2432

32

2

Cum ∑ = 0t (col. 0, tab. 8.5.), sistemul devine:

=+

=

=+

∑ ∑∑

∑∑

∑∑

yttcta

tytb

ytcna

242

2

2

Fig. 8.1 ÌQ�XUPD�UH]ROY ULL�VLVWHPXOXL��UH]XOW ��D� ��E� �-1,25; c = 0,32. (FXD LD�GH�DMXVWDUH�HVWH�

232,025,189 ttYt +−= (col.2 tab. 8.5)

iar ( )∑ =− 642tt Yy (col. 3 tab. 8.5)

��

� �

��

��

��

��

��

��

Ani

Pro

d (

mii

m3)

Tabelul 8.4. t yt t2 t4 ty t2y 0 1 2 3 4 5 -7 110 49 2401 -770 5390 -5 104 25 625 -520 2600 -3 100 9 81 -330 900 -1 88 1 1 -88 88 1 84 1 1 84 84 3 90 9 81 270 810 5 93 25 625 465 2325 7 97 49 2401 679 4753

Total 766 168 6216 -210 16950 Tabelul 8.5.

t yt Yt (mii m3)

( )2tt Yy − Y’t ( )2’

tt Yy −

0 1 2 3 4 5 -7 110 113 9 104 36 -5 104 105 1 102 4 -3 100 95 25 100 0 -1 88 90 4 97 81 1 84 88 16 94 100 3 90 88 4 92 4 5 93 91 4 90 9 7 97 96 1 87 100

Total 766 766 64 766 334

$MXVWDUHD�SULQ�OLQLD�GUHDSW

=+

=+

∑∑∑

∑∑tytbta

ytbna2

−=−==

===⇒

∑∑

∑

25,1168

210

75,95

2t

tyb

yn

ya

(FXD LD�GH�DMXVWDUH�HVWH�

tYt ⋅−= 25,175,95’ (col. 4 tab. 8.5)

LDU�VXPD�S WUDWHORU�DEDWHULORU

( ) 3342’ =−∑ tt Yy (col. 5 tab. 8.5)

&HD�PDL�SRWULYLW �PHWRG �GH�DMXVWDUH�HVWH�FHD�ED]DW �SH�SDUDEROD�GH�gradul II.

c) 3UHYL]LRQDUHD�YROXPXOXL�SURGXF LHL�vQ��úL�� Y1998 = 89 – 1,25 ⋅ 9 + 0,32 ⋅ 92 = 103,6 mii m3 Y1999 = 89 – 1,25 ⋅ 11 + 0,32 ⋅ 112 = 113,9 mii m3

2. Se cunosc urmatoarele date, privind emisiile radio (mii ore program) in perioada 1990 – 1995:

Tabelul 8.6

Anii Nr. emisii radio program public (mii ore program) 1990 52 1991 59 1992 64 1993 67 1994 71 1995 73

Se cere:

a) 6 �VH�UHSUH]LQWH�JUDILF�VHULD� b) 6 �VH�FDOFXOH]H�LQGLFDWRULL�FH�FDUDFWHUL]HD] �VHULD�FURQRORJLF � c) 6 �VH�DMXVWH]H�VHULD�SULQ�PHWRGH�PHFDQLFH�úL�DQDOLWLFH� d) 6 �VH�SUHYL]LRQH]H�QXP UXO�GH�RUH-SURJUDP�vQ�XUP WRULL�GRL�DQL�� vQ� LSRWH]D�F �VHULD� vúL�YD�

S VWUD�DFHHDúL�WHQGLQ �GH�HYROX LH�FD�FHD�GHWHUPLQDW �SULQ�DMXVWDUH� 6ROX LH�

a) (VWH� R� VHULH� FURQRORJLF � GH� LQWHUYDOH�� UHSUH]HQWDUHD� JUDILF � VH� UHDOL]HD] � SULQ� LQWHUPHGLXO�

cronogramei.

Fig. 8.2

b) ,QGLFDWRULL�DEVROX L�úL�UHODWLYL�VH�UHJ VHVF�vQ�WDEHOHOH��úL�� Tabelul 8.7 Anii ty 1/t∆ 1/ −∆ tt 1/tI 1/ −ttI 1990 52 0 - 1 - 1991 59 7 7 1,13 1,13 1992 64 12 5 1,23 1,08 1993 67 15 3 1,28 1,05 1994 71 19 4 1,36 1,06 1995 73 21 2 1,40 1,03 Total 386 Σ=21 Π=1,40

��

��

��

��

��

��

��

$QL

1U�

�HP

LVLL

�UD

GLR

��P

LL�R

UH�S

URJ

U��

Tabelul 8.8 Anii ty 1/tR 1/ −ttR 1/tA 1/ −ttA 1990 52 0 - 0,52 - 1991 59 13 13 0,52 0,52 1992 64 23 8 0,52 0,59 1993 67 28 5 0,52 0,64 1994 71 36 6 0,52 0,67 1995 73 40 3 0,52 0,71 Total 386 - - - -

Indicatorii medii:

� nivelul mediu: 33,646

386 === ∑n

yy t mii ore program

� PRGLILFDUHD�PHGLH�DEVROXW �� 2,45

21

111/2

1/

==−

∆=

−

∆=∆

∑=

−

nnn

n

ttt

mii ore program

� indicele mediu de modificare: 07,152

73511/

11/ ==== −− −∏ n

nn

tt III

� ritmul mediu de modificare: ( ) %71001 =⋅−= IR

c1) Ajustarea prin metode mecanice 0HWRGD�PRGLILF ULL�PHGLL�DEVROXWH

∆⋅−+= )1(1 tyYt (col. 3 tab. 8.9) 6XPD�S WUDWHORU�DEDWHULORU�vQWUH�WHUPHQLL�HPSLULFL�úL�WHUPHQLL�WHRUHWLFL�HVWH��

( )∑ =− 4,312tt Yy (col. 4 tab. 8.9)

0HWRGH�LQGLFHOXL�PHGLX�GH�GLQDPLF

( ) 11

’ −⋅= tt IyY (col. 5 tab. 8.9)

6XPD� S WUDWHORU� DEDWHULORU� vQWUH� WHUPHQLL� HPSLULFL� úL� WHUPHQLL� WHRUHWLFL�

este: ( )∑ =− 55,502’

tt Yy (col. 6 tab. 8.9) &XP�VXPD�S WUDWHORU�DEDWHULORU�vQWUH�WHUPHQLL�HPSLULFL�úL�WHUPHQLL�WHRUHWLFL�HVWH�PDL�PLF �vQ�

cazul apOLF ULL�PHWRGHL�PRGLILF ULL�PHGLL� DEVROXWH��GHFkW� vQ�FHO�DO� LQGLFHOXL�PHGLX�� UH]XOW � F �PDL�

SRWULYLW �DU�IL��SHQWUX�DMXVWDUH��PHWRGD�PRGLILF ULL�PHGLL�DEVROXWH� Tabelul 8.9

Ani t yt Yt (yt-Yt)2 ’

tY ( )2’tt Yy −

0 1 2 3 4 5 6 1990 1 52 52,0 0 52,0 0 1991 2 59 56,2 7,84 55,6 11,56 1992 3 64 60,4 12,96 59,5 20,25 1993 4 67 64,6 5,76 63,7 10,89 1994 5 71 68,8 4,84 68,2 7,84 1995 6 73 73,0 0 72,9 0,01 Total - 386 386 31,4 386 50,55

c) Ajustarea prin metode analitice DuS �FXP�VH�REVHUY �GLQ�UHSUH]HQWDUHD�JUDILF ��)LJ��YRP�UHDOL]D�DMXVWDUHD�SULQ�OLQLD�GUHDSW � Cum Σ t = 0

=+

=+

∑∑∑

∑∑tytbta

ytbna2

===

===⇒

∑∑

∑

057,270

144

33,64

2t

tyb

yn

ya

mii ore program

&DOFXOHOH�LQWHUPHGLDUH�VH�J VHVF�vQ�WDEHOXO�� Tabelul 8.10

t yt t2 ty Yt 0 1 2 4 5 -5 52 25 -260 54,08 -3 59 9 -177 58,18 -1 64 1 -64 62,28 1 67 1 67 66,38 3 71 9 213 70,48 5 73 25 365 74,58

Total 386 70 144 386

tYt ⋅+= 057,233,64 (col. 5 tab. 8.10)

d) 3UHYL]LRQDUHD�QXP UXOXL�GH�RUH-program: � SULQ�PHWRGD�PRGLILF ULL medii absolute:

Y1996 = 52 + (7 – 1) ⋅ 4,2 = 77,2 mii ore program Y1997 = 52 + (8 – 1) ⋅ 4,2 = 81,4 mii ore program

� prin metoda trendului liniar: Y1996 = 64,33 + 2,057 ⋅ 7 = 78,68 mii ore program Y1997 = 64,33 + 2,057 ⋅ 9 = 82,78 mii ore program

3. DesprH�SURGXF LD�GH�FHUDPLF �D�5RPkQLHL�vQ�SHULRDGD��-1996 se cunosc datele: Tabelul 8.11

Anii 1991 1992 1993 1994 1995 1996 0RGLILFDUHD�DEVROXW �FX�ED] �vQ�

ODQ ��PLL�P2) 110 93 460 558 1158 685

ùWLLQG�F �vQ�DQXO��SURGXF LD�GH�FHUDPLF �D�FUHVFXW�FX��ID �GH�DQXO��V �VH�

UHFRQVWLWXLH� VHULD� GH� YDORUL� DEVROXWH� úL� V � VH� FDOFXOH]H� LQGLFDWRULL� PHGLL� FH� FDUDFWHUL]HD] � VHULD�

FURQRORJLF � 6ROX LH� 'LQ�GDWHOH�GLVSRQLELOH�VH�REVHUY �F �

R96 / 93 = (I96 / 93 – 1) 100� ��UH]XOW �GHFL�F

=+= 1100

93/9693/96

RI

93

963343,1y

y=

úL� 939696

941/93/96 24016851158558 yy

ttt −==++=∆=∆ ∑

=−

71822401

3343,193

9396

9396 =⇒

=−⋅=

⇒ yyy

yy mii m2

Atunci, valorile absolute sunt:

y92 = 7182 – 460 = 6722 mii m2 y91= 6722 – 93 = 6629 mii m2 y90 = 6629 – 110 = 6519 mii m2 y94 = 7182 – 460 = 7740 mii m2 y95 = 7740 + 1158 = 8898 mii m2 y96 = 8898 + 685 = 9583 mii m2

Indicatorii medii ai seriei cronologice sunt:

� nivelul mediu al seriei:

4,76107

53273 === ∑n

yy t mii m2

� PRGLILFDUHD�PHGLH�DEVROXW �

67,5106

65199583

1111/ =−=

−−

=−

∆=∆

n

yy

nnn mii m2/an

� indicele mediu de modificare:

0663,16519

9583771

11/ ==== −

y

yII nn

n (106,63%)

� ritmul mediu de modificare:

( ) ( ) %63,610010663,11001 =−=⋅−= IR

ÌQ�PHGLH�SURGXF LD�GH�FHUDPLF �vQ�5RPkQLD��vQ�SHULRDGD��-1996 a crescut cu 510,67 mii m2 anual sau cu 6,63% pe an (de 1,0663 ori).

4. În perioada 1991-��SURGXF LD�UHDOL]DW �GH�6�&��³;´�65/�OD�SURGXVXO�³$´�D�FUHVFXW�GH�OD��

WRQH��OD��WRQH��vQ��ÌQ�LSRWH]D�F �HYROX LD�SURGXF LHL�VH�SRDWH�DMXVWD�SULQ�PHWRGD�

VSRUXOXL�PHGLX� úL� F � vQ� SHULRDGD�XUP WRDUH� VH�YD�PHQ LQH�DFHHDúL� WHQGLQ � GH�HYROX LH�� V � VH�

determLQH�DQXO�vQ�FDUH�SURGXF LD�DQXOXL��VH�YD�GXEOD� 6ROX LH�

to/an43,57

846

7190981/ =−=

−=

−∆

=∆yy

nn

1RW P�yt QLYHOXO�SURGXF LHL�vQ�DQXO�t, în care volumul se va dubla.

947,8

43,5

4643,5462 989898

≅=

==⇒⋅=⇒∆⋅=⇒∆⋅+=⋅= tttytyyyt

$QXO�vQ�FDUH�SURGXF LD�VH�YD�GXEOD�HVWH�DQXO��

5. 1XP UXO�ELOHWHORU�GH�RGLKQ �vQWU-R�VWD LXQH�PRQWDQ ��YkQGXWH�GH�R�DJHQ LH�WXULVWLF ��D�FXQRVFXW�

în perioada 1996-��XUP WRDUHD�HYROX LH� Tabelul 8.12

Perioada Nr. bilete Trim. I ‘96 32 Trim. II ‘96 48 Trim. III ‘96 64 Trim. IV ‘96 58 Trim. I ‘97 40 Trim. II ‘97 52 Trim. III ‘97 74 Trim. IV ‘97 66 Trim. I ‘98 44 Trim. II ‘98 60 Trim. III ‘98 82 Trim. IV ‘98 74

Se cere:

a) 6 �VH�UHSUH]LQWH�JUDILF�VHULD�FURQRORJLF �SUH]HQWDW � b) 6 �VH�GHWHUPLQH�DEDWHULOH�VH]RQLHUH�úL�FRHILFLHQ LL�VH]RQLHUL� c) 6 �VH�SUHYL]LRQH]H�QXP UXO�ELOHWHORU�FH�VH�YRU�YLQGH�vQ�WULP��,,,�¶��vQ�LSRWH]D�F �VHULD�vúL�YD�

S VWUD�DFHHDúL�HYROX LH�FD�úL�vQ�SHULRGD��-1998. 6ROX LH�

a) 'LQ� JUDILF� SUH]HQWDW� vQ� ILJXUD� �� VH� REVHUY � DWkW� H[LVWHQ D� WUHQGXOXL� FUHVF WRU�� FkW� úL�

afectarea valorilor trimestriale de�F WUH�IDFWRUXO�VH]RQLHU� b) 3HQWUX� GHWHUPLQDUHD� DEDWHULORU� úL� FRHILFLHQ LORU� VH]RQLHUL� VH� HOLPLQ � WUHQGXO� SULQ�metoda

mediilor mobile:

ijjijij SYy ε+=− ),1( 4,1

),1( 3,1

mjj

nii

==

== (tab. 8.13)

Fig. 8. 3

��

�! ��" ! ��

! ��

#! ��

�! � � ! � �

! � �

#! � �

�! � �$ ! � �

! � �

#! ��

%'&)( *�+

, -)./01232

Pentru determinarea abaterilor sezoniere, datele se vor sistematiza astfel: Tabelul 8.13

Perioada yij Medii mobile provizorii din

4 termeni

Yij MM

Abateri yij-Yij= Sj+εij

6HULH�FRUHFWDW yij-Sj

I 96 32 - - - 48

II 96 48 - - 52 50,5

III 96 64 51,5 12,5 50 52,5

IV 96 58 53 5 52 53,5

I 97 40 54,75 -14,75 56 56

II 97 52 57 -5 56 58

III 97 74 58,5 15,5 60 59

IV 97 66 60 6 60 61

I 98 44 62 -18 60 63

II 98 60 64 -4 64 65

III 98 82 - - 68

IV 98 74 - - - 68 Tabelul 8.14

Trim. Anii

yij-Yij

I II III IV 1996 - - 12,5 5 1997 -14,75 -5 15,5 6 1998 -18 -4 - - Total -32,75 -9 28 11

s’j -16,375 -4,5 14 5,5 sj -16,032≈ -16 -4,156≈ -4 14,344≈14 5,844≈6

unde: j

jj n

Ss

∑=’ VXQW�HVWLPDWRUL�EUX L�DL abaterilor sezoniere;

344,04

375,1

4

1128975,32’

−=−=++−−=∑

m

s j�HVWH�P ULPHD�FH�FRUHFWHD] �YDORULOH�EUXWH��

6H�RE LQ�DVWIHO�DEDWHULOH�VH]RQLHUH�QHWH�

∑∑

=−= )0( ’

’j

jjj s

m

sss

6H�REVHUY �F �vQ�WULPHVWUHOH�,�úL�,,�DEDWHULOH�VH]RQLHUH�GH�OD�WUHQG�VXQW�QHJDWLYH��YDORULOH�VH�

sitXHD] �VXE�OLQLD�GH�WUHQG��SHQWUX�FD�vQ�WULPHVWUHOH�,,,�úL�,9�GHYLHULOH�V �ILH�SR]LWLYH� 6HULD�FRUHFWDW �GH�VH]RQDOLWDWH�VH�YD�GHWHUPLQD�FD� yij-sj(col. 5 tab. 8.13). Pentru determinarea indicilor de sezonalitate (în ipoteza unui model multiplicativ) vom

caOFXOD�UDSRUWXO�vQWUH�YDORULOH�UHDOH�úL�YDORULOH�DMXVWDWH�SULQ�PHGLL�PRELOH��FRO��WDE�� Tabelul 8.15.

Perioada yij Yij MM

yij/Yij 6HULH�FRUHFWDW yij/s*j

0 1 2 3 4 I 96 32 - - 43,48≈43 II 96 48 - - 50,78≈51 III 96 64 51,5 1,2427 49,94≈50 IV 96 58 53 1,0943 51,72≈52 I 97 40 54,75 0,7306 54,34≈54 II 97 52 57 0,9123 55,01≈55 III 97 74 58,5 1,2650 57,74≈58 IV 97 66 60 1,1000 58,86≈59 I 98 44 62 0,7097 59,78≈60 II 98 60 64 0,9375 63,47≈63 III 98 82 - - 63,98≈64 IV 98 74 - - 65,99≈66

Tabelul 8.16

Trim. Anii

yij/Yij

I II III IV 1996 - - 1,2427 1,0943 1997 0,7306 0,9123 1,2650 1,1000 1998 0,7097 0,9375 - -

’ *js 0,7200 0,9248 1,2538 1,0971

*js 0,736 0,9453 1,2816 1,1214

Indicele brut de sezonalitate pentru sezonul j se poate calcula ca:

jnjj ss **’ = ; úL�DQXPH�

7200,07097,07306,0*’ =⋅=Is

9248,09375,09123,0*’ =⋅=IIs

2538,12650,12427,1*’ =⋅=IIIs

0971,11000,10943,1*’ =⋅=IVs &RUHF LD�LQGLFLORU�EUX L�VH�YD�IDFH�FX�

9783,00971,12538,19248,072,04*’ =⋅⋅⋅=∏mjs

Atunci, valorile nete (corectate) ale indicilor de sezonalitate vor fi:

)1( *

*’

*’* ∏

∏== j

mj

jj s

s

ss

736,09783,0

7200,0* ==Is

9453,09783,0

9248,0* ==IIs

2816,19783,0

2538,1* ==IIIs

1214,19783,0

0971,1* ==IVs

Corectarea seriei cronologice de sezonalitate pe baza indicilor se va face:

*j

ij

s

y (col. 4 tab. 8.16)

c) 3HQWUX�VHULD�FRUHFWDW �GH�VH]RQDOLWDWH�FX�DMXWRUXO�LQGLFLORU�GH�VH]RQDOLWDWH��FRO��WDE��

determinarea trendului se poate face prin metode mecanice sau analitice. Tabelul 8.17.

Per. yt )(∆tY )(ItY t t2

0 1 2 3 4 5 96 I 43 43 43 -11 121 96 II 51 45,09 44,71 -9 81 96 III 50 47,18 46,48 -7 49 96 IV 52 49,27 48,33 -5 25 97 I 54 51,36 50,25 -3 9 97 II 55 53,45 52,25 -1 1 97 III 58 55,54 54,32 1 1 97 IV 59 57,63 56,48 3 9 98 I 60 59,72 58,72 5 25 98 II 63 61,81 61,05 7 49 98 III 64 63,9 63,48 9 81 98 IV 66 66 66 11 121 Total 675 - - 0 572

Tabelul 8.18 Per. ty Yt lin ( )2)(∆− tt Yy ( )2)(Itt Yy − ( )2tt Yy −

0 1 2 3 4 5 96 I -473 46,27 0 0 10,7 96 II -459 48,09 34,9 39,6 8,5 96 III -350 49,90 7,9 12,4 0,01 96 IV -260 51,72 7,5 13,5 0,08 97 I -162 53,53 7,0 14,1 0,22 97 II -55 55,34 2,4 7,6 0,12 97 III 58 57,16 6,1 13,5 0,71 97 IV 177 58,97 1,9 6,4 0 98 I 300 60,78 0,1 1,6 0,61 98II 441 62,60 1,4 3,8 0,16

98 III 576 64,41 0,1 0,3 0,17 98 IV 726 66,23 0 0 0,06 Total 519 675 69,1 112,6 21,3

0HWRGD�PRGLILF ULL�PHGLL�DEVROXWH�

bilete 09,211

4366

11111/2

1/

=−=−−

=−

∆=

−

∆=∆

∑=

−

n

yy

nnnn

n

ttt

∆⋅−+=∆ )1(1)( tyYt (col. 2 tab. 8.17)

Atunci:

( ) ( ) bilete 932816,109,23663 *981999 ≅⋅+=∆+= IIIIVIII syy

Metoda indicelui mediu de modificare:

04,143

66511/

11/ ==== −− −∏ n

nn

tt III

( ) 11

’ −⋅= tt IyY (col. 3 tab. 8.17)

Atunci:

( ) ( ) bilete 952816,104,166 3*3981999 ≅=⋅= IIIIVIII sIyy

Modelul liniar: tbaYt ⋅+=

=+

=+

∑∑∑

∑∑tytbta

ytbna

2

===

===⇒

∑∑

∑

907,0572

519

25,56

2t

tyb

yn

ya

tYt ⋅+= 907,025,56 (col. 2 tab. 8.18)

iar previzionarea pentru trim. III 1999, în ipoteza modelului multiplicativ se face astfel: ( ) bilete 922816,117907,025,561999 ≅⋅+=IIIy

Alegerea celei mai potrivite metode de determinare a trendului se va face prin criteriul

PLQLPL] ULL�VXPHL�S tratelor abaterilor termenilor seriei de la valorile de trend:

( ) PLQLP→−∑ 2tt Yy ��FRO��WDE��úL�VH�DOHJH�PHWRGD�OLQLDU ��SHQWUX�FDUH� ( ) 3,212 =−∑ tt Yy

6. 6�&��³;´�65/D�vQUHJLVWUDW�XUP WRDUHOH�VWRFXUL�SHQWUX�PDUID�³$´�vQ�DQXO�� Tabelul 8.19

Data Stoc (kg) 1.01.98 120 (y1) 31.01.98 210 (y2) 28.02.98 80 (y3) 31.03.98 185 (y4) 30.06.98 110 (y5) 31.10.98 160 (y6) 31.12.98 60 (y7)

Se cere:

a) 6 �VH�UHSUH]LQWH�JUDILF�VHULD�GH�PRPHQWH� b) 6 �VH�GHWHUPLQH�VWRFXO�PHGLX�vQ�WULPHVWUXO�,��úL�VWRFXO�PHGLX�SH�DQXO��

6ROX LH�

a) 5HSUH]HQWDUHD� JUDILF � VH� YD� IDFH� LQkQG� FRQW� GH� IDSWXO� F � VHULD� HVWH� GH� PRPHQWH�� QX� GH�

LQWHUYDOH��6H�ULGLF �EDU �EDWRQ�GLQ�ILHFDUH�PRPHQW�SkQ �vQ�GUHSWXO�QLYHOXOXL�FDUDFWHULVWLFLL�

b) Pentru trimestrul I 1998 se poDWH�DSOLFD�PHGLD�FURQRORJLF �VLPSO ��GDF �LQWHUYDOXO�GH�WLPS�

luat în considerare este luna):

( ) ( ) ( )=

++

+++++=

1112

1

2

1

2

1433221

I cr trim

yyyyyyy

kg 5,1473

5,442

32

18580210

2

120

3224

321

==+++

=+++

=

yyy

y

3HQWUX� vQWUHJXO�DQ�� VH�YD�FDOFXOD�PHGLD�FURQRORJLF �SRQGHUDW � �LQWHUYDOHOH� vQWUH�PRPHQWH� VH�

H[SULP �vQ�OXQL�úi anume: t1=1, t2=1, t3=1, t4=3, t5=4, t6=2):

( ) ( ) ( ) ( )+

+++++++=

122

3

2

1

2

1

2

154433221

1998.

yyyyyyyyycr

( ) ( )

+

++

++⋅

=+++

+12

2

1

2

1

2

1

2

1

2

1

122

2

2

43217665 yyyyyyy

=⋅+

++

++

+

+12

2

2

2

2

2

4

2

4

2

3

2

3

2

17654 yyyy

kg 08,13712

602

6160

2

7110

2

418580210

2

120

=+

+

+

+++

=

Probleme propuse

1. &RVXPXO� PHGLX� ]LOQLF� GH� DS � SRWDELO � �O�ORFXLWRU�� D� FXQRVFXW� XUP WRDUHD� HYROX LH� vQWU-o

localitate, în perioada 1991-1999: Tabelul 8.20

Ani ‘91 ‘92 ‘93 ‘94 ‘95 ‘96 ‘97 ‘98 ‘99 Consum (l) 134 156 191 197 205 217 230 240 246

Se cere:

a) 6 �VH�UHSUH]LQWH�JUDILF�VHULD�FURQRORJLF � b) 6 �VH�FDOFXOH]H�VLVWHPXO�GH�LQGLFDWRUL�FH�FDUDFWHUL]HD] �DFHDVW �VHULH; c) 6 �VH�DMXVWH]H�VHULD�IRORVLQG�PHWRGH�PHFDQLFH�úL�DQDOLWLFH� d) 6 �VH�DOHDJ �PHWRGD�GH�WUHQG�FHD�PDL�SRWULYLW �úL�V �VH�H[WUDSROH]H�VHULD�SHQWUX�XUP WRULL�GRL�

ani.

2. 6H�FXQRVF�XUP WRDUHOH�GDWH�SULYLQG�HYROX LD�QXP UXOXL�GH�DERQDPHQWH�WHOHIRQLFH�SDUWLFXODUH�vn

MXGH XO�³;´��vQ�SHULRDGD��-1998: Tabelul 8.21

Anul 1993 1994 1995 1996 1997 1998 0RGLILF UL�ID �GH�DQXO�DQWHULRU�

(nr. abonamente) +825 +740 +1207 +648 +727 +1003

ùWLLQG�F �vQ�DQXO��QXP UXO�DERQDPHQWHORU�WHOHIRQLFH�vQ�MXGH XO�³;´�D�FUHVFXW�FX��ID �GH�

1992, se cere:

a) 6 �VH�UHFRQVWLWXLH�VHULD�FURQRORJLF �GH�YDORUL�DEVROXWH� b) 6 �VH�UHSUH]LQWH�JUDILF�VHULD�RE LQXW � c) 6 �VH�FDOFXOH]H�LQGLFDWRULL�FH�FDUDFWHUL]HD] �VHULD� d) 6 �VH�DMXVWH]H�VHULD�FURQRORJLF �UHFRQVWLWXLW �

3. Stocurile existente în depR]LWHOH�XQHL�VRFLHW L�FRPHUFLDOH��SHQWUX�PDUID�³$´�DX�IRVW� Tabelul 8.22

Data 1.01 15.01 31.01 28.02 10.03 24.03 31.03 Stoc (kg) 140 20 80 30 70 10 50

6 �VH�GHWHUPLQH�VWRFXO�PHGLX�SHQWUX�WULPHVWUXO�,�

4. 6H�FXQRVF�XUP WRDUHOH�GDWH��SULYLQG�HYROX LD�QXP UXOXL�GH�HOHYL�GLQWU-R�XQLWDWH�GH�vQY PkQW� Tabelul 8.23

Ani 1995 1996 1997 1998 1999 5LWP�GH�PRGLILFDUH�D�QU��GH�HOHYL�ID �GH�DQXO�

precedent (%) - 5 6 10 8

ùWLLQG�F �YDORDUHD�DEVROXW �D�XQXL�SURFHQW�GLQ�ULWPXO�GH�PRGLILFDUH�DO�QXP UXOXL�GH�HOHYL�GLn anul ��ID �GH��D�IRVW�GH��SHUVRDQH��se cere:

a) 6 �VH�UHFRQVWLWXLH�VHULD�GH�YDORUL�DEVROXWH�SULYLQG�QXP UXO�GH�HOHYL�GLQ�XQLWDWH��vQ�SHULRDGD�

DQDOL]DW � b) 6 �VH�UHSUH]LQWH�JUDILF�HYROX LD�QXP UXOXL�GH�HOHYL�GLQ�XQLWDWHD�GH�vQY PkQW�DQDOL]DW ��vQ�

perioada 1995-1999; c) 6 �VH�FDOFXOH]H�VLVWHPXO�GH�LQGLFDWRUL�FH�FDUDFWHUL]HD] �VHULD�FURQRORJLF � d) 6 �VH�DMXVWH]H�VHULD�IRORVLQG�PHWRGD�DGHFYDW �úL�V �VH�H[WUDSROH]H�VHULD�SH�XUP WRULL�GRL�DQL�

5. 'HVSUH� HYROX LD� QXP UXOXL� GH� FDGUH� PHGLFDOH� GLQWU-un spital, în perioada 1994-1999 se se

cunosc datele: Tabelul 8.24

Ani 1995 1996 1997 1998 1999 0RGLILFDUHD�DEVROXW �ID �GH�DQXO�SUHFHGHQW��SHUV�� 50 10 60 30 10 ùWLLQG�F � vQ�DQXO��YDODRUHD�DEVROXW �D�XQXL�SURFHQW�GH�FUHúWHUH� ID �GH��D�QXP UXOXL�GH�

cadre medicale a fost de 1,2 persoane, se cere: a) 6 � VH� UHFRQVWLWXLH� VHULD�GH�YDORUL� DEVROXWH� SULYLQG�QXP UXO� GH�FDGUH�PHGLFDOH� vQ� SHULRDGD�

DQDOL]DW � b) 6 �VH�UHSUH]LQWH�JUDILF�HYROX LD�QXP UXOXL�GH�FDGUH�PHGLFDOH�vQ�SHULRDGD��-1999; c) 6 �VH�FDOFXOH]H�VLVWHPXO�GH�LQGLFDWRUL�FH�FDUDFWHUL]HD] �VHULD�FURQRORJLF � d) 6 �VH�DMXVWH]H�VHULD�IRORVLQG�PHWRGD�DGHFYDW �úL�V �VH�H[WUDSROH]H�VHULD�SH�XUP WRULL�GRL�DQL�

6. 2�VRFLHWDWH�FRPHUFLDO �D�UHDOL]DW�vQ�OXQD�IHEUXDULH��YkQ] UL�vQ�YDORDUH�GH��POG��OHL��LDU�vQ�

perioada februarie 1999 –�GHFHPEULH��YDORDUHD�OXQDU �PHGLH�D�YkQ] ULORU�D�FUHVFXW�FX��

POG��OHL��'DF �VH�SUHVXSXQH�F �vQ�SHULRDGD�LDQXDULH��–�PDUWLH��YDORDUHD�YkQ] ULORU�YD�

vQUHJLVWUD� DFHHDúL� HYROX LH� FD� FHD� GLQ� SHULRDGD� IHEUXDULH� �� –� GHFHPEULH� �� V se GHWHUPLQH�YDORULOH�HVWLPDWH�DOH�YkQ] ULORU�vQ�OXQLOH�LDQXDULH��IHEUXDULH�úL�PDUWLH��

99

ÍQWUHE UL� ¾ Care sunt principalele probleme teoretice de construire a indicilor? ¾ Care sunt principalele sisteme de ponderare folosite la construirea indicilor de grup factoriali? ¾ Principalele tipuri de indici sintetici. ¾ 'HWHUPLQDUHD�LQGLFLORU�FDOFXOD L�FD�UDSRUW�D�GRX �PHGLL� ¾ 3RVLELOLW L� GH� GHVFRPSXQHUH� SH� IDFWRUL� D� YDULD LHL� XQXL� IHQRPHQ� FRPSOH[� IRORVLQG� PHWRGD�

indicilor.

INDICII STATISTICI

([HUFL LL�üL probleme

1. 'HVIDFHULOH�GH�P UIXUL�DOH�XQHL�VRFLHW L�FRPHUFLDOH�VH�FDUDFWHUL]HD] �SULQ�GDWHOH��

Tabelul 9.1 Marfa &DQWLW L�YkQGXWH 3UH XUL�XQLWDUH��PLL�OHL�

vQ�SHULRDGD�GH�ED] �

(0) vQ�SHULRDGD�FXUHQW �

(1) vQ�SHULRDGD�GH�ED] �

(0) vQ�SHULRDGD�FXUHQW �

(1) M1 (kg) 4.000 7.000 32 38 M2 (l) 50.000 40.000 5 6

M3 (kg) 10.000 10.000 40 50

Se cere: a) 6 �VH�VWXGLH]H�GLQDPLFD��PRGLILFDUHD�DEVROXW �úL�UHODWLY �D�FDQWLW LORU��SUH XULORU�úL�YDORULL�

GHVIDFHULORU�SHQWUX�ILHFDUH�PDUI � b) 6 � VH� DQDOL]H]H� GLQDPLFD� YDORULL� GHVIDFHULORU� SH� DQVDPEOXO� VRFLHW LL� FRPHUFLDOH�� FX�

HYLGHQ LHUHD�LQIOXHQ HL�IDFWRULORU� c) 6 � VH� DQDOL]H]H� PRGLILFDUHD� DEVROXW � D� GHVIDFHULORU� GH� P UIXUL� SH� DQVDPEOXO� 6�&�� FX�

HYLGHQ LHUHD�LQIOXHQ HL�IDFWRULORU�FH�SRW�IL�LGHQWLILFD L�

6ROX LH�

Pentru analiza datelor disponibile vom nota: q0i� �FDQWLWDWHD�FRPHUFLDOL]DW �vQ�SHULRDGD�T0, pentru marfa i ( )3,2,1 MMMi = ) q1i� �FDQWLWDWHD�FRPHUFLDOL]DW �vQ�SHULRDGD�T1, pentru marfa i ( )3,2,1 MMMi = ) p0i� �SUH XO�XQLWDU�DO�P UILL�i (mii lei/u.m.) în perioada T0 p1i� �SUH XO�XQLWDU�DO�P UILL�i (mii lei/u.m.) în perioada T1 v0i = valoarea desfacerilor în perioada T0, pentru marfa i ( )3,2,1 MMMi = ) v1i = valoarea desfacerilor în perioada T1, pentru marfa i ( )3,2,1 MMMi = )

iii qpv 000 ⋅= ; iii qpv 111 ⋅= 2E LQHP�GDWHOH�GLQ�WDEHOXO�� Tabelul 9.2.

Marfa

&DQWLW L 3UH XUL�XQLWDUH (mii lei)

Valoarea mf. (mil. lei)

qi 0/1

q0 q1 p0 p1 v0 v1 0 1 2 3 4 5 6 7

M1 4000 7000 32 38 128 266 1,75 M2 50.000 40.000 5 6 250 240 0,8 M3 10.000 10.000 40 50 400 500 1,0

- - - + 778 1006 - Tabelul 9.2 – continuare

pi 0/1 vi 0/1 ∆q ∆p ∆v ∆v(p) ∆v(q) p0 q1

8 9 10 11 12 13 14 15 1,1875 2,0781 3000 6 138 42 96 224

1,2 0,96 -10.000 1 -10 40 -50 200 1,25 1,25 0 10 100 100 0 400

- + - - 228 182 46 824

a) Pentru marfa M1 vom calcula: a1) Dinamica:

� dinamica factorului extensiv (cantitatea) (col. 7, tab. 9.2.)

75,14000

7000

10

1110/1 ===

M

MqM q

qi (175%)

&DQWLWDWHD�YkQGXW �GLQ�PDUID�0��D�FUescut, în T1�ID �GH�70, de 1,75 ori, sau cu 75%. � GLQDPLFD�SUH XOXL��IDFWRUXO�LQWHQVLY��FRO��WDE��

1875,132

38

10

1110/1 ===

M

MpM p

pi (118,75%)

3UH XO� XQLWDU� GH� YkQ]DUH� SHQWUX� PDUID� 0�� D� FUHVFXW� vQ� 71� ID � GH� 70 de 1,1875 ori, sau cu 18,75%.

� dinamica valorii desfacerilor (col. 9 tab. 9.2)

0781,2128

266

10

1110/1 ===

M

MvM v

vi (207,81%)

'XS �FXP�VH�REVHUY ��p

Mq

MMM

MM

M

MvM ii

pq

pq

v

vi 11

1010

1111

10

111 0/10/10/1

⋅=⋅⋅

==

(2,0781 = 1,75 Â��

a2) 0RGLILF ULOH�UHODWLYH�

( ) ( ) %75100175,1100111 0/10/1 =−=−= qq

MMir

( ) ( ) %75,1810011875,1100111 0/10/1 =−=−= pp

MMir

( ) ( ) %81,10710010781,2100111 0/10/1 =−=−= vv

MMir

$úDGDU�� YDORDUHD� GHVIDFHULORU� SHQWUX�PDUID�0�� D� FUHVFXW� FX� �� SH� VHDPD� FDQWLW LL� FX�

��LDU�SH�VHDPD�SUH XOXL�XQLWDU�FX��

a3) 0RGLILF ULOH�DEVROXWH� � PRGLILFDUHD�DEVROXW �D�FDQWLW LL�YkQGXWH��FRO��WDE��

30004000700010110/1 1=−=−=∆ MM

q qqM

kg

� modificarea absolut �D�SUH XOXL�XQLWDU��FRO��WDE��

6323810110/1 1=−=−=∆ MM

p ppM

mii lei/kg

� PRGLILFDUHD�DEVROXW �D�YDORULL�GHVIDFHULORU��FRO��WDE��

138128266101110/1 =−=−=∆ MMv

M vv mil. lei

0RGLILFDUHD� DEVROXW � D� YDORULL� GHVIDFHULORU� VH� SRDWH� GHVFRPSXQH� SH� VHDPD� IDFWRULORU� Ge LQIOXHQ �

1010111110110/1 1 MMMMMMv qpqpvv

M−=−=∆

lei) mil. (42 lei mii 000.426700011 0/11111101111

)(0/1

=⋅=

=∆⋅=−=∆ pMMMMM

pv

MMqqpqp

lei) mil. (96 lei mii 000.9630003211 0/11010101110

)(0/1

=⋅=

=∆⋅=−=∆ qMMMMM

qv

MMpqpqp

'XS �FXP�VH�REVHUY ��FRO��úL��WDE��

)(0/1

)(0/10/1 111

qvpvvMMM

∆+∆=∆ (138 mil. lei = 42 mil. lei + 96 mil. lei)

Valoarea desfacerilor la marfa M1 a crescut cu 138 mil lei, din care cu 42 mil. lei pe seama

PRGLILF ULL�SUH XULORU�XQLWDUH�úL�FX��PLO��OHL�SH�VHDPD�PRGLILF ULL�FDQWLW LL�YkQGXWH� $QDOL]D�VH�HIHFWXHD] �vQ�PRG�VLPLODU�SHQWUX�P UIXULOH�0��úL�0��

6H�REVHUY �F �OD�PDUID�0��FDQWLWDWHD�GHVIDFHULORU�D�VF ]XW�FX��0% în T1�ID �GH�70, astfel vQFkW��GHúL�SUH XO�XQLWDU�D�FUHVFXW�GH��RUL��YDORDUHD�GHVIDFHULORU�V-a situat în T1 la nivelul a 96% din T0�� ÌQ�P ULPL�DEVROXWH��YDORDUHD�GHVIDFHULORU�D�VF ]XW�FX��PLO�� OHL� vQ�H[FOXVLYLWDWH�SH�VHDPD�

FDQWLW LL��-50 mil. lei��LDU�FUHúWHUHD�YDORULL�SH�VHDPD�SUH XOXL�XQLWDU��PLO��OHL��QX�D�FRPSHQVDW�VF GHUHD�YDORULL�GDWRUDWH�IDFWRUXOXL�H[WHQVLY�

b) Pe ansamblul S.C. vom analiza dinamica indicatorilor astfel:

293,1778

1006

00

11

0

10/1 ====

∑∑

∑∑∑

qp

qp

v

vI v (129,3%)

Folosind datele din col. 15, tabelul 9.2:

221,1824

1006

10

11)(0/1 ===

∑∑∑

qp

qpI pv (122,1%)

059,1778

824

00

10)(0/1 ===

∑∑∑

qp

qpI qv (105,9%)

6H�REVHUY �F � ∑∑∑ ⋅= )(

0/1)(

0/10/1qvpvv III (1,293 = 1,221 ⋅1,059)

9DORDUHD�GHVIDFHULORU�D�FUHVFXW�GH��RUL��FX��FUHúWHUH�FH�SRDWH�IL�GHVFRPSXV �SH�

seama factorului calitativ (pUH XO�� GH� �� RUL� �FX� �� LDU� SH� VHDPD� IDFWRUXOXL� H[WHQVLY�(cantitativ) de 1,059 ori (cu 5,9%).

c) 0RGLILF ULOH�DEVROXWH�VXQW�

22877810060011010/1 =−=−=−=∆ ∑∑∑∑∑ qpqpvvv mil. lei

18282410061011)(

0/1 =−=−=∆ ∑∑∑ qpqppv mil. lei

467788240010)(

0/1 =−=−=∆ ∑∑∑ qpqpqv mil. lei

∑∑∑ ∆+∆=∆ )(0/1

)(0/10/1

qvpvv (228 mil. lei = 182 mil. lei + 46 mil. lei)

6H�REVHUY �úL�vQ�WDEHOXO��WRWDO�FRO�� WRWDO�FRO��WRWDO�FRO�� 9DORDUHD�GHVIDFHULORU�GH�P UIXUL�SH�DQVDPEOXO�6�&��D�FUHVFXW�vQ�71�ID �GH�70 cu 228 mil. lei, cu 182 PLO��OHL�GDWRULW �FUHúWHULL�SUH XULORU�XQLWDUH��úL�FX��PLO��OHL�SH�VHDPD�IDFWRUXOXL�FDQWLWDWLY�

2. /D�R�VRFLHWDWH�FRPHUFLDO �V-au înregistrat datele: Tabelul 9.3 Produsul Valoarea desfacerilor în

SHULRDGD�GH�ED] ��LDQ��

(mil. lei)

0RGLILFDUHD�DEVROXW �D�YDORULL�

desfacerilor în perioada curenW ��IHEU��ID �GH�SHULRDGD�GH�ED]

0RGLILFDUHD�UHODWLY �D�

SUH XULORU��

P 50 -10 +14 Q 70 +20 -2 R 30 -15 +10 S 100 +30 +7

Se cere:

a) 6 �VH�VWXGLH]H�GLQDPLFD�FDQWLW LORU��SUH XULORU�úL�YDORULL�GHVIDFHULORU�SHQWUX�ILHFDUH�SURGXV�úL�

modificarea absoluW �D�YDORULL�GHVIDFHULORU� b) 6 �VH�DQDOL]H]H�GLQDPLFD�YDORULL�GHVIDFHULORU�úL�PRGLILFDUHD�DEVROXW �D�YDORULL�GHVIDFHULORU�SH�

DQVDPEOXO�VRFLHW LL�FRPHUFLDOH��FX�HYLGHQ LHUHD�LQIOXHQ HL�IDFWRULORU� 6ROX LH�

a) /D� QLYHO� GH� SURGXV� �LQGLYLGXDO�� FDOFXO P� GLQDPLFD� YDORULL� GHVIDFHULORU�� YROXPXOXL� IL]LF� úL�

SUH XULORU� Tabelul 9.4

Produs v0 (mil. lei) v0/1∆ (mil. lei) pr 0/1 (%) v1 (mil. lei)

0 1 2 3 4

P 50 -10 +14 40 Q 70 +20 -2 90 R 30 -15 +10 15 S 100 +30 +7 130

Total 250 +25 - 275 Tabelul 9.4 – continuare

vi 0/1 (%) pi 0/1 (%) qi 0/1 (%) p0q1 (mil. lei) )( pv∆ (mil. lei)

)(qv∆ (mil. lei)

5 6 7 8 9 10 0,8 1,14 0,702 35,09 4,91 -14,91

1,286 0,98 1,312 91,84 -1,84 21,84 0,5 1,10 0,454 13,64 1,36 -16,36 1,3 1,07 1,215 121,50 8,50 21,50 - - - 262,07 12,93 12,07

Pentru produsul P: 9DORDUHD�GHVIDFHULORU�vQ�SHULRDGD�FXUHQW ��IHEU��VH�GHWHUPLQ �SRUQLQG�GH�OD�UHOD LD�

lei mil. 40)10(500/101

010/1010/1

=−+=∆+=⇒

⇒−=∆⇒−=∆v

PPP

PPv

Pv

vv

vvvv

SimLODU�VH�GHWHUPLQ �YDORDUHD�GHVIDFHULORU�SHQWUX�FHOHODOWH�SURGXVH��FRO��WDE�� Dinamica valorii desfacerilor va fi deci:

8,050

40

0

10/1

0

10/1 ===⇒=

P

PvP

v

v

vi

v

vi (80%)

Valoarea desfacerilor pentru produsul P s-D� VLWXDW� vQ� OXQD� IHEUXDULH� OD� QLYHOXO� D� �� ID � GH� OXna LDQXDULH��6LPLODU��VH�GHWHUPLQ �GLQDPLFD�YDORULL�GHVIDFHULORU�SHQWUX�FHOHODOWH�SURGXVH��FRO��WDE�� 'LQDPLFD�SUH XOXL�XQLWDU�OD�QLYHOXO�XQXL�SURGXV�VH�GHWHUPLQ �GLQ�FRO��

P

Pp

PpP

pp

p

pri

ri

0

10/10/1

0/10/1 14,11

100

141

1001

100==+=+=⇒+=

3UH XO�XQLWDU�GH�GHVIDFHUH�DO�SURGXVXOXL�3�D�FUHVFXW�vQ�OXQD�IHEUXDULH�ID �GH�OXQD�LDQXDULH�GH��RUL�

(col. 6, tab. 9.4). 3HQWUX�D�GHWHUPLQD�LQGLFHOH�YROXPXOXL�IL]LF��FDQWLW LL��OD�QLYHOXO�ILHF UXL�SURGXV��YRP�SRUQL�GH�OD�

UHOD LD�

p

vqqpv

i

iiiiiqpv

0/1

0/10/10/10/10/1 =⇒⋅=⇒⋅=

6H�RE LQH�SHQWUX�SURGXVXO�3�

702,014,1

8,0

0/1

0/10/1 ===

pP

vPq

Pi

ii � �� LDU� SHQWUX� FHOHODOWH� SURGXVH� LQGLFLL� FDQWLW LL� VXQW�

SUH]HQWD L�vQ�FRO��WDE�� 0RGLILFDUHD�DEVROXW �D�YDORULL�GHVIDFHULORU� VH�SRDWH�GHVFRPSXQH�SH�VHDPD�IDFWRULORU�GH� LQIOXHQ ��

Astfel, la nivelul unui produs:

0011010/1 qpqpvvv −=−=∆

0RGLILFDUHD�DEVROXW �D�YDORULL�GHVIDFHULORU�GDWRUDW �PRGLILF ULL�SUH XULORU�YD�IL�

ppv

i

qpqpqp

p

pqpqpqp

0/1

111111

1

0111011

)(0/1 −=−=−=∆

ÌQ�FRO��WDE��VXQW�SUH]HQWDWH�YDORULOH�GHVIDFHULORU�GLQ�SHULRDGD�FXUHQW ��IHEU��FDUH�V-ar fi RE LQXW�GDF �VH�PHQ LQHDX�SUH XULOH�SHULRDGHL�GH�ED] ��p0q1) Pentru produsul P:

09,3514,1

40

0/1

1

0/1

1110 ====

pP

Pp

P

PPPP

i

v

i

qpqp mil. lei

Atunci 91,409,35401011)(

0/1 =−=−=∆ PPPPpv

P qpqp mil. lei

(col. 9 tab. 9.4) 0RGLILFDUHD�DEVROXW �D�GHVIDFHULORU�SH�VHDPD�LQIOXHQ HL�IDFWRUXOXL�FDQWLWDWLY�YD�IL�

0010)(

0/1 qpqpqv −=∆ , ori prin rela LLOH�

)(0/10/1

)(0/1

)(0/1

)(0/10/1

pvvqvqvpvv ∆−∆=∆⇒∆+∆=∆

sau 00/10000

100

)(0/1 vivqp

q

qqp qqv −⋅=−=∆ (vezi col. 10 tab. 9.4)

Pentru produsul P:

−=−−=∆−∆

−=−=−=∆

lei mil. 91,1491,410

lei mil. 91,145009,35)(

0/10/1

0010)(0/1 pv

Pv

P

PPPPqvP

qpqp

b) 3H�DQVDPEOXO�6�&��GLQDPLFD�YDORULL��YROXPXOXL�IL]LF�úL�SUH XULORU�HVWH�

1,1250

275

0

10/1 ===

∑∑∑

v

vI v (110%)

Valoarea desfaceriloU�D�FUHVFXW�vQ�OXQD�IHEUXDULH��SHULRDGD�FXUHQW ��ID �GH�OXQD�LDQXDULH��SHULRDGD�

GH�ED] ��GH��RUL��GH��VDX�FX�� $FHVW�LQGLFH�VH�SRDWH�GHWHUPLQD�úL�FD�R�PHGLH�DULWPHWLF �SRQGHUDW �D�LQGLFLORU�LQGLYLGXDOL�

250

1003,1305,070286,1508,0

0

00/1

0

0

1

0

10/1

⋅+⋅+⋅+⋅=⋅

===∑

∑∑

∑

∑∑∑

v

vi

v

v

v

v

vI

vv 1,1

250

275 ==

,QGLFHOH� GH� JUXS� DO� SUH XULORU� �LQGLFHOH� GH� JUXS� DO� YDORULL� GHVIDFHULORU� SH� VHDPD� PRGLILF ULL�

SUH XULORU��HVWH�

05,107,262

275

10

1

10

11)(0/1 ====

∑∑

∑∑∑

qp

v

qp

qpI pv (105%)

VDX�FD�PHGLH�DUPRQLF �D�LQGLFLORU�LQGLYLGXDOL�

05,1

07,1

1300

10,1

15

98,0

90

14,1

40275

11

1

111

0

11)(0/1 =

+++===

∑∑

∑∑∑

vi

v

qpp

p

qpI

p

pv

Indicele de grup al valorii desfDFHULORU�SH�VHDPD�FDQWLW LL��LQGLFHOH�VLQWHWLF�DO�FDQWLW LL��HVWH�

048,1250

262

0

10

00

10)(0/1 ====

∑∑

∑∑∑

v

qp

qp

qpI qv (104,8%)

FDUH�VH�SRDWH�GHWHUPLQD�úL�SULQ�UHOD LD�

048,105,1:1,1)(

0/1

0/1)(0/1

)(0/1

)(0/10/1 ===⇒⋅=

∑

∑∑∑∑∑

pv

vqvqvpvv

I

IIIII

VDX�FD�PHGLH�DULWPHWLF �D�LQGLFLORU�LQGLYLGXDOL�DL�YROXPXOXL�IL]LF�

=⋅

===∑

∑∑

∑

∑∑∑

0

00/1

00

0

100

00

10)(0/1 v

vi

qp

q

qqp

qp

qpI

qqv

048,1250

07,262

250

215,1100454,030312,170702,050 ==⋅+⋅+⋅+⋅=

0RGLILF ULOH�DEVROXWH�YRU�IL�

252502750011010/1 =−=−=−=∆ ∑∑∑∑∑ qpqpvvv mil. lei

(total col. 2 tab. 9.4)

93,1207,2622751011011)(

0/1 =−=−=−=∆ ∑ ∑∑∑∑ qpvqpqppv mil. lei (total col. 9 tab. 9.4)

07,1225007,2620100010)(

0/1 =−=−=−=∆ ∑∑∑∑∑ vqpqpqpqv mil. lei (total col. 10 tab. 9.4)

sau

07,1293,1225)(0/10/1

)(0/1 =−=∆−∆=∆ ∑∑∑ pvvqv mil. lei

$úDGDU��YDORDUHD�GHVIDFHULORU�SH�DQVDPEOXO�6�&��D�FUHVFXW� vQ�OXQD�IHEUXDULH�ID �GH�LDQXDULH�FX��

PLO��OHL��GLQ�FDUH�FX��PLO��OHL�SH�VHDPD�PRGLILF ULL�SUH XULORU�XQLWDUH�DOH�SURGXVHORU�úL�FX��

PLO��OHL�GDWRULW �IDFWRUXOXL�H[WHQVLY��FDntitatea).

3. 6H�FXQRVF�XUP WRDUHOH�GDWH�SULYLQG�YDORDUHD�YkQ] ULORU�OD�WUHL�SURGXVH��$��%��&��OD�R�VRFLHWDWH�

FRPHUFLDO �� vQ� OXQD� GHFHPEULH� �� SUHFXP� úL� GLQDPLFD� SUH XULORU� ID � GH� OXQD� DQWHULRDU �

(noiembrie 1998): Tabelul 9.5

Produsul 9DORDUHD�YkQ] ULlor în decembrie

1998 (mil. lei)

'LQDPLFD�SUH XULORU�XQLWDUH�vQ�

GHFHPEULH�ID �GH�QRLHPEULH��

A 24,5 107

B 31,2 110

C 43,3 104

Total 99,0 -

6 � VH� GHWHUPLQH� PRGLILFDUHD� DEVROXW � D� YDORULL� YkQ] ULORU� GHWHUPLQDW � GH� FUHúWHUHD� SUH XULORU� vQ�

decembrie fD �GH�OXQD�DQWHULRDU �

6ROX LH� 9DORDUHD�YkQ] ULORU�SHQWUX�XQ�SURGXV�HVWH� iii qpv ⋅= (i = A, B, C) iar pe ansamblul celor trei produse: ∑∑ ⋅= iii qpv 0RGLILFDUHD�DEVROXW �D�YDORULL�YkQ] ULORU�GDWRUDW �FUHúWHULL�SUH XULORU�HVWe:

∑∑∑∑ =−=−=∆∑11

1

0111011

)(0/1 qp

p

pqpqpqppv

=

⋅+⋅+⋅−=−= ∑∑ 3,43

04,1

12,31

10,1

15,24

07,1

10,99

111

0/1

11 qpi

qpp

99,0 – 92,9 = 6,1 milioane lei

4. 9DORDUHD� YkQ] ULORU� SHQWUX� SDWUX� SURGXVH� FRPHUFLDOL]DWH� GH� R� VRFLHWDWH� FRPHUFLDO � vQ� GRX �

SHULRDGH�VH�SUH]LQW �DVWIHO� Tabelul 9.6

Produsul 9DORDUHD�YkQ] ULORU�vQ�

perioDGD�GH�ED] ��PLO��

lei)

9DORDUHD�YkQ] ULORU�vQ�

SHULRDGD�FXUHQW (mil. lei)

0RGLILFDUHD�UHODWLY �D�

FDQWLW LL�YROXPXOXL�IL]LF��

(%) 1 120 140 10 2 80 95 0 3 75 120 22 4 115 110 -5

Total 390 465 - 6 � VH� GHWHUPLQH� PRGLILFDUHD� UHODWLY � D� SUH XULORU�� SUHFXP� úL� PRGLILFDUHD� DEVROXW � D� YDORULL�

YkQ] ULORU�GHWHUPLQDW �GH�PRGLILFDUHD�SUH XULORU�OD�QLYHOXO�ILHF UXL�SURGXV�úL�SH�DQVDPEOXO�VRFLHW LL�

comerciale. 6ROX LH� Datele disponibile se pot prezenta astfel (tab. 9.7): Tabelul 9.7

Produsul 000 qpv ⋅= (mil. lei) 111 qpv ⋅= (mil. lei) qi 0/1

1 120 140 1,1 2 80 95 1,0 3 75 120 1,22 4 115 110 0,95

Total 390 465 - $WXQFL��OD�QLYHOXO�ILHF UXL�SURGXV��VH�SRDWH�FDOFXOD�GLQDPLFD�YDORULL�YkQ] ULORU�

0

10/1 v

viv = (col. 2, tab. 9.8)

LDU�LQGLFHOH�SUH XULORU�XQLWDUH�HVWH�

qvp iii 0/10/10/1 := (col. 3, tab. 9.8)

0RGLILFDUHD�UHODWLY �D�SUH XULORU�SHQWUX�ILHFDUH�SURGXV�HVWH�

( )10010/10/1 −= pp ir (col. 4, tab. 9.8)

0RGLILFDUHD�DEVROXW �D�YDORULL�YkQ] ULORU�GHWHUPLQDW �GH�PRGLILFDUHD�SUH XULORU�HVWH�

1

0/1

1111

0111011

)(0/1

1v

ivqp

p

pqpqpqp

ppv −=−=−=∆ (col. 5 tab. 9.8)

3H�DQVDPEOXO�FHORU�SDWUX�SURGXVH��LQGLFHOH�SUH XULORU�HVWH�

13,1413

4651

11

0/1

11

10

1)(0/1 ====

∑

∑∑

∑∑

qpi

qp

qp

vI

p

pv (113%)

0RGLILFDUHD�UHODWLY �D�SUH XULORU�HVWH�

( ) %13100113,1)(0/1 +=−=∑ pvR

ModificaUHD�DEVROXW �D�YDORULL�YkQ] ULORU�SH�VHDPD�PRGLILF ULL�SUH XULORU�

524134651011)(

0/1 =−=−=∆ ∑∑∑ qpqppv mil. lei

$úDGDU��SH�DQVDPEOXO�VRFLHW LL�FRPHUFLDOH�YDORDUHD�YkQ] ULORU�D�FUHVFXW�vQ�SHULRDGD�FXUHQW �ID �GH�

SHULRDGD�GH�ED] �FX��PLO��OHL��– 390), din carH�FX��PLO��OHL�GDWRULW �PRGLILF ULL�SUH XULORU�úL�

FX��PLO��OHL�GDWRULW �PRGLILF ULL�YROXPXOXL�IL]LF��- 390). Tabelul 9.8

Produs vi 0/1 pi 0/1 pr 0/1 (%) )(0/1pv∆

(mil. lei) 1110

1qp

iqp

p=

(mil. lei) 0 1 2 3 4 5 1 1,17 1,06 6 8 132 2 1,19 1,19 19 15 80 3 1,60 1,31 31 28 92 4 0,96 1,01 1 1 109

Total - - - 52 413

5. 'HVSUH�SURGXFWLYLWDWHD�PXQFLL�vQUHJLVWUDW �vQWU-R�VRFLHWDWH�FRPHUFLDO �úL�vQ�FHOH�GRX �ILOLDOH�DOH�

sale, penWUX�GRX �SHULRDGH��VH�FXQRVF�GDWHOH� Tabelul 9.9

Filiala Productivitatea muncii (mil. lei / pers.) 3HULRDGD�GH�ED] 3HULRDGD�FXUHQW

A 11 13 B 14 16

Total S.C. 12,5 15 Se cere:

a) ùWLLQG�F �PRGLILFDUHD�DEVROXW �D�SURGXF LHL�SH�vQWUHDJD�VRFLHWDWH�FRPHUFLDO ��GHWHUPLQDW �GH�

PRGLILFDUHD� SURGXFWLYLW LL� PHGLL� D� PXQFLL� D� IRVW� GH� �� PLOLRDQH� OHL�� V � VH� GHWHUPLQH�

YDORDUHD�SURGXF LHL�UHDOL]DW �vQ�SHULRDGD�FXUHQW ��SH�ILHFDUH�ILOLDO �úL�SH�WRWDO� b) ùWLLQG�F �QXP UXO�WRWDO�DO�VDODULD LORU�VRFLHW LL�FRPHUFLDOH�vQ�SHULRDGD�GH�ED] �D�IRVW�GH��

SHUVRDQH��V �VH�GHWHUPLQH�PRGLILFDUHD�DEVROXW �úL�GLQDPLFD�YDORULL�SURGXF LHL�SH�DQVDPEOXO�

VRFLHW LL�FRPHUFLDOH��FX�HYLGHQ LHUHD�IDFWRULORU�GH�LQIOXHQ �

6ROX LH� a) La nivelul unei filiale, productivitatea muncii este:

i

ii N

Qw = , unde Qi� �SURGXF LD�ILOLDOHL�³L´

Ni� �QU��VDODULD LORU�ILOLDOHL�³L´ /D�QLYHOXO�VRFLHW LL�FRPHUFLDOH�

∑∑

∑∑∑ === N

iii

ii

i

i gwN

Nw

N

Qw

ùWLP�F �

30011011 =⋅∆=−=∆ ∑∑∑∑ NwNwNwwQ milioane lei

5,25,121501 =−=−=∆ www milioane lei/pers. Atunci:

1205,2

3001 ==

∆

∆=∑ ∑

wN

wQ

persoane

Cum NBB

NAA

Nii gwgwgww 1111111 ⋅+⋅== ∑

úL� NB

NA

NB

NA gggg 1111 11 −=⇒=+

5H]XOW �F �

( ) NBB

NBA gwgww 11111 1 ⋅+−⋅=

de unde:

67,01316

1315

11

111 =

−−=

−−

=AB

ANB ww

wwg (67%), deci 33,01 =N

Ag (33%)

Datele se pot sistematiza în tabelul 9.10. Tabelul 9.10

Filiala w0 (mil. lei/pers.)

w1 (mil. lei/pers.)

Ng1 N1 (pers.) Q1 (mil. lei)

0 1 2 3 4 5 A 11 13 0,33 40 520 B 14 16 0,67 80 1280

Total 12,5 15 1,0 120 1800

∑⋅= 111 NgN Nii (col. 4 tab. 9.10)

iar 111 iii NwQ ⋅= (col. 5 tab. 9.10)

b) 12501005,12100 0000 =⋅==⇒= ∑∑∑ NwQN mil. lei

5501250180001 =−=−=∆ ∑∑∑ QQQ mil. lei

250300550 =−=∆−∆=∆ ∑∑∑∑

wQQ

NQ mil. lei

44,11250

1800

0

1 ===∑∑

∑ Q

QI Q

2,15,12

15

10

11 ====∑∑

∑ww

Q INw

NwI

2,1100

120

00

10 ==== ∑∑∑ ∑

∑ NNQ I

Nw

NwI

6. Se cunosc datele: Tabelul 9.11

Firma 1U��VDODULD L��SHUV�� Salariu mediu (mil. lei) dec. ’98 (0) ian. ’99 (1) dec. ’98 (0) ian. ’99 (1) I 500 400 1,2 1,32 II 300 300 1,4 1,19 III 250 150 1,3 1,30 IV 350 400 1,25 1,10

Total 1400 1250 - - Se cere: ¾ 6 �VH�DQDOL]H]H�GLQDPLFD� úL�PRGLILFDUH�DEVROXW �D�QXP UXOXL�GH� VDODULD L�� VDODULXOXL�PHGLX� úL�

IRQGXOXL�GH�VDODUL]DUH�OD�QLYHOXO�ILHF UHL�ILUPH��FX�HYLGHQ LHUHD�LQIOXHQ HL�IDFWRULORU� ¾ 6 �VH�DQDOL]H]H�GLQDPLFD� úL�PRGLILFDUH�DEVROXW �D�QXP UXOXL�GH� VDODULD L�� VDODULXOXL�PHGLX� úL�

fondului de salarizare la nivelul ansambluluL�FHORU��ILUPH��FX�HYLGHQ LHUHD�LQIOXHQ HL�IDFWRULORU� 6ROX LH�

a) 3HQWUX�UH]ROYDUHD�SUREOHPHL��VH�FDOFXOHD] �IRQGXO�GH�VDODUL]DUH�SH�ED]D�UHOD LHL��FS = S ⋅ N (col. 1,2 tab. 9.12).

Pentru firma I dinamica indicatorilor este:

(80%) 8,0500

400

0

10/1 ===

N

Ni N

1XP UXO�DQJDMD LORU�D�VF ]XW�vQ�DFHDVW �ILUP �FX��

(110%) 1,12,1

32,1

0

10/1 ===

S

Si S

6DODULXO�PHGLX�DO�DQJDMD LORU�GLQ�ILUPD�,�D�FUHVFXW�FX��

(88%) 88,0600

528

0

10/1 ===

FS

FSi FS

)RQGXO�GH�VDODUL]DUH�D�VF ]XW�vQ�LDQXDULH��ID �GH�GHFHPEULH��FX�� 0RGLILFDUHD�DEVROXW �D indicatorilor este:

100500400010/1 −=−=−=∆ NNN persoane

(IHFWLYXO�VDODULD LORU�V-a redus cu 100 de persoane.

12,02,132,1010/1 =−=−=∆ SSS (120 mii lei)

6DODULXO�PHGLX�SH�DQJDMDW�D�FUHVFXW�vQ�LDQXDULH��FX��PLL�OHL�ID �GH�GHFHPEULH��

72600528010/1 −=−=−=∆ FSFSFS mil. lei.

)RQGXO� GH� VDODUL]DUH� D� VF ]XW� FX� �� PLO�� OHL�� $FHDVW � VF GHUH� VH� SRDWH� GHVFRPSXQH� SH� VHDPD�

IDFWRULORU�GH�LQIOXHQ ��VDODULXO�PHGLX�SH�ILUP �úL�QXP UXO�GH�DQJDMD L�

726005280011010/1 −=−=−=−=∆ NSNSFSFSFS mil. lei

leimilNNSNS SSFS .4840012,010/11011)(

0/1 =⋅=⋅∆=−=∆

leimilSNSNS NNFS .120)100(120/100010)(

0/1 −=−⋅=∆⋅=−=∆

FSNFSSFS0/1

)(0/1

)(0/1 ∆=∆+∆ (48 + (-120) = - 72 mil. lei)

'HúL�VDODULXO�PHGLX�SH�ILUPD�,�D�FUHVFXW�� OD�IRQGXO�GH�VDODUL]DUH�V-D�vQUHJLVWUDW�R�UHGXFHUH�GDWRUDW �VF GHULL�QXP UXOXL�GH�DQJDMD L� $QDOL]D�VH�HIHFWXHD] �VLPLODU�SHQWUX�ILUPHOH�,,��,,,�úL�,9��FRO. 3-10 tab. 9.12)

b) La nivel de grup, se pot cumula: fondul de salarizare ( ∑ FS ��úL�QXP UXO�GH�DQJDMD L��∑ N ). 6DODULXO�PHGLX�VH�FDOFXOHD] �

∑∑

∑∑∑ ⋅=⋅== NgS

N

NS

N

FSS

leimilN

FSS .216,1

1250

1520

1

11 ===

∑∑

leimilN

FSS .273,1

1400

5,1782

0

00 ===

∑∑

Atunci, dinamica indicatorilor este:

(85,3%) 853,05,1782

1520

0

10/1 ===

∑∑∑

FS

FSI FS

(89,3%) 893,01400

1250

0

10/1 ===

∑∑∑

N

NI N

(95,5%) 955,0273,1

216,1

0

10/1 ===

S

SI S

Dinamica salariului mediu pe ansamblul celor 4 firme se poate descompune pe seama celor doi IDFWRUL�GH�LQIOXHQ ��VDODULXO�PHGLX�OD�QLYHOXO�ILUPHORU�úL�VUWXFWXUD�DQJDMD LORU��FRO��WDE��

(95,5%) 955,0273,1

216,1

00

11

0

10/1 ====

∑

∑N

NS

gS

gS

S

SI

(95,3%) 953,0273,1

216,1

10

1

10

11)(0/1 ====

∑∑

∑NN

NSS

gS

S

gS

gSI

(100,2%) 002,1273,1

276,1

0

10

00

10)(0/1 ====

∑

∑

∑S

gS

gS

gSI

N

N

NgS N

SgSSS IIIN

0/1)(

0/1)(

0/1 =⋅ (0,953⋅1,002 = 0,955)

'LQDPLFD�IRQGXOXL�GH�VDODUL]DUH�VH�SRDWH�GHVFRPSXQH�SH�VHDPD�D�GRL�IDFWRUL�GH� LQIOXHQ ��

( S úL�∑ N ) sau a trei factori: (S, Ng úL�∑ N )

∑ ∑

∑ ∑∑∑

∑∑ ===∑

000

111

00

11

0

10/1

NgS

NgS

NS

NS

FS

FSI

N

NFS

953,0)(0/1

110

111)(0/1 ===

∑ ∑

∑ ∑∑ SSN

NSFS I

NgS

NgSI

002,1)(0/1

100

110)(0/1 ===

∑ ∑

∑ ∑∑ NN gSN

NgFS I

NgS

NgSI

)(0/1

)(0/1

)(0/1

SFSgFSSFS IIIN ∑∑∑ =⋅

893,00/1000

100)(0/1 === ∑∑ ∑

∑ ∑

∑ ∑ NN

NNFS I

NgS

NgSI

∑∑∑∑∑∑∑∑ =⋅=⋅⋅ FSNFSSFSNFSgFSSFS IIIIIIN

0/1)(

0/1)(

0/1)(

0/1)(

0/1)(

0/1

0,953⋅ 1,002 ⋅ 0,893 = 0,955 ⋅ 0,893 = 0,853

0RGLILFDUHD� DEVROXW � D� LQGLFDWRULORU� DQDOL]D L� OD� QLYHOXO� DQVDPEOXOXL�� vQ� LDQXDULH� �� ID � GH�

decembrie 1998 este:

5,2625,17821520010/1 −=−=−=∆ ∑∑∑ FSFSFS mil. lei

15014001250010/1 −=−=−=∆ ∑∑∑ NNN persoane

057,0273,1216,1010/1 −=−=−=∆ SSS mil. lei / pers. (- 57 mii lei/pers.)

0RGLILFDUHD�DEVROXW �D�VDODULXOXL�PHGLX�SH�DQVDPEOX�VH�SRDWH�GHVFRPSXQH�SH�VHDPD�IDFWRULORU

057,000110/1 −=−=∆ ∑∑ NNS gSgS mil. lei / pers.

=−=−=−=∆ ∑∑∑ 276,1216,11011011)(

0/1NNNSS gSSgSgS

= - 0,06 mil. lei / pers.

=−=−=−=∆ ∑∑∑ 273,1276,10100010)(

0/1 SgSgSgS NNNgS N

= 0,003 mil. lei / pers. $úDGDU��VDODULXO�PHGLX�SH�VDODULDW�D�VF ]XW�FX��PLL�OHL�SHUV��3H�VHDPD�VDODULLORU�OD�QLYHO�GH�ILUP �D�

VF ]XW� FX��PLL� OHL�SHUV�� GDU� D�FUHVFXW� FX��PLL� OHL�SHUV��GDWRULW �GHSODV ULL� VWUXFWXULL�DQJDMD LORU�

F WUH�ILUPHOH�FX salarii mari.

SgSSS N

0/1)(

0/1)(

0/1 ∆=∆+∆

- 0,06+ 0,003= - 0,057 mil. lei/pers.

3HQWUX�IRQGXO�GH�VDODUL]DUH�DQDOL]D�PRGLILF ULL�DEVROXWH�SH�VHDPD�D��VDX��IDFWRUL�GH�LQIOXHQ �VH�

poate face astfel:

−=−=−=∆ ∑ ∑∑∑ ∑∑∑1110011010/1 NgSNSNSFSFS NFS

5,262000 −=− ∑ ∑ NgS N mil. lei

=⋅∆=−=∆ ∑∑ ∑∑ ∑∑1

)(0/1110111

)(0/1 NNgSNgS SSNNSFS

75125006,0 −=⋅−= mil. lei (total col. 9 tab. 10.12)

=⋅∆=−=∆ ∑∑ ∑∑ ∑∑1

)(0/1100110

)(0/1 NNgSNgS

NN gSNNgFS

= 0,003 ⋅ 1250 ≅ 3,5 mil. lei

=∆⋅=−=∆ ∑∑ ∑ ∑ ∑∑ ∑ NNNNFS SNgSNgS 0/10000100)(

0/1

= 1,273 ⋅ (-150) ≅ -191 mil. lei

∑∑∑ ∆=∆+∆ )(0/1

)(0/1

)(0/1

SFSgFSSFS N (-75 + 3,5 = - 71,5 mil. lei)

∑∑ ∑∑ ∆=∆+∆ )(0/1

)(0/1

)(0/1

NFSNFSgFS N (3,5 - 191 = - 187,5 mil. lei) (total col. 10 tabel 9.12)

=−++−=∆+∆+∆ ∑ ∑∑∑ )191(5,3)75()(0/1

)(0/1

)(0/1

NFSgFSSFS N

= - 262,5 mil. lei.

Tabelul 9.12 Fond salarizare (mil. lei) dec. ‘98 ian. ‘99

Ni 0/1 Si 0/1 FSi 0/1 N0/1∆

(pers.) 1 2 3 4 5 6

600 528 0,8 1,1 0,88 -100 420 357 1,0 0,85 0,85 0 325 195 0,6 1,0 0,6 -100

437,5 440 1,143 0,88 1,01 50 1782,5 1520 - - - -150

Tabelul 9.12 – continuare

S0/1∆

(mil. lei)

FS0/1∆

(mil. lei)

)(0/1

SFS∆

(mil. lei)

)(0/1

NFS∆

(mil. lei)

Ng1 NgS 10

(mil. lei) 7 8 9 10 11 12

0,12 -72 48 -120 0,32 0,384 -0,21 -63 -63 0 0,24 0,336

0 -130 0 -130 0,12 0,156 -0,15 2,5 -60 62,5 0,32 0,400

- -262,5 -75 -187,5 1,00 1,276

7. 6H�FXQRVF�XUP WRDUHOH�GDWH�SULYLQG�LQGLFLL�SUH XULORU�GH�FRQVXP�SH�WLSXUL�GH�P UIXUL�úL�VWUXFWXUD�

cheltuielilor familiilor:

Tabelul 9.13 7LSXUL�GH�P UIXUL Structura cheltuielilor (%) IndiciL�SUH XULORU�vQ

úL�VHUYLFLL 3HU��ED] �� 3HU��FXUHQW �� SHULRDGD�FXUHQW �ID �GH�SHULRDGD�GH�

ED] �� 0 UIXUL�DOLPHQWDUH 57 64 130 0 UIXUL�QHDOLPHQWDUH 30 ... 120

Servicii ... 12 140 Total 100 100 -

6 �VH�FDOFXOH]H�LQGLFHOH�SUH XULORU�GH�FRQVXP�Se ansamblu. 6ROX LH� ,QGLFHOH� SUH XULORU� GH� FRQVXP� �,3&�� HVWH� XQ� LQGLFH� GH� WLS� /DVSH\UHV� �FX� SRQGHUL� FRQVWDQWH� GLQ�

SHULRDGD�GH�ED] ��$úDGDU�

100

0(%)0/1

00

000/1

00

000

1

00

01vi

pp gi

qp

qpi

qp

qpp

p

qp

qpIPC

∑

∑∑

∑

∑

∑∑ ⋅

=⋅

===

(128,3%) 283,1100

1340,13020,15730,1 =⋅+⋅+⋅=

3UH XULOH�SH�DQVDPEOX� DX� FUHVFXW�� DúDGDU�� vQ� SHULRDGD�FXUHQW � ID � GH� SHULRDGD�GH� ED] �FX�

28,3%.

Probleme propuse

1. La nivelul a trei puncte de desfacere se cunosc datele: Tabelul 9.14

3LD D Valoarea desfacerilor (mil. lei) Modificarea valorii desfacerilor 3HU��GH�ED] 3HU��FXUHQW GDWRULW �PRGLILF ULL�SUH XULORU��PLO��OHL�

A 600 690 100 B 420 500 80 C 1000 1400 510

Se cere:

a) 6 �VH�FDOFXOH]H�LQGLFLL�LQGLYLGXDOL�DL�YDORULL��SUH XULORU�úL�YROXPXOXL�IL]LF� b) 6 �VH�FDOFXOH]H�LQGLFLL�GH�JUXS�FRUHVSXQ] WRUL� c) 6 �VH�GHVFRPSXQ �PRGLILFDUHD�YDORULL�GHVIDFHULORU�vQ�IXQF LH�GH�IDFWRULL�GH�LQIOXHQ �

2. &X�SULYLUH�OD�FRQVXPXO�D�WUHL�SURGXVH�OD�QLYHOXO�XQHL�IDPLOLL�VH�FXQRVF�XUP WRDUHOH�GDWH�SHQWUX�

GRX �OXQL�FRQVHFXWLYH� Tabelul 9.15

Produs Cantitate 3UH �XQLWDU��OHL�X�P�� aprilie mai aprilie mai

Pâine (buc.) 40 35 1200 1300 $S �PLQHUDO ��O� 25 20 1000 1500

Pantofi (per.) 1 1 217.000 320.000 Se cere:

a) 6 �VH�FDOFXOH]H�LQGLFLL�YDORULL�SHQWUX�ILHFDUH�SURGXV� b) 6 �VH�GHWHUPLQH�SHQWUX�ILHFDUH�SHULRDG �VWUXFWXUD�FRQVXPXOXL�úL�V �VH�UHSUH]LQWH�JUDILF� c) 6 �VH�GHWHUPLQH�PRGLILFDUHD�YDORULL�FRQVXPXOXL�SH�IDFWRUL�GH�LQIOXHQ �

3. 6H�FXQRVF�GDWHOH�SULYLQG�YDORDUHD�GHVIDFHULORU�XQXL�SURGXV�úL�LQGLFLL�SUH XULORU� Tabelul 9.16 Punct de desfacere 9DORDUHD�GHVIDFHULORU�vQ�SHULRDGD�FXUHQW ��PLO��OHL� ,QGLFLL�SUH XULORU��

A 410 140 B 470 135

6H�FHUH�V �VH�FDOFXOH]H�PRGLILFDUHD�YDORULL�GHVIDFHULORU�GDWRUDW �PRGLILF ULL�SUH XULORU�LQGLYLGXDOH�

4. 6H�FXQRVF�XUP WRDUHOH�GDWH� Tabelul 9.17

Unitate HFRQRPLF

3URGXF LD�IL]LF ��EXF�� Nr. de muncitori 3UH XUL

3HU��ED] 3HU��FXUHQW Per. ED] 3HU��FXUHQW comparabile (lei/um) A 690 600 80 75 4300 B 2000 2100 22 20 1960 C 500 512 10 10 1200

Se cere:

a) 6 � VH� FDUDFWHUL]H]H� GLQDPLFD� SURGXFWLYLW LL� PXQFLL�� YROXPXOXL� IL]LF� DO� SURGXF LHL�� YDORULL�

SURGXF LHL�úL�QXP UXOXL�GH�PXQFLWRUL�SH�ILHFDUH�XQLWDWH�HFRQRPLF � b) ,QGLFLL�GH�JUXS�DL�SURGXFWLYLW LL�PXQFLL�úL�PRGLILFDUHD�DEVROXW �úL�UHODWLY �D�SURGXFWLYLW LL�

PHGLL�D�PXQFLL�SH�VHDPD�SURGXFWLYLW LL�PXQFLL�SH�ILHFDUH�XQLWDWH�HFRQRPLF �úL�D�VWUXFWXULL�

QXP UXOXL�GH�PXQFLWRUL� c) 0RGLILFDUHD�UHODWLY �úL�DEVROXW �D�YDORULL�SURGXF LHL�SH�WRWDO�úL�GHVFRPSXQHUHD�SH�IDFWRUL�GH�

LQIOXHQ �SULQ�PHWRGD�LQGLFLORU�

5. $FWLYLWDWHD�XQHL�VRFLHW L�FRPHUFLDOH�VH�FDUDFWHUL]HD] �SULQ�GDWHOH�

Tabelul 9.18 Marfa 9DORDUHD�P UIXULORU�YkQGXWH��PLO��OHL�

3HULRDGD�ED] 3HULRDGD�FXUHQW 0RGLILFDUHD�DEVROXW �D�YDORULL�P UIXULORU�

SH�VHDPD�SUH XULORU��PLO��OHL� A 20 30 15 B 90 80 10 C 60 60 10

Se cere: a) ,QGLFLL�LQGLYLGXDOL�DL�YDORULL��YROXPXOXL�IL]LF�úL�SUH XULORU� b) ,QGLFLL�GH�JUXS�DL�YDORULL��YROXPXOXL�IL]LF�úL�SUH XULORU� c) RepUH]HQWDUHD�JUDILF �D�YDORULL�P UIXULORU�YkQGXWH�

6. 3HQWUX�R�VRFLHWDWH�FRPHUFLDO �VH�FXQRVF�GDWHOH�

Tabelul 9.19 Firma 9kQ] UL�GH�P UIXUL�DOLPHQWDUH�FX�

DP QXQWXO��PLO��OHL�

1U��PHGLX�GH�VDODULD L�GLQ�FRPHU

ian. feb. ian. feb.

A 21 31 50 41

B 44 108 108 153

Se cere: a) 3URGXFWLYLWDWHD�PXQFLL�SH�ILHFDUH�ILUP �úL�SH�WRWDO�VRFLHWDWH�FRPHUFLDO �vQ�LDQXDULH� b) 'LQDPLFD�YkQ] ULORU��QXP UXOXL�PHGLX�GH�VDODULD L�úL�D�SURGXFWLYLW LL�PXQFLL�

E��SH�ILHFDUH�ILUP � E�� SH� WRWDO� VRFLHWDWH� FRPHUFLDO � �SHQWUX� SURGXFWLYLtatea muncii se vor calcula indicii ca

UDSRUW�D�GRX �PHGLL�� c) 0RGLILF ULOH�UHODWLYH�úL�DEVROXWH�SH�WRWDO�VRFLHWDWH�FRPHUFLDO �D�

F��SURGXFWLYLW LL�PHGLL�D�PXQFLL�úL�GHVFRPSXQHUHD�DFHVWRUD�SH�FHL��IDFWRUL��SURGXFWLYLWDWHD�

PXQFLL�úL�VWUXFWXUD�VDODULD LORU�; c2) YkQ] ULORU� GH� P UIXUL� FX� DP QXQWXO� úL� GHVFRPSXQHUHD� DFHVWRUD� SH� FHL� �� IDFWRUL�

(productivitatea�PXQFLL�úL�QXP UXO�PHGLX�GH�VDODULD L�SH�ILHFDUH�ILUP ��

7. 5HIHULWRU�OD�R�XQLWDWH�FRPHUFLDO �VH�FXQRVF�GDWHOH�

Tabelul 9.20 Marfa 9DORDUHD�P UIXULORU�YkQGXte în

SHULRDGD�FXUHQW ��PLO��OHL� 'LQDPLFD�SUH XULORU�vQ�

SHU�FUW��ID �GH�SHU��ED] 0RGLILFDUHD�SURFHQWXDO �D�

volumului fizic %

A 5400 3,0 12,5 B 15500 3,0 -6 C 4400 2,0 -30

Se cere: a) ,QGLFLL�LQGLYLGXDOL�DL�YDORULL��YROXPXOXL�IL]LF�úL�SUH XULORU� b) IndiciL�GH�JUXS�DL�YDORULL��YROXPXOXL�IL]LF�úL�SUH XULORU� c) 0RGLILFDUHD�DEVROXW �D�YDORULL�P UIXULORU�úL�GHVFRPSXQHUHD�DFHVWHLD�SH�IDFWRUL�GH�LQIOXHQ �

prin metoda indicilor; d) *UDILFXO�YDORULL�P UIXULORU�YkQGXWH�vQ�SHULRDGD�GH�ED] �úL�SHULRDGD�FXUHQW �

8. Despre 3 vQWUHSULQGHUL�FH�IDEULF �DFHODúL�SURGXV�VH�FXQRVF�GDWHOH� Tabelul 9.21

3URGXF LD�PDUI ��PLO��OHL��

SUH XUL�FRQVWDQWH 1U��GH�VDODULD L��SHUV�� Întreprinderea

3HU��ED] 3HU��FXUHQW

Procentul de modificare a SUH XULORU 3HU��ED] 3HU��FXUHQW

A 1560 1600 +16 1800 1800 B 2670 2700 +10 3500 3400 C 1200 1230 +8 820 800 5430 5530 - 6120 6000

Se cere:

a) ,QGLFLL�LQGLYLGXDOL�úL�GH�JUXS�DL�YDORULL��YROXPXOXL�IL]LF�úL�SUH XULORU� b) ,QGLFLL� GH� JUXS� DL� SURGXFWLYLW LL� PXQFLL�� PRGLILFDUHD� DEVROXW � D� SURGXFWLYLW LL�medii pe

VHDPD�SURGXFWLYLW LL�PXQFLL�SH�ILHFDUH�vQWUHSULQGHUH� c) 0RGLILFDUHD� DEVROXW � D� YDORULL� SURGXF LHL�PDUI � úL� GHVFRPSXQHUHD� SH� IDFWRUL� GH� LQIOXHQ �

prin metoda indicilor; d) *UDILFXO�SURGXF LHL�PDUI �SH�WRWDO�vQ�IXQF LH�GH�SURGXFWLYLWDWHD�PHGLH�D�PXQFLL�úL�GH�QXP UXO�

GH�VDODULD L��LQ�SHULRDGD�GH�ED] �úL�SHULRDGD�FXUHQW �

9. 9kQ] ULOH�GH�P UIXUL�UHDOL]DWH�GH�R�VRFLHWDWH�FRPHUFLDO �DX�vQUHJLVWUDW�XUP WRDUHD�HYROX LH� Tabelul 9.22

Marfa UM 9DORDUHD�P UIXULORU�

YkQGXWH�vQ�SHU��GH�ED] �

(mil. lei)

Modificarea abVROXW �D�valorii pe seama SUH XULORU��PLO��OHL�

Modificarea UHODWLY �D�FDQWLW LL�

(%) 1 to 100 +30 +5 2 kg 200 -50 +10 3 buc. 150 -20 -7 4 l 70 +50 +10

6 �VH�GHWHUPLQH�

a) 'LQDPLFD�YDORULL�YkQ] ULORU��D�FDQWLW LORU� úL�D�SUH XULORU�P UIXULORU��SH� ILHFDUH�PDUI �úL�SH�

WRWDO�VRFLHWDWH�FRPHUFLDO � b) 0RGLILF ULOH� DEVROXWH� DOH� YDORULORU� YkQ] ULORU�� SH� ILHFDUH� PDUI � úL� SH� WRWDO� VRFLHWDWH�

FRPHUFLDO ��FX�HYLGHQ LHUHD�LQIOXHQ HL�IDFWRULORU� c) 'HWHUPLQDUHD� LQIOXHQ HL� FRQFRPLWHQWH� D� DPELORU� IDFWRUL� DVXSUD� PRGLILF ULL� DEVROXWH� úL�

UHODWLYH� D� YkQ] ULORU� OD� QLYHO� GH� VRFLHWDWH� FRPHUFLDO �� vQ� FD]XO� XWLOL] ULL� PHWRGHL� UHVWXOXL�

nedescompus.

LUCRARI APLICATIVE PROPUSE SPRE REZOLVARE

1. 'LQ�WUHL�HúDQWLRDQH�HIHFWXDWH�vQ�WUHL�MXGH H�$��%�úL�&��V-DX�RE LQXW�GDWHOH�XUP WRDUH�UHIHULWRDUH�OD�

GHVIDFHULOH�XQXL�VRUWLPHQW�GH�PDUI �vQ�FDQWLW L�HJDOH�GDU�OD��SUH XUL�GLIHULWH��X�P��SDFKHW��

3UH XO -XGH H (u.m./pac) A… B… C…

1 13,5 13,2 13,4 2 14,2 13,3 13,3 3 14,1 13,1 14,0 4 13,4 13,5 14,2 5 13,3 13,4 14,1

Se cere:

a) 6 �VH�FDOFXOH]H�úL�V �VH�LQWHUSUHWH]H�PHGLLOH�úL�GLVSHUVLLOH�FRQGL LRQDWH� b) 3HQWUX� XQ� SUDJ� GH� VHPQLILFD LH� α �� V � VH� HIHFWXH]H� DQDOL]D� GLVSHUVLRQDO �

( F2 12 0 95 3 89; ; , ,= ��úL�V �VH�LQWHUSUHWH]H�UH]XOWDWHOH�

c) 6 �VH�DQDOL]H]H�JUDGXO�GH�GHWHUPLQDUH�úL�GH�QHGHWHUPLQDUH�úL�V �VH�IRUPXOH]H�FRQFOX]LL�

2. Dintr-R�DQFKHW � VWDWLVWLF �HIHFWXDW � vQ� UkQGXO�HOHYLORU�GLQ�JUXSD�GH�YkUVW ��DQL� - 16 ani s-au

VLVWHPDWL]DW�GDWHOH�UHIHULWRDUH�OD�VXPHOH�V SW PkQDOH�GH�³EDQL�GH�EX]XQDU´�

Anii 12 12 15 14 16 14 12 13 11 11 *

Mii lei 4,9 4,1 13,6 12,2 13,8 8,6 7,2 9,4 4,2 3,6 *

Nr.elevi 15 13 10 11 9 18 20 30 15 19 150

Se cere:

a) 6 � VH� UHSUH]LQWH� JUDILF� GHSHQGHQ D� GLQWUH� FHOH� GRX � YDULDELOH� SUHFL]kQGX-VH�� H[LVWHQ D��GLUHF LD�úL�IRUPD�OHJ WXULL��FRRUGRQDWHOH�FHQWUXOXL�GH�JUHXWDWH�DO�GLDJUDPHL�QRUXOXL�GH�SXQFWH�

b) 6 � VH� FDOFXOH]H� úL� V � VH� DQDOL]H]H� LQGLFDWRULL� VLQWHWLFL� DL� GLVWULEX LHL��PHGLLOH�� GLVSHUVLLOH� úL�

FRYDULDQ D� c) 6 �VH�HVWLPH]H�SDUDPHWULL�IXQF LHL�GH�UHJUHVLH�DOHDV �FRUHVSXQ] WRU�JUDILFXOXL�WUDVDW�úL�V �VH�

FDOFXOH]H�YDORULOH�DMXVWDWH�úL�YDORULOH�YDULDELOHL�GH�SHUWXUED LH� d) 3HQWUX� XQ� SUDJ� GH� VHPQLILFD LH� α �� V � VH� WHVWH]H� VHPQLILFD LD� SDUDPHWULORU� HVWLPD L� úL�

validitatea modelului adoptat; e) 3HQWUX� DFHODúL� SUDJ� GH� VHPQLILFD LH� V � VH� VWDELOHDVF � LQWHUYDOHOH� GH� vQFUHGHUH� SHQWUX��

SDUDPHWULL� úL� YDORULOH� IXQF LHL� GH� UHJUHVLH� -� DFHVWH� XOWLPH� YDORUL� UHSUH]HQWD L-le pe graficul trasat la punctul a);

f) 6 �VH�SURJQR]H]H� VXPHOH�GH�³EDQL�GH�EX]XQDU´�SHQWUX�HOHYLL�GH��DQL�GLQ�HúDQWLRQ� úL�GLQ�

FROHFWLYLWDWHD�JHQHUDO � g) 6 � VH� FDUDFWHUL]H]H� LQWHQVLWDWHD� OHJ WXULL� GLQWUH� FHOH� GRX � YDULDELOH� �P VXUD� vQ� FDUH�

PRGLILFDUHD� VXPHORU� GH� EDQL� GHSLQGH� GH� YkUVWD� HOHYLORU�� V � VH� WHVWH]H� VHPQLILFD LD�

UH]XOWDWXOXL�RE LQXW�úL�V �VH�GHWHUPLQH�LQWHUYDOXO�GH�vQFUHGHUH�FRUHVSXQ] WRU��SHQWUX�XQ�SUDJ�

GH�VHPQLILFD LH�α=5%).

3. Analizându-VH� FRPSRUWDPHQWXO� FHORU� �� PHPEUL� DL� HFKLSHL� GH� úDK� ³&OXE� úDK� DFWLY´� vQ�

campionatul din 1996 s-D�HODERUDW�XUP WRDUHD�VLWXD LH� -XF WRUXO A… C… V… D… F… G… L… M… N… R… Puncte în tur 8,3 7,6 9,1 9,5 8,4 6,9 9,2 7,8 8,6 8,2 Puncte în retur 7,9 7,4 9,1 9,3 8,4 7,5 9,0 7,2 8,2 8,1

Se cere:

a) 6 �VH�YHULILFH�GDF �vQWUH�SXQFWDMXO�RE LQXW�vQ�WXU�úL�UHWXU�H[LVW �R�GHSHQGHQ �VWDWLVWLF ��ÌQ�FD]�

DILUPDWLY��P VXUD L�LQWHQVLWDWHD�OHJ WXULL�FX�DMXWRUXO�XQRU�FRHILFLHQ L�FRUHVSXQ] WRUL� b) ÌQ� UDSRUW� FX� SXQFWDMXO� PHGLDQ� vQUHJLVWUDW� GH� FRPSRQHQ LL� HFKLSHL� ³&OXE� úDK� DFWLY´�� vQ�

FDPSLRQDWXO�GLQ��V �VH�HODERUH]H�WDEHOXO�GH�DVRFLHUH��&DOFXOD L�úL�LQWHUSUHWD L�FRHILFLHQWXO�

GH�DVRFLHUH�úL�χ2 (cu α=5%).

4. ÌQ�0XQLFLSLXO�%XFXUHúWL�V-D�HIHFWXDW�R�DQFKHW �VWDWLVWLF �SHQWUX�D�WHVWD�DWLWXGLQHD�SRSXOD LHL�ID �

GH�IHQRPHQHOH�vQUHJLVWUDWH�GH�SLD D�ILQDQFLDU ��'LQ�GDWHOH�vQUHJLVWUDWH�úL�VLVWHPDWL]DWH s-D�RE LQXW�XUP WRDUHD�VLWXD LH�

Atitudini

)DYRUDELO 1HIDYRUDELO ,QGLIHUHQW Masculin 21 15 9

Feminin 39 13 3

Se cere:

a) 6 �VH�SUHFL]H]H�QDWXUD�VHULHL�úL�D�WDEHOXOXL�SUH]HQWDW� b) 6 �VH�FDOFXOH]H�úL�V �VH�DQDOL]H]H�IUHFYHQ HOH�UHODWLYH�SDU LDOH��FRQGL LRQDWH�úL�PDUJLQDOH� c) 6 � VH� WHVWH]H� LSRWH]D� UHIHULWRDUH� OD� LQGHSHQGHQ D� GLQWUH� FHOH� GRX � YDULDELOH� SHQWUX� α=5%

( χ teoret2 5 99= , );

d) 6 �VH�HVWLPH]H�SHQWUX�R�SUREDELOLWDWH�GH��] ��SURFHQWXO�PD[LP�DO�FHORU�FX�DWLWXGLQH�

QHIDYRUDELO �úL�LQGLIHUHQW �

5. 8Q�MXUQDOLVW�vQ�VWXGLXO�SLH HL�GH�DXWRPRELOH�GH�RFD]LH�XUP UHúWH�R�DQXPLW �PDUF ��ÌQ�DFHVW�VHQV�

HO�D�IRUPDW�XQ�HúDQWLRQ�GH��DXWRPRELOH� OD�FDUH�D� vQUHJLVWUDW�QXP UXO�GH�NLORPHWUL�SDUFXUúL�

SkQ � vQ�PRPHQWXO� YkQ] ULL� úL� SUH XO� GH� YkQ]DUH�� vQDLQWHD� QHgocierii, echivalent în U.S.D. la cursul oficial al zilei.

*UXSH�GH�DXWRPRELOH�GXS �SUH XO�GH�YkQ]DUH��PLL�86'� *UXSH�GH�DXWRPRELOH�GXS

NP�SDUFXUúL��PLL�NP� 1 - 1,8 1,8 - 2,6 2,6 - 3,4 3,4 - 4,2 4,2 - 5 TOTAL

60 - 80 - - - 10 8 18 80 - 100 - - 16 18 7 41

100 - 120 - 22 27 27 8 84 120 - 140 16 24 10 7 - 57 TOTAL 16 46 53 62 23 200

Se cere:

a) 6 �VH�UHSUH]LQWH�JUDILF�OHJ WXUD�GLQWUH�FHOH�GRX �YDULDELOH�úL�FHOH�GRX �FXUEH��VDX�GUHSWH��GH�

regresie; b) 6H� SRDWH� HIHFWXD� R� DMXVWDUH� OLQLDU � D� QRUXOXL� GH� SXQFWH� REVHUYDW"� ÌQ� FD]� DILUPDWLY�� V � VH�

SDUFXUJ �HWDSHOH�QHFHVDUH�úL�V �VH�LQWHUSUHWH]H�UH]XOWDWHOH� c) 6 � VH� FDUDFWHUL]H]H� LQWHQVLWDWHD� OHJ WXULL� GLQWUH� FHOH� GRX � YDULDELOH� SULQWU-un indicator

FRUHVSXQ] WRU� d) 6 � VH� FDOFXOH]H� P VXUD� XQJKLXOXL� GLQWUH� FHOH� GRX � GUHSWH� GH� UHJUHVLH� úL� V � VH� FRPSDUH�

UH]XOWDWXO�RE LQXW�FX�FHO�GH�OD�SXQFWXO�F��([SOLFD L�HYHQWXDOHOH�GLIHUHQ H�

6. Pentru a proba valoarea unui test psihotehnic propus de serviciul “resurse umane”, dintr-o

LPSRUWDQW �VRFLHWDWH�FRPHUFLDO �SURILODW �SH�IDEULFDUea pieselor de schimb necesare motoarelor auto, s-D� IRUPDW� XQ� HúDQWLRQ� GH� �� GH� PXQFLWRUL�� 'LVWULEX LD� PXQFLWRULORU� GXS � UH]XOWDWHOH�

RE LQXWH� OD� WHVW�úL�GXS �SURGXF LD�RE LQXW � vQWU-R�V SW PkQ �GH� OXFUX�GH��GH�RUH�VH�SUH]LQW �

astfel:

Grupe de muncitori 1RWH�RE LQXWH�OD�WHVW GXS �QXP UXO�GH�SLHVH�IDEULFDWH��EXF�� 15 14 13 12 11 10 9 8 Total

130-139 2 2 120-129 1 3 1 5 110-119 1 6 6 2 1 16 100-109 4 22 24 4 4 58 90-99 2 18 44 28 5 1 98 80-89 2 19 50 14 1 86 70-79 1 6 19 3 29 60-69 1 2 3 50-59 2 1 3

TOTAL 2 17 49 90 89 43 9 1 300 Se cere:

a) 6 �VH�WUDVH]H�JUDILF�úL�V �VH�VWXGLH]H�GLDJUDPD�QRUXOXL�GH�SXQFWH�úL�FHQWUXO�V X�GH�JUHXWDWH� b) 6WXGLD L� GLVWULEX LLOH� PDUJLQDOH� DOH� PXQFLWRULORU� GXS � QRWHOH� RE LQXWH� úL� GXS � QXP UXO� GH� SLHVH�

fabricate calculându-VH�PHGLLOH�úL�GLVSHUVLLOH�DFHVWRUD��7HVWD L�QRUPDOLWDWHD�GLVWULEX LLORU�PDUJLQDOH� c) 3HQWUX�ILHFDUH�GLVWULEX LH�FRQGL LRQDW �GH�R�DQXPLW �YDORDUH�D�YDULDELOHL�;�FDOFXOD L�úL�DQDOL]D L�

mediile xy úL�σyx

2��&DOFXOD L�úL�DQDOL]D L�UDSRUWXO�GH�FRUHOD LH�DO�OXL�<�vQ�IXQF LH�GH�;�

d) &DOFXOD L��WHVWD L�úL�LQWHUSUHWD L�FRHILFLHQWXO�GH�FRUHOD LH�GLQWUH�FHOH�GRX �YDULDELOH��5H]XOWDWXO�

RE LQXW�GLIHU �GH�YDORDUHD�UDSRUWXOXL�GH�FRUHOD LH"�$UJXPHQWD L� e) 6WDELOL L� UHOD LD� FDUH� HVWLPHD] � FHO� PDL� ELQH� WHQGLQ D� OHJ WXULL� GLQWUH� UDQGDPHQWXO� XQXL�

PXQFLWRU�vQ�IXQF LH�GH�QRWHOH�RE LQXWH�OD�WHVW��7UDVD L�GUHDSWD�GH�UHJUHVLH�SH�DFHODúL�JUDILF�vQ�

FDUH�DX�IRVW�UHSUH]HQWDWH�PHGLLOH�FRQGL LRQDWH� f) (VWLPD L�OLPLWele probabile, în 95% din cazuri, ale timpului necesar pentru 10 muncitori care

V �H[HFXWH��EXF�� I��GDF �FHL��PXQFLWRUL�DX�IRVW�VHOHFWD L�DOHDWRU� I��GDF �ILHFDUH�GLQ�FHL��PXQFLWRUL�DX�RE LQXW�OD�WHVW��SXQFWH�

g) )RUPXOD L�FRQFOX]LL�FX�SULYLUH�OD�RSRUWXQLWDWHD�VWXGLXOXL�OHJ WXULL�GLQWUH�FHOH�GRX �YDULDELOH�

7. &RQVLGHU P�XQ�WDEHO�GH�FRQWLQJHQ ��FRUHOD LH�

( )[ ]{ }x y n j Jj k jk, ; ∈ ∈ si k K

a) $U WD L�F �

( )( )Cov x yn

x x y y nj kk

jkj

( , ) = − −∑∑1se poate scrie astfel:

Cov x yn

x y n x y xy x yj k jkkj

( , ) = − ⋅ = − ⋅∑∑1;

b) 'DF �Ux x

jj

x=

− 0

σ�úL��V

y yk

k

y=

− 0

σ�DU WD L�F

Cov x y x yCov u v( , ) ( , )= σ σ ;

8. a) Pentru ca expresia

( )F y xi ii

( , )β β β β0 1 0 12= − −∑ � V � ILH� PLQLP � WUHEXLH� FD� GHULYDWHOH� VDOH�

SDU LDOH�vQ�IXQF LH�GH�β0�úL�β1�V �ILH�HJDOH�FX�]HUR�

∂∂β∂∂β

F

F0

1

0

0

=

=

$U WD L�F �VROX LD�DFHVWHL�VLVWHP�GH�HFXD LL�QRUPDOH�HVWH�GDW �GH�YDORULOH�b0�úL�b1 ale lui β0�úL�β1 definite prin:

bCov x y

x1 2

= ( , )

σ�úL� xbyb 10 −=

E��ùWLLQG�F � ( )σ y iin

y y2 21= −∑ , ( )σ y x iin

y y. �2 21= −∑ úL� ( )σ Y i

iny y2 21= −∑ � unde

ii xbbY 10ˆ += DU WD L�F �

b1) ( )σ σy x y r.2 2 21= −

b2) σ σ σy y x Y2 2 2= +.

9. /X P� vQ� FRQVLGHUDUH� WDEHOXO� GH�FRQWLQJHQ � �GH� FRUHOD LH�� ( )[ ]{ }x y n j Jj k jk, ; ∈ ∈ si k K � úL�

GLVWULEX LLOH�FRQGL LRQDWH�DOH�OXL�<�GH�;� ( ){ }y n j k Kk jk, , , fixat ∈ ��3HQWUX�DFHVWHD�GLQ�XUP �VH�

cuQRVF�PHGLLOH�úL�GLVSHUVLLOH�

y xn

n yjj

jk kk

( ).

= ∑1; [ ]σ y j

jk j jk

k

xn

y y x n2 21( ( ))

.= −∑ unde n nj jk

k. = ∑

HVWH�HIHFWLYXO��IUHFYHQ D��PDUJLQDO� ��DVRFLDW �YDULDQWHL�xj.

a) $U WD L�F �GLVSHUVLD�YDULDELOHL�<�DUH�XUP WRDUHD�H[SUHVLH�

( ) [ ]σ σy j y jj

j jjn

n xn

n y x y2 2 21 1= + −∑ ∑. . ( )

b) 3URSRU LD�YDULD LHL��GLVSHUVLHL��PHGLLORU�FRQGL LRQDWH�vQ�GLVSHUVLD��YDULD LD��JHQHUDO �GHILQHúWH�

JUDGXO�GH�GHWHUPLQD LH�D�F UXL�H[SUHVLH�HVWH�

[ ]2

.2

2/

)(1

y

jjj

xy

nyxyn

Rσ

−

=∑

'HPRQVWUD L�XUP WRDUHOH�SURSULHW L�

b1) 10 2/ ≤≤ xyR

b2) 02/ =xyR �GDF �WRDWH�PHGLLOH�FRQGL LRQDWH�DOH�OXL�<�vQ�IXQF LH�GH�;�VXQW�HJDOH�vQWUH�HOH

b3) 12/ =xyR �GDF �ILHF UHL�YDORUL�[j�vL�FRUHVSXQGH�R�DQXPLW ��VLQJXU ��YDORDUH�D�OXL�<�

10. 6RFLHWDWHD� FRPHUFLDO � ³/8;´� 6�$�� HVWH� SURILODW � SH� IDEULFDUHD� GH� vQF O PLQWH�� 'DWHOH�

UHIHULWRDUH� OD�FRVWXULOH� WRWDOH� GH� SURGXF LH� úL�P ULPHD� FRPHQ]LORU�SHQWUX� XQ�DQXPLW�PRGHO� VH�

SUH]LQW �vQ�XUP WRDUHD�VHULH�

Mii perechi 3,1 4,3 0,1 0,6 1,2 0,35 1,6 4,72 2,4 3,8 2,95 1,75 Sute mii lei 326 422 10 62 124 40 209 582 259 414 310 198

Se cere:

a) 6 �VH�UHSUH]LQWH�JUDILF�GLDJUDPD�QRUXOXL�GH�SXQFWH b) &XUED�GH�UHJUHVLH�DGHFYDW �SRDWH�IL�UHSUH]HQWDW �GH�IXQF LLOH�

y abx= ; y axb= ; y ax

b= + 1; y a bx cx= + + 2 ? $UJXPHQWD i.

c) 6 �VH�FDUDFWHUL]H]H�LQWHQVLWDWHD�OHJ WXULL�

11. 3HQWUX�D�FHUFHWD�OHJ WXUD�GLQWUH�FRVWXO�WRWDO�DO�XQHL�SHUHFKL�GH�SDQWRIL�SHQWUX�E UED L�vQ�IXQF LH�

GH�FRVWXO�GLUHFW�úL�LQGLUHFW�GLVSXQHP�GH�XUP WRDUHOH�GDWH�

Costul total (mii lei) Costul direct (mii lei) Costul indirect (mii lei) 105,16 95,80 9,36 94,14 88,61 9,53 100 90 10

103,33 91,69 11,64

Costul total (mii lei) Costul direct (mii lei) Costul indirect (mii lei) 103,33 99,17 4,16 114,28 99,99 14,29 130,62 116,87 13,70 123,31 112,08 11,23 107,91 97,50 10,41 108,95 99,22 9,73 105,07 96,60 8,47 113,14 101,15 11,99

Se cere:

a) 6 � VH� UHSUH]LQWH� JUDILF� WHQGLQ D� OHJ WXULORU� VLPSOH� SRVLELOH�� SUHFL]kQGX-VH� GLUHF LD� úL� IRUPD�OHJ WXULORU�

b) 6 �VH�HVWLPH]H�SDUDPHWULL�PRGHOHORU�VLPSOH�GH�UHJUHVLH�úL�V �VH�WHVWH]H�SHQWUX�XQ�SUDJ�GH��

VHPQLILFD LLOH�SDUDPHWULORU� c) 6 �VH�WHVWH]H�YDOLGLWDWHD�Podelelor de regresie pentru un prag de 5%; d) 6 �VH�FDUDFWHUL]H]H�LQWHQVLWDWHD�OHJ WXULORU�GLQWUH�YDULDELOH�úL�V �VH�WHVWH]H�DFHVWHD�SHQWUX�XQ�

prag precizat mai sus; e) 6 �VH�H[SOLFLWH]H�PRGHOXO�GH�UHJUHVLH�PXOWLSO �úL�V �VH�WHVWH]H�VHPQLILFD LD�SDUDPHWULORU� f) 6 �VH�FDUDFWHUL]H]H�LQWHQVLWDWHD�SDU LDO �úL�PXOWLSO � g) ùWLLQG�F �GDWHOH�SUH]HQWDWH�VH�UHIHU �OD�SHULRDGD�70��V �VH�HVWLPH]H�SXQFWXDO�úL�vQ�LQWHUYDO�GH�

vQFUHGHUH�FRVWXO� WRWDO�úWLLQG�F � vQ�SHULRDGD�70-T1�FRVWXO�GLUHFW�YD�FUHúWH� vQ�PHGLH�FX��LDU�

costul indirect se va diminua în medie cu 2%.

12. 6H�FXQRVF�XUP WRDUHOH�GDWH�

Structura cifrei de afaceri

Departamente GH�VRFLHW L comerciale

Procentul mediu al sarcinii de plan al

cifrei de afaceri (%)

Procentul mediu al îndeplinirii pl.

cifrei de afaceri (%) P. ED]

P. FXUHQW

CoeficieQW�GH�YDULD LH�al îndeplinirii

planului cifrei de afaceri (%)

A 100 90 15 20 25

B 100 100 35 30 20

C 102 101 40 30 18

D 108 110 10 20 15

TOTAL …. …. 100 100 ….

Se cere:

a) 6 � VH� UHSUH]LQWH� JUDILF� JUDGXO� vQGHSOLQLULL� SODQXOXL� FLfrei de afaceri pe departamente si structura cifrei de afaceri;

b) 6 � VH� FDOFXOH]H� SURFHQWXO�PHGLX�DO� VDUFLQLL� GH� SODQ�� DO� vQGHSOLQLULL� SODQXOXL� si al dinamicii cifrei de afaceri;

c) 6 � VH� FDUDFWHUL]H]H� RPRJHQLWDWHD� PXO LPLL� GHSDUWDPHQWHORU� GXS � �� vQGHSOLQLULL� Slanului cifrei de afaceri;

d) ùWLLQG�F �îQ�PXO LPHD�VRFLHW LORU�FRPHUFLDOH�GLQ�FHOH�SDWUX�GHSDUWDPHQWH��SURFHQWXO�PRGDO�DO�

vQGHSOLQLULL�SODQXOXL�HVWH��FH�VH�SRDWH�VSXQH�GHVSUH�DVLPHWULD�GLVWULEX LHL"�'DU�GHVSUH�

procentul median al îndeplinirii planului?

13. 'LVWULEX LD�DQJDMD LORU�XQHL�VRFLHW L�FRPHUFLDOH�GXS �YHFKLPH�úL�VDODULX�PHGLX�VH�SUH]LQW �DVWIHO�

în luna ianuarie:

*UXSH�GH�DQJDMD L�GXS �

vechime (ani)

NumaU�DQJDMD L Salariu mediu (mii u.m./pers.)

Sub 5 10 250 5-10 30 370 10-15 40 400 15-20 50 425 20-25 40 460 25-30 35 485

Peste 30 15 500 Se cere:

a) 6 �VH�UHSUH]LQWH�JUDILF�GLVWULEX LD�DQJDMD LORU�GXS �YHFKLPH�úL�IRQGXO�GH�VDODULL� b) 6 �VH�FDOFXOH]H�YHFKLPHD�PHGLH�úL�V �VH�YHULILFH�UHSUH]HQWDWLYLWDWHD�VD� c) 6 �VH�FDOFXOH]H�QXP UXO�GH�DQJDMD L�FDUe au salariile în intervalul intercuartilic (între Q1 �úL�

Q3 ) care de asemenea se va calcula; d) ÎQ�SURJUDPXO�GH�UHIRUP �DGRSWDW�îQ�OXQD�PDUWLH�VH�SUHYHGH�UHGXFHUHD�OD�MXP WDWH�D�QXP UXOXL�

GH� DQJDMD L� úL� LQGH[DUHD� VDODULXOXL� FX� �� &DUH� YD� IL� QRXO� VDODULX� PHGLX� úL� FXP� VH� YD�

PRGLILFD�DFHVWD�ID �GH�VDODULXO�PHGLX�GLQ�OXQD�LDQXDULH" e) 6 �VH�FDUDFWHUL]H]H�SULQWU-XQ�LQGLFDWRU�VLQWHWLF�vPSU úWLHUHD�VDODULLORU�PHGLL�GLQ�FHOH�GRX �OXQL� f) 6H�SRDWH�DSOLFD�FRUHF LD�OXL�6KHSSDUG�OD�GLVSHUVLD�VDODULLORU�OXQLL�LDQXDULH"�$UJXPHQWD L�

14. 'HVSUH�YkQ] ULOH�GH�FDOFXODWRDUH�UHDOL]DWH�GH�PDL�PXOWH�PDJD]LQH�DOH�XQHL�ILUPH�GH�FRPSXWHUH se

cunosc datele:


YkQ] ULORU��EXF�

)UHFYHQ H�UHODWLYH�FXPXODWH�

(%)

Valoarea vâQ] ULORU�WRWale realizate

(mil lei)

0-2 10 70 2-4 25 378 4-6 44 855 6-8 68 1270

8-10 85 1467 10-12 95 924

��úL�SHVWH 100 650 Se cere: a) 6 �VH�FDOFXOH]H�QXP UXO�PHGLX�GH�FRPSXWHUH�YkQGXWH�GH�XQ�PDJD]LQ�îQ�SHULRDGD�DQDOL]DW �úL�V �

VH�DUDWH�GDF �PHGLD�HVWH�R�YDORDUH�UHSUH]HQWDWLY � b) 6 �VH�UHSUH]LQWH�JUDILF�GLVWULEX LD�GH�IUHFYHQ H�úL�V �VH�VWXGLH]H�DVLPHWULD�DFHVWHLD� c) 6 �VH�DQDOL]H]H�VWUXFWXUD�YDORULL�YkQ] ULORU� d) 'DF �QXP UXO�GH�FRPSXWHUH�YkQGXWH�D�IRVW�GH��EXF L�îQ�SHULRDGD�DQDOL]DW ��V �VH�FDOFXOH]H�

SUH XO�Pediu de vânzare pe grupe.

15. Se cunosc datele:

Firma 6WUXFWXUD�SURGXF LHL�

realizate ianuarie (%)

0 ULPH�UHODWLY �GH�FRRUGRQDUH�D�

SURGXF LHL�SODQLILFDWH�IHEUXDULH Indice de realizare a planului

februarie (%)

1 10 1,4 120 2 20 2,4 80 3 25 2,8 100 4 40 4,0 75 5 5 1,0 150

ùWLLQG�F �SURGXF LD�UHDOL]DW �GH�ILUPD�QU��în luna februarie a fost de 75 mil. lei, ceea ce a vQVHPQDW�R�GHYDQVDUH�D�SURGXF LHL�GLQ�OXQD�LDQXDULH�FX��VH�FHU�

a) 1LYHOXULOH�SURGXF LHL�UHDOL]DWH�în luna ianuarie sL�DOH�SURGXF LHL�planificate pentru februarie, SH�ILHFDUH�ILUP �úL�SH�WRWDO�

b) 0 ULPLOH� UHODWLYH�DOH�SODQXOXL�SH� ILHFDUH� ILUP � si pe total, cu specificarea tipului de medie XWLOL]DW �SHQWUX�FDOFXOXO�LQGLFDWRULORU�SH�WRWDO�

c) 3URGXF LD�PHGLH�UHDOL]DW �în luna ianuarie pe întreaJD�VRFLHWDWH�HVWH�R�YDORDUH�UHSUH]HQWDWLY "�

([SOLFD L� d) 6 �VH�FRQVWUXLDVF �XQ�JUDILF�SHQWUX�UHSUH]HQWDUHD�P ULPLORU�UHODWLYH�DOH�SODQXOXL��

16. 2� PDUH� DJHQ LH� GH� vQFKLULDW� DXWRPRELOH� KRW U úWH� V -úL� YkQG � DXWRPRELOHOH� GXS � XWLOL]DUHD�

acestora timp de un an. MDQDJHUXO�ILUPHL�SUHVXSXQH�F �GLVWDQ D�SDUFXUV �GH�PDúLQL�LQIOXHQ HD] �

FRVWXO�GH� vQWUH LQHUH�DO�DXWRYHKLFXOHORU� úL�GHFL�SUH XO�GH�YkQ]DUH�DO�DFHVWRUD��3HQWUX�D�YHULILFD�

SUHVXSXQHUHD�VD��VH�vQUHJLVWUHD] ��SHQWUX�XQ�QXP U�GH��GH�PDúLQL��GLVWDQ D�SDUFXUV �în ultimul an (îQ�PLL�NP��úL�FRVWXO�GH�vQWUH LQHUH�DO�DFHVWRUD��îQ�VXWH�PLL�OHL��6H�DOF WXLHVF�DVWIHO��VWUDWXUL�

GXS �GLVWDQ D�SDUFXUV ��VXE��PLL�NP��-60 mii km, 60-��PLL�NP�úL��VDX�SHVWH��PLL�NP��'DWHOH�VXQW�XUP WRDUHOH�

&RVW�vQWUH LQHUH��VXWH�PLL�OHL� 6WUDWXUL�GXS �GLVWDQ D

SDUFXUV ��PLL�NP� 5-7 7-9 9-11 11-13

TOTAL

20-40 4 7 4 - 15 40-60 2 5 7 6 20

ùLLQG�F �

� pentru stratul al treilea (60-80 mii km) s-DX�vQUHJLVWUDW�GDWH�SHQWUX��PDúLQL��FRVWXO�PHGLX�GH�vQWUH LQHUH�HVWH�GH��OHi cu un coeficient de variatLH�GH��úL

� SHQWUX�VWUDWXO�DO�SDWUXOHD��úL�SHVWH��PLL�NP��FKHOWXLHOLOH�WRWDOH�GH�vQWUH LQHUH�DX�IRVW�GH�

��OHL��YDORDUHD�PRGDO �D�FKHOWXLHOLORU��PLO��OHL��FX�XQ�&DV� ��se cere: a) 6 � VH� P VRDUH� LQIOXHQ D� GLVWDQ HL� SDUFXUVH� DVXSUD� YDULD LHL� FKHOWXLHOLORU� GH� vQWUH LQHUH��

XWLOL]kQG�FRHILFLHQWXO�GH�GHWHUPLQD LH� b) 6 �VH�DQDOL]H]H�VWUXFWXUD�FKHOWXLHOLORU�GH�vQWUH LQHUH�SH�JUXSH�GXS �GLVWDQ D�SDUFXUV �úL�V �VH�

UHSUH]LQWH�JUDILF�DFHDVW �VWUXFWXU � c) 6 � VH� DUDWH� GDF � YDORDUHD� PHGLH� D� FKHOWXLHOLORU� SHQWUX� FHOH� �� GH� PDúLQL� HVWH� R� YDORDUH�

UHSUH]HQWDWLY ��

17. 8Q� MXUQDOLVW�HIHFWXHD] �XQ� VWXGLX�GH�SLD �SHQWUX�D�XUP UL�R�DQXPLW �PDUF �GH�DXWRPRELO�� ÌQ�

DFHVW�VFRS��HO�D�IRUPDW�XQ�HúDQWLRQ�GH��DXWRPRELOH�OD�FDUH�D�vQUHJLVWUDW�QXP UXO�GH�NLORPHWUL�

SDUFXUúL� SkQ � vQ� PRPHQWXO� YkQ] ULL� úL� SUH XO� ORU� GH� YkQ]DUH� �HFKLYDOHQWXO� 86'�� 'DWHOH�

REVHUYDWH�VXQW�VLQWHWL]DWH�vQ�XUP WRDUHD�VHULH�

*UXSH�GH�DXWRPRELOH�GXS �SUH XO�GH�YkQ]DUH��PLL�86'� Total Grupe de automobile

GXS �NP��PLL�NP� 2 - 2,4 2,4 - 2,8 2,8 - 3,2 3,2 - 3,6

60 - 80 - - 10 8 18 80 - 100 - 16 18 7 41

100 - 120 - 49 27 8 84 120 - 140 40 10 7 - 57 TOTAL 40 75 62 23 200

Se cere:

a) 6 �VH�YHULILFH�LQIOXHQ D�QXP UXOXL�GH�NLORPHWUL�SDUFXUúL�DVXSUD�PRGLILF ULL�SUH XOXL��utilizând UHJXOD�DGXQ ULL�GLVSHUVLLORU�

b) 3HQWUX�R�SUREDELOLWDWH�GH��] ��V �VH�GHWHUPLQH�LQWHUYDOXO�vQ�FDUH�VH�vQFDGUHD] �SUH XO�

mediu; c) 6 �VH�UHSUH]LQWH�JUDILF� WHQGLQ D� OHJ WXULL�GLQWUH�FHOH�GRX �YDULDELOH�SUHFL]kQGX-VH�GLUHF LD�úL�

IRUPD�OHJ WXULL� d) S �VH�HVWLPH]H�SDUDPHWULL�IXQF LHL�GH�UHJUHVLH�úL�V �VH�FDOFXOH]H�YDORULOH�DMXVWDWH� e) 6 �VH�FDUDFWHUL]H]H�LQWHQVLWDWHD�OHJ WXULL�GLQWUH�FHOH�GRX �YDULDELOH�

18. 'HVSUH�HYROX LD�FLIUHL�GH�DIDFHUL�D�XQHL�FRPSDQLL�VH�FXQRVF�XUP WRDUHOH�GDWH�

Anii Procentul modifLF ULL�UHODWLYH�D�&�$��ID �GH�DQXO�SUHFHGHQW�

(%)

1988 +1 1989 +3 1990 +2 1991 +5 1992 +4 1993 +3 1994 +5 1995 +10

Se cere:

a) 6 �VH�FDOFXOH]H�LQGLFLL�FX�ED] �IL[ �úL�PRELO � b) ùWLLQG�F �vQ��ID �GH��FLIUD�GH�DIDFHUL�FRUHVSXQ] WRDUH�XQXL�SURFHQW�GH�FUHúWHUH�D�IRVW�

GH��POG��OHL�V �VH�UHFRQVWLWXLH�VHULD�GH�GDWH�DEVROXWH� c) 6 �VH�UHSUH]LQWH�JUDILF�VHULD�UHFRQVWLWXLW �úL�V �VH�FDOFXOH]H�LQGLFDWRULL�PHGLL�DL�DFHVWHLD� d) 6 � VH� DMXVWH]H� VHULD� SULQ� PHWRGH� PHFDQLFH� úL� DQDOLWLFH� FRUHVSXQ] WRDUH� úL� V � VH� DSUecieze

FDOLWDWHD�DMXVW ULL� e) 3ULQ�PHWRGD�FHD�PDL�SRWULYLW �V �VH�H[WUDSROH]H�GDWHOH�SHQWUX�SHULRDGD��-1999.

19. Dintr-XQ�VRQGDM�GH��QHUHSHWDW�HIHFWXDW�vQ��VH�FXQRVF�XUP WRDUHOH�GDWH�

Grupe de muncitori GXS �YHFKLPH��DQL�

1XP U�GH�PXQFLWRUL Salariul mediu orar (mii lei/pers)

Coeficientul de YDULD LH��

sub 10 90 6 25 10 - 20 150 10 18

��úL�SHVWH� 60 12 20 TOTAL 300 … …

Se cere:

a) 6 � VH� HVWLPH]H�� SHQWUX� R� SUREDELOLWDWH� GH� �� ] �� OLPLWHOH� vQWUH� FDUH� VH� vQFDGUHD] �

VDODULXO� RUDU�PHGLX� úL� IRQGXO� GH� VDODULL� RUDUH� SHQWUX� FD]XO� VRQGDMXOXL� VLPSOX� úL� VWUDWLILFDW��

H[SOLFD L�HYHQWXDOHOH�GLIHUHQ H� b) 3HQWUX� R� HURDUH� OLPLW � GH� ± �� PLL� OHL� V � VH� FDOFXOH]H� YROXPXO� QRXOXL� HúDQWLRQ� úL� V � VH�

UHSDUWL]H]H�RSWLP�DFHVWD�SH�VXEHúDQWLRDQH� c) ùWLLQG�F �HYROX LD�VDODUiului mediu orar în perioada 1990 -��VH�FDUDFWHUL]HD] �SULQ�GDWHOH�

1991 1992 1993 1994 1995 ��ID �GH�DQXO�SUHFHGHQW +2 +8 +10 +9 +10

F��6 �VH�DMXVWH]H�SULQ�PHWRGH�DGHFYDWH�VHULD�VDODULXOXL�SHQWUX�SHULRDGD��-1995. c2) Se poate utiliza metoda mHGLLORU�PRELOH�GH�DMXVWDUH"�$UJXPHQWD L�GDF �GD�VDX�QX�

20. 6H� FXQRVF� XUP WRDUHOH� GDWH� GHVSUH� H[SRUWXO� QHW� úL� YDORDUHD� DMXVWDW � EUXW � DOH� VRFLHW LORU�

comerciale dintr-un departament (în T0): Nr. crt. 1 2 3 4 5 6 7 8

Exportul net (mil . lei) 10 5 7 10 11 8 12 12 9DORDUHD�DG XJDW �

(mld. lei) 22 10 20 23 24 28 30 32

Se cere: a) 6 �VH�UHSUH]LQWH�JUDILF�WHQGLQ D�OHJ WXULL�GLQWUH�FHOH�GRX �YDULDELOH� b) 6 �VH�HVWLPH]H�SDUDPHWULL�IXQF LHL�GH�UHJUHVLH�úL�V �VH�FDOFXOH]H�YDORULOH�DMXVWDWH� c) 6 �VH�HVWLPH]H�SHQWUX�α=5% (t ��VHPQLILFD LD�SDUDPHWULORU�FDOFXOD L� d) 3HQWUX�DFHHDúL�YDORDUH�D�OXL t�V �VH�FDUDFWHUL]H]H�úL�V �VH�WHVWH]H�LQWHQVLWDWHD�OHJ WXULL� e) ùWLLQG�F �SH�vQWUHJXO�GHSDUWDPHQW�V-D�SUHFRQL]DW�FUHúWHUHD�H[SRUWXOXL�QHW�vQ�71/T0�FX��V �

se extrapoleze datele, utili]kQG�PRGHOXO�DGRSWDW��YDORULOH�DG XJDWH�DOH�vQWUHJXOXL�GHSDUWDPHQW�

pentru T1+1, T1+2, T1+3.

21. Dintr-un sondaj aleator nerepetat de 10% s-DX�RE LQXW�XUP WRDUHOH�GDWH�

*UXSH�GH�DQJDMD L�GXS �

vechime (ani) 1XP U�DQJDMD L Salariul mediu net (u.m.)

sub 5 10 100 5 - 10 30 1130 10 - 15 40 135 15 - 20 50 145 20 - 25 40 160 25 - 30 35 165 peste 30 15 175

Se cere: a) 6 �VH�UHSUH]LQWH�JUDILF�GLVWULEX LD�DQJDMD LORU�GXS �YHFKLPH�úL�IRQGXO�GH�VDODULL� b) 'XS �FDUH�YDULDELO �HúDQWLRQXO�HVWH�PDL�RPRJHQ" c) Pentru o probabLOLWDWH� GH� �� ]� � �� V � VH� FDOFXOH]H� OLPLWHOH� vQWUH� FDUH� VH� YD� vQFDGUD�

VDODULXO�PHGLX��IRQGXO�GH�VDODULL�úL�SURFHQWXO�PD[LP�DO�SHUVRQDOXOXL�FDUH�DUH�R�YHFKLPH�PDL�

PDUH�VDX�HJDO �FX��DQL� d) 6 � VH� UHSUH]LQWH� JUDILF� WHQGLQ D� OHJ WXULL� GLQWUH� FHOH� GRX � YDULDELOH� úL� V � VH� FDUDFWHUL]H]H�

LQWHQVLWDWHD�OHJ WXULL�


Produse 9DORDUHD�SURGXF LHL�

vQ�SHULRDGD�GH�ED] �

(mii lei)

Dinamica volumului

fizic (%)

0RGLILFDUHD�SUH XULORU�vQ�SHULRDGD�

FXUHQW �ID �GH�SHULRDGD�GH�ED] ��

A 85000 100 +40 B 95000 105 +80 C 30000 80 +20

Se cere: a) ,QGLFLL�LQGLYLGXDOL�DL�YDORULL�úL�DL�SUH XULORU� b) ,QGLFLL�GH�JUXS�DL�YDORULL��YROXPXOXL�IL]LF�úL�SUH XULORU��SUHFXP�úL�UHOD LD�GLQWUH�HL� c) 0RGLILFDUHD�DEVROXW �D�YDORULL�FD�XUPDUH�D�PRGLILF ULL�SUH XULORU�úL�FD�XUPDUH�D�PRGLILF ULL�

volumului fizic; d) ùWLLQG�F �QXP UXO�SHUVRQDOXOXL�FDUH�D� OXFUDW� OD� WRDWH�FHOH� WUHL�SURGXVH�V-a redus în perioada

FXUHQW �ID �GH�SHULRDGD�GH�ED] �FX��V �VH�FDOFXOH]H�LQGLFHOH�SURGXFWLYLW LL�PHGLL� e) )RORVLQG�GDWHOH�SUHFHGHQWH�úL�FXQRVFkQG�F �vQ�SHULRDGD�GH�ED] �QXP UXO�WRWDO�DO�SHUVRQDOXOXL�

PXQFLWRU�D�IRVW�GH��SHUVRDQH��V �VH�GHVFRPSXQ �SH�IDFWRUL�GH�LQIOXHQ ��SURGXFWLYLWDWH�úL�

QXP U�GH�DQJDMD L��PRGLILFDUHD�DEVROXW �WRWDO �D�YROXPXOXL�IL]LF�DO�SURGXVHORU�

23. Dintr-XQ�HúDQWLRQ�GH��QHUHSHWDW��VH�FXQRVF�XUP WRDUHOH�GDWH�

*UXSH�GH�PXQFLWRUL�GXS �P ULPHD�VDODULXOXL�OXQDU�

(u.m.)

Total *UXSH�GH�PXQFLWRUL�GXS �

vechime (ani)

sub 68 68-72 72-76 ��úL�SHVWH� sub 15 ani 10 10 - - 20

��DQL�úL�SHVWH - 5 40 15 60 TOTAL 10 15 40 15 80

Se cere:

a) 6 �VH�UHSUH]LQWH�JUDILF�UHSDUWL LD�PXQFLWRULORU�GXS �P ULPHD�VDODULXOXL�OXQDU� b) 6 �VH�YHULILFH�GDF �YHFKLPHD�vQ�PXQF �HVWH�XQ�IDFWRU�GHWHUPLQDQW�SHQWUX�P ULPHD�VDODULXOXL� c) 6 �VH�HIHFWXH]H�DQDOL]D�GLVSHUVLRQDO �SHQWUX�XQ�SUDJ�α=5% (F=10,2); d) PentrX�R�SUREDELOLWDWH�GH��] ��V �VH�HVWLPH]H�OLPLWHOH�vQWUH�FDUH�VH�YD�vQFDGUD�VDODULXO�

PHGLX�úL�IRQGXO�GH�VDODULL� e) 6 �VH�FDUDFWHUL]H]H�LQWHQVLWDWHD�OHJ WXULL�GLQWUH�FHOH�GRX �YDULDELOH�

24. Despre consumul de gaz metan în mil. m3, în perioada T0 - T0+3 pe trimestre, se cunosc

XUP WRDUHOH�GDWH�

Trim. T0 T0+1 T0+2 T0+3 I 7397 7468 7693 8793 II 5512 5604 5663 5694 III 5406 5453 5453 5473 IV 7685 7795 7940 8040

Se cere:

a) 6 �VH�UHSUH]LQWH�JUDILF�HYROX LD�FRQVXPXOXL�GH�JD]�PHWDQ�vQ�SHULRDGD�70 - T0+3; b) 6 �VH�GHWHUPLQH�WHQGLQ D�GH�HYROX LH�úL�V �VH�GHWHUPLQH�VH]RQDOLWDWHD�VHULHL�XWLOL]kQG�PHWRGD�

PHGLHL�DULWPHWLFH�úL�PHWRGD�PHGLLORU�PRELOH� c) ùWLLQG�F �vQWUH�SURGXF LD�GH�JD]�PHWDQ�úL�FRQVXPXO�GH�JD]�PHWDQ�H[LVW �R�HYROX LH�OLQLDU ��F �

DEDWHUHD�PHGLH�S WUDWLF �D�SURGXF LHL�D�IRVW�GH��PLO��P3, iar coeficientul de regresie este de ��V �VH�FDUDFWHUL]H]H�LQWHQVLWDWHD�OHJ WXULL�GLQWUH�FHOH�GRX �YDULDELOH�

25. 3URGXFWLYLWDWHD�PXQFLL�úL�VWUXFWXUD�SHUVRQDOXOXL�PXQFLWRU�vQUHJLVWUDW�vQ�WUHL�vQWUHSULQGHUL�vQ�70�úL�

T0+2�VH�FDUDFWHUL]HD] �SULQ�XUP WRDUHOH�GDWH�

Întreprinderea Productivitatea muncii (u.m.) Structura personalului muncitor (%)

T0 T0+2 T0 T0+2

I 80 82 30 25 II 83 85 32 35 III 84 87 38 40

TOTAL - - 100 100 Se cere:

a) ,QGLFLL�LQGLYLGXDOL�DL�SURGXFWLYLW ii muncii; b) &UHúWHUHD�UHODWLY �úL�DEVROXW �SH�WRWDO�D�SURGXFWLYLW LL�PXQFLL�úL�LQIOXHQ HOH�IDFWRULORU�UHVSHFWLYL� c) ùWLLQG� F � SH� WRWDO� QXP UXO� SHUVRQDOXOXL� PXQFLWRU� D� IRVW� vQ� 70+2� GH� �� SHUVRDQH�� V � VH�

FDOFXOH]H�VSRUXO�DEVROXW�DO�SURGXF LHL�SH�VHDPD�PRGLILF ULL�VWUXFWXULL�SHUVRQDOXOXL�PXQFLWRU�úL�

SH�VHDPD�PRGLILF ULL�SURGXFWLYLW LL�PXQFLL�

26. ÌQ� XUPD� VLVWHPDWL] ULL� úL� SUHOXFU ULL� GDWHORU� SULYLQG� SURGXFWLYLWDWHD� RUDU � úL� VDODULXO� RUDU�

vQUHJLVWUDW�SHQWUX�XQ�HúDQWLRQ�GH��GH�PXQFLWRUL��GLQ�FROHFWLYLWDWHD�JHQHUDO ��D�UH]XOWDW�

XUP WRDUHD�VLWXD LH�

*UXSH�GH�PXQFLWRUL�GXS �

SURGXFWLYLWDWHD�RUDU ��PLL�OHL�

1XP U�GH�PXQFLWRUL

(persoane)

Salariul mediu

orar (mii lei)

Dispersia

sub 1,00 40 0,5 16 1,00-3,00 100 1,6 25 ��úL�SHVWH 60 2,8 49

TOTAL 200

Se cere: a) 6 �VH�UHSUH]LQWH�JUDILF�VWUXFWXUD�PXQFLWRULORU�vQ�IXQF LH�GH�SURGXFWLYLWDWHD�RUDU �D�PXQFLL� b) 6 �VH�YHULILFH�SH�ED]D�UHJXOLL�DGXQ ULL�GLVSHUVLLORU�GDF �:h este un factor determinant pentru

GHYLD LD�VDODULXOXL�RUDU� c) Pentru o probabilitatea de 0,9973 (z=�� V � VH� HVWLPH]H� OLPLWHOH� vQWUH� FDUH� VH� YRU� vQFDGUD�

VDODULXO�PHGLX�RUDU�úL�IRQGXO�GH�VDODULL�PHGLX�vQ�FROHFWLYLWDWHD�JHQHUDO �

27. 6H�FXQRVF�XUP WRDUHOH�GDWH�GHVSUH�WUHL�VRFLHW L�FRPHUFLDOH�

6RFLHW L�

comerciale Modificarea SUH XULORU�ID �

1XP UXO�PHGLX�

de muncitori

3URGXF LD�vQ�W1 (mil. lei în

pr.crt) de t0 (t0=100%)

Dinamica volumului IL]LF�DO�SURGXF LHL�

t1/t0 (%) t0 t1 A 500 +50 880 500 450 B 1200 +80 100 1500 1500 C 300 +100 90 600 580

Se cere: a) 6 �VH�FDOFXOH]H�LQGLFLL�LQGLYLGXDOL�úL�GH�JUXS�DL�YDORULL��YROXPXOXL�IL]LF�úL�SUH XULORU� b) 6 �VH�FDOFXOH]H�LQGLFLL�GH�JUXS�DL�SURGXFWLYLW LL�PXQFLL�úL�LQIOXHQ D�IDFWRULORU� c) &XQRVFkQG�F �vQ�SHULRDGD�W0-t1, pe 5 ani, ritmul mediu anual al volumului fizic pe total a fost

GH��V �VH�H[WUDSROH]H�YROXPXO�Iizic pentru perioada t1+1, t1+2, t1+3, t1+4.


0 UIXUL Structura valorii desf. (%)

T0 T1

0RGLILFDUHD�DEVROXW �D�YDORULL�GHVI��vQ�

T1/T0�FD�XUPDUH�D�PRGLILF ULL�SUH XULORU�

(mil. lei)

0RGLILFDUHD�SURFHQWXDO �

D�SUH XULORU�vQ�71/T0

A … 25 200 +60 B 20 … 180 +40 C 50 40 100 +50

Se cere: a) 6 �VH�UHSUH]LQWH�JUDILF�VWUXFWXUD�YDORULL�P UIXULORU� b) ùWLLQG� F � YkQ] ULOH� GLQ� PDUID� $� vQ� 70� DX� IRVW� GH� �� PLO�� OHL�� V � VH� FDOFXOH]H� LQGLFLL�

LQGLYLGXDOL�úL�GH�JUXS�DL�YDORULL��YROXPXOXL�IL]LF�úL�DL�SUH XULORU� c) 6 �VH�GHVFRPSXQ �SH�IDFWRUL�GH�LQIOXHQ �PRGLILFDUHD�DEVROXW �D�YDORULL�GHVIDFHULORU�WRWDOH� d) 6 � VH� FDOFXOH]H� SH� WRWDO� YDORDUHD� DEVROXW � D� XQXL� SURFHQW� GH� FUHúWHUH� ID � GH� 70� úL� ULWPXO�

PHGLX�DQXDO�GH�FUHúWHUH�DO�YROXPXOXL�IL]LF�

29. Se cunosc dintr-XQ�VRQGDM�GH��QHUHSHWDW��XUP WRDUHOH�GDWH�

*UXSH�GH�PXQFLWRUL�GXS �VDODULXO�PHGLX�QHW��PLL�OHL� TOTAL *UXSH�GH�PXQFLWRUL�GXS �

vechime (ani) 800-900 900-1000 1000-1100 1100-1200 1200-1300 1300-1400 sub 15 ani 15 20 5 - - - 40

��DQL�úL�SHVte - - 25 45 40 30 140 TOTAL 15 20 30 45 40 30 180

Se cere: a) 6 �VH�UHSUH]LQWH�JUDILF�GLVWULEX LD�PXQFLWRULORU�GXS �VDODULXO�PHGLX�QHW�SH�WRWDO�úL�IRQGXO�WRWDO�

de salarii; b) 6 �VH�FDUDFWHUL]H]H�RPRJHQLWDWHD�GLVWULEX LHL�WRWDOH� c) 8WLOL]kQG�GDWHOH�GH�VHOHF LH�úL�DSOLFkQG�UHJXOD�DGXQ ULL�GLVSHUVLLORU�V �VH�HVWLPH]H�P VXUD�vQ�

FDUH�YDULD LD�VDODULXOXL�PHGLX�QHW�HVWH�GHWHUPLQDW �GH�YHFKLPH� d) ùWLLQG�F ��PXQFLWRULORU�FDUH�úL-DX�vQGHSOLQLW�QRUPHOH�D�IRVW�GH��OD�FHL�VXE��DQL�úL�GH�

94% la cei peste 15 ani vechime V � VH� FDOFXOH]H� SURFHQWXO�PHGLX� úL�GLVSHUVLD� SHUVRQDOXOXL�FDUH�úL-au îndeplinit normele, pe total;

e) 3HQWUX� R� SUREDELOLWDWH� GH� �� ] �� V � VH� GHWHUPLQH� SHQWUX� FD]XO� VRQGDMXOXL� VWUDWLILFDW�VDODULXO�PHGLX� úL� IRQGXO� GH� VDODULL� SHQWUX� vQWUHDJD� FROHFWLYLWDWH��([SOLFD L� GLIHUHQ HOH� GLQWUH�

UH]XOWDWHOH�RE LQXWH�

30. 'HVSUH� UHFROWD� PHGLH� OD� KHFWDU� OD� R� DQXPLW � FXOWXU � FXOWLYDW � vQ� �� GH� MXGH H� VH� FXQRVF�

XUP WRDUHOH�GDWH�

*UXSH�GH�MXGH H�GXS �VXSUDID D�FXOWLYDW ��KD� Recolta la ha (q/ha) 1XP U�GH�MXGH H

sub 1000 10,8,13,5,13,12,11,10,13 9 1000-10000 12,10,13,17,17,10,3,12,11,12 10

10000-20000 17,7,7,8,10 5 peste 20000 12,17,10 3

Se cere:

a) Efectuându-VH� DQDOL]D� GLVSHUVLRQDO �� V � VH� YHULILFH� SHQWUX� XQ� QLYHO� GH� VPQLILFD LH� α=5% (Fα;4;23 ��GDF �SURGXF LD�FXOWXULL�DQDOL]DWH�VH�PRGLILF �vQ�IXQF LH�GH�VXSUDID D�vQV PkQ DW �

b) 6 � VH� FDUDFWHUL]H]H� LQWHQVLWDWHD� OHJ WXULL� GLQWUH� FHOH� GRX � YDULDELOH� XWLOL]kQG� XQ� LQGLFDWRU�

sintetic.

31. 'HVSUH�GHVIDFHULOH�D�SDWUX�SURGXVH�VH�FXQRVF�XUP WRDUHOH�GDWH�

Produse Valoarea desfacerilor

din T0�OD�SUH XULOH�GLQ�70

��PRGLILF ULL�

SUH XULORU�vQ�70+2�ID �

de T0

Sporul absolut al valorii desfacerilor

în T0+2�ID �GH�70 ca urmare a

PRGLILF ULL�SUH XULORU��PLL�X�P��

A 12000 +30 2600 B 7000 +50 8200 C 6500 -2 -100 D 0 0

Se cere V �VH�FDOFXOH]H�úL�V �VH�DQDOL]H]H�

a) ,QGLFLL�LQGLYLGXDOL�DL�YDORULL��YROXPXOXL�IL]LF�úL�SUH XULORU� b) ,QGLFLL�GH�JUXS�DL�YDORULL��YROXPXOXL�IL]LF�úL�SUH XULORU�SUHFXP�úL�UHOD LD�GLQWUH�HL� c) 0RGLILFDUHD�DEVROXW �D�YDORULL�SURGXVHORU��SH�WRWDO��FD�XUPDUH�D�LQIOXHQ HL�IDFWRULORU��SUH �úL�

volum fizic); d) ùWLLQG�F �SH�DQVDPEOX�GLQDPLFD�QXP UXOXL�WRWDO�GH�SHUVRQDO�PXQFLWRU�FDUH�D�FUHDW�FHOH�SDWUX�

produse a fost de 95% în T0+2�ID �GH�70��V �VH�GHWHUPLQH�VSRUXO�DEVROXW�DO�YROXPXOXL�IL]LF�VXE�

LQIOXHQ D�PRGLILF ULL�SURGXFWLYLW Li medii a muncii.

32. Dintr-XQ�VRQGDM�GH��vQWkPSO WRU�úL�QHUHSHWDW�VH�FXQRVF�XUP WRDUHOH�GDWH�

&ROHFWLYLWDWHD�JHQHUDO &ROHFWLYLWDWHD�GH�VHOHF LH Grupe de muncitori GXS �YHFKLPH��DQL�

1XP U�GH�

muncitori Timpul mediu efectiv

nelucrat (min) Abatere medie p WUDWLF ��PLQ�

��PXQFLWRUL�FDUH�QX�úL-au îndeplinit normele

Sub 10 800 60 9 20 10-20 1600 50 8 10 20-30 1000 40 7 10

��úL�SHVWH 600 30 6 9 TOTAL 4000 - - -

Se cere: a) 6 �VH�FDUDFWHUL]H]H�RPRJHQLWDWHD�vQWUHJXOXL�HúDQWLRQ� b) 8WLOL]kQG�GDWHOH�GH� VHOHF LH� V � VH�YHULILFH�FX�DMXWRUXO� UHJXOLL�GH�DGXQDUH�D�GLVSHUVLLORU�GDF �

YHFKLPHD�HVWH�XQ�IDFWRU�VHPQLILFDWLY�GH�LQIOXHQ �SHQWUX�WLPSXO�OXFUDW� c) 3HQWUX�R�SUREDELOLWDWH�GH��] ��V �VH�FDOFXOH]H�OLPLWHOH�vQWUH�FDUH�VH�YD�vQFDGUD�WLPSXO�

mediu nelucrat, fonduO�GH�WLPS�QHOXFUDW�SUHFXP�úL�SURFHQWXO�PD[LP�DO�PXQFLWRULORU�FDUH�QX�úL-au îndeplinit normele;

d) 6 �VH�UHSUH]LQWH�JUDILF�VWUXFWXUD�IRQGXOXL�GH�WLPS�QHOXFUDW�SH�JUXSH�GH�YHFKLPH�

33. 'HVSUH�UH]XOWDWHOH�HFRQRPLFH�DOH�XQHL�VRFLHW L�FRPHUFLDOH�VH�FXQRVF�XUP WRDUHle date:

0 UIXUL Valoarea desfacerilor de P UIXUL�vQ�70 (mil. lei)

&UHúWHUHD�SUH XULORU�vQ�

T0+3�ID �GH�70 0RGLILFDUHD�YDORULL�P UIXULORU�vQ�

T0+3�ID �GH�70 ca urmare a PRGLILF ULL�SUH XULORU�PLO��OHL

A 4600 +80 +216 B 6200 +60 +140 C 2600 +40 +150

Se cere: a) ,QGLFLL�LQGLYLGXDOL�úL�GH�JUXS�DL�SUH XULORU��YROXPXOXL�IL]LF�úL�DL�YDORULL�GHVIDFHULORU�GH�P UIXUL� b) 6 �VH�FDOFXOH]H�FRQWULEX LD�PRGLILF ULL�SUH XULORU�úL�YROXPXOXL�IL]LF�OD�PRGLILFDUHD�DEVROXW �SH�

WRWDO�D�YDORULL�P UIXULORU� c) ùWLLQG�F �LQGLFHOH�PHGLX�DQXDO�DO�QXP UXOXL�GH�PXQFLWRUL�vQ�SHULRDGD�70-T0+3�D�IRVW�GH��V �

VH�FDOFXOH]H�LQGLFHOH�PHGLX�úL�ULWPXO�PHGLX�DQXDO�DO�SURGXFWLYLW LL�PXQFLL� d) ùWLLQG�F �ULWPXO�PHGLX�DQXO�GH�FUHúWHUH�D�YROXPXOXL�IL]LF�WRWDO�vQ�SHULRDGD�70-T0+3�VH�PHQ LQH�

QHVFKLPEDW�úL în perioada T0+4-T0+6�V �VH�HVWLPH]H�YROXPXO�IL]LF�WRWDO�GLQ�70+4, T0+5,T0+6.

34. Printr-un sondaj, repetat, de 7%, efectuat într-R�XQLWDWH�LQGXVWULDO �V-DX�RE LQXW�XUP WRDUHOH�GDWH�

&ROHFWLYLWDWHD�JHQHUDO &ROHFWLYLWDWHD�GH�VHOHF LH Grupe de muncitori dup �YHFKLPH��DQL�

1XP U�GH�

muncitori Timpul mediu nelucrat

pe grupe (min) Dispersia pe grupe

��PXQFLWRUL�FDUH�QX�úL-au îndeplinit normele

Sub 10 800 60 81 20 10-15 1600 50 64 18 15-20 1000 40 49 10 20-25 400 30 36 10

peste 25 200 20 649 9 TOTAL 4000 … … …

Se cere: a) 6 �VH�UHSUH]LQWH�JUDILF�IRQGXO�WRWDO�GH�WLPS�QHOXFUDW�GH�PXQFLWRULL�FXSULQúL�vQ�HúDQWLRQ� b) 3H�ED]D�GDWHORU�GLQ�HúDQWLRQ�úL�XWLOL]kQG�UHJXOD�DGXQ ULL�GLVSHUVLLORU�� V � VH�VWDELOHDVF �GDF �

YHFKLPHD�HVWH�XQ�IDFWRU�VHPQLILFDWLY�GH�LQIOXHQ �SHQWUX�YDULD LD�WLPSXOXL�QHOXFUDW� c) 3HQWUX� R� SUREDELOLWDWH� GH� �� ] �� V � VH� FDOFXOH]H� OLPLWHOH� vQWUH� FDUH� VH� YRU� SODVD�

WLPSXO�PHGLX� úL� WLPSXO� WRWDO� QHOXFUDW�� SUHFXP� úL�SURFHQWXO�PD[LP�DO� SHUVRQDOXOXL� FDUH� úL-a îndeplinit normele pentru întreaga colectivitate;

d) ÌQ�FRQGL LLOH�XQHL�HURUL�OLPLW �GH�± ��PLQXWH��V �VH�FDOFXOH]H�YROXPXO�QRXOXL�HúDQWLRQ�

35. 'HVSUH�YDORDUHD�D�SDWUX�P UIXUL�úL�GLQDPLFD�SUH XULORU�DFHVWRUD��VH�FXQRVF�GDWHOH�XUP WRDUH�

Produse Valoarea produselor (mii lei) ModifiFDUHD�SUH XULORU�vQ�71/T0 T0 T1

A 200 380 +130 B 300 600 +280 C 250 200 - D 280 350 +80

Se cere: a) 6 �VH�FDOFXOH]H�úL�V �VH�DQDOL]H]H�LQGLFLL�LQGLYLGXDOL�DL�YDORULL��YROXPXOXL�IL]LF�úL�DL�SUH XULORU� b) ,QGLFLL�GH�JUXS�DL�YDORULL��YROXPXOXL�IL]LF�úL�SUH XULORU�SUHFXP�úL�UHOD LD�GLQWUH�HL� c) 0RGLILFDUHD�DEVROXW �D�YDORULL�SH�WRWDO�úL�GHVFRPSXQHUHD�VD�SH�IDFWRULL�GHWHUPLQDQ L� d) 3UHVXSXQkQG�F �ULWPXO�PHGLX��SH�XOWLPLL��DQL��DO�YROXPXOXL� IL]LF�U PkQH�QHVFKLPEDW�úL� vQ�

YLLWRU��V �VH�H[WUDSROH]H�GDWHOH�FRUHVSXQ] WRDUH�DQLORU�71+1, T1+2, T1+3.

36. 'HVSUH�SURGXF LD�]LOQLF �D��GH�XQLW L�GH�REVHUYDUH�VH�FXQRVF�XUP WRDUHOH�GDWH��vQ�X�P��

21, 33, 57, 55, 54, 58, 38, 47, 55, 59, 56, 81, 50, 52, 61, 21, 77, 63, 47, 59, 37, 38, 83, 59, 72, 66, 86, 45, 50, 48, 63, 46, 79, 67, 43, 52, 57, 37, 35, 71, 65, 65, 54, 75, 42, 56, 48, 56, 67, 64, 58, 33, 36, 43, 44, 53, 61, 75, 59, 24, 45, 28, 83, 58, 21, 81, 51, 39, 54, 53, 72, 71, 59, 37, 56, 65, 38, 40, 61.

Se cere: a) 6 �VH�JUXSH]H�GDWHOH�SH��LQWHUYDOH�HJDOH��GXS �SURGXF LD�]LOQLF �RE LQXW � b) 6 �VH�UHSUH]LQWH�JUDILF�UHSDUWL LD�RE LQXW �OD�SXQFWXO�D�� c) 6 � VH� FDOFXOH]H� SURGXF LD� PHGLH� ]LOQLF � SH� ED]D� VHULHL� GH� OD� SXQFWXO� D�� FkW� úL� SH� ED]D�

YDULDQWHORU�SUH]HQWDWH��([SOLFD L�HYHQWXDOHOH�GLIHUHQ H� d) 6 �VH�FDUDFWHUL]H]H�RPRJHQLWDWHD�úL�DVLPHWULD�GLVWULEX LHL�RE LQXWH� e) 3UHVXSXQkQG�F �GDWHOH�SURYLQ�GLQWU-XQ�HúDQWLRQ�GH��VHOHFWDW� vQWkPSO WRU�úL�QHUHSHWDW��V �VH�

FDOFXOH]H�SHQWUX�R�SUREDELOLWDWH�GH��] ��OLPLWHOH�vQWUH�FDUH�VH�YRU�vQFDGUD�SURGXF LD�PHGLH�

]LOQLF � úL� SURGXF LD� ]LOQLF � WRWDO � DOH� vQWUHJLL� FROHFWLYLW L�� SUHFXP� úL� SURFHQWXO� PD[LP� DO�

QXP UXOXL�GH�XQLW L�GLQ�FROHFWLYLWDWHD�JHQHUDO �FDUH�DX�R�SURGXF LH�PDL�PDUH�VDX�HJDO �FX��X�P.

37. 6H�FXQRVF�XUP WRDUHOH�GDWH�UHIHULWRDUH�OD�R�FROHFWLYLWDWH�GH��GH�PXQFLWRUL�GLQWU-o sHF LH�


vechime (ani) 1XP U�GH�PXQFLWRUL 3URGXF LD�PHGLH�

]LOQLF ��EXF�� Coeficientul de YDULD LH��

sub 10 90 2 23 10-20 150 43 15

��úL�SHVWH 60 45 20 TOTAL 300 ? ?

ùWLLQG�F �SHQWUX�JUXSD�VXE��DQL�SURGXF LD�]LOQLF �PD[LP �D fost de 85 buc., iar abaterea medie SR]LWLY �D�IRVW�GH��EXF��V �VH�DQDOL]H]H�RPRJHQLWDWHD�FROHFWLYLW LL�VWDWLVWLFH��GH��GH�SHUVRDQH��

GXS � FDUDFWHULVWLFD� XUP ULW �� 6SHFLILFD L� VHPQLILFD LD� IDFWRUXOXL� GH� JUXSDUH� úL� WHVWD L� DFHDVW �

VHPQLILFD LH�SHQWUX un prag de 5%.

38. 'HVSUH�GLVWULEX LD�PXQFLWRULORU�GH�R�DQXPLW �VSHFLDOLWDWH�GXS �YHFKLPH�úL�GXS �VDODULXO�PHGLX�

net se cunosc datele din T0:


vechime (ani) 1XP U�GH�PXQFLWRUL Salariul mediu net (u.m.)

sub 5 10 2100 5-10 30 2700 10-15 50 2800 15-20 60 3000 20-25 40 3100 25-30 10 3300

Se cere:

a) 6 �VH�GHWHUPLQH�SDUDPHWULL�IXQF LHL�FDUH�H[SULP �WHQGLQ D�OHJ WXULL�GLQWUH�FHOH�GRX �YDULDELOH�

în conformitate cu graficul adecvat; b) 3HQWUX� XQ� QLYHO� GH� VHPQLILFD LH� GH� α=5% (Fα=3,89), s � VH� YHULILFH� YDOLGLWDWHD� PRGHOXOXL�

DGRSWDW�úL�V �VH�FDUDFWHUL]H]H�LQWHQVLWDWHD�OHJ WXULL� c) ùWLLQG�F �SHVWH�WUHL�DQL�VDODULXO�PHGLX�D�FUHVFXW�FX��ID �GH�70�� ULWP�FH�VH�YD�S VWUD�úL� vQ�

perioada T0+4–T0+6��V �VH�H[WUDSROH]H�VDODULXO�PHGLX�SHQWUX�DFHDVW �SHULRDG �úL�V �VH�GHWHUPLQH�

VSRUXULOH�DQXDOH�vQUHJLVWUDWH�OD�IRQGXO�GH�VDODULL�FD�XUPDUH�D�PRGLILF ULL�VDODULXOXL�PHGLX�

39. 6H�FXQRVF�XUP WRDUHOH�GDWH�SHQWUX�HFRQRPLD�5RPkQLHL�SHQWUX�SHULRDGD��-1997:

Anii 3,%��POG��OHL�SUH XUL�

curente) ,QGLFLL�3,%�ID �Ge 1990

(%) 0 1 2

1990 857,9 100 1991 2203,9 87,1 1992 6029,2 79,4 1993 20035,7 80,6 1994 49773,2 83,8 1995 72135,5 89,8 1996 108919,6 93,3 1997 250480,2 86,9

Sursa: Anuarul Statistic al României, anul 1997, pag. 354-357; anul 1998, pag. 342-343 Se cere: a) 'HIODWRUXO�3,%�vQ�ILHFDUH�DQ�ID �GH��úL�ID �GH�DQXO�SUHFHGHQW� b) 5DWD�LQIOD LHL�vQ�ILHFDUH�DQ�ID �GH��úL�ID �GH�DQXO�SUHFHGHQW� c) 3,%�UHDO�ID �GH��úL�ID �GH�DQXO�SUHFHGHQW� d) 'LQDPLFD�3,%�UHDO�vQ�ILHFDUH�DQ�ID �GH�DQXO�SUHFHGHQW� e) DinaPLFD�3,%�UHDO�úL�UDWD�LQIOD LHL�vQ��ID �GH��

40. 6H�FXQRVF�XUP WRULL�LQGLFDWRUL�SHQWUX�HFRQRPLD�5RPkQLHL�SHQWUX�DQLL��úL��PLOLDUGH�OHL�

SUH XUL�FXUHQWH��

Indicator 1994 1997

0 1 2

��3URGXF LD�EUXW 100339,4 - 2. Consumul intermediar 54384,7 - ��9DORDUHD�DG XJDW �EUXW - 229445,3 ��)RUPDUHD�EUXW �GH�FDSLWDO�IL[ 10095,7 55074,2 5. Consumul final 38452,4 213772,8 ��&RQVXPXO�ILQDO�DO�SRSXOD LHL 31442,0 188416,2 7. Impozite pe produs (inclusiv TVA) 3848,2 19306,7 8. Taxe vamale 648,9 3808,2 ��6XEYHQ LL�SH�SURGXV 678,6 2080,0 9. Modificarea stocurilor 2252,6 -501,3 10. Export net -1027,5 -17865,5 Sursa: Anuarul Statistic al României, anul 1998, pag. 340-343

Se cere:

a) 3,%�QRPLQDO�IRORVLQG�PHWRGHOH�FH�SRW�IL�DSOLFDWH�vQ�IXQF LH�GH�datele disponibile; b) 3,%� UHDO� GDF � SUH XULOH� SURGXVHORU� úL� VHUYLFLLORU� ILQDOH� DX� FUHVFXW� vQ� �� ID � GH� �� FX�

385,3%; c) 'LQDPLFD�3,%�UHDO�ID �GH��

41. 6H�FXQRVF�GDWHOH��PLOLDUGH�OHL�SUH XUL�FXUHQWH��

Indicator 1994 1997

0 1 2

��5HPXQHUDUHD�VDODULD Llor 22250 82000

2. Amortirarea capitalului fix 4870 20400

��)RUPDUHD�EUXW �GH�FDSLWDO�IL[ 12200 53000

4. Excedentul brut de exploatare 19750 122500

5. Impozite indirecte 10300 55500

��6XEYHQ LL�GH�H[SORDWDUH 2600 10000

��5DWD�LQIOD LHL�� 385

Se cere:

a) 3,%�QRPLQDO�vQ�FHOH�GRX �SHULRDGH� b) PIB real ; c) 'LQDPLFD�3,%�UHDO�vQ��ID �GH��


SUH XUL�FXUHQWH�� Indicator 1994 1996

0 1 2 1. Consumul final 38452,4 89939,4 ��&RQVXPXO�ILQDO�DO�SRSXOD LHL 31442,0 75288,8 ��&RQVXPXO�ILQDO�DO�DGPLQLVWUD LHL�SXEOLFH 6851,8 14273,9 ��&RQVXPXO�ILQDO�DO�DGPLQLVWUD LHL�SULYDWH 158,6 376,7 ��)RUPDUHD�EUXW �GH�FDSLWDO�IL[ 10095,7 24998,5 6. Modificarea stocurilor 2252,6 3161,4 7. Export net -1027,5 -9179,7 6. Export 12394,2 30651,1 7. Import 13421,7 39830,8 ��0RGLILFDUHD�SUH XULORU�ID �GH��OD�

• Consumul final • )RUPDUHD�EUXW �GH�FDSLWDO�IL[ • Modificarea stocurilor • Export • Import

+97,36 +119,12 -16,6

+88,13 +84,41

Sursa: Anuarul Statistic al României, anul 1998, pag. 343, 347, 536

Se cere: a) 3,%�QRPLQDO�vQ�FHOH�GRX �SHULRDGH� b) 3,%�UHDO�vQ��ID �GH�� c) Dinamica PIB real; d) 'HIODWRUXO�3,%�úL�UDWD�LQIOD LHL�vQ��ID �GH��; e) 6WUXFWXUD�3,%��GXS �PHWRGD�FKHOWXLHOLORU��vQ�FHOH�GRX �SHULRDGH�úL�DQDOL]D�PXWD LLORU�VWUXFWXUDOH�


SUH XUL�FXUHQWH�� Indicator 1994 1996

0 1 2 ��3URGXF LD�EUXW 100339,4 240598,8 2. Consumul intermediar 54384,7 138744,6 3. Impozite pe produs (inclusiv TVA) 3848,2 7458,7 4. Taxe vamale 648,9 1852,3 ��6XEYHQ LL�SH�SURGXV 678,6 2245,6 ��0RGLILFDUHD�SUH XULORU�ID �GH��OD�

• 3URGXF LD�EUXW • Consumul intermediar • Impozite pe produs • Taxe vamale • 6XEYHQ LL�SH�SURGXV

+132,84 +141,59 +73,38

+205,52 +171,49

Sursa: Anuarul Statistic al României, anul 1998, pag. 354, 360

Se cere: a) 3,%�QRPLQDO�vQ�FHOH�GRX �SHULRDGH� b) 3,%�UHDO�vQ��ID �GH��; c) 'LQDPLFD�3,%�UHDO�vQ��ID �GH��

44. 6H�FXQRVF�GDWHOH��PLOLDUGH�OHL�SUH XUL�FXUHQWH��

Indicator 3HULRDGD�GH�ED] 3HULRDGD�FXUHQW 0 1 2

��9HQLWXO�QD LRQDO 3800 18200 2. Amortizarea capitalului fix 5800 20400 3. Impozite indirecte nete 5000 12000 4. Excedentul net de exploatare 22500 82000 ��6ROGXO�YHQLWXULORU�IDFWRULORU�vQ�UDSRUW�FX�VWU LQ WDWHD -900 -35600 6. Exportul net -1000 -17000 ��6ROGXO�WUDQVIHUXULORU�FXUHQWH�vQ�UDSRUW�FX�VWU LQ WDWHD +1400 +2000

Se cere: a) PIB nominal în cHOH�GRX �SHULRDGH� b) 3,1�OD�SUH XULOH�SLH HL� c) 5HPXQHUDUHD�VDODULD LORU� d) 9HQLWXO�QD LRQDO�GLVSRQLELO�

45. 6H� FXQRVF� XUP WRDUHOH� GDWH� SULYLQG� SRSXOD LD� RFXSDW � úL� YDORDUHD� DG XJDW � EUXW �� SH� UDPXUL��

pentru economia României: Ramura VAB (miliarde lei

SUH XUL�FXUHnte) 3RSXOD LD�RFXSDW

(mii persoane) 0RGLILFDUHD�SUH XULORU�vQ�

1996/1994 1994 1996 1994 1996 (%) 0 1 2 3 4 5

Industrie + FRQVWUXF LL

21269,6 43248,9 3520 3200 +81,48

$JULFXOWXU ��

VLOYLFXOWXU 9897,6 20949,2 3630 3290 +110,82

Alte ramuri 14787,5 37656,1 2880 2940 +118,5 Total 45954,7 101854,2 10030 9430

Sursa: Anuarul Statistic al României, anul 1998, pag. 118, 340-342

Se cere: a) 3URGXFWLYLWDWHD�PXQFLL�SH�UDPXUL�úL�SH�vQWUHDJD�HFRQRPLH� b) 6 � VH� GHWHUPLQH� LQIOXHQ D� IDFWRULORU� DVXSUD� PRGLILF ULL� SURGXFWLYLW LL� PXQFLL� SH� vQWUHDJD�

economie; c) 6 �VH�GHWHUPLQH�FRQWULEX LD�ILHF UXL�IDFWRU� OD�PRGLILFDUHD�DEVROXW �D�SURGXFWLYLW LL�PHGLL�D�

muncii pe întreaga economie.

46. 6H�FXQRVF�XUP WRDUHOH�GDWH�SULYLQG�FRQVXPXO�GH�HQHUJLH�HOHFWULF �úL�YDORDUHD�DG XJDW �EUXW , pe

UDPXUL��SHQWUX��úL�� Ramura VAB*��PLOLDUGH�OHL�SUH XUL�

constante) (QHUJLH�HOHFWULF �FRQVXPDW �

(mil. kWh) 1994 1996 1994 1996

0 1 2 3 4 ,QGXVWULH��FRQVWUXF LL 21269,6 23831,2 34583 39759 $JULFXOWXU ��VLOYLFXOWXU 9897,6 9937,0 1834 1332 Alte ramuri 14787,5 17233,9 20509 22501 Total 45954,7 51002,1 56926 63592

Sursa: Anuarul Statistic al României, anul 1997, pag. 543; anul 1998, pag. 118, 340-342

Se cere: a) &RQVXPXO�GH�HQHUJLH�HOHFWULF �FRQVXPDW �FH�UHYLQH�OD��OHL�9$%* la nivelul fiec UHL�UDPXUL�

úL�SH�DQVDPEOXO�HFRQRPLHL� b) 6 �VH�GHWHUPLQH�LQIOXHQ D�IDFWRULORU�DVXSUD�PRGLILF ULL�FRQVXPXOXL�PHGLX�GH�HQHUJLH�HOHFWULF �

pe întreaga economie; c) &RPSDUD L� GLQDPLFD� FRQVXPXOXL� PHGLX� GH� HQHUJLH� HOHFWULF � SH� DQVDPEOXO� HFRQRPLHL� FX�

GLQDPLFD�3,%�úWLLQG�F �3,%�QRPLQDO�D�IRVW�GH��POG��OHL�vQ��GH��POG��OHL�

vQ��LDU�UDWD�LQIOD LHL�vQ��ID �GH��D�IRVW�GH�� d) 'HWHUPLQD L� FRQWULEX LD� ILHF UXL� IDFWRU� OD� PRGLILFDUHD� DEVROXW � D� FRQVXPXOXL� � PHGLX� GH�

HQHUJLH�HOHFWULF �SH�vQWUHDJa economie.


Ramura 3URGXFWLYLWDWHD�PXQFLL��PLL�OHL�SHUVRDQ � 6WUXFWXUD�SRSXOD LHL�RFXSDWH�� 1994 1996 1994 1996

0 1 2 3 4 Industrie + FRQVWUXF LL

6042,5 7447,25 35,09 33,93

$JULFXOWXU ��

VLOYLFXOWXU 2726,6 3020,36 36,19 34,89

Alte ramuri 5134,5 5861,87 28,72 31,18 Total 100,0 100,0

Se cere:

a) 'LQDPLFD�SURGXFWLYLW LL�PXQFLL�SH�ILHFDUH�UDPXU � b) 3URGXFWLYLWDWHD�PHGLH�D�PXQFLL�OD�QLYHOXO�vQWUHJLL�HFRQRPLL�vQ�FHOH�GRX �SHULRDGH� c) Indicii de grup ai prodXFWLYLW LL�PHGLL�D�PXQFLL�FX�VWUXFWXU �YDULDELO ��FX�VWUXFWXU �IL[ �úL�DO�

YDULD LHL�VWUXFWXULL� d) 'HWHUPLQD L� FRQWULEX LD� ILHF UXL� IDFWRU� OD� PRGLILFDUHD� DEVROXW � D� SURGXFWLYLW LL� � PHGLL� D�

muncii pe întreaga economie.

48. Se cunosc datele:

Indicator 1996 1997 0 1 2

1. Salariul mediu net (lei) 321169 632086 2. Structura cheltuielilor de consum (%):

• 0 UIXUL�DOLPHQWDUH • 0 UIXUL�QHDOLPHQWDUH • Servicii

56,4 28,5 15,1

58,1 25,5 16,4

��0RGLILFDUHD�SUH XULORU�ID �GH��OD� • 0 UIXUL�DOLPHQWDUH • 0 UIXUL�QHDOLmentare • Servicii

+151,36 +152,52 +176,47

Sursa: Anuarul Statistic al României, anul 1998, pag. 168, 184

Se cere: a) ,QGLFHOH�SUH XULORU�GH�FRQVXP� b) Salariul real; c) Dinamica salariului real; d) 5DWD�LQIOD LHL�vQ�GRPHQLXO�FRQVXPXOXL�

49. 6H�FXQRVF�XUP WRDUHOe date:

Ramura (ILFLHQ D�IRORVLULL�PLMORDFHORU�IL[H

(lei VAB*/1000 lei F) Structura mijloacelor fixe (%)

1994 1996 1994 1996 0 1 2 3 4

Industrie + FRQVWUXF LL

1088 980 60,5 59,0

$JULFXOWXU ��

VLOYLFXOWXU 3533 2474 9,0 12,4

Alte ramuri 1563 1455 30,5 28,6 Total 100,0 100,0

Se cere:

a) 'LQDPLFD�HILFLHQ HL��IRORVLULL�PLMORDFHORU�IL[H�SH�ILHFDUH�UDPXU � b) (ILFLHQ D�PHGLH�D�IRORVLULL�PLMORDFHORU�IL[H�OD�QLYHOXO�vQWUHJLL�HFRQRPLL�vQ�FHOH�GRX �SHULRDGH� c) ,QGLFLL� GH� JUXS� DL� HILFLHQ HL� PHGLL� D� IRORVLULL� PLMORDFHORU� IL[H� FX� VWUXFWXU � YDULDELO �� FX�

VWUXFWXU �IL[ �úL�DO�YDULD LHL�VWUXFWXULL� d) 'HWHUPLQD L�FRQWULEX LD� ILHF UXL� IDFWRU� OD�PRGLILFDUHD�DEVROXW �D� HILFLHQ HL�PHGLL� D� IRORVLULL�

mijloacelor fixe pe întreaga economie.

Anexa 1

)XQF LD��*DXVV-Laplace

( ) ∫−

=Φz u

duez0

2

2

2

1

π

z 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,0 0,0000 0,0040 0,0080 0,0120 0,0160 0,0199 0,0239 0,0279 0,0319 0,0359 0,1 0,0398 0,0438 0,0478 0,0517 0,0557 0,0596 0,0636 0,0675 0,0714 0,0753 0,2 0,0793 0,0832 0,0871 0,0910 0,0948 0,0987 0,1026 0,1064 0,1103 0,1141 0,3 0,1179 0,1217 0,1255 0,1293 0,1331 0,1368 0,1406 0,1443 0,1480 0,1517 0,4 0,1554 0,1591 0,1628 0,1664 0,1700 0,1736 0,1772 0,1808 0,1844 0,1879

0,5 0,1915 0,1950 0,1985 0,2019 0,2054 0,2088 0,2123 0,2157 0,2190 0,2224 0,6 0,2257 0,2291 0,2324 0,2357 0,2389 0,2422 0,2454 0,2486 0,2517 0,2549 0,7 0,2580 0,2611 0,2642 0,2673 0,2704 0,2734 0,2764 0,2794 0,2823 0,2852 0,8 0,2881 0,2910 0,2939 0,2967 0,2995 0,3023 0,3051 0,3078 0,3106 0,3133 0,9 0,3159 0,3186 0,3212 0,3238 0,3264 0,3289 0,3315 0,3340 0,3365 0,3389

1,0 0,3413 0,3438 0,3461 0,3485 0,3508 0,3531 0,3554 0,3577 0,3599 0,3621 1,1 0,3643 0,3665 0,3686 0,3708 0,3729 0,3749 0,3770 0,3790 0,3810 0,3830 1,2 0,3849 0,3869 0,3888 0,3907 0,3925 0,3944 0,3962 0,3980 0,3997 0,4015 1,3 0,4032 0,4049 0,4066 0,4082 0,4099 0,4115 0,4131 0,4147 0,4162 0,4177 1,4 0,4192 0,4207 0,4222 0,4236 0,4251 0,4265 0,4279 0,4292 0,4306 0,4319

1,5 0,4332 0,4345 0,4357 0,4370 0,4382 0,4394 0,4406 0,4418 0,4429 0,4441 1,6 0,4452 0,4463 0,4474 0,4484 0,4495 0,4505 0,4515 0,4525 0,4535 0,4545 1,7 0,4554 0,4564 0,4573 0,4582 0,4591 0,4599 0,4608 0,4616 0,4625 0,4633 1,8 0,4641 0,4649 0,4656 0,4664 0,4671 0,4678 0,4686 0,4693 0,4699 0,4706 1,9 0,4713 0,4719 0,4726 0,4732 0,4738 0,4744 0,4750 0,4756 0,4761 0,4767

2,0 0,4772 0,4778 0,4783 0,4788 0,4793 0,4798 0,4803 0,4808 0,4812 0,4817 2,1 0,4821 0,4826 0,4830 0,4834 0,4838 0,4842 0,4846 0,4850 0,4854 0,4857 2,2 0,4861 0,4864 0,4868 0,4871 0,4875 0,4878 0,4881 0,4884 0,4887 0,4890 2,3 0,4893 0,4896 0,4898 0,4901 0,4904 0,4906 0,4909 0,4911 0,4913 0,4916 2,4 0,4918 0,4920 0,4922 0,4925 0,4927 0,4929 0,4931 0,4932 0,4934 0,4936

2,5 0,4938 0,4940 0,4941 0,4943 0,4945 0,4946 0,4948 0,4949 0,4951 0,4952 2,6 0,4953 0,4955 0,4956 0,4957 0,4959 0,4960 0,4961 0,4962 0,4963 0,4964 2,7 0,4965 0,4966 0,4967 0,4968 0,4969 0,4970 0,4971 0,4972 0,4973 0,4974 2,8 0,4974 0,4975 0,4976 0,4977 0,4977 0,4978 0,4979 0,4979 0,4980 0,4981 2,9 0,4981 0,4982 0,4982 0,4983 0,4984 0,4984 0,4985 0,4985 0,4986 0,4986

3,0 0,4987 0,4987 0,4987 0,4988 0,4988 0,4989 0,4989 0,4989 0,4990 0,4990

Anexa 2

Tabel cu valorile reparti LHL��6WXGHQW�vQ�IXQF LH�GH�SUREDELOLWDWHD� ( )P t t≤ α úL�QXP UXO�³I´�DO�

gradelor de libertate 1LYHO�GH�VHPQLILFD LH�SHQWUX�WHVWXO�ELODWHUDO

f

0,50

0,20 0,10 0,05 0,02 0,01 0,002 0,001 0,0001

1 1,000 3,078 6,314 12,706 31,821 63,657 318,309 636,618 6366,198 2 0,816 1,886 2,920 4,303 6,965 9,925 22,327 31,598 99,992 3 0,765 1,638 2,353 3,482 4,541 5,841 10,214 12,924 28,000 4 0,741 1,533 2,132 2,776 3,747 4,604 7,173 8,610 15,544 5 0,727 1,476 2,015 2,571 3,365 4,032 5,893 6,869 11,178 6 0,718 1,440 1,943 2,447 3,143 3,707 5,208 5,959 9,082 7 0,711 1,415 1,895 2,365 2,998 3,499 4,785 5,408 7,885 8 0,706 1,397 1,860 2,306 2,896 3.355 4,501 5,041 7,120 9 0,703 1,383 1,833 2,262 2,821 3,250 4,297 4,781 6,594

10 0,700 1,372 1,812 2,228 2,764 3,169 4,144 4,587 6,211 11 0,697 1,363 1,796 2,201 2,718 3,106 4,025 4,437 5,921 12 0,695 1,356 1,782 2,179 2,681 3,055 3,930 4,318 5,694 13 0,694 1,350 1,771 2,160 2,650 3,102 3,852 4,221 5,513 14 0,692 1,345 1,761 2,145 2,624 2,977 3,787 4,140 5,363 15 0,691 1,341 1,753 2,131 2,602 2,947 3,733 4,073 5,239 16 0,690 1,337 1,746 2,120 2,583 2,921 3,686 4,015 5,134 17 0,689 1,333 1,740 2,110 2,567 2,898 3,646 3,965 5,014 18 0,688 1,330 1,734 2,101 2,552 2,878 3,610 3,922 ‘4,966 19 0,688 1,328 1,729 2,093 2,539 2,861 3,579 3,883 4,897 20 0,687 1,325 1,725 2,086 2,528 2,845 3,552 3,850 4,837 21 0,686 1,323 1,721 2,080 2,518 2,831 3,527 3,819 4,784 22 0,686 1,321 1,717 2,074 2,508 2,819 3,505 3,792 4,736 23 0,685 1,319 1,714 2,069 2,500 2,807 3,485 3,767 4,693 24 0,685 1,318 1,711 2,064 2,492 2,797 3,467 3,745 4,654 25 0,684 1,316 1,708 2,060 2,485 2,787 3,450 3,725 4,619 26 0,684 1,315 1,706 2,056 2,479 2,779 3,435 3,707 4,587 27 0,684 1,314 1,703 2,052 2,473 2,771 3,421 3,690 4,558 28 0,683 1,313 1,701 2,048 2,467 2,763 3,408 3,674 4,530 29 0,683 1,312 1,699 2,045 2,462 2,756 3,396 3,659 4,506 30 0,683 1,310 1,697 2,042 2,457 2,750 3,385 3,646 4,482 35 0,682 1,306 1,690 2,030 2,438 2,724 3,340 3,591 4,389 40 0,681 1,303 1,684 2,021 2,423 2,704 3,307 3,551 4,321 45 0,680 1,301 1,679 2,014 2,412 2,690 3,281 3,520 4,269 50 0,679 1,299 1,676 2,009 2,403 2,678 3,261 3,496 4,288 60 0,679 1,296 1,671 2,000 2,390 2,660 3,232 3,460 4,169 70 0,678 1,294 1,667 1,994 2,381 2,648 3,211 3,435 4,127 80 0,678 1,292 1,664 1,990 2,374 2,639 3,195 3,416 4,096 90 0,677 1,291 1,662 1,987 2,368 2,632 3,183 3,402 4,072

100 0,677 1,290 1,660 1,984 2,364 2,626 3,174 3,390 4,053 120 0,677 1,289 1,658 1,980 2,358 2,617 3,160 3,373 4,025 200 0,676 1,286 1,653 1,972 2,345 2,601 3,131 3,340 3,970 500 0,675 1,283 1,648 1,965 2,334 2,586 3,107 3,310 3,922

1000 0,675 1,282 1,646 1,962 2,330 2,581 3,098 3,300 3,906 � 0,675 1,282 1,645 1,960 2,326 2,576 3,090 3,290 3,891

f

0,25 0,10 0,05 0,025 0,01 0,005 0,001 0,0005 0,00005

1LYHO�GH�VHPQLILFD LH�SHQWUX�WHVWXO�XQLODWHUDO

Anexa 3 5HSDUWL LD�)��)LVKHU�6QHGHFRU�

9DORULOH�IXQF LHL�)��22

21 ssF = ) pentru l1, l2�JUDGH�GH�OLEHUWDWH�úL�D = 0,05, D = 0,01 nivel de

VHPQLILFD LH

l1 = 1 l1 = 2 l1 = 3 l1 = 4 l1 = 5 l2 α =

0,05 α = 0,01 α =

0,05 α = 0,01 α =

0,05 α = 0,01 α =

0,05 α = 0,01 α =

0,05 α = 0,01

1 161,40 4052,00 199,50 4999,00 215,70 5403,00 224,60 5625,00 230,20 5764,00 2 18,51 98,49 19,00 99,00 19,16 99,17 19,25 99,25 19,30 99,30 3 10,13 34,12 9,55 30,81 9,28 29,46 9,12 28,71 9,01 28,24 4 7,71 21,20 6,94 18,00 6,59 16,69 6,39 15,98 6,26 15,52 5 6,61 16,26 5,79 13,27 5,41 12,06 5,19 11,39 5,05 10,97 6 5,99 13,74 5,14 10,91 4,76 9,78 4,53 9,15 4,39 8,75 7 5,59 12,25 4,74 9,55 4,35 8,45 4,12 7,85 3,97 7,45 8 5,32 11,26 4,46 8,65 4,07 7,59 3,84 7,01 3,69 6,63 9 5,12 10,56 4,26 8,02 3,86 6,99 3,63 6,42 3,48 6,06

10 4,96 10,04 4,10 7,56 3,71 6,55 3,48 5,99 3,33 5,64 11 4,84 9,65 3,98 7,20 3,59 6,22 3,36 5,67 3,20 5,32 12 4,75 9,33 3,88 6,93 3,49 5,95 3,26 5,41 3,11 5,06 13 4,67 9,07 3,80 6,70 3,41 5,74 3,18 5,20 3,02 4,86 14 4,60 8,86 3,74 6,51 3,34 5,56 3,11 5,03 2,96 4,69 15 4,54 8,68 3,68 6,36 3,29 5,42 3,06 4,89 2,90 4,56 16 4,49 8,53 3,63 6,23 3,24 5,29 3,01 4,77 2,85 4,44 17 4,45 8,40 3,59 6,11 3,20 5,18 2,96 4,67 2,81 4,34 18 4,41 8,28 3,55 6,01 3,16 5,09 2,93 4,58 2,77 4,25 19 4,38 8,18 3,52 5,93 3,13 5,01 2,90 4,50 2,74 4,17 20 4,35 8,10 3,49 5,85 3,10 4,94 2,87 4,43 2,71 4,10 21 4,32 8,02 3,47 5,78 3,07 4,87 2,84 4,37 2,68 4,04 22 4,30 7,94 3,44 5,72 3,05 4,82 2,82 4,31 2,66 3,99 23 4,28 7,88 3,42 5,66 3,03 4,76 2,80 4,26 2,64 3,94 24 4,26 7,82 3,40 5,61 3,01 4,72 2,78 4,22 2,62 3,90 25 4,24 7,77 3,38 5,57 2,99 4,68 2,76 4,18 2,60 3,86 26 4,22 7,72 3,37 5,53 2,98 4,64 2,74 4,14 2,59 3,82 27 4,21 7,68 3,35 5,49 2,96 4,60 2,73 4,11 2,57 3,78 28 4,20 7,64 3,34 5,45 2,95 4,57 2,71 4,07 2,56 3,75 29 4,18 7,60 3,33 5,42 2,93 4,54 2,70 4,04 2,54 3,73 30 4,17 7,56 3,32 5,39 2,92 4,51 2,69 4,02 2,53 3,70 40 4,08 7,31 3,23 5,18 2,84 4,31 2,61 3,83 2,45 3,51 60 4,00 7,08 3,15 4,98 2,76 4,13 2,52 3,65 2,37 3,34

120 3,92 6,85 3,07 4,79 2,68 3,95 2,45 3,48 2,29 3,17 ∞ 3,84 6,64 2,99 4,60 2,60 3,78 2,37 3,32 2,21 3,02

Anexa 4

Valorile variabilei χα2

�vQ�IXQF LH�GH�SUREDELOLWDWHD�α χ χα= >P( )2 2

l

0,990

0,975 0,950 0,900 0,100 0,050 0,025 0,010 0,001

1 0,0002 0,0010 0,0039 0,0158 2,71 3,84 5,02 6,63 10,83 2 0,02 0,05 0,10 0,21 4,61 5,99 7,38 9,21 13,82 3 0,12 0,22 0,35 0,58 6,25 7,81 9,35 11,34 16,27 4 0,30 0,48 0,71 1,06 7,78 9,49 11,14 13,28 18,47 5 0,55 0,83 1,15 1,61 9,24 11,07 12,83 15,09 20,52 6 0,87 1,24 1,64 2,20 190,64 12,59 14,45 16,81 22,46 7 1,24 1,69 2,17 2,83 12,02 14,07 16,01 18,47 24,32 8 1,65 2,18 2,73 3,49 13,36 15,51 17,53 20,09 26,13 9 2,09 2,70 3,33 4,17 14,68 16,92 19,02 21,67 27,88

10

2,56 3,25 3,94 4,87 15,99 18,31 20,48 23,21 29,59

11 3,05 3,82 4,57 5,58 17,27 19,67 21,92 24,72 31,26 12 3,57 4,40 5,23 6,30 18,55 21,03 23,34 26,22 32,91 13 4,11 5,01 5,89 7,04 19,81 22,36 24,74 27,69 34,53 14 4,66 5,63 6,57 7,79 21,06 23,68 26,12 29,14 36,12 15 5,23 6,26 7,26 8,55 22,31 25,00 27,49 30,58 37,70 16 5,81 6,91 7,96 9,31 23,54 26,30 28,84 32,00 39,25 17 6,41 7,56 8,67 10,08 24,77 27,59 30,19 33,41 40,79 18 7,01 8,23 9,39 10,86 25,99 28,87 31,53 34,80 42,31 19 7,63 8,91 10,12 11,65 27,20 30,14 32,85 36,19 43,82 20

8,26 9,59 10,85 12,44 28,41 31,41 34,17 37,57 45,32

21 8,90 10,28 11,59 13,24 29,61 32,67 35,48 38,93 46,80 22 9,54 10,98 12,34 14,04 30,81 33,92 36,78 40,29 48,27 23 10,20 11,69 13,09 14,85 32,01 35,17 38,08 41,64 49,73 24 10,86 12,40 13,85 15,66 33,20 36,41 39,37 42,98 51,18 25 11,52 13,12 14,61 16,47 34,38 37,65 40,65 44,31 52,62 26 12,20 13,84 15,38 17,29 35,56 38,88 41,92 45,64 54,05 27 12,88 14,57 16,15 18,11 36,74 40,11 43,19 46,96 55,48 28 13,57 15,31 16,93 18,94 37,92 41,34 44,46 48,28 56,89 29 14,26 16,05 17,71 19,77 39,09 42,56 45,72 49,59 58,30 30

14,95 16,79 18,49 20,60 40,26 43,77 46,98 50,89 59,70

Anexa 5 Tabel cu numere aleatoare

Coloana Rândul

1 2 3 4 5 6 7 8 9 10 1 87 35 67 44 51 49 18 98 97 84 75 22 53 29 10 52 26 87 54 92 2 25 52 29 67 35 99 48 88 40 08 63 68 82 39 38 47 91 39 11 00 3 87 17 83 31 25 59 87 48 25 80 24 08 81 45 21 32 90 08 44 31 4 05 04 40 35 72 95 48 56 77 57 63 19 80 16 48 52 06 47 64 98 5 81 16 09 21 91 71 29 76 54 01 53 17 30 67 62 95 56 58 10 91 6 54 85 78 88 57 91 11 69 10 22 71 87 24 92 52 64 42 82 78 95 7 44 78 19 18 35 40 27 66 89 72 21 17 71 69 95 17 97 17 62 60 8 97 20 98 97 37 33 93 75 18 88 35 85 46 05 07 20 08 17 66 24 9 98 77 57 51 40 41 76 21 18 54 60 61 79 13 94 57 50 73 89 68

10 78 12 77 30 83 30 59 28 73 33 47 07 60 07 45 38 82 10 73 19 11 41 19 70 62 43 46 06 13 22 38 31 18 61 60 07 14 49 16 28 16 12 70 64 30 55 67 46 95 79 63 66 82 56 67 10 76 77 03 22 42 18 13 06 56 09 89 68 87 79 19 35 94 66 18 17 91 72 81 72 77 92 39 14 29 46 18 28 08 88 48 56 49 44 67 82 72 67 28 83 10 26 58 13 15 42 14 55 51 72 95 29 25 15 18 25 68 48 92 87 16 78 43 17 47 16 33 75 87 15 15 23 13 79 62 73 76 69 09 77 82 65 72 47 59 56 17 09 80 99 61 98 08 34 11 88 79 08 32 46 78 35 58 44 16 12 23 18 98 31 57 50 85 80 53 39 05 92 54 42 29 01 35 23 09 81 96 64 19 51 70 52 55 83 12 95 02 79 11 49 79 87 95 98 48 88 08 64 77 20 27 83 61 07 49 05 46 20 35 78 31 34 42 50 68 11 42 14 29 77 21 78 84 69 15 64 42 92 39 36 08 56 39 35 02 92 78 46 63 82 98 22 22 12 89 66 49 09 99 10 62 53 19 31 81 83 50 43 37 42 10 00 23 69 41 59 54 82 72 44 66 64 03 76 59 12 12 41 56 34 90 26 06 24 54 99 46 54 51 38 59 07 64 21 81 17 88 47 23 05 63 43 08 67 25 99 91 82 79 92 62 44 24 01 34 45 16 33 56 17 78 42 86 70 94 26 54 96 58 70 68 38 48 37 08 25 51 28 42 09 45 96 78 84 23 59 27 88 37 31 01 58 97 65 10 96 97 32 75 65 73 06 01 60 19 68 10 28 49 40 84 90 19 91 75 41 55 63 98 25 01 69 29 57 72 77 75 95 29 21 18 87 99 98 48 38 12 64 18 78 50 43 97 04 07 85 18 98 87 30 06 31 99 51 27 13 43 21 07 55 90 35 29 84 08 70 66 03 18 96 31 58 56 75 14 10 32 35 15 47 05 38 54 92 08 08 72 79 66 65 20 32 83 00 61 74 76 68 58 67 31 58 77 93 92 44 94 16 01 29 32 19 33 51 45 05 35 16 73 80 43 44 64 19 14 52 22 81 62 79 68 13 37 34 90 55 27 93 33 06 76 42 16 08 16 81 86 15 09 79 30 86 74 77 35 68 65 05 79 82 57 78 22 37 41 82 44 07 43 98 89 78 23 96 04 36 51 84 34 98 03 38 87 12 79 88 57 88 46 81 62 56 94 91 06 59 37 85 23 92 01 84 05 83 64 12 95 51 44 60 64 68 86 53 05 26 16 38 31 95 14 27 80 20 36 96 81 84 88 55 50 47 22 39 70 56 66 41 39 91 55 02 31 42 31 54 72 45 32 43 76 89 95 74 04 79 33 46 91 40 46 80 10 25 92 61 83 83 58 58 71 46 29 95 72 38 89 05 26 05 41 07 78 66 31 63 37 30 38 95 57 84 99 85 32 91 28 06 85 67 13 42 48 69 61 99 15 93 98 43 38 84 46 49 84 89 44 51 75 42 45 17 43 70 69 68 47 02 10 60 99 02 84 41 36 81 65 44 08 49 93 49 62 44 98 63 62 58 95 62 61 53 28 49 20 20 21 49 45 35 01 57 45 64 45 76 50 34 97 87 14 65 07 59 45 02 56 89 50 85 03 42 65 07 35 46 23 47 83 35 73 08 83 52 47 19 79 85 26 07 53 80 00 33 18 07 47 18 38 27 88 50 09 58 11 96 83 05 86 49 61 35 03 46 63 88 48 48 93 93 51 63 96 33 82 81 58 02 19 36 55 80 74 84 14 63 79 22 49 65 23 13 22 01 10 20 01 11 15 36 43 49 49 19 24 77 70 95 08 50 77 19 27 34 65 55 62 52 47 73 09 34 58 38 28 28 18 15 82 65

Anexa 5 (continuare) Coloana Rândul

1 2 3 4 5 6 7 8 9 10 51 61 39 57 28 82 11 49 79 26 68 85 55 27 69 41 27 70 74 51 72 52 20 79 22 91 99 39 15 98 39 00 87 41 31 42 37 79 94 48 90 37 53 96 28 37 52 96 74 84 48 85 77 65 63 36 72 11 78 83 00 01 39 54 35 25 98 71 48 26 14 39 73 31 81 15 76 22 17 76 03 72 32 70 55 65 29 37 18 09 40 08 51 59 45 46 94 29 57 71 74 74 19 16 83 56 30 06 86 39 75 15 85 61 83 12 60 77 88 14 67 62 50 00 59 77 57 62 63 22 21 37 17 88 28 54 66 45 52 75 40 99 32 81 19 22 04 58 44 86 78 61 36 08 79 10 56 33 15 56 66 39 74 69 73 49 36 74 59 70 30 42 32 63 10 22 08 32 39 43 53 02 33 45 90 62 92 50 23 60 53 31 94 17 71 65 36 71 69 62 96 13 24 22 91 24 36 89 98 05 61 01 19 00 22 89 97 50 76 89 94 09 25 09 40 22 55 00 11 94 82 62 59 53 30 97 09 82 69 01 51 35 69 28 68 18 68 62 34 37 36 16 63 12 64 27 42 66 42 47 13 10 10 52 61 34 19 91 68 00 50 19 99 64 83 91 19 54 89 49 54 79 30 64 16 09 07 34 69 84 09 30 11 44 65 16 84 22 33 12 54 06 08 44 90 77 47 41 03 68 80 32 94 60 48 66 64 20 55 02 45 72 18 37 53 54 01 07 13 90 41 61 99 14 79 21 67 83 21 49 79 27 82 84 32 91 86 21 35 30 65 29 07 42 09 69 16 68 58 34 75 23 57 30 02 73 03 17 06 18 71 83 98 61 47 77 12 32 69 49 74 56 30 49 30 04 11 31 75 07 02 03 56 36 78 66 59 03 66 70 14 36 53 40 17 30 00 12 32 71 07 35 92 67 32 69 77 14 47 44 71 56 77 08 01 57 35 91 01 34 17 82 26 17 74 73 63 28 82 81 52 72 62 47 85 70 90 70 96 78 94 75 63 83 92 83 15 17 22 98 66 54 73 62 33 40 58 01 57 14 36 29 50 42 99 77 10 90 09 65 93 78 59 74 00 51 35 07 12 40 90 41 69 86 17 61 75 05 63 77 50 23 06 87 75 55 30 12 69 80 94 51 03 79 62 86 41 57 05 71 98 13 08 82 17 76 15 05 72 68 21 08 61 57 62 54 74 83 24 07 86 09 21 14 40 95 77 24 56 81 69 20 97 25 19 59 55 80 77 81 34 56 35 38 42 33 01 78 32 97 31 52 09 41 18 77 88 79 38 43 07 74 33 39 54 26 02 66 79 55 11 78 50 47 47 17 94 44 98 16 93 09 55 53 73 54 00 52 26 80 48 11 03 79 08 84 00 32 26 94 50 67 98 35 98 68 68 04 86 09 81 74 24 53 19 95 66 61 98 45 46 89 64 44 73 56 57 91 52 39 56 82 62 35 99 91 01 37 76 51 83 10 63 92 61 96 48 59 39 75 60 18 83 90 14 34 82 90 53 25 80 96 50 63 41 52 21 41 51 15 10 44 53 84 35 19 70 16 35 40 68 61 42 32 39 52 93 16 18 67 05 37 21 44 85 10 34 98 89 60 50 54 77 33 50 03 89 09 02 83 11 46 82 26 31 86 95 88 45 42 65 85 77 56 59 71 96 40 49 10 70 55 02 98 64 44 87 93 90 00 60 22 99 86 12 00 21 48 74 74 90 99 82 75 08 93 39 88 68 22 99 19 65 37 71 50 69 31 56 45 59 63 50 92 63 52 02 35 89 78 44 32 84 79 93 84 63 86 57 13 70 87 01 26 91 55 91 58 67 90 03 58 35 81 62 40 53 21 63 24 83 66 37 48 94 67 11 66 21 73 91 28 91 01 98 65 56 38 27 09 45 15 13 41 60 38 99 89 78 94 29 92 00 40 05 09 71 05 35 60 25 77 19 17 30 54 38 37 82 14 86 42 93 85 61 58 86 74 13 13 69 23 03 61 36 52 86 27 92 45 59 61 63 94 06 35 53 06 68 30 23 00 65 54 30 13 69 65 61 28 61 28 12 73 95 39 69 61 11 02 81 98 55 93 20 65 13 75 06 27 21 27 21 45 71 96 12 26 23 24 42 18 34 64 08 47 39 64 96 42 49 20 28 10 91 80 97 76 93 55 27 27 17 55 98 17 07 77 29 29 88 32 20 87 18 90 95 98 52 08 40 32 40 33 67 70 25 18 45 79 33 53 29 99 88 35 97 59 99 91 49 00 88 14 76 96 02 50 36 93 96 07 29 97 18 55 79 84 63

100 16 78 05 19 94 15 12 18 25 87 93 58 30 60 49 28 05 57 47 17 101 02 22 85 19 48 74 55 24 89 69 15 53 00 20 88 48 95 08 00 47 102 85 76 34 51 40 44 62 93 65 99 72 64 09 34 01 13 09 74 90 65 103 00 88 96 79 38 24 77 00 70 91 47 43 43 82 71 67 49 90 37 00

$QH[D��VIkUúLW� Coloana Rândul

1 2 3 4 5 6 7 8 9 10 104 64 29 81 85 50 47 36 50 91 19 09 15 98 75 60 58 33 15 51 44 105 94 03 80 04 21 49 54 91 77 85 00 45 68 23 12 94 23 44 36 88 106 42 28 52 73 06 41 37 47 47 31 52 99 89 92 22 81 86 55 99 09 107 09 27 52 72 49 11 30 93 33 29 54 17 54 48 47 42 04 79 18 64 108 54 68 64 07 85 32 05 96 54 79 57 43 96 97 30 72 12 19 41 70 109 25 04 92 29 71 11 64 10 42 23 23 67 01 19 20 58 35 93 39 46 110 28 58 32 91 95 28 42 36 98 59 66 32 15 51 46 63 57 10 83 55 111 64 35 04 62 24 87 44 85 45 68 41 66 19 17 13 09 63 37 15 33 112 61 05 55 88 25 01 15 77 12 90 69 34 36 93 52 39 36 23 59 73 113 98 93 18 93 86 98 99 04 75 28 30 05 12 09 57 35 90 15 98 07 114 61 89 35 47 16 32 20 16 78 52 82 37 26 33 67 42 11 93 35 61 115 94 40 82 18 06 61 54 67 03 66 76 82 90 31 71 90 39 27 97 85 116 54 38 58 65 27 70 93 57 59 00 63 56 18 79 85 52 21 03 03 16 117 63 70 89 23 76 46 97 70 00 62 15 35 97 42 47 54 60 60 78 12 118 61 58 65 62 81 29 69 71 95 53 53 69 20 95 66 60 50 70 22 97 119 51 68 98 15 05 64 43 32 74 03 44 63 52 38 67 59 56 69 11 14 120 59 25 41 48 64 79 62 26 87 86 94 30 43 54 26 98 61 38 53 44 121 85 00 02 24 67 85 88 10 34 01 54 53 23 77 33 11 19 68 13 50 122 01 46 87 56 19 19 19 43 70 25 24 29 48 22 44 81 35 40 33 23 123 42 41 25 10 87 27 77 28 05 90 73 03 95 46 88 82 25 02 05 00 124 03 57 14 03 17 80 47 85 94 49 89 55 10 27 19 50 20 37 02 71 125 18 95 93 40 45 53 04 57 17 03 34 54 83 91 69 02 90 72 98 45 126 74 11 04 66 68 52 70 11 97 01 55 36 63 49 42 68 82 15 48 64 127 31 54 98 82 61 64 40 50 42 48 96 84 82 42 55 15 72 34 90 96 128 85 51 93 55 89 63 47 92 88 42 00 08 21 52 27 28 77 48 02 42 129 19 95 97 55 27 91 15 20 96 25 48 75 49 95 88 68 36 09 66 17 130 75 74 55 98 33 02 36 99 11 84 07 71 40 65 95 54 01 90 44 32

Anexa 6

9DORULOH�IXQF LHL�GH�UHSDUWL LH�D�DPSOLWXGLQLL�vPSU úWLHULL

≤

−= t

xxPP

n

σ)1()(

t n 5 6 7 8 9

4,00 0,9623 0,9469 0,9300 0,9120 0,8929 4,10 0,9693 0,9566 0,9425 0,9273 0,9112 4,20 0,9752 0,9647 0,9530 0,9404 0,9269 4,25 0,9777 0,9682 0,9576 0,9461 0,9338 4,35 0,9821 0,9744 0,9657 0,9562 0,9460 4,45 0,9857 0,9795 0,9724 0,9647 0,9563 4,50 0,9873 0,9817 0,9754 0,9684 0,9608 4,55 0,9887 0,9837 0,9780 0,9917 0,9649 4,60 0,9899 0,9855 0,9804 0,9747 0,9686 4,65 0,9911 0,9871 0,9825 0,9775 0,9719 4,70 0,9921 0,9885 0,9845 0,9799 0,9750 4,75 0,9930 0,9898 0,9862 0,9822 0,9777 4,80 0,9938 0,9910 0,9878 0,9842 0,9802 4,85 0,9945 0,9920 0,9892 0,9860 0,9824 4,90 0,9952 0,9930 0,9904 0,9876 0,9844 4,95 0,9958 0,9938 0,9916 0,9890 0,9862 5,00 0,9963 0,9946 0,9926 0,9903 0,9876 5,05 0,9967 0,9952 0,9935 0,9915 0,9893 5,10 0,9971 0,9958 0,9942 0,9925 0,9906 5,15 0,9975 0,9963 0,9950 0,9934 0,9917 5,20 0,9978 0,9968 0,9956 0,9942 0,9927 5,25 0,9981 0,9972 0,9961 0,9949 0,9936 5,30 0,9983 0,9975 0,9966 0,9956 0,9944 5,35 0,9985 0,9979 0,9971 0,9961 0,9951 5,40 0,9987 0,9980 0,9974 0,9966 0,9957 5,45 0,9989 0,9984 0,9978 0,9971 0,9963 5,50 0,9994 0,9986 0,9981 0,9975 0,9968 5,70 0,9995 0,9992 0,9989 0,9986 0,9989 6,00 0,9998 0,9997 0,9996 0,9994 0,9993

BIBLIOGRAFIE

1. Andrei, T.

Stancu, S. 6WDWLVWLF ��WHRULH�úL�DSOLFD LL��(GLWXUD�$//��%XFXUHúWL��

2. Baron, T. Anghelache, C.

L DQ��(�

6WDWLVWLF ��(GLWXUD�(FRQRPLF ��%XFXUHúWL��

3. Baron T. L DQ�(�

Matache S.

6WDWLVWLF ��/XFU UL�DSOLFDWLYH, A.S.E., %XFXUHúWL��

4. % GL ��0� Baron, T. Korka, M.

Statistica pentru afaceri��(GLWXUD�(ILFLHQW��%XFXUHúWL��

5. Biji, E. Baron, T. (coordonatori):

6WDWLVWLF � WHRUHWLF � úL� HFRQRPLF �� (GLWXUD� 'LGDFWLF � úL� 3HGDJRJLF ��

%XFXUHúWL��

6. Biji, E. Lilea, E. Wagner, P.

6WDWLVWLF ��(GLWXUD�8QLYHUVLWDWHD�³7LWX�0DLRUHVFX´��%XFXUHúWL��

7. Capanu, I. Wagner, P. 6HF UHDQX��&��

6WDWLVWLF �PDFURHFRQRPLF ��(GLWXUD�(FRQRPLF ��%XFXUHúWL��

8. Craiu, V. %kVF ��2��

Teste de omogenitate��(GLWXUD�(FRQRPLF ��%XFXUHúWL� 1998.

9. Isaic-Maniu, Al. *U GLQDUX��$� Voineagu, V. 0LWUX ��&�

6WDWLVWLF �WHRUHWLF �úL�HFRQRPLF ��(GLWXUD�7HKQLF ��&KLúLQ X��

10. Isaic-Maniu, Al. 0LWUX ��&� Voineagu, V.

0DFURHFRQRPLH�úL�DQDOL] �PDFURHFRQRPLF , Editura Univ. “Constantin %UkQFRYHDQX´��%XFXUHúWL��


Statistica pentru managementul afacerilor�� (GLWXUD� (FRQRPLF ��

%XFXUHúWL��


Statistica pentru managementul afacerilor�� HGL LD� D� GRXD�� (GLWXUD�

(FRQRPLF ��%XFXUHúWL��

13. Pecican, E. Econometrie��(GLWXUD�$//��%XFXUHúWL��

14. Porojan, D. 6WDWLVWLFD�úL�WHRULD�VRQGDMXOXL��(GLWXUD�ùDQVD�65/��%XFXUHúWL��

15. Voineagu V. 0LWUX �&� Isaic-Maniu Al.

L DQ�(� Baron T. Matache S. Isaic-Maniu I. ùHUEDQ�'� Voineagu M.

6WDWLVWLF �WHRUHWLF �úL�PDFURHFRQRPLF ��7HVWH��OXFU UL�SUDFWLFH��VWXGLL�GH�

caz��(G��(FRQRPLF ��%XFXUHúWL��

16. Biji, E. Wagner, P. Lilea, E. Petcu N. 9 WXL�0��

6WDWLVWLF , Editura Universitatea “Titu 0DLRUHVFX´��%XFXUHúWL��

TESTUL 1 *)

1) 1X�FRQVWLWXLH�R�FRPSRQHQW �D�DYX LHL�QD LRQDOH�

a) fondul funciar; b) mijloacele fixe; c) 9HQLWXO�1D LRQDO��3URGXVXO�,QWHUQ�%UXW�� d) resursele minerale atrase în circuitul economic; e) stocurile de materiale.

2) 0XO LPHD�YDORULORU�FRHILFLHQWXOXL�GH�FRUHOD LH�FDOFXODW�vQ�FD]XO�GHSHQGHQ HORU�OLQLDUH�GLUHFWH�HVWH�

a) [-1, 1] ; b) [-1, 0) ; c) PXO LPHD�QXPHUHORU�UHDOH� d) (0, 1] ; e) [-3, 3] .

3) Într-R�SRSXOD LH� VWDWLVWLF � VWUXFWXUDW �SH�JUXSH� �FODVH�� UHODWLY�RPRJHQH�� VH�FDOFXOHD] ��GLVSHUVLD�

general (σ 2 ); media dispersiilor (σ 2 ��úL�GLVSHUVLD�GLQWUH�JUXSH��δ 2 ��(VWH�DGHY UDW � vQ�RULFH�VLWXD LH�UHOD LD�

a) δ σ2 2> ;

b) δ σ2 2≤ ;

c) δ σ σ2 2 2= + ;

d) δ 2 0< ;

e) comparabilitatea dintre δ 2 úL�σ 2 nu are sens. 4) În cazul tabelelor Input - Output, ramurile se delimiteaz dup :

a) FULWHULXO�DFWLYLW LL�SUHSRQGHUHQWH� b) FULWHULXO�IXQF LRQDO� c) XQLWDWHD�LQVWLWX LRQDO � d) VHFWRUXO�LQVWLWX LRQDO� e) GXS �IOXxurile monetare din economie.

5) 5DWD� úRPDMXOXL� vQ� VHQVXO� %LURXOXL� ,QWHUQD LRQDO� DO�0XQFLL� VH� FDOFXOHD] � SURFHQWXDO�� UDSRUWkQG�

1XP UXO�GH�úRPHUL la:

a) 3RSXOD LD�FXUHQW�DFWLY � b) 3RSXOD LD�RFXSDW � c) 5HVXUVHOH�GH�PXQF � d) 5H]HUYHOH�GH�PXQF � e) 3RSXOD LD�vQ�YkUVW �GH�PXQF �

*)�6XELHFW�OD�H[DPHQXO�GH�OLFHQ ��)DFXOWDWHD�&LEHUQHWLF ��6WDWLVWLF �úL�,QIRUPDWLF �(FRQRPLF ��VHVLXQHD�LXOLH�

1997

6) Fenomenele social-HFRQRPLFH�GH�PDV �VWXGLDWH�GH�VWDWLVWLF �QX�VH�FDUDFWHUL]HD] �

a) SULQ�YDULDELOLWDWH�vQ�WLPS�úL�vQ�VSD LX� b) printr-R� OHJH� GH� DSDUL LH� FDUH� VH� PDQLIHVW � FD� WHQGLQ � FH� QX� SRDWH� IL� FXQRVFXW � úL�

YHULILFDW �GHFkW�OD�QLYHOXO�DQVDPEOXOXL�úi nu în fiecare caz în parte; c) printr-R�OHJH�GH�DSDUL LH�FDUH�VH�PDQLIHVW �FD�WHQGLQ �FH�SRDWH�IL�FXQRVFXW �úL�YHULILFDW �vQ�

fiecare caz în parte; d) SULQ�IRUPH�LQGLYLGXDOH�GH�DSDUL LH�DVHP Q WRDUH� e) SULQ� IDSWXO� F � HOH� VXQW� VSHFLILFH� OHJLORU� VWDWLVWLFH� - legi FDUH� VH�PDQLIHVW � VXE� IRUP �GH�

WHQGLQ � ID � GH� FDUH� DEDWHULOH� vQWkPSO WRDUH� vQWU-XQ� VHQV� VDX� DOWXO� VH� FRPSHQVHD] �reciproc.

7) 5HSUH]HQWDWLYLWDWHD�HVWH�XUP ULW �vQ�PRG�GHRVHELW�vQ�FD]XO�FXOHJHULL�GDWHORU�SULQ�

a) UHFHQV PkQW� b) sondaje statistice; c) anchete statistice; d) rapoarte statistice; e) monografii statistice;

8) Deflatorul Produsului Intern Brut este:

a) LQGLFHOH�GH�SUH XUL�GH�WLS�/DVSH\UHV� b) indicele volumului fizic de tip Paasche; c) LQGLFHOH�GH�SUH XUL�GH�WLS�)LVKHU� d) LQGLFHOH�GH�SUH XUL�GH�WLS�3DDVFKH� e) indice al valorii.

9) ([SUHVLD�VLQWHWL] ULL�YDORULORU�LQGLYLGXDOH�DOH�XQHL�YDULDELOH�VWDWLVWLFH�D�WRW�FHHD�FH�H�HVHQ LDO��WLSLF�

úL�RELHFWLY�vQWU-XQ�VLQJXU�QLYHO�UHSUH]HQWDWLY�HVWH�GDW �GH��

a) medie; b) PHGLDQ � c) YDORDUH�PRGDO � d) FRHILFLHQWXO�GH�FRUHOD LH� e) coeficientul de varia LH�

10) &DUH�LQGLFDWRU�VWDWLVWLF�QX�DUH�VHQV�V �VH�FDOFXOH]H�SHQWUX�DQDOL]D�VHULLORU�GH�UHSDUWL LH�IRUPDWH�

GXS �R�YDULDELO �DOWHUQDWLY " a) IUHFYHQ HOH�UHODWLYH� b) media; c) dispersia; d) coeficientul de asimetrie.

11) Într-R� SRSXOD LH� VWDWLVWLF � V-au cules date despre doX � YDULDELOH� QXPHULFH� GLVWLQFWH�� 6HULLOH�

IRUPDWH�vQ�XUPD�VLVWHPDWL] ULL�VXQW��

{ } { }xi i= =1 7

2 2 2 10181818,

; ; ; ; ; ; �úL�{ } { }yi i = =1 7

9 9 9 10 111111,

; ; ; ; ; ; .

2EVHUYkQG�YDULDQWHOH�FHORU�GRX �VHULL�VH�FRQVWDW �F �

a) VHULD�IRUPDW �GXS �<�HVWH�PDL�RPRJHQ �GHFkW�FHD�IRUPDW �GXS �;�

b) serLD�IRUPDW �GXS �;�HVWH�PDL�RPRJHQ �GHFkW�FHD�IRUPDW �GXS �<� c) FHOH�GRX �VHULL�SUH]LQW �DFHHDúL�RPRJHQLWDWH��GHRDUHFH�DX�DFHHDúL�PHGLH�úL�PHGLDQ �HJDOH�

cu 10; d) QX�DUH�VHQV�FRPSDUDELOLWDWHD�RPRJHQLW LL�GLQ�FHOH�GRX �VHULL��GHRDUHFH�VXQW�IRUPDWH�GXS �

variabile distincte; e) VHULD� IRUPDW �GXS �;�HVWH�PDL�RPRJHQ �GHRDUHFH�DEDWHULOH� LQGLYLGXDOH�ID �GH�YDORDUHD�

PHGLDQ �VXQW�PDL�PDUL�

12) Un circuit economic este considerat circuit închis pentru sectorul A când:

a) QX�H[LVW �LQWU UL�vQ�$� b) QX�H[LVW �LHúLUL�GLQ�$� c) QX�H[LVW �QLFL�LQWU UL�úL�QLFL�LHúLUL�SHQWUX�VHFWRUXO�$� d) VXPD�IOX[XULORU�GH�LQWUDUH�HVWH�HJDO �FX�VXPD�IOX[XULORU�GH�LHúLUH� e) fluxurile monetare sunt identice cu fluxurile de bunuri.

13) ÌQ�XUPD�REVHUY ULL�XQHL�YDULDELOH�QXPHULFH�;��vQWU-R�SRSXOD LH�VWDWLVWLF �V-au înregistrat variantele: x1,

x2, …, xn��0HGLD�DULWPHWLF �D�DFHVWRUD� x HVWH�FDOFXODW �vQWU-R�SULP �HWDS �GXS �UHOD LD�1

1nxi

i

n

=∑ (1). În

SURFHVXO�SUHOXFU ULL�YDULDQWHOH�FXOHVH�VXQW�JUXSDWH�vQ�³U´�LQWHUYDOH��RE LQkQGX-se o serie�GH�UHSDUWL LH�GH�

IUHFYHQ H��3H�ED]D�VHULHL�IRUPDWH�V-D�UHFDOFXODW�QLYHOXO�PHGLX��XWLOL]kQG�UHOD LD�1

1

1nx n

ii

r i ii

r

=

=∑∑

��5H]XOWDWXO�XWLOL] ULL�FHORU�GRX �UHOD LL�HVWH�LGHQWLF�

a) vQ�RULFH�VLWXD LH� b) QLFLRGDW � c) DWXQFL�FkQG�SHQWUX�DSOLFDUHD�UHOD iei (2) s-au luat în considerare mijloacele intervalelor; d) DWXQFL�FkQG�LQWHUYDOHOH�GH�JUXSDUH�SUH]LQW �IUHFYHQ H�HJDOH� e) DWXQFL� FkQG� IUHFYHQ HOH� VXQW� QRUPDO� UHSDUWL]DWH� vQ� ILHFDUH� LQWHUYDO�� FkQG� IUHFYHQ HOH�

LQWHUYDOHORU� VXQW� HJDOH� vQWUH� HOH� úL� FkQG� vQ� UHOD La (2) se iau în considerare centrele intervalelor.

14) 6ROGXO�FRQWXOXL��³0RGLILFDUHD�SDWULPRQLXOXL´�UHSUH]LQW �

a) economiile nete; b) VROGXO�ILQDQ ULL� c) amortizarea; d) LQYHVWL LLOH�EUXWH� e) venitul disponibil

15) $PSOLWXGLQHD�LQWHUFXDUWLOLF �vQWU-R�UHSDUWL LH�QRUPDO �FRQ LQH�

a) ��GLQ�QXP UXO�REVHUYD LLORU� b) ��GLQ�QXP UXO�REVHUYD LLORU� c) ��GLQ�QXP UXO�REVHUYD LLORU� d) WRDWH�REVHUYD LLOH� e) QXPDL�YDULDQWHOH�H[WUHPH��DO�F URU�QXP U�HVWH�PDL�PDUH�GHFkW��GLQ�PDVD�REVHUYD LLORU�

16) În cadrul conturilor macroeconomice, trecereD�GH�OD�FRQFHSWXO�³LQWHUQ´�OD�FRQFHSWXO�³QD LRQDO´�

se realizeaz în cadrul contului:

a) contul 2 - “Crearea veniturilor”; b) contul 1 -�³3URGXF LH´� c) contul 3 -�³5HSDUWL LD�YHQLWXULORU´� d) contul 4 - “Redistribuirea veniturilor”; e) vQ�QLFL�XQXO�GLQ�FRQWXULOH�PHQ LRQate.

17) Într-R� SRSXOD LH� VWDWLVWLF � VH� XUP UHVF� YDULDELOHOH� ;�� <� úL� =�� vQWUH� FDUH� H[LVW � UHOD LD� < ;=��

0HGLD�DULWPHWLF �D�YDULDELOHL�<�HVWH�HJDO �FX�SURGXVXO�PHGLLORU�YDULDELOHORU�;�úL�=��

a) vQ�RULFH�VLWXD LH� b) FkQG�vQWUH�;�úL�=�H[LVW �R�GHSHQGHQ �OLQLDU �GH�forma Z X= +α β ;

c) FkQG�vQWUH�;�úL�=�H[LVW �R�GHSHQGHQ �OLQLDU �GH�IRUPD� X Z= +α β ;

d) FkQG�FRYDULDQ D�GLQWUH�;�úL�=�HVWH�QXO � e) FkQG�VHULLOH�IRUPDWH�GXS �;��<��=�VXQW�GLVWULEX LL�GH�IUHFYHQ H�UHODWLYH�

18) ,QGLFDWRUXO�³3RSXOD LD�FXUHQW�DFWLY ´�QX�FXSULQGH�

a) VDODULD LL�SUH]HQ L�OD�OXFUX� b) VDODULD LL�DEVHQ L�WHPSRUDU�GH�OD�OXFUX� c) úRPHULL� d) patronii; e) HOHYLL�úL�VWXGHQ LL�GH�OD�FXUVXULOH�GH�]L�

19) ÌQ�XWLOL]DUHD�PHWRGHL�UHVWXOXL�QHGHVFRPSXV�SHQWUX�P VXUDUHD� LQIOXHQ HORU� L]RODWH�DOH�IDFWRULORU�

unui fenomen complex sunt luate în considerare:

a) ponderile perioadei curente ale factorului calitativ; b) SRQGHULOH�GLQ�SHULRDGD�GH�ED] �DOH�IDFWRUXOXL�FDQWLWDWLY� c) SRQGHULOH�GLQ�SHULRDGD�GH�ED] �DOH�IDFWRUXOXL�FDOLWDWLY� d) SRQGHULOH�GLQ�SHULRDGD�FXUHQW �DOH�IDFWRUului calitativ; e) SRQGHULOH�GLQ�SHULRDGD�GH�ED] ��LQGLIHUHQW�GH�QDWXUD�FDQWLWDWLY �VDX�FDOLWDWLY �D�IDFWRUXOXL�

izolat. 20) Într-R� VHULH� GH� YDORUL� LQGLYLGXDOH� DOH� XQHL� YDULDELOH� QXPHULFH�� REVHUYDW � vQWU-R� SRSXOD LH�

VWDWLVWLF ��YDORDUHD�PRGDO �HVWH�

a) varianta pozLWLY �FHD�PDL�PDUH� b) YDULDQWD�QHJDWLY �GDU�PD[LP �vQ�YDORDUH�DEVROXW � c) YDULDQWD��SR]LWLY �VDX�QHJDWLY ��FX�FHD�PDL�PDUH�IUHFYHQ �GH�DSDUL LH� d) YDULDQWD�FDUH�PLQLPL]HD] �GLVSHUVLD� e) YDULDQWD�FDUH�vQUHJLVWUHD] �FHD�PDL�PDUH�DEDWHUH�DEVROXW �ID �GH�PHGLH�

21) Se cunosc datele:

INDICATORII 3HU��ED] �� 3HU��FXUHQW �� - Salariul nominal (mii lei)

600 900

- Structura cheltuielilor familiilor (%):

-�P UIXUL�DOLPHQWDUH 60 62 -�P UIXUL�nealimentare

25 20

- servicii 15 18 - Modificarea SUH XULORU�vQ��

ÌQ�SHULRDGD�FXUHQW �ID �GH� SHULRDGD� GH� ED] �� VDODULXO�real: ��D��FUHúWH�FX�� b) scade cu 18,6 %; ��F��FUHúWH�FX�� G��FUHúWH de 18,6 ori; ��H�� FUHúWH� FX� �� PLL�

lei.

-�P UIXUL�DOLPHQWDUH 0 +25 -�P UIXUL�nealimentare

0 +40

- servicii 0 +10

22) &DUH�GLQ�XUP WRDUHOH�UHOD LL�H[SULP �³)RUPDUHD�EUXW �D�FDSLWDOXOXL´�

a) ,QYHVWL LL�QHWH�GH�FDSLWDO�IL[�SOXV�DPRUWL]DUHD� b) ,QYHVWL LL�EUXWH�GH�FDSLWDO�IL[�SOXV�SURILW�SOXV�PRGLILFDUHD�VWRFXULORU� c) )RUPDUHD�QHW �D�FDSLWDOXOXi fix plus modificarea stocurilor; d) )RUPDUHD�EUXW �D�FDSLWDOXOXL�IL[�SOXV�PRGLILFDUHD�VWRFXULORU� e) )RUPDUHD�EUXW �D�FDSLWDOXOXL�IL[��SOXV�DPRUWL]DUHD��SOXV�PRGLILFDUHD�VWRFXULORU�

23) /D� RSW� PXQFLWRUL� FDUH� HIHFWXHD] � DFHOHDúL� RSHUD LL� vQ� FDGUXO� XQXL� VFKLPE� V-au observat variabilele timp nelucrat�úL�QXP UXO�GH�RSHUD LL��'DWHOH�VXQW�XUP WRDUHOH�

Nr. curent 1 2 3 4 5 6 7 8 Timp nelucrat (u.m.) 7 1 2 5 8 6 3 4 1U��RSHUD LL 4 10 9 6 4 5 9 8

7HQGLQ D�OHJ WXULL�GLQWUH�YDULDELOH�HVWH�H[SULPDW �SULQ�IXQF LD�

a) ix xYi

96,02,11 −= ;

b) ix xYi

96,02,11 += ;

c) ix xYi

96,0−= ;

d) 204,096,02,11 iix xxYi

++= ;

e) ix xYi

96,0= .

24) ÌQ�6�&�1��FKHOWXLHOLOH�SHQWUX�vQO WXUDUHD�GDXQHORU�DGXVH�PHGLXOXL�VXQW�FRQVLGHUDWH�

a) &RVW�GH�SURGXF LH� b) ,QYHVWL LL�QHWH� c) ,QYHVWL Li brute; d) Venituri ale consumatorilor; e) Venituri ale statului.

25) 6H�FXQRVF�GDWHOH��FUHúWHUHD�SURGXFWLYLW LL�PHGLL��LQGLFHOH�SURGXFWLYLW LL�PHGLL�SH�VHDPD�

SURGXFWLYLW LL�LQGLYLGXDOH��,QGLFHOH�LQIOXHQ HL�VWUXFWXULL�DVXSUD�SURGXFWLYLW LL�PHGLL�HVWH:

a) 100%; b) 91,7%; c) 132%; d) 110%; e) 85%;

26) ÌQ�DFWLYLWDWHD�GH�VWDWLVWLF �DQDOL]D�GLVSHUVLRQDO �QX�VH�XWLOL]HD] �SHQWUX��

a) YHULILFDUHD�QRUPDOLW LL�UHSDUWL LLORU� b) HYLGHQ LHUHD� P VXULL� vQ� FDUH� XQXO� VDX� PDL� PXO L� IDFWRUL� �VDX� R� FRPELQD LH� D� DFHVWRUD��

LQIOXHQ HD] �vQ�PRG�HVHQ LDO�YDULD LD�XQHL�YDULDELOH�UH]XOWDWLYH� c) DQDOL]D�P VXULL�vQ�FDUH�YDORULOH�UHDOH�DOH�XQHL�FDUDFWHULVWLFL�VH�DEDW�GH�OD�YDORULOH�WHRUHWLFH��

FDOFXODWH�GH�UHJXO �VXE�IRUP �GH�P ULPL�PHGLL�VDX�HFXD LL�GH�UHJUHVLH� d) DQDOL]D�P VXULL�vQ�FDUH�YDULD LLOH�XQHL�variabile rezultative sunt sau nu sunt dependente de

factorul de grupare; e) YHULILFDUHD�YDOLGLW LL�PRGHOHORU�GH�UHJUHVLH�VLPSO �úL�PXOWLSO �

27) 'LVWULEX LD�D��GH�IDPLOLL�GXS �QXP UXO�GH�FRSLL�VH�SUH]LQW �DVWIHO��GDWH�FRQYHQ LRQDOH��

Nr. copii 0 1 2 3 4 5 Total Nr. familii 20 65 70 30 10 5 200

9DORDUHD�PHGLDQ �vQ�VHULD�SUH]HQWDW �HVWH�

a) 1 copil; b) 2 copii; c) 70 familii; d) 3 copii; e) 5 copii;

28) 'DF � VWUXFWXUD� SH� UDPXUL� D� 3URGXVXOXL� ,QWHUQ� %UXW� VH� PRGLILF � vQ� IDYRDUHD� UDPXULORU� FX�“necesarul de mijloace fixe pentru� RE LQHUHD� XQHL� XQLW L� GH� SURGXF LH´�PDL�PLF� GHFkW� QLYHOXO�mediu al acestui indicator, atunci, ca urmare a modific ULL�IDFWRUXOXL�VWUXFWXUDO��HILFLHQ D�XWLOL] rii PLMORDFHORU�IL[H�OD�QLYHOXO�HFRQRPLHL�QD LRQDOH�

a) FUHúWH� b) scade; c) U PkQH�QHPRGLILFDW � d) nu se SRDWH�SUHFL]D�VHQVXO�PRGLILF ULL� e) SRDWH�V �VFDG �

29) 2� VRFLHWDWH� FRPHUFLDO � D� vQUHJLVWUDW� OD� FLIUD� GH� DIDFHUL� vQ� SHULRDGD� 6HSW�� µ� �� - Dec. ‘ 96 o

FUHúWHUH�PHGLH�OXQDU �FX��ÌQ�SHULRDGD�'HF��µ��-�0DL�µ��DFHODúL�LQGLFDWRU�V-a modificat în PHGLH�SH�OXQ de 1,17 ori. În perioada Sept. ‘ 96 - Mai ‘ 97 cifra medie de afaceri a crescut în PHGLH�GH�OD�R�OXQ �OD�DOWD�FX�

a) 13,5% ; b) 2,98% ; c) 3,2% ; d) 113,4% ; e) 27%.

30) 7UHL�DQJDMD L�GH�DFHHDúL�SURIHVLH�úL�FDOLILFDUH�SHQWUX�HIHFWXDUHD�XQHL�RSHUD LL� LGHQWLFH�FRQVXP �

timpiL��vQ�PLQXWH��XUP WRUL��7LPSXO�PHGLX�FRQVXPDW�HVWH�

a) 21,7 min./pers. ;

b) 20 min./pers. ; c) 22,5 min./pers. ; d) 18,5 min./pers. ; e) 19 min./pers.

31) &XQRDúWHP� GDWHOH�� 3URGXVXO� JOREDO� EUXW� �� X�P�� SRQGHUHD� FRQVXPXOXL� LQWHUPHGLDU� vQ�produsul global bruW��FRQVXPXO�SULYDW� ��FRQVXPXO�SXEOLF��IRUPDUHD�EUXW �D�FDSLWDOXOXL� �½ consumul privat=10.000 u.m. În aceste condi LL�

a) Exportul = Importul; b) Exportul > Importul; c) PIB > 100.000 u.m.; d) PIN > 80.000 u.m.; e) Exportul < Importul.

32) În perioada T0 - T1 Produsul Intern Brut a crescut cu 20%, mijloacele fixe cu 30%, iar ponderea PLMORDFHORU� IL[H� DFWLYH� vQ�PLMORDFHOH� IL[H� VFDGH� GH� OD� �� OD� �� (ILFLHQ D�PLMORDFHORU� IL[H�active:

a) scade cu 105,4% ; b) scade cu 92,3% ; c) FUHúWH�FX�� d) FUHúWH�FX�� e) scade cu 7,7%.

33) Într-R�SRSXOD LH�VWDWLVWLF �GH��XQLW L�FRPHUFLDOH�V-DX�REVHUYDW�YDULDELOHOH��YDORDUHD�DG XJDW �úL�YROXPXO�PLMORDFHORU�IL[H�SHQWUX�FDUH�V-DX�GHWHUPLQDW�GLVSHUVLLOH��úL�UHVSHFWLY��6H�FXQRDúWH��GH�DVHPHQHD��F �WHQGLQ D�OHJ WXULL�HVWH�H[SULPDWD�SULQ�IXQF LD� ix xY

i44,042,119 += .

0 ULPHD�FRHILFLHQWXOXL�GH�FRUHOD LD�OLQLDU �HVWH�

a) 0,72 ; b) -0,75 ; c) 0,27 ; d) 0,95 ; e) 0,44 ;

34) 'HVSUH�HYROX LD�FLIUHL�GH�DIDFHUL�D�XQHL�VRFLHW L�FRPHUFLDOH�VH�FXQRVF�GDWHOH�

ANI 1993 1994 1995 1996 ��PRGLILF ULL�UHODWLve a cifrei GH� DIDFHUL� ID � GH� DQXO�precedent

+3 +4 +2 +4

ùWLLQG� F � FLIUD� GH� DIDFHUL� GLQ� �� D� IRVW� GH� �� POG�� OHL�� PRGLILFDUHD� PHGLH� DQXDO � DEVROXW � vQ�perioada 1992 - 1996 a fost de:

a) 1,2 mld. lei/an; b) 1,2 %/an; c) 2,5 mld. lei/an; d) 210 %/an; e) 3 mld. lei/an;

35) 7HQGLQ D�OHJ WXULL�GLQWUH�GRX �YDULDELOH�VH�H[SULP �SULQ�IXQF LD�Y a bx cxx = + + 2 . Intensitatea

OHJ WXULL�GLQWUH�FHOH�GRX �YDULDELOH�VH�FDUDFWHUL]HD] �SULQ�

a) FRHILFLHQWXO�GH�FRUHOD LH�OLQLDU � b) coeficientul lui Spearman;

c) coeficientul lui Kendall; d) coeficientul lui Bowley; e) UDSRUWXO�GH�FRUHOD LH�

36) 5H]XOWDWHOH� H[DPLQ ULL� XQHL� VHPLJUXSH� OD� SUREHOH� WHRUHWLFH� úL� SUDFWLFH� DOH� XQHL� GLVFLSOLQH� GH�VSHFLDOLWDWH�VXQW�XUP WRDUHOH�

Studentul A B C D E F G H I J TEORIE 8 3 9 2 7 10 4 6 1 5

35$&7,& 9 5 10 1 8 7 3 4 2 6 'HSHQGHQ D� GLQWUH� YDORULOH� FHORU� GRX � YDULDELOH� VH� P VRDU � FX� DMXWRUXO� FRHILFLHQWXOXL� GH�FRUHOD LH�D�UDQJXULORU��DO�OXL�6SHDUPDQ��D�F UXL�P ULPH�HVWH�

a) 0,855; b) -0,855; c) 1,00; d) -1,00; e) 0,225;

37) 6H�FXQRVF�GDWHOH��3URGXVXO�1D LRQDO�%UXW�OD�SUH XULOH�SLH HL��X�P��DORFD LD�SHQWUX�FRQVXPXO�GH�FDSLWDO�IL[��X�P��úL�YHQLWXO�QD LRQDO��X�P��0 ULPHD�,PSR]LWHORU�,QGLUHFWH�1HWH�HVWH�

a) 300 u.m.; b) 400 u.m.; c) 200 u.m.; d) 700u.m.; e) 500 u.m.

38) 3HQWUX�RE LQHUHD�9HQLWXOXL�SHUVRQDO�DO�PHQDMHORU�GLQ�3URGXVXO�1D LRQDO�1HW�OD�SUH XULOH�SLH HL�QX�se scad:

a) &RQWULEX LLOH�OD�$VLJXU ULOH�6RFLDOH� b) Impozitele Indirecte Nete; c) $MXWRDUHOH�GH�úRPDM� d) 3URILWXO�QHGLVWULEXLW�DO�VRFLHW LORU� e) ,PSR]LWXO�SH�YHQLWXO��SURILWXO��VRFLHW LORU��ILUPHORU��

39) 6H�FXQRVF�XUP WRDUHOH�GDWH� Departamente Structura cifrei de afaceri (%)

Procentul

programat (al sarcinii de plan) al

cifrei de afaceri (%)

Procentul îndeplinirii

programului la cifra de afaceri

(%)

Perioada de ED]

Perioada FXUHQW

A 105 110 40 35 B 115 118 60 65

&DUH�LQIRUPD LH�VLQWHWLF �GLQ�FHOH�SUH]HQWDWH�PDL�MRV�HVWH�IDOV " a) Procentul mediu al îndeplinirii programului la cifra de afaceri a fost de 115,1%; b) Procentul mediu programat (al sarcinii de plan) al cifrei de afaceri a fost de 111%; c) Pe ansamblu cifra de afaceri a crescut în� SHULRDGD�FXUHQW � ID �GH�SHULRDGD�GH�ED] �FX�

27,8%; d) 3H� DQVDPEOX� FLIUD� GH� DIDFHUL� UHDOL]DW � vQ� SHULRDGD� FXUHQW � D� IRVW� PDL� PDUH� GHFkW� FHD�

SURJUDPDW �FX�� e) ,QGLFHOH�GH�GLQDPLF �SH�DQVDPEOXO�GHSDUWDPHQWHORU�D�IRVW�GH��

40) /D�R�VRFLHWDWH�FRPHUFLDO ��GLQ vânzarea unui produs s-a realizat în luna mai 1997 o încasare de ��PLOLRDQH� OHL��&D�XUPDUH� D�PDMRU ULL� SUH XOXL� ID � GH�PDL� �� vQFDV ULOH� OXQLL�PDL� �� DX�FUHVFXW�FX��OHL��&DUH�DU�IL�IRVW�vQFDV ULOH�OXQLL�PDL��GDF �SUH XO�QX�V-ar fi modificat?

a) 4,5 mil. lei; b) 3,5 mil. lei; c) 4 mil. lei; d) 8 mil. lei; e) 5 mil. lei.

TESTUL 2 *)

1) Pe baza unei serii cronologice din perioada 1987-�� IRUPDW � GLQ� WHUPHQL� FRQVHFXWLYL�� V-a

DMXQV� OD� FRQFOX]LD� F � WHQGLQ D� GH� HYROX LH� D� XQHL� YDULDELOH� HVWH� H[SULPDW � SULQ� IXQF LD�

Y tt = +400 60 �vQ�FRQGL LLOH� vQ�FDUH� t =∑ 0 ). Valorile estimate ale variabilei analizate pentru

��úL��vQ�DFHDVW �RUGLQH��VXQW�

a) 1300; 1480; 1600; b) 1120; 1180; 1240; c) 340; 280; 220; d) 460; 520; 580; e) 760; 820; 880.

2) 3HQWUX� GHVFRPSXQHUHD� XQXL� IHQRPHQ� FRPSOH[� SH� IDFWRUL� GH� LQIOXHQ � FX� DMXWRUXO� LQGLFLORU�� VH�

SRDWH�DSHOD� OD�PHWRGD�VXEVWLWX LHL� vQ� ODQ �VDX� OD�PHWRGD� UHVWXOXL�QHGHVFRPSXV��0HWRGD� UHVWXOXL�QHGHVFRPSXV��VSUH�GHRVHELUH�GH�PHWRGD�VXEVWLWX LHL�vQ�ODQ �

a) exagerHD] �LQIOXHQ D�IDFWRUXOXL�FDOLWDWLY� b) H[DJHUHD] �LQIOXHQ D�IDFWRUXOXL�FDQWLWDWLY� c) HVWH�PDL�FRUHFW �GHRDUHFH�QX� LQH�VHDPD�GH�QDWXUD�FDOLWDWLY �VDX�FDQWLWDWLY �D�IDFWRUXOXL�

izolat; d) HVWH�PDL�FRUHFW �GHRDUHFH�L]ROHD] �QXPDL�IDFWRUXO�FDOLWDWLY�OD�QLYHOXO�SHULRDGHL�GH�ED] � e) HVWH�PDL� FRUHFW � GHRDUHFH� HD� RSHUHD] � DWkW� FX�PRGLILF UL� DEVROXWH� FkW� úL� FX�PRGLILF UL�

relative.

3) 5HSUH]HQWDWLYLWDWHD�PHGLHL��FDOFXODW �SHQWUX�XQ�úLU�GH�YDORUL�LQGLYLGXDOH�HVWH�DVLJXUDW �

a) GDF � vQ� GHWHUPLQDUHD�PHGLHL� VH� LDX� vQ� FRQVLGHUDUH� IUHFYHQ HOH� DEVROXWH� GH� DSDUL LH� DOH�valorilor individuale;

b) GDF �YDORULOH�LQGLYLGXDOH�GLQ�FDUH�VH�FDOFXOHD] �PHGLD�VXQW�RPRJHQH� c) GDF �VH�XWLOL]HD] �PHGLD�DULWPHWLF � d) GDF �úLUXO�GH�YDORUL�LQGLYLGXDOH�HVWH�VWUXFWXUDW�SH�LQWHUYDOH�HJDOH� e) GDF �VXPD�DOJHEULF �D�DEDWHULORU�LQGLYLGXDOH�GH�OD�PHGLH�HVWH�QXO �

4) 6H�FXQRVF�GDWHOH��31%� OD�SUH XO�SLH HL��X�P�� DORFD LD�SHQWUX�FRQVXPXO�GH�FDSLWDO� IL[��

X�P��LPSR]LWHOH�LQGLUHFWH��X�P��úL�VXEYHQ LL�GH�H[SORDWDUH��X�P� 9HQLWXO�QD LRQDO�HVWH�

a) 700 u.m.; b) 600 u.m.; c) 660 u.m.; d) 960 u.m.; e) 1340 u.m.

*)� 6XELHFW� OD� H[DPHQXO� GH� OLFHQ �� )DFXOWDWHD� &LEHUQHWLF �� 6WDWLVWLF � úL� ,QIRUPDWLF � (FRQRPLF �� VHVLXQHD�

februarie 1998.

5) ÌQ�VWXGLXO�GHSHQGHQ HORU�VWDWLVWLFH�GLQWUH�YDULDELOH�VH�FXQRDúWH�IDSWXO�F �VXQW�DQDOL]DWH��DEDWHULOH�

GLQWUH�YDORULOH�HPSLULFH�úL�FXUED��GUHDSWD��GH�UHJUHVLH�� y yi x− ) evaluate sintetic prin dispersia

σ y r/2 �� DEDWHULOH� GLQWUH� YDORULOH� DMXVWDWH� úL� PHGLD� FDUDFWHULVWLFLL� GHSHQGHQWH� � y yx − �� HYDOXDW �

sintetic prin dispersia σ y x/2 ��GLVSHUVLD� WRWDO �D�YDULDELOHL� � UH]XOWDWLYH�HVWH�σ y

2 . În orice sitXD LH�

HVWH�DGHY UDW �UHOD LD�

a) σ σy x y/2 2> ;

b) σ σy x y/2 2< ;

c) σ σy r y/2 2> ;

d) σ σ σy y r y x2 2 2= −/ / ;

e) σ

σy x

y

/.

2

21>

6) )HQRPHQHOH�GH�PDV �� vQ�JHQHUDO��DSDU�FD�R�PXO LPH�GH�IRUPH� LQGLYLGXDOH�GLIHULWH��FX�H[LVWHQ �diferiW �� DSDUHQW� I U � QLFL� R� OHJ WXU � GH� OD� R� IRUP � OD� DOWD�� GDU� FDUH�� DQDOL]DWH� FRPSDUDWLY�� VH�FRQVWDW �F �DX�DFHHDúL�HVHQ ��&HHD�FH�QX�H[SOLF �DFHDVW �DILUPD LH�HVWH�F �

a) IRUPHOH�LQGLYLGXDOH�VXQW�JHQHUDWH�GH�FDX]H�FDUH�VH�PDQLIHVW �GH�UHJXO �vQ�FRQGL LL�GLIHrite; b) IRUPHOH�LQGLYLGXDOH�GH�PDQLIHVWDUH�GLIHU �GH�OD�R�XQLWDWH�OD�DOWD��vQ�IXQF LH�GH�PRGXO�FXP�

VH� DVRFLD] � úL� VH� FRPELQ � IDFWRULL� VLVWHPDWLFL� FX� FHL� vQWkPSO WRUL�� FHL� RELHFWLYL� FX� FHL�subiectivi;

c) formele individuale de manifestare analizate la nivelul anVDPEOXOXL� SDU� DVHP Q WRDUH�vQWUH�HOH��ILLQG�JHQHUDWH�GH�FDX]H�HVHQ LDOH�FRPXQH�

d) IRUPHOH� LQGLYLGXDOH� GH� PDQLIHVWDUH� VH� VXSXQ� DFHOHLDúL� OHJL� GH� DSDUL LH� úL� GH]YROWDUH�FXQRVFXW �úL�YHULILFDW �OD�QLYHOXO�ILHF UHL�XQLW L�GH�REVHUYDUH�

e) formele individuale de maQLIHVWDUH� VH� VXSXQ� DFHOHLDúL� OHJL� VWDWLVWLFH� FDUH� OD� QLYHOXO�DQVDPEOXOXL�VH�PDQLIHVW �FD�WHQGLQ �

7) 6WDELOLUHD� SUHFLV � úL� FODU � D� VFRSXOXL� XQHL� REVHUY UL� VWDWLVWLFH�� WRWDO � VDX� SDU LDO �� QX� SUH]LQW �

LPSRUWDQ �SHQWUX�

a) delimitarea obiectului de observare; b) GHILQLUHD�XQLWD LORU�GH�REVHUYDUH� c) VWDELOLUHD�VFRSXOXL�FHUFHW ULL�VWDWLVWLFH� d) stabilirea programului propriu-]LV�DO�REVHUY ULL� e) GHOLPLWDUHD�RELHFWLYHORU�SDU LDOH�DOH�FHUFHW ULL��FDUH�vQ�FD]XO�REVHUY ULL��VH�UHIHU �FRQFUHW�

OD�YROXPXO�úL�FDOLWDWHD�GDWHORU�QHcesare.

8) Contul sintetic de bunuri:

a) are ca sold consumul intermediar; b) are ca sold consumul final; c) DUH�FD�VROG�YROXPXO�LQYHVWL LLORU�EUXWH� d) nu are sold; e) are ca sold exportul net.

��$QDOL]D L� XUP WRDUHOH� QR LXQL�� P VXUDUHD�� DQDOL]D� úL� VLQWH]D�� HVWimarea; 4) verificarea LSRWH]HORU�úL�VDX�WHVWDUHD�VHPQLILFD LLORU��HODERUDUHD�GHFL]LLORU�

3ULQFLSDOHOH�IXQF LL�DOH�LQGLFDWRULORU�VWDWLVWLFL�VXQW�

a) ��úL�� b) ��úL�� c) ��úL�� d) ��úL�� e) ��úL��

10) 6H�QRWHD] �FX�(i (i = 1 80, ��P ULPHD�³H[WUDVXOXL�GH�FRQW´�VROLFLWDW�OD��GH��GH�FOLHQ L�OD�JKLúHXO�$-%�DO�E QFLL�$OID��9DULDELOD�XUP ULW �SUH]LQW �YDORULOH�SH�R�VFDO �

a) de rapoarte; b) de intervale; c) FRQWLQX � d) GLVFUHW � e) RUGLQDO �

11) 'DF � vQWU-R� GLVWULEX LH� XQLGLPHQVLRQDO � IRUPDW � GXS � R� YDULDELO � QXPHULF � ;� YDORULOH�LQGLYLGXDOH�VXQW�HYLGHQW�GHSODVDWH�F WUH�YDORULOH�PDUL��DWXQFL�

a) x Me Mo< < ; b) x Me Mo> > ; c) x Me Mo= = ; d) x Mo Me< < ; e) x Me Mo x− = − .

12) 9DORDUHD�PHGLDQ � vQ� FRPSDUD LH� FX� YDORDUHD�PHGLH�� LQGLFDWRUL� FDOFXOD L� SHQWUX� FDUDFWHUL]DUHD�WHQGLQ HL�FHQWUDOH�vQWU-R�VHULH�GH�UHSDUWL LH��SUH]LQW �DYDQWDMXO�XUP WRU�

a) vQ�FDOFXOXO�V X�VH�LDX�WRDWH�YDORULOH�LQGLYLGXDOH�vQUHJLVWUDWH� b) YDORDUHD�VD�QX�HVWH�LQIOXHQ DW �GH�SUH]HQ D�YDORULORU�LQGLYLGXDOH�H[WUHPH��DEHUDQWH�� c) HD�VH�SUHWHD] �XúRU�OD�HIHFWXDUHD�RSHUD LLORU�DOJHEULFH� d) GHWHUPLQDUHD�VD�VH�UHDOL]HD] �vQ�IXQF LH�GH�OLPLWD�LQIHULRDU �D�LQWHUYDOXOXL�PHGLDQ�úL�QX�vQ�

IXQF LH�GH�FHQWUXO�LQWHUYDOXOXL��HYLWkQGX-se astfel ipoteza nRUPDOLW LL�UHSDUWL LHL� e) HD�SRDWH�IL�XWLOL]DW �vQ�FRPSDUDUHD�WHQGLQ HORU�FHQWUDOH�DOH�YDULDELOHORU�GLVWLQFWH�vQ�GLIHULWH�

SRSXOD LL�VWDWLVWLFH�

13) ,QGLFDWRULL� VWDWLVWLFL� SRW� IL� RE LQX L� � SULQ� P VXUDUH� úL� HVWLPDUH�� 1R LXQHD� GH� HVWLPDUH� QX� VH�FDUDFWHUL]HD] �SULQ�IDSWXO�F �

a) HVWLPDUHD�SUHVXSXQH�R�FDUDFWHUL]DUH�VWDWLVWLF �XWLOL]kQG�XQ�PRGHO�VDX�R�LSRWH] �VWDWLVWLF �SULQ�FDUH�SH�ED]D�GDWHORU�SDU LDOH�GH�FDUH�VH�GLVSXQH�VH�GHWHUPLQ �R�VHULH�GH�LQGLFDWRUL�JHQHUDOL]DWRUL�

b) HVWH�OHJDW �vQWRWGHDXQD�vQ�VWDWLVWLF �GH�XWLOLzarea unui interval de eroare, ceea ce face ca P ULPHD�LQGLFDWRUXOXL�HVWLPDW�V �QX�DLE �R�YDORDUH�FHUW ��FL�SODVDW �vQWU-un interval;

c) HVWH�OHJDW �QHPLMORFLW�GH�QR LXQHD�GH�PRGHO�VWDWLVWLF� d) HVWH� OHJDW � vQWRWGHDXQD�GH�RE LQHUHD�YDORULL� LQGLFDWRULORU� VLQWHWLFi pe baza datelor culese

printr-R�PHWRG �GH�REVHUYDUH�WRWDO � e) HVWH�XWLOL]DW �SHQWUX�FD]XULOH� vQ�FDUH�QX�H[LVW �XQ�PRGHO�GH� WUHFHUH�GH� OD�XQ�HúDQWLRQ� OD�

SRSXOD LD� JHQHUDO �� LDU� SHQWUX� GHWHUPLQDUHD� YDORULL� XQXL� LQGLFDWRU� VH� IDF� UD LRQDPHQWH�logice, calculul direct nefiind posibil.

14) 'HVSUH� HYROX LD� PLMORDFHORU� IL[H� �OD� VIkUúLWXO� DQXOXL�� vQ� 5RPkQLD� vQ� SHULRDGD� ��-1997 se FXQRVF�GDWHOH��FRQYHQ LRQDOH��

Anii 1994 1995 1996 1997 0RGLILFDUHD�ID �GH�DQXO�SUHFHGHQW��POG��OHL� 194,3 189,8 177,2 166,9

ùWLLQG�F �vQ��YDORDUHD�DEVROXW �D�XQXL�SURFHQW�GH�FUHúWHUH�ID �GH��D�PLMORDFHORU�IL[H�D�IRVW�GH��POG��OHL��PRGLILFDUHD�PHGLH�UHODWLY �DQXDO �vQ�SHULRDGD��- 1997 a fost de :

a) 114,7 %; b) 182,05 mld. lei/an; c) 4,7 %; d) - 4,7 %; e) 14,7 %.

15) Ajustarea (determinarea trendului) seriilor cronologice prin metoda mediilor mobile se UHDOL]HD] �DWXQFL�FkQG�

a) IHQRPHQXO�SUH]LQW �R�HYROX LH�DSUR[LPDWLY�H[SRQHQ LDO � b) LQGLFLL�GH�GLQDPLF �FX�ED] �PRELO �SUH]LQW �YDORUL�DSURSLDWH�vQWUH�HL� c) YDULD LD�WHUPHQLORU�VHULHL�SUH]LQW �HYLGHQWH�UHJXODULW L�FLFOLFH� d) nu se pot aplica metodele analitice de ajustare; e) YDULD LD�WHUPHQLORU�VHULHL�SUH]LQW �R�HYLGHQW �OLQLDULWDWH�

16) ÌQ�DQDOL]D�GLVSHUVLRQDO �VH�XWLOL]HD] �SHQWUX�XQ�SUDJ�GH�VHPQLILFD LH� �úL�QXP UXO�JUDGHORU�GH�libertate (r-1��úL��n-r��WHVWXO�)LVKHU��'DF � rnrcalculat FF −−> ,1,α (din tabel), atunci:

a) OHJ WXUD� HVWH� VHPQLILFDWLY � úL� FDUDFWHULVWLFD� GXS � FDUH� V-a efectuat gruparea este într-DGHY U�GHWHUPLQDQW �SHQWUX�YDULD LD�FDUDFWHULVWLFLL�UH]XOWDWLYH�

b) se respinge ipoteza cu privire la eJDOLWDWHD�PHGLLORU�GH�JUXS � c) WR L� FHLODO L� IDFWRUL�GH� LQIOXHQ � �FX�H[FHS LD�DFHOXLD�GH�JUXSDUH��GHWHUPLQ � VHPQLILFDWLY�

YDULD LD�YDULDELOHL�UH]XOWDWLYH� d) IXQF LD�GH�UHJUHVLH�HVWH�OLQLDU � e) SHQWUX�WHVWDUHD�VHPQLILFD LHL�IDFWRUXOXL�GH�LQIOXHQ �DVXSUD�YDULD LHi variabilei rezultative

VH�XWLOL]HD] �7HVWXO�6WXGHQW��³W´��

17) ÌQ� GHFXUVXO� XQHL� SHULRDGH� SURGXFWLYLWDWHD� PXQFLL� D� FUHVFXW� FX� �� LDU� vQ]HVWUDUHD� WHKQLF � D�PXQFLL�D�FUHVFXW�FX��(ILFLHQ D�XWLOL] ULL�PLMORDFHORU�IL[H�

a) a crescut cu 1,2%; b) a crescut cu 23,2%; c) D�VF ]XW�FX�� d) D�VF ]XW�FX�� e) a crescut cu 1,8%.

18) ,QGLFHOH�PHGLX�GH�GLQDPLF �VH�FDOFXOHD] �FD�

a) PHGLH�JHRPHWULF ��LQGLIHUHQW�GH�YDORULOH�LQGLFLORU�GH�GLQDPLF �FX�ED] �PRELO � b) PHGLH�JHRPHWULF �QXPDL�GDF �LQGLFLL�GH�GLQDPLF �FX�ED] �IL[ �SUH]LQW �YDlori apropiate; c) PHGLH� JHRPHWULF � QXPDL� GDF � LQGLFLL� GH� GLQDPLF � FX� ED] � PRELO � SUH]LQW � YDORUL�

aproximativ egale între ele; d) PHGLH�DULWPHWLF �D�LQGLFLORU�GH�GLQDPLF �FX�ED] �PRELO � e) PHGLH�DUPRQLF �D�LQGLFLORU�GH�GLQDPLF �FX�ED] �PRELO �

19) Un circuit economic este considerat circuit închis pentru sectorul A când:

a) QX�H[LVW �LQWU UL�vQ�$� b) QX�H[LVW �LHúLUL�GLQ�$�

c) QX�H[LVW �QLFL�LQWU UL�úL�QLFL�LHúLUL�SHQWUX�VHFWRUXO�$� d) fluxurile financiare sunt egale cu fluxurile reale; e) VXPD�IOX[XULORU�GH�LQWUDUH�HVWH�HJDO �FX�VXPD�IOX[XULORU�GH�LHúLUH�

20) Într-R� SRSXOD LH� VWDWLVWLF � V-au înregistrat pentru variabila X valorile x1, x2, …, xn�� 5HOD LD�

RELHFWLY � GLQWUH� DFHVWH� GDWH� FRQGXFH� OD� FRQFOX]LD� F � WHQGLQ D� FHQWUDO � SRDWH� IL� FHO� PDL� ELQH�

H[SULPDW �SULQ�PHGLD�DULWPHWLF �D�F UHL�YDORDUH�FDOFXODW �HVWH� x �� ÌQ�XUPD�YHULILF ULL�GDWHORU� úL�FDOFXOHORU� VH� FRQVWDW � F � ILHFDUH� YDORDUH� LQGLYLGXDO � vQUHJLVWUDW �� GLQ� DQXPLWH� PRWLYH�� D� IRVW�PLFúRUDW �FX��$GHY UDWD�YDORDUH�D�PHGLHL�HVWH�

a) 0,85 x ;

b) x

x0 15

6 67,

,= ;

c) 1,15 x ; d) 1,51 x ; e) HVWH�HJDO �FX�PHGLD�FDOFXODW �DQWHULRU�

21) Într-R� SRSXOD LH� VWDWLVWLF � V-DX� XUP ULW� GRX � YDULDELOH�� XQD� QXPHULF � [� úL� XQD� QHQXPHULF �$��6LVWHPDWL]DUHD�GDWHORU�GXS �FHOH�GRX �YDULDELOH�VH�SUH]LQW �vQ�IRUPD�JHQHUDO �DVWIHO�

X

A

x1

...

xj

...

xp

Total

A1 ... ... ... ... ... ... ... ... Ai ni1 ... nij ... nip ni. ... ... ... ... ... ... Ak ... ...

Total ... ... n.j ... ... n..

'LVWULEX LD�DO WXUDW �FRQ LQH�

D��GRX �GLVWULEX LL�PDUJLQDOH� E��GRX �GLVWULEX LL�PDUJLQDOH�úL�S�GLVWULEX LL�FRQGL LRQDWH�GH�$� F��GRX �GLVWULEX LL�PDUJLQDOH�úL�N�GLVWULEX LL�FRQGL LRQDWH�GH�;� G��GRX �GLVWULEX LL�PDUJLQDOH�úL�N�S�GLVWULEX LL�FRQGL LRQDWH�

e) GRX �GLVWULEX LL�PDUJLQDOH�GXS �$�úL�GXS �;��N�S�GLVWULEX LL�XQLGLPHQVLRQDOH�FRQGL LRQDWH�GH�;�úL�$�úL�R�GLVWULEX LH�ELGLPHQVLRQDO �

22) O întreprindere de transport auto-F O WRUL�DORF ��DXWREX]H�SHQWUX�WUDVHXO�0-1�D�F UXL�GLVWDQ �este D. Datele referitoare la vitezele medii înregistrate de autovehicule într-R�]L�VXQW�XUP WRDUHOH�

Viteza medie (km/h) 40 60 70 Nr. de autovehicule 3 5 2

ÌQ�PHGLH�XQ�DXWRYHKLFXO�D�SDUFXUV�vQ�]LXD�FRQVLGHUDW �WUDVHXO�0-1�GH�GLVWDQ �'�FX�R�YLWH] �GH� a) 56 km/h; b) 53,5 km/h; c) 60 km/h; d) 70 km/h; e) 56,7 km/h.

23) ÌQ�FD]XO�DJUHJ ULL� WUDQ]DF LLORU� vQ�6�&�1�� WUDQ]DF LLOH�HIHFWXDWH� vQWUH� VXELHFWHOH� HFRQRPLFH�GLQ�sectoare diferite:

D��VH�FRQVROLGHD] � E��VH�vQVXPHD] � c) se scad; G��XQHOH�VH�vQVXPHD] �LDU�R�DOW �FDWHJRULe se scad; H��DJUHJDUHD�QX�VH�UHIHU �OD�WUDQ]DF LL� 24) 6WUXFWXUD� FRQVXPXOXL� SRSXOD LHL� vQ� SHULRDGD� GH� ED] � HVWH�� P UIXUL� DOLPHQWDUH� �� P UIXUL�

QHDOLPHQWDUH� �� VHUYLFLL� �� ÌQ� SHULRDGD� FXUHQW � SUH XULOH� DX� FUHVFXW� DVWIHO�� P UIXUL�DOLPHQWDUH��P UIXUL�QHDOLPHQWDUH��VHUYLFLL��5DWD�LQIOD LHL�D�IRVW�

a) 25%; b) 23,25%; c) 125%; d) 123,25%;

f) 3,25%. 25) 3HQWUX�RFXSDUHD�XQXL� ORF�GH�PXQF �YDFDQW�GH�VWHQRGDFWLORJUDI ��V-au prezentat la concurs opt

candidate. Testul propus a constat din dactilografieUHD� XQXL� WH[W� �GH� DFHHDúL� P ULPH� úL�GLILFXOWDWH��'DWHOH�FX�SULYLUH�OD�QXP UXO�GH�HURUL�FRPLVH�úL�WLPSXO�GH�GDFWLORJUDILHUH�VH�SUH]LQW �astfel:

Candidata A B C D E F G H Nr. erori 9 8 6 4 5 10 12 7 Timp dactilo (minute) 13 11 12 15 14 9 8 10

'HSHQGHQ D�GLQWUH�FHOH�GRX �YDULDELOH�VH�P VRDU �FX�DMXWRUXO�FRHILFLHQWXOXL�GH�FRUHOD LH�DO�UDQJXULORU��DO�OXL�.HQGDOO��D�F UXL�P ULPH�HVWH� a) 0,71; b) -0,30; c) -1,00; d) -0,71; e) 1,00. 26) De la un punct de desfacere s-DX�FXOHV�GDWHOH�XUP WRDUH�

Produse 9DORDUHD�YkQ] ULORU�vQ�LDQXDULH�� (mil. lei)

��PRGLILF ULL�YROXPXOXL�IL]LF�vQ�LDQXDULH��ID �GH�LDQXDULH��

A 18 +5 B 20 -2

0RGLILFDUHD� DEVROXW � WRWDO � D� YkQ] ULORU� vQ� LDQXDULH� �� ID � GH� LDQXDULH� �� FD� XUPDUH� D�PRGLILF ULL�YROXPXOXL�fizic este: a) 500 mii lei; b) 1,30 mil. lei; c) -500 mii lei; d) 38 mil. lei; e) 20,76 mil. lei.

27) &RQVWLWXLH�HOHPHQW�DO�FRPSHQV ULL�IDFWRUXOXL�PXQF �FRPSRQHQW �D�3,%� a) profitul;

E��VXPHOH�SO WLWH�DWkW�vQ�EDQL�FkW�úL�vQ�QDWXU �

c) dividendele;

d) renta;

H��QXPDL�VDODULLOH�SO WLWH�vQ�EDQL�

28) 6H�FXQRDúWH�IDSWXO�F � vQ�VWXGLXO�GHSHQGHQ HORU�VWDWLVWLFH�GLQWUH�YDULDELOHOH�QXPHULFH�<�úL�;�VH�XWLOL]HD] � úL� LQGLFDWRUXO�PRPHQW� FHQWUDW� GH� RUGLQXO� �� QXPLW� úL� FRYDULDQ �� QRWDW � FRY�[�\��Valoarea absoluW �PD[LP �D�DFHVWXL�LQGLFDWRU�HVWH�

a) 1; b) σ x ;

c) σσ

x

y

2

2;

d) σ y ;

e) σ σx y .

29) ÌQ�FDOFXOXO�3URGXVXOXL�,QWHUQ�%UXW��H[SRUWXO�QHW�VH�FDOFXOHD] � a) Export-Impozite;

b) Export-Import;

c) Export-Amortizarea;

d) Export+Impozite indirecte-6XEYHQ LL�

e) Export+Import.

30) $QFKHWD�VWDWLVWLF �� D��HVWH�R�PHWRG �GH�REVHUYDUH�WRWDO �

b) HVWH� R� PHWRG � GH� REVHUYDUH� SDU LDO �� FDUH� QXPDL� vQWkPSO WRU� SRDWH� V � vQGHSOLQHDVF �

FRQGL LD�GH�UHSUHzentativitate;

c) HVWH� R� PHWRG � GH� REVHUYDUH� SDU LDO �� FDUH� vQ� PRG� REOLJDWRULX� WUHEXLH� V � vQGHSOLQHDVF �

FRQGL LD�GH�UHSUH]HQWDWLYLWDWH�

d) HVWH� R� PHWRG � GH� REVHUYDUH� SDU LDO �� FDUH� VH� ED]HD] � SH� REOLJDWLYLWDWHD� FRPSOHW ULL�

chestionarelor;

H��PDL�HVWH�GHQXPLW �úi sondaj statistic.

31) 2�SRSXOD LH�VWDWLVWLF �HVWH�VWUXFWXUDW �GXS �R�DQXPLW �YDULDELO �vQ�GRX �VXESRSXOD LL�$�úL�%��ÌQ�

ILHFDUH� GLQ� FHOH� GRX � VXESRSXOD LL�� V-D� XUP ULW� DFHHDúL� YDULDELO � QXPHULF � ;�� LDU� FHOH� GRX �GLVWULEX LL�VH�SUH]LQW �JUDILF�DVWIHO�

x

Observându-VH�JUDILFXO�DO WXUDW��VH�SRDWH�DILUPD�

a) LQGLFDWRULL� WHQGLQ HL� FHQWUDOH� SUH]LQW � YDORUL� GLIHULWH� vQ� FHOH� GRX � VXESRSXOD LL�� LDU�GLVWULEX LD�JHQHUDO �HVWH�DVLPHWULF �

b) H[LVW � YDORUL� PRGDOH� SDU LDOH� GLIHULWH�� GDU� UHOD LD� GH� Dditivitate a acestora permite GHWHUPLQDUHD�YDORULL�PRGDOH�GLQ�GLVWULEX LD�JHQHUDO �

c) YDORULOH�LQGLYLGXDOH�GLQ�FHOH�GRX �VXESRSXOD LL�DX�DFHODúL�JUDG�GH�GLVSHUVDUH� d) YDORDUHD�PRGDO �D�GLVWULEX LHL�JHQHUDOH�HVWH�FXSULQV �vQWUH�0RA�úL�0RB; e) valorile mediane ale FHORU�GRX �VXESRSXOD LL��0HA�úL�0HB) sunt identice.

32) $SOLFDUHD�XQXL�PRGHO�GH�DQDOL] �GLVSHUVLRQDO �FX�XQ�VLQJXU�IDFWRU�SHQWUX�VHULL�VLPSOH�SUHVXSXQH�

FDOFXOXO�úL�LQWHUSUHWDUHD�LQGLFDWRULORU�XUP WRUL�� ( )S y y ni ii

r

12

1

= −=∑ SHQWUX�D�HYDOXD�FD�VXUV �GH�

YDULD LH� LQIOXHQ D� vQ� H[FOXVLYLWDWH� D� IDFWRUXOXL� VLVWHPDWLF� úL� ( )S y yij ii

r

j

n

22

11

= −==∑∑ pentru a

HYDOXD�LQIOXHQ D�WXWXURU�FHORUODO L�IDFWRUL��FX�H[FHS LD�IDFWRUXOXL�VLVWHPDWLF�L]RODW��3HQWUX�D�HYDOXD�FRQWULEX LD� IDFWRUXOXL� VLVWHPDWLF� OD� YDULD LD� YDULDELOHL� GHSHQGHQWH� VH� FDOFXOHD] � XQ� LQGLFDWRU� GH�forma:

a) S

S1

2;

b) S

S2

1;

c)

( )S

y yiji

r

j

n1

2

11

−==∑∑

;

d)

( )S

y yiji

r

j

n2

2

11

−==∑∑

;

A

B

MoA MoB

ni

e)

( ) ( )y y n

r

y y

n r

i ii

r

ij ij

n

i

r

−

−

−

−= ==∑ ∑∑2

1

2

11

1: .

33) 9HQLWXO�QD LRQDO�HVWH�HJDO�FX�

a) 31%�OD�SUH XULOH�SLH HL� b) 311�OD�SUH XULOH�SLH HL� c) PNN la�SUH XULOH�IDFWRULORU� d) 3,1�OD�SUH XULOH�IDFWRULORU� e) 3,1�OD�SUH XULOH�SLH HL�

34) 3HQWUX� LGHQWLILFDUHD� úL� GLPLQXDUHD� HURULORU� GH� REVHUYDUH� HVWH� QHFHVDU� FRQWUROXO� GDWHORU� FXOHVH��Acest control nu presupune:

a) FD�SULQ�VRQGDM�V �VH�UHIDF ��DQXPLWH�FDOFXOH�GH�RE LQHUH�D�YDORULORU�XQRU�LQGLFDWRUL�vQVFULúL�în formulare;

b) FD�OD�FHQWUHOH�GH�SUHOXFUDUH�V �VH�YHULILFH�GDF �DX�VRVLW�WRDWH�IRUPXODUHOH�FX�WRDWH�UXEULFLOH�completate;

c) XWLOL]DUHD� XQRU� PRGHOH� GH� YHULILFDUH� D� LSRWH]HORU� VWDWLVWLFH� úL� DSOLFDUHD� GH� WHVWH� GH�verificaUH�DOH�VHPQLILFD LLORU�LQGLFDWRULORU�

d) HIHFWXDUHD�GH�FRPSDU UL� e) VXE�QLFL�R�IRUP �XWLOL]DUHD�XQRU�SURJUDPH�LQIRUPDWLFH�VSHFLDO�HODERUDWH�

35) 'HQVLWDWHD�XQHL�UHSDUWL LL�VWDWLVWLFH�HVWH�YL]XDOL]DW �SULQ�

a) KLVWRJUDP � b) FXUED�FXPXODWLY �GH�IUHFYHQ H� c) diagrama prin benzi; d) diagrama prin coloane; e) curba de concentrare a lui Lorentz.

36) ÌQ�GRX �SRSXOD LL� VWDWLVWLFH�$� úL�%�GH�DFHODúL� YROXP�R� YDULDELO � QXPHULF �;�SUH]LQW � DFHOHDúL�

YDULDQWH��3HQWUX�FHOH�GRX �GLVWULEX LL�GH�IUHFYHQ H�V-au calculat mediile x A úL� x B . Aceste medii sunt egale deoarece:

a) YDORULOH�LQGLYLGXDOH�GLQ�$�úL�%�VXQW�LGHQWLFH� b) YDORDUHD�ORU�GHSLQGH�QX�QXPDL�GH�YDORULOH�LQGLYLGXDOH�FL�úL�GH�IUHFYHQ HOH�ORU�GLIHULWH�GH�

DSDUL LH� c) VH�FDOFXOHD] �GXS �DFHHDúL�PHWRG ��LDU�YROXPXO�SRSXOD LLORU�VWDWLVWLFH�HVWH�DFHODúL� d) IUHFYHQ HOH�GH�DSDUL LH�DOH�YDORULORU�LQGLYLGXDOH�GLQ�$�úL�%�VXQW�LGHQWLFH��GDU�GLIHULWH�GH�OD�

R�SRSXOD LH�OD�DOWD� e) vQ�$�úL�%�YDORULOH�LQGLYLGXDOH�VXQW�LGHQWLFH�úL�DX�DFHOHDúL�IUHFYHQ H�GH�DSDUL LH�

37) ,QGLFHOH�SUH urilor consumatorului este calculat ca un indice de tip:

a) Paasche; b) Laspeyres; c) Fischer; d) indice al valorii; e) indice al volumului fizic.

38) 'RL� VWDWLVWLFLHQL� DX� DQDOL]DW� DFHOHDúL� GDWH� UHIHULWRDUH� OD� ³�� vQGHSOLQLULL� QRUPHORU´� D� �� GH�PXQFLWRUL�GH�DFHHDúL�SURIHVLH��3HQWUX�D�FDUDFWHUL]D�WHQGLQ D�FHQWUDO �D�YDULDELOHL��XQXO�D�FDOFXODW�PHGLD� DULWPHWLF �� RE LQkQG� �� LDU� FHO ODOW� D� FRQVLGHUDW� RSRUWXQ �PHGLD� S WUDWLF �� RE LQkQG�107% (amândoi au efectuat corect calculele). Pentru a caracteriza omogenitatea datelor este QHFHVDU �GHWHUPLQDUHD�FRHILFLHQWXOXL�GH�YDULD LH��9DORDUHD�DFHVWXLD�HVWH�

a) 19,6%; b) 25%; c) -75%; d) ��SHUVRDQ � e) 50%.

39) Într-XQ� HúDQWLRQ� GH� �� GH� DJHQ L� HFRQRPLFL� V-DX� XUP ULW� GRX � YDULDELOH� GHSHQGHQWH� VWDWLVWLF��SURILWXO� QHW� �PLO�� OHL�� úL� QXP UXO� PHGLX� GH� VDODULD L�� 'HVSUH� GLVWULEX LD� FRPELQDW � IRUPDW � SH�SDWUX� LQWHUYDOH�HJDOH�GXS �SURILW�úL�SH� WUHL� LQWHUYDOH�GXS �QXP UXO�PHGLX�GH�VDODULD L�VH�FXQRVF�XUP WRDUHOH�GDWH��SH� vQWUHJXO�HúDQWLRQ�SURILWXO�PHGLX�D�IRVW�GH��PLO�� OHL�� LDU�FRHILFLHQWXO�GH�YDULD LH�D�IRVW�GH��SURILWXO�PHGLX�D�IRVW�GH��PLO��OHL�OD�JUXSD�DJHQ LORU�HFRQRPLFL�FX�QU��PHGLX�GH�VDODULD L�VXE��SHUVRDQH��PLO��OHL�OD�JUXSD��-��QU��PHGLX�GH�VDODULD L�úL��PLO��OHL�OD�JUXSD��úL�SHVWH�QU��PHGLX�GH�VDODULD L��QXP UXO�GH�DJHQ L�HFRQRPici pe cele trei grupe sunt ��úL��0 VXUD�vQ�FDUH�PRGLILFDUHD�QXP UXOXL�PHGLX�GH�VDODULD L�FRQWULEXLH�OD�YDULDELOLWDWHD�SURILWXOXL�DJHQ LORU�HFRQRPLFL�D�IRVW�GH�

a) 46,6%; b) 53,4%; c) -10%; d) 110%; e) 100%.

40) ÌQ�DQDOL]D�VHULLORU�GH�GDWH�VWDWLVWLFH�VH�XWLOL]HD] �QR LXQLOH�

1. YDULDELOLWDWHD��LQGHSHQGHQ D��LQWHUGHSHQGHQ D��FRQFHQWUDUHD�� RPRJHQLWDWHD��IRUPD�GH�UHSDUWL LH��SHULRGLFLWDWHD�

3URSULHW LOH��SDUWLFXODULW LOH��VHULLORU�FURQRORJLFH�VXQW�

a) ��úL�� b) ��úL�� c) ��úL�� d) ��úL 7.

TESTUL 3

1) 6H� GDX� GDWHOH� UHIHULWRDUH� OD� UHSDUWL LD� VDODULD LORU� XQHL� VRFLHW L� FRPHUFLDOH� GXS � YHFKLPHD� vQ�

PXQF �

Grupe (ani) 1U��VDODULD L

( ]0 5− 3

( ]5 10− 5

( ]10 15− 9

( ]15 20− 15

( ]20 25− 9

( ]25 30− 5

peste 30 3 TOTAL n=49

9HFKLPHD�PHGLH�D�XQXL�VDODULDW�vQ�FDGUXO�VRFLHW LL�FRPHUFLDOH�HVWH�

a) 12,5 ani; b) 21,5 ani; c) 9,75 ani; d) 17,5 ani; e) 24,5 ani.

2) $JUHJDWHOH�PDFURHFRQRPLFH�VH�GHWHUPLQ �OD�SUH XULOH�SLH HL�VDX�OD�SUH XULOH�IDFWRULORU��7UHFHUHD�

GH�OD�R�FDWHJRULH�GH�SUH XUL�OD�DOWD�VH�UHDOL]HD] �

a) SUH XO�IDFWRULORU� �SUH XO�SLH HL�- impozitele indirecte; b) SUH XO�IDFWRULORU� �SUH XO�SLH HL��LPSR]LWHOH�LQGLUHFWH� c) SUH XO�IDFWRULORU� �SUH XO�SLH HL�- impozitele indirecte nete; d) SUH XO�IDFWRULORU� �SUH XO�SLH HL��VXEYHQ LLOH� e) SUH XO�IDFWRULORU� �SUH XO�SLH HL�-�VXEYHQ LLOH�

3) Indicatorii de rezultate macroeconomice (agregatele macroeconomice) se pot calcula ca

LQGLFDWRUL�GH�QDWXU �EUXW �úL�GH�QDWXU �QHW ��&DOFXOXO�LQGLFDWRULORU�vQ�YDULDQWD�QHW �VH�IDFH�

a) ,QGLFDWRULL� GH� QDWXU � QHW � � ,QGLFDWRULL� GH� QDWXU � EUXW � ��$ORFD LD� SHQWUX� FRQVXPXO� GH�capital fix;

b) ,QGLFDWRULL�GH�QDWXU �QHW � �,QGLFDWRULL�GH�QDWXU �EUXW �-�6ROGXO�YDORULL�DG XJDWH�EUXWH�vQ�UDSRUW�FX�VWU LQ WDtea;

c) ,QGLFDWRULL�GH�QDWXU �QHW � �,QGLFDWRULL�GH�QDWXU �EUXW ��6ROGXO�YDORULL�DG XJDWH�EUXWH�vQ�UDSRUW�FX�VWU LQ WDWHD�

d) ,QGLFDWRULL� GH� QDWXU � QHW � � ,QGLFDWRULL� GH� QDWXU � EUXW � -� $ORFD LD� SHQWUX� FRQVXPXO� GH�capital fix;

e) ,QGLFDWRULL�GH�QDWXU �QHW � �,QGLFDWRULL�GH�QDWXU �EUXW �- Exportul net. 4) 9HQLWXO� SHUVRQDO� GLVSRQLELO� �93'�� H[SULP � YHQLWXULOH� JRVSRG ULLORU� �PHQDMHORU�� FH� SRW� IL�

XWLOL]DWH�SHQWUX�DFRSHULUHD�FKHOWXLHOLORU�SHUVRQDOH�úL�SHQWUX�HFRQRPLVLUH� 9HQLWXO�3HUVRQDO�'LVSRQLELO�VH�FDOFXOHD] �

a) VPD = Veniturile personale + Impozitele personale; b) 93'� �9HQLWXO�1D LRQDO�-�6ROGXO�YDORULL�DG XJDWH�EUXWH�vQ�UDSRUW�FX�VWU LQ WDWHD� c) VPD = Veniturile personale - Impozitele personale; d) 93'� �9HQLWXO�1D LRQDO�- Impozitele pe veniturile firmelor + Transferurile de la Guvern

úL�ILUPH� e) 93'� �9HQLWXO�1D LRQDO��,PSR]LWHOH�SHQWUX�DVLJXU UL�VRFLDOH��'REkQ]LOH�SULPLWH�GH�SRSXOD LH�

5) ,QGLFDWRULL� PDFURHFRQRPLFL� GH� UH]XOWDWH� �DJUHJDWHOH�� VH� GHWHUPLQ � FD� LQGLFDWRUL� QRPLQDOL� �vQ�SUH XUL� FXUHQWH�� úL�VDX� FD� LQGLFDWRUL� UHDOL� �vQ� SUH XUL� FRPSDUDELOH� VDX� FRQVWDQWH�� &DOFXOXO� vQ�H[SUHVLH�UHDO �VH�HIHFWXHD] �

a) Agregat real = Agregat nominal -�$ORFD LD�SHQWUX�FRQVXPXO�GH�FDSLWDO�IL[��$&&)��

b) )Laspeyres(ilorproducatorpreturilorIndicele

realAgregatQRPLQDO$JUHJDW=

c) )PIBdeflatorul(pretuluiIndicele

realAgregatLDJUHJDWXOX

QRPLQDO$JUHJDW=

d) $JUHJDW�UHDO� �$JUHJDW�QRPLQDO�[�,QGLFHOH�SUH XOui agregatului (deflatorul PIB); e) Agregat real = Agregat nominal x Indicele Paasche.

6) 2�VHULH�GH�WLPS�VH�UHSUH]LQW �JUDILF�SULQWU-o:

a) FRUHORJUDP � b) GLDJUDP �GH�VWUXFWXU � c) FURQRJUDP ��KLVWRULRJUDP �� d) FDUWRJUDP � e) FDUWRGLDJUDP �

7) 'DF �VH�QRWHD] �� x a -�PHGLD�DULWPHWLF �� x g -�PHGLD�JHRPHWULF �� x h -�PHGLD�DUPRQLF �� x p - PHGLD�S WUDWLF ��UHOD LD�GH�RUGLQH�GLQWUH�PHGLL�HVWH�XUP WRDUHD�

a) x x x xp a h g≤ ≤ ≤ ;

b) x x x xa h g p≤ ≤ ≤ ;

c) x x x xh a g p≤ ≤ ≤ ;

d) x x x xg p a h≤ ≤ ≤ ;

e) x x x xh g a p≤ ≤ ≤ .

8) 'DF �VH�QRWHD] � x a -�PHGLD�DULWPHWLF ��[mo -�YDORDUHD�PRGDO ��[me -�YDORDUHD�PHGLDQ ��vQWU-o

VHULH�GH�UHSDUWL LH��FHL�WUHL�LQGLFDWRUL�VXQW�HJDOL�� x a =xmo = xme) atunci când:

a) seria este de tip calitativ; b) VHULD�HVWH�OHSWRFXUWLF � c) VHULD�HVWH�SODWLFHQWLF � d) VHULD�HVWH�SHUIHFW�VLPHWULF � e) seria este de tip normal.

9) 9HULILFDUHD� FRQFRUGDQ HL� GLQWUH� UHSDUWL LD� H[SHULPHQWDO � �³HPSLULF ´�� RE LQXW � vQWU-o cercetare FRQFUHW � úL� UHSDUWL LD� WHRUHWLF � SUHVXSXV � �GH� H[HPSOX� QRUPDO �� VH� HIHFWXHD] � FX� DMXWRUXO�PDL�multor teste.

Expresia:

( )n n p

n pi i

ii

k − ⋅⋅=

∑2

1

UHSUH]LQW �VWDWLVWLFD�WHVWXOXL� a) Shapiro - Wilk; b) Lalliefors; c) Hi-S WUDW��χ2); d) Kolmogorov-Smirnov; e) Gnedenko.

10) 3HQWUX�D�GLPLQXD�HURDUHD�VLVWHPDWLF �LQWURGXV �OD�FDOFXOXO�GLVSHUVLHL�vQWU-R�VHULH�GH�UHSDUWL LH�SH�

LQWHUYDOH�� SULQ� IRORVLUHD� FHQWUHORU� GH� LQWHUYDO��:�)�6KHSSDUG� D� SURSXV� R� FRUHF LH�� DVWIHO� vQFkW�UHOD LD�GH�FDOFXO�HVWH�

a) σ ’ ( )’

2 2 2= ⋅ − −∑∑

x f

fh x hi i

i;

b) σ σ’2 22

12= − h

;

c) ( )σ ’2 1= −p p ;

d) σ ’22

2

= −

∑∑

∑∑

x fi

fi

x fi

fii i

;

e) ( )

σ ’22

100=

−∑ x x fi i.

Unde σ ’2 -� YDORDUHD� UHFDOFXODW � D� GLVSHUVLHL�� h -� P ULPHD� LQWHUYDOXOXL� GH� JUXSDUH�� xi - valorile

caracteristicii; x - media; fi -�IUHFYHQ HOH��p - media caracteristicii binare. 11) 'HWHUPLQDUHD� YROXPXOXL� HúDQWLRQXOXL� vQ� YDULDQWD�� ³DOHDWRU� VLPSOX� UHSHWDW´� VH� HIHFWXHD] �

XWLOL]kQG�UHOD LD�

a) nz

z

Nx

=⋅

+⋅

α

α

σσ

2 2

22 2

∆

;

b) nn N

N

j j

j jj

k=

⋅ ⋅

⋅=∑

2 2

2

1

σ

σ;

c) nz z

z z

k

p p=

+

−

⋅ +

α β

1 2

22

12

;

d) n

p

pp

p

p

p

=

−−

−−−

lg

lg lg

1

11

1

1

2

2

1

2

1

;

e) nz

x

=⋅α σ2 2

2∆.

12) ,QGLFLL� SUH XULORU� PHGLL� OD� SULQFLSDOHOH� SURGXVH� YkQGXWH� SH� SLD D� U QHDVF � VH� GHWHUPLQ � vQ�

varianta:

a) Laspeyres; b) Edgeworth; c) Fischer; d) Paasche; e) Marschal.

13) 3H�ED]D�XQXL�VRQGDM�Q ��VWXGHQ L��HIHFWXDW�DOHDWRU�vQ�FDGUXO�IDFXOW LL��V-au înregistrat diferite

FDUDFWHULVWLFL�� 3UHOXFUkQG� GDWHOH� UHIHULWRDUH� OD� GRX � GLQWUH� DFHVWH� FDUDFWHULVWLFL� V-DX� RE LQXW�XUP WRDUHOH�UH]XOWDWH�

Culoarea ochilor GHVFKLV vQFKLV

Culoarea deschis 17 6 23 S UXOXL închis 9 19 28

Total 26 25 51

&RHILFLHQWXO�GH�DVRFLHUH��FRQFRUGDQ ��vQWUH�FHOH�GRX �FDUDFWHULVWLFL�DUH�YDORDUHD�

a) 0,243; b) 0,713; c) 0,980; d) 0,01; e) 0,901.

14) 9DORDUHD�UDSRUWXOXL�GH�FRUHOD LH�HVWH�HJDO �FX�YDORDUHD�FRHILFLHQWXOXL�GH�FRUHOD LH�DWXQFL�FkQG�

a) UHSDUWL LD�HVWH�XQLPRGDO � b) UHSDUWL LD�HVWH�VLPHWULF � c) OHJ WXUD�GLQWUH�YDULDELOH�HVWH�OLQLDU � d) OHJ WXUD�GLQWUH�YDULDELOH�HVWH�LQYHUV � e) OHJ WXUD�GLQWUH�YDULDELOH�HVWH�GLUHFW �

15) Produsul Intern Brut a fost 109.515 mld. lei în anul 1996, respectiv de 249.750 mld. lei în 1997,

VF GHUHD� �UHFDOFXODW � vQ� SUH XUL� FRPSDUDELOH�� ILLQG� � GH� -�� 5DWD� LQIOD LHL� vQ� DQXO� ��comparativ cu 1996 a fost:

a) 244,16%; b) 109,20%;

c) 344,16%; d) 57,20%; e) 144,16%.

16) ,QGLFLL�SUH XULORU�GH�FRQVXP�DO�SRSXOD LHL�vQ�DQXO��ID �GH��DX�IRVW�

• OD�P UIXUL�DOLPHQWDUe 251,4%; • OD�P UIXUL�QHDOLPHQWDUH�� • la servicii 276,5%.

Ponderile celor trei grupe în totalul cheltuielilor au fost: • P UIXUL�DOLPHQWDUH�� • P UIXUL�QHDOLPHQWDUH�� • servicii 12%;

&XQRVFkQG� IDSWXO� F � vQ� �� ID � GH� �� FkúWLJXO� VDODULDO� PHGLX� QRminal net a fost de ��GLQDPLFD�VDODULLORU�UHDOH�D�IRVW�vQ��HJDO �FX�

a) 77,6%; b) 112,5%; c) 177,6%; d) 100,0%; e) 47,6%.

17) /D�R�ILUP ��vQ�GHFXUVXO�XQHL�SHULRDGH��V-DX�vQUHJLVWUDW�XUP WRDUHOH�GDWH�

• LQGLFHOH�vQ]HVWU ULL�WHKQLFH�D�PXQFLL�� • modificarea prRFHQWXDO �D�HILFLHQ HL�FDSLWDOXOXL�IL[�-5%; • LQGLFHOH�QXP UXOXL�GH�VDODULD L��

0RGLILFDUHD�SURGXF LHL�HVWH�HJDO �FX�

a) 100%; b) 0,9576; c) -7,5%; d) +12,3%; e) 112,3%.

18) ,QGLFHOH�VLQWHWLF��GH�JUXS��VWDELOLW�FD�PHGLH�DUPRQLF �D�LQGLFLORU�LQGLYLGXDOL�HVWH�

a) x f

f

x f

f1 1

1

0 0

0

∑∑

∑∑

: ;

b) ( )( )

x f f

x f f1 1 0

0 1 0

++

∑∑

;

c) x f

x f

x f

x f0 1

0 0

1 1

1 0

∑∑

∑∑

⋅ ;

d) x f

ix f

x

1 1

1 11

∑∑

;

e) q

q

q

q

q

qn n n

n0

1

0 1: −

−= .

19) Într-R�VHULH�GH�LQGLFL��SHQWUX�IDFWRUXO�FDOLWDWLY��[��YDULDQWD�³ED] �PRELO ´�úL�³SRQGHUH�YDULDELO ´�

este:

a) x f

x f

x f

x f

x f

x f

x f

x fi i

i i

n n

n n

1 0

0 0

2 1

1 1

1

1 1

1

1 1

∑∑

∑∑

∑∑

∑∑

−

− −

−

− −; ; ...; ; ...; ;

b) x f

x f

x f

x f

x f

x f

x f

x fi

i i

n n

n n

1 1

0 0

2 2

1 1

1

1 1 1 1

∑∑

∑∑

∑∑

∑∑− − − −

; ; ...; ; ...; ;

c) x f

x f

x f

x f

x f

x f

x f

x fi i

i i

n n

n n

0 0

1 1

1 1

2 2

1 1 1 1∑∑

∑∑

∑∑

∑∑

− − − −; ; ...; ; ...; ;

d) x f

x f

x f

x f

x f

x f

x f

x fi i

i i

n n

n n

1 1

0 1

2 2

1 2 1 1

∑∑

∑∑

∑∑

∑∑− −

; ; ...; ; ...; ;

e) x f

x f

x f

x f

x f

x f

x f

x fi

i

n

n

1 1

1 1

1 2

1 2

1

1

1

1

∑∑

∑∑

∑∑

∑∑

; ; ...; ; ...; .

20) 5HIHULWRU�OD�DFWLYLWDWHD�HFRQRPLF �D�XQHL�VRFLHW L�FRPHUFLDOH�VH�FXQRVF�XUP WRDUHOH�GDWH�

Unitatea 9DORDUHD�SURGXF LHL��PLO��OHL� 3HU��GH�ED] 3HU��FXUHQW

Modificarea valorii pe seama modiILF ULL�

SUH XULORU��PLO��OHL� A 1200 1850 350 B 875 1600 275

0RGLILFDUHD�UHODWLY �D�YDORULL�OD�QLYHOXO�vQWUHJLL�VRFLHW L��FDX]DW �GH�PRGLILFDUHD�YROXPXOXL�IL]LF�DO�SURGXF LHL�vQWUH�FHOH�GRX �SHULRDGH�D�IRVW�HJDO �FX�

a) -12,5%; b) +100,0%; c) +36,1%; d) +136,1%; e) -112,5%.

21) ÌQ� VWDELOLUHD� WUHQGXOXL� �³DMXVWDUHD´� VHULHL� SHQWUX� LGHQWLILFDUHD� WHQGLQ HL� FHQWUDOH�� PHWRGD�

PRGLILF ULL�PHGLL�DEVROXWH�VH�DSOLF �DWXQFL�

a) FkQG�HYROX LD�HVWH�H[SRQHQ LDO � b) FkQG�PRGLILF ULOH�FX�ED] �PRELO �VXQW�DSUR[Lmativ egale; c) FkQG�VHULD�HVWH�IRUPDW �GLQWU-XQ�QXP U�LPSDU�GH�WHUPHQL� d) FkQG�VHULD�HVWH�IRUPDW �GLQ�P ULPL�UHODWLYH� e) FkQG�VHULD�HVWH�GH�PRPHQWH�LQHJDO�GLVWDQ DWH�

22) 0HWRGD�LQIOXHQ HORU�L]RODWH��³UHVWXOXL�QHGHVFRPSXV´��QX�VH�SRDWH�DSOLFD�DWXQFL�FkQG�

a) x x f f1 0 1 0= =; ;

b) x x f f1 0 1 0< <; ;

c) x x f f1 0 1 0< >; ;

d) x x f f1 0 1 0> <; ;

e) x x f f1 0 1 0> =; .

23) 6H�FXQRVF�XUP WRDUHOH�GDWH�

Ramuri economice 'LQDPLFD�9$%�ID �GH��

(%) Structura VAB pe ramuri

(%) Industrie 95,0 36,8 $JULFXOWXU ��6LOYLFXOWXU 101,4 20,8 &RQVWUXF LL 78,2 7,9 Servicii 88,8 34,5

0RGLILFDUHD�SRSXOD LHL�RFXSDWH�vQ��ID �GH��D�IRVW�GH�-2%. 'LQDPLFD�SURGXFWLYLW LL�VRFLDOH�D�PXQFLL��D�IRVW�HJDO �FX�

a) 94,76%; b) 102,51%; c) 104,76%; d) 2,51%; e) 194,76%.

24) Cunoscând valorile observate (Yi��úL�FHOH�DMXVWDWH��UHFDOFXODWH�SULQWU-un model analitic ( �Yi ):

�Yi : 98; 109; 121; 145; 152 Yi: 99; 108; 119; 147; 152 &RHILFLHQWXO�GH�GHWHUPLQD LH�DO�PRGHOXOXL�GH�UHJUHVLH�SULQ�FDUH�V-a realizat ajustarea este egal cu:

a) 1,215; b) 0,095; c) 0,875; d) 0,962; a) 0,715.

25) Pentru estimarea fondului de salarizare necesar într-R�OXQ � OD�R�6&�FX��GH�DQJDMD L��GLQ�FDUH�

�� DQJDMD L� SHUPDQHQW� úL� �� FRODERUDWRUL�� V-a organizat un sondaj aleator stratificat SURSRU LRQDO��I U �UHYHQLUH��GH�YROXP��GLQ�FROHFWLYLWDWHD�JHQHUDO ��'LQ�SUHOXFUDUHD�GDWHORU��GLQ�HúDQWLRQ�D�UH]XOWDW�XQ�VDODULX�PHGLX�DO�DQJDMD LORU�FX�FRQWUDFW�GH�PXQF �GH��PLO��OHL�úL�XQ�VDODULX�PHGLX�DO�DQJDMD LORU�vQ�UHJLP�GH�FRODERUDUH�GH��PLO� lei, iar dintr-R�FHUFHWDUH�DQWHULRDU �VH�úWLH�F �IDFWRUXO�ÄUHJLPXO�GH�DQJDMDUH´� LQIOXHQ HD] �YDULD LD�VDODULXOXL� vQ�SURSRU LH�GH��)RQGXO�GH�salarizare estimat pentru întreaga S.C. (pentru o probabilitate de 0,9973, z � � =3), este:

a) (3,32 mil. lei; 4,28 mil. lei); b) (2568 mil. lei; 3504 mil. lei); c) (2656 mil. lei; 3424 mil. lei); d) (3,21 mil. lei; 4,38 mil. lei); e) (2800 mil. lei; 3280 mil. lei).

26) 2�FUHúWHUH�D�HILFLHQ HL�DFWLYLW LL�HFRQRPLFH�QX�VH�UHDOL]HD] �GDF � a) LQGLFHOH�GH�GLQDPLF �D�HILFLHQ HL�IRORVLULL fondurilor fixe este supraunitar; b) GLQDPLFD�SURGXF LHL�PDUI �GHYDQVHD] �GLQDPLFD�FDSLWDOXOXL�IL[� c) LQGLFHOH�GH�GLQDPLF �D�QHFHVDUXOXL�GH�PLMORDFH�IL[H�HVWH�VXSUDXQLWDU� d) indicele de dinaPLF �D�SURGXFWLYLW LL�PXQFLL�HVWH�VXSUDXQLWDU� e) GLQDPLFD�SURGXF LHL�PDUI �GHYDQVHD] �GLQDPLFD�QXP UXOXL�GH�VDODULD L�

27) ÌQ�OXQD�LDQXDULH��vQ�GRX �ILOLDOH�DOH�XQHL�ILUPH�V-DX�vQUHJLVWUDW�XUP WRDUHOH�GDWH�

- vQ�ILOLDOD�$��FHL��DQJDMD L�UHSUH]HQWkQG��GLQ�WRWDOXO�DQJDMD LORU�ILUPHL�PDX�UHDOL]DW�R�

SURGXF LH�GH��PLO��OHL;

- în filiala B s-a realizat o productivitate medie a muncii de 3,1 mil. lei.

ùWLLQG�F �vQ�OXQD�IHEUXDULH�SURGXFWLYLWDWHD�PHGLH�D�PXQFLL�SH�DQVDPEOX�FHORU�GRX �ILOLDOH�D�IRVW�GH�

��PLOLRDQH� OHL�� LDU�QXP UXO� WRWDO�GH�DQJDMD L�D� IRVW�GH��PRGLILFDUHD�DEVROXW �D�SURGXF LHL� vQ�

OXQD�IHEUXDULH�ID �GH�LDQXDULH�SH�DQVDPEOXO�ILUPHL��GDWRUDW �PRGLILF ULL�QXP UXOXL�WRWDO�GH�DQJDMD L�

a fost:

a) 178,44 mil. lei;

b) 195,96 mil. lei;

c) 138 mil. lei;

d) 170,4 mil. lei;

e) 374,4 mil lei.

28) ÌQ�DQDOL]D�VWDWLVWLF �D�VH]RQDOLW LL�XQHL�6&5��LQGLFLL�GH�VH]RQDOLWDWH�VH�GHWHUPLQ �DWXQFL�FkQG�

a) IHQRPHQXO�DUH�R�HYROX LH�SH�WHUPHQ�OXQJ�DVHP Q WRDUH�XQHL�SURJUHVLL��JHRPHWULFH�

b) vQ� VHULD� FURQRORJLF � VH� FRQVWDW � F �PRGLILF ULOH� DEVROXWH� FX� ED] �PRELO � VXQW� DSUR[LPDWLY�

egale;

c) s-a utilizat pentru deteUPLQDUHD�WUHQGXOXL�R��PHWRG �PHFDQLF �

d) s-D�XWLOL]DW�SHQWUX�GHWHUPLQDUHD�WUHQGXOXL�R�PHWRG �DQDOLWLF �

e) FRPSRQHQWHOH�GH�WUHQG��RVFLODWRULH�úL�DOHDWRDUH�VH�FRPSXQ�GXS �XQ�PRGHO�PXOWLSOLFDWLY�

29) ,QGLFDWRUXO�VWDWLVWLF�ÄHQHUJLD�LQIRUPD LRQDO �2QLFHVFX´��HVWH�XWLOL]DW�vQ�DQDOL]D�VWDWLVWLF �D�

a) REOLFLW LL�

b) WHQGLQ HL�FHQWUDOH�

c) YDULD LHL�

d) FRQFHQWU ULL�GLYHUVLILF ULL�

e) EROWLULL�DSODWL] ULL�

30) ÌQ�FRQVWUXF LD�FRQWXULORU�PDFURHFRQRPLFH�GLQ�FDGUXO�6&1��QX�DUH�VROG�

a) FRQWXO�GH�UHSDUWL LH�D�YHQLWXULORU�

b) contul de creare a veniturilor;

c) FRQWXO�GH�SURGXF LH�

d) contul sintetic de bunuri;

e) contul de modificare a patrimoniului.

31) 'DF �GLVWULEX LD�XQXL�ORW�GH�PLQJL��FRQIHF LRQDWH�GLQ�DFHODúL�PDWHULDO��vQ�IXQF LH�GH�GLPHQVLXQH��

HVWH�QRUPDO �úL�SHUIHFW�VLPHWULF ��DWXQFL�GLVWULEX LD�ORWXOXL�GXS �YROXP�HVWH�

a) SHUIHFW�VLPHWULF �

b) DVLPHWULF ��vQ�FDUH�SUHGRPLQ �YDORULOH�PDUL�

c) DVLPHWULF ��vQ�FDUH�SUHGRPLQ �YDORULOH�PLFL�

d) vQ�IRUP �GH�Ä8´�

e) QX�VH�SRDWH�SUHFL]D�IRUPD�GLVWULEX LHL�

32) ÌQ�GHWHUPLQDUHD�DJUHJDWHORU�PDFURHFRQRPLFH��LQGLFDWRULL�FDUH�H[SULP �Srodusul domestic iau în

calcul:

a) UH]XOWDWXO�DFWLYLW LL�SURGXFWLYH�D�WXWXURU�DJHQ LORU�GLQ�LQWHUiRUXO� ULL��QD LRQDOL�VDX�VWU LQL�

b) UH]XOWDWXO�DFWLYLW LL�SURGXFWLYH�D�WXWXURU�DJHQ LORU�QD LRQDOL�FDUH�úL-DX�GHVI úXUDW�DFWLYLWDWHD�

SH�WHULWRULXO� ULL�VDX�vQ�VWU LQ WDWH�

c) UH]XOWDWXO�DFWLYLW LL�SURGXFWLYH�D�DJHQ LORU�QD LRQDOL�FDUH�úL-DX�GHVI úXUDW�DFWLYLWDWHD�QXPDL�

SH�WHULWRULXO� ULL�

d) UH]XOWDWXO� DFWLYLW LL� SURGXFWLYH� D� DJHQ LORU� QD LRQDOL� FDUH� úL-DX� GHVI úXUDW� DFWLYLWDWHD� SH�

WHULWRULXO� ULL��FHO�SX in 11 luni, în cadrul unui an;

e) SURGXF LD�PHQDMHORU�SHQWUX�FRQVXPXO�SURSULX�

TESTUL 4

1) ÌQ�6LVWHPXO�&RQWXULORU�1D LRQDOH��LPSR]LWHOH�LQGLUHFWH�QHWH�IDF�RELHFWXO�

a) GHELWXOXL�FRQWXOXL�³3URGXF LH´� b) FUHGLWXOXL�FRQWXOXL�´3URGXF LH´� c) VROGXOXL�FRQWXOXL�³3URGXF LH´� d) soldului contului “Venituri”; e) debitului contului de “Modificare a patrimoniului sectorului firme”.

2) 3UHVXSXQHP�F �VXQWH L�LQWHUHVD L�vQ�HIHFWXDUHD�WHVWXOXL�VWDWLVWLF�

H m

H ma

0 200

200

:

:

=>

úL�XWLOL]D L� UHJXOD�GH�GHFL]LH��³6H� UHVSLQJH�+0� GDF �PHGLD�HúDQWLRQXOXL�GH��GH�XQLW L� HVWH�PDL�PDUH�GH��´��'HYLD LD�VWDQGDUG�vQ�SRSXOD LH�HVWH�GH��3UREDELOLWDWHD�FRPLWHULL�XQHL�HURUL�GH�JHQXO�întâi este:

a) 13,36%; b) 6,68%; c) 43,32%; d) 0,4332; e) 3,34%.

3) 6DODULD LL� XQHL� VRFLHW L� FRPHUFLDOH� DX� XQ� VDODULX� PHGLX� GH� �� PLL� OHL�� 3DWURQXO� KRW U úWH� V �

P UHDVF �VDODULXO�ILHF UXL�DQJDMDW�GH��RUL��1RXO�VDODULX�PHGLX�YD�IL�

a) 750 mii lei; b) 780 mii lei; c) 975 mii lei; d) 576,9 mii lei; e) 880 mii lei.

4) ,QFOXGHUHD� UHVXUVHORU�PLQHUDOH� vQ�DYX LD�QD LRQDO � VH� UHDOL]HD] �SULQ�FRQVWLWXLUHD�XQRU trepte de

DWUDJHUH�vQ�FLUFXLWXO�HFRQRPLF�DO�DFHVWRU�UHVXUVH��GLQ�SXQFW�GH�YHGHUH�DO�JUDGXOXL�GH�FXQRDúWHUH�D�DFHVWRUD� úL� DO�SRVLELOLW LORU�GH�H[SORDWDUH��1XP UXO�GH� WUHSWH� vQ� FDUH� VH�FXSULQG�DFHVWH� UHVXUVH�minerale sunt:

a) 2; b) 4; c) 3; d) 5; e) 8.

5) 1XP UXO�DQJDMD LORU�XQHL�FRPSDQLL�FDUH�DEVHQWHD] �OXQHD�DUH�DSUR[LPDWLY�R�GLVWULEX LH�3RLVVRQ��

LDU�QXP UXO�PHGLX�GH�DEVHQ L�GLQ�DFHDVW �]L�D�V SW PkQLL�HVWH��3UREDELOLWDWHD�FD�PDL�SX LQ�GH��DQJDMD L�V �DEVHQWH]H�vQWU-R�]L�GH�OXQL�D�V SW PkQLL�HVWH�

a) 25,10%; b) 19,31%; c) 2,6%;

d) 51,84; e) 26,74%.

6) )LH�R�FROHFWLYLWDWH�VWDWLVWLF �VLVWHPDWL]DW �vQ�r�JUXSH�GXS �YDORULOH�FDUDFWHULVWLFLL�GH�JUXSDUH�;�úL�

în m�JUXSH�GXS �YDORULOH�YDULDELOHL�DQDOL]DWH�<�úL�SHQWUX�FDUH�V-au calculat dispersiile: σ 2 =

GLVSHUVLD� WRWDO �� σ i2 � GLVSHUVLLOH� GH� JUXS � � i r= 1, ), σ 2

�PHGLD� GLVSHUVLLORU� GH� JUXS �� δ 2 = dispersia dintre grupe.

&RQWULEX LD�IDFWRUXOXL�GH�JUXSDUH�;�OD�YDULD LD�JHQHUDO �D�YDULDELOHL�<�VH�P VRDU �FX�LQGLFDWRUXO�

a) R22

2= σ

σ;

b) R22

2= δ

σ;

c) R22

21= − δ

σ;

d)

( )R

n

n

y y n

n

i ii

r

ii

r

jj

m

j

jj

m2

2

1

1

1

2

1

=

−=

=

=

=

∑

∑

∑

∑

σ .

.

.

.

: ;

e)

( ) ( )R

y y n

n

y y n

n

j ij

m

i

r

ij

ijj

m

i

r

j jj

m

jj

m2 11

2

11

2

1

1

=

− −==

==

=

=

∑∑

∑∑

∑

∑:

.

.

.

��/HJ WXULOH�GLQWUH�IHQRPHQHOH�HFRQRPLFR-VRFLDOH�GH�PDV �VXQW�

a) OHJ WXUL�IXQF LRQDOH� b) OHJ WXUL�FDUH�VH�VXSXQ�DF LXQLL�OHJLORU�VWRFKDVWLFH� c) OHJ WXUL�FH�SRW�IL�SXVH�vQ�HYLGHQ �OD�QLYHOXO�ILHF UHL�XQLW L�vQ�SDUWH� d) OHJ WXUL�FDUH�VH�VXSXQ�DF LXQLL�OHJLORU�GLQDPLFH� e) OHJ WXUL�XQLYRF�GHWHUPLQDWH�

��6H�FRQVLGHU �XQ�WDEHO�GH�FRQWLQJHQ ��FRUHOD LH��vQ�IRUPD�VD�JHQHUDO :

( )[ ]{ }x y n j J k Kj k jk, ; ,∈ ∈

&RYDULDQ D�GLQWUH�YDULDELOHOH�;�úL�<��&RY�;�<��FDOFXODW �SH�ED]D�WDEHOXOXL�HVWH�

a) ( ) ( )( )Cov X Yn

x x y y njk

k jj

, .= − −∑∑1;

b) Cov X Yn

x y n xyj k jkkj

( , ) = −∑∑1;

c) ( ) ( )( )Cov X Yn

x x y yjk

kj

, = − −∑∑1;

d) Cov X Yn

x x

y yn

j

kij

kj

( , ) =−

−∑∑1;

e)

( )( )Cov X Y

x x y y n

n

j j ijkj

x y( , ) =

− −∑∑

σ σ.

9) 'DF � vQ� WDEHOXO� GH� FRQWLQJHQ � GHILQLW� OD� SUREOHPD� �� QRW P� Ux x

jj

x=

−σ

� úL� Vy y

kk

y=

−σ

,

atunci: a) Cov X Y Cov U Vx y( , ) ( , )= σ σ ;

b) Cov X Y Cov U Vx y( , ) ( , )≠ σ σ ;

c) Cov X YCov U V

x y( , )

( , )=σ σ

;

d) Cov X Y Cov U V x y( , ) ( , )− = σ σ ;

e) Cov X Y Cov U VU V( , ) ( , )= σ σ . 10) 3UHFL]LD�HVWLP ULL�SDUDPHWULORU�SRSXOD LHL�JHQHUDOH�SH�ED]D�GDWHORU�GH�VRQGDM�GHSLQGH�GH�HURULOH�

GH�vQUHJLVWUDUH�úL�GH�HURULOH�GH�UHSUH]HQWDWLYLWDWH��3HQWUX�D�PD[LPL]D�SUHFL]LD�HVWLP ULORU�

a) HVWH� VXILFLHQW� V � VH� HOLPLQH� VXUVHOH� HURULORU� GH� vQUHJLVWUDUH� SULQ� GLIHULWH� PRGDOLW L� GH�verifLFDUH�D�YHULGLFLW LL�GDWHORU�FXOHVH�

b) HVWH�VXILFLHQW�V �DFRUG P�XQLW LORU�SRSXOD LHL�VWDWLVWLFH�úDQVH�HJDOH�GH�D�IL�LQFOXVH�vQ�HúDQWLRQ� c) HVWH� QHFHVDU� V � HOLPLQ P� HURULOH� GH� UHSUH]HQWDWLYLWDWH� SULQ� UHVSHFWDUHD� SULQFLSLLORU�

VHOHF LHL�DOHDWRDUH� d) este necesar� V � HOLPLQ P� HURULOH� GH� vQUHJLVWUDUH�� VXUVHOH� HURULORU� VLVWHPDWLFH� GH�

UHSUH]HQWDWLYLWDWH�úL�V �PLQLPL] P�HURULOH�DOHDWRDUH�GH�UHSUH]HQWDWLYLWDWH� e) HVWH� QHFHVDU� V � HOLPLQ P� HURULOH� GH� vQUHJLVWUDUH� úL� �QRQU VSXQVXULOH�� OD� vQWUHE ULOH� GLQ�

chestionare.

11) ClasifiFDUHD�DFWLYLW LORU�HFRQRPLFR-VRFLDOH�GLQ�HFRQRPLD�QD LRQDO � VH� UHDOL]HD] �SH�FDWHJRULL��GLYL]LXQL��JUXSH�úL�VDX�FODVH�

a) 2�FDWHJRULH�HVWH�PDL�RPRJHQ �GHFkW�R�GLYL]LXQH��JUXS �VDX�FODV � b) *UXSD�GH�DFWLYLW L�SUH]LQW �FHD�PDL�PDUH�RPRJHQLWDWH�vQ�UDSRUW�FX�FHOelalte; c) 2�JUXS �HVWH�PDL�RPRJHQ �GHFkW�R�FDWHJRULH�VDX�GLYL]LXQH��GDU�PDL�HWHURJHQ �GHFkW�R�FODV � d) &ODVD�GH�DFWLYLW L�HVWH�PDL�HWHURJHQ �GHFkW�R�FDWHJRULH��GLYL]LXQH�VDX�JUXS � e) 'LYL]LXQHD� HVWH� PDL� RPRJHQ � GHFkW� R� JUXS � VDX� FODV �� GDU� PDL� HWHURJHQ � GHFkW� R

FDWHJRULH�GH�DFWLYLW L�

&RQGL LD� QHFHVDU � FD� H[SUHVLD� ( ) ( )F y xi ii

β β β β0 1 0 12, = − −∑ V � ILH� PLQLP � HVWH� DFHHD� FD�

GHULYDWHOH�VDOH�SDU LDOH�vQ�IXQF LH�GH�β0�úL�β1�V �ILH�HJDOH�FX�]HUR�

dF

d

dF

d

β

β

0

1

0

0

=

=

12) 6ROX LD�DFHVWXL�VLVWHP�GH�HFXD LL�HVWH�FXSOXO�GH�YDORri (b0, b1)�FRUHVSXQ] WRU�SDUDPHWULORU��β0, β1)

definit prin:

a) ( )

bCov x y

x1 2

=,

σ�úL�b y b x0 1= − ;

b) ( )

bCov x y

n x y1 =

,

σ σ�úL�b y b x0 1= − ;

c) b y b x1 0= − �úL�( )

bCov x y

x0 2

=,

σ;

d) b b y1 0= = ;

e) ( )

b bCov x y

x1 0 2

= =,

σ.

13) 6H� úWLH� F � ( ) ( )σ σy ii

y x i iir

y yn

y Y2 2 2 21 1= − = −∑ ∑; .� úL� ( )σ Y i

inY y2 21= −∑ � unde

�Y b b xi i= +0 1 ��'DF �U�HVWH�FRHILFLHQWXO�GH�FRUHOD LH�GLQWUH�YDULDELOHOH�<�úL�;�DWXQFL�

a) ( )σ σy x y r.2 2 21= − �úL�σ σ σy y x Y

2 2 2< +. ;

b) ( )σ σy x y r.2 2 21> − �úL�σ σ σy y x Y

2 2 2= +. ;

c) ( )σ σy x y r.2 2 21= − �úL�σ σ σy y x Y

2 2 2= +. ;

d) ( )σ σy x y r.2 2 21> − �úL�σ σ σy y x Y

2 2 2< +. ;

e) ( )σ σy x y r.2 2 21< − �úL�σ σ σy y x Y

2 2 2> +. .

14) 2� YDULDELO � FRPSOH[ �<� HVWH� H[SULPDW � vQ� IXQF LH� GH� IDFWRULL� D�E�F�G�H�I�J�K� D� F URU� LQIOXHQ �

WUHEXLH� L]RODW ��5HOD LD� GLQWUH�<� úL� IDFWRULL� V L� ILLQG�PXOWLSOLFDWLY �� SHUPLWH� R� GHVFRPSXQHUH� vQ�

trepte de forma:

b) 'DF �S�HVWH� SDU�� HVWH� QHFHVDU� V � FDOFXO P�PHGLL�PRELOH� SDU LDOH� GH� RUGLQXO�� DSRL� SH�ED]D� DFHVWRUD�� PHGLL� PRELOH� SDU LDOH� GH� RUGLQXO� �� ú�D�P�G�� SkQ � DMXQJHP� OD� PHGLLOH�finale (valorile ajustate) plasate în dreptul termenilor reali;

c) 'DF �S�HVWH�LPSDU��VH�SLHUG�SULQ�DMXVWDUH�XQ�QXP U�GH�p +

1

2 termeni reali;

d) 'DF �S�HVWH�SDU��VH�SLHUG�SULQ�DMXVWDUH�XQ�QXP U�GH�S�WHUPHQL�UHDOL� e) 1XP UXO�YDORULORU�DMXVWDWH�HVWH�DFHODúL��LQGLIHUHQW�GH�QDWXUD�OXL� p N∈ .

17) ,QGLFHOH�PHGLX�GH�GLQDPLF �SHQWUX�XQ�RUL]RQW�de timp 1,T �VH�FDOFXOHD] �FD�PHGLH�JHRPHWULF �D�

LQGLFLORU�GH�GLQDPLF �FX�ED] �PRELO �GLQ�RUL]RQWXO�1,T ��FX�FRQGL LD�FD�DFHúWLD�V �SUH]LQWH�YDORUL�DSURSLDWH��V �ILH�RPRJHQL��$FHDVW �FRQGL LH�

a) HVWH�IDFXOWDWLY � b) este LPSRUWDQW �úL�WUHEXLH�UHVSHFWDW �GHRDUHFH�LQGLFHOH�PHGLX�VH�SRDWH�FDOFXOD�úL�vQ�IXQF LH�

doar de termenii extremi (y1� úL� \T�� I U � V � VH� LQ � VHDPD� GH� HYROX LD� IHQRPHQXOXL� vQ�interiorul orizontului de timp;

c) QX�HVWH�QHFHVDU �GHRDUHFH�HD�H[FOXGH�QHFHVLWDWHD�UHSUH]HQWDWLYLW LL�PHGLHL�FDOFXODWH�vQWU-o serie de date statistice;

d) QX� HVWH� QHFHVDU � GHRDUHFH� vQ� RULFH� VHULH� FURQRORJLF � � PHGLD� JHRPHWULF � VW � OD� ED]D�GHWHUPLQ ULL�LQGLFHOXL�PHGLX�GH�GLQDPLF �

e) WUHEXLH�UHVSHFWDW �QXPDL�DWXQFL�FkQG�RUL]RQWXO�GH�WLPS�1,T al seriei cronologice nu este mai mic de 5 ani.

18) 3HQWUX� GHWHUPLQDUHD� DJUHJDWHORU� PDFURHFRQRPLFH� VH� XWLOL]HD] � GDWH� GLQ� XUP WRDUHOH� FRQWXUL�

QD LRQDOH��FRQWXO�VLQWHWLF�GH�EXQXUL��FRQWXO�PRGLILF ULL�SDWULPRQLXOXL��FRQWXO�GH�FUHDUH�D�vHQLWXULORU��FRQWXO�SURGXF LH��FRQWXO�GH�UHSDUWL LH�D�YHQLWXULORU��FRQWXO�GH�UHGLVWULEXLUH�D�YHQLWXULORU��FRQWXO�GH�ILQDQ DUH�D�PRGLILF ULL�SDWULPRQLXOXL��FRQWXO�XWLOL] ULL�YHQLWXULORU�

&DUH�GLQWUH�YDULDQWHOH�GH�PDL�MRV�HVWH�IDOV �

a) Soldul contului 4) este produsul intern brut; b) 3URGXVXO�LQWHUQ�EUXW�OD�SUH XO�IDFWRULORU�HVWH�VROGXO�FRQWXOXL�� c) 9HQLWXO�QD LRQDO�HVWH�VROGXO�FRQWXOXL��

$IODUHD�QXP UXOXL�GH�termeni din care se FDOFXOHD] �PHGLLOH�mobile (ordinul de

filtraj = p∈ N)

P este par

Calcularea unei a doua sume mobile

(de ordinul 2)

Divizarea prin 2 P

Calculul sumelor mobile

Aflarea valorilor DMXVWDWH�úL�WUDVDUHD�trendului pe grafic

P este impar

Divizarea prin 2 P

d) 9HQLWXO�QD LRQDO�GLVSRQLELO�HVWH�VROGXO�FRQWXOXL�� e) 9HQLWXO�QD LRQDO�GLVSRQLELO�HVWH�UHVXUV �SHQWUX�FRQWXO��.

19) Pe baza tabelului de date punctuale prezentat mai jos:

i 1 2 3 4 5 6 7 8 xi 1 2 1 3 2 1 1 2

FDOFXOD L�FXDUWLOHOH�Q1, Q2�úL�Q3. Valorile acestora sunt:

a) Q1=1; Q2=2; Q3=3; b) Q1=1; Q2=(1,2); Q3=2; c) Q1=3; Q2=(1,2); Q3=2; d) Q1=1; Q2=3; Q3=2; e) altele decât cele prezentate în variantele anterioare;

20) Într-o colectivitate s-DX�FXOHV�GDWHOH�SHQWUX�GRX �YDULDELOH�VWDWLVWLFH��RE LQkQGX-se: { } { }1;2;3;6;5;1;1;48,1 ==iix

{ } { }70;100;50;70;80;40;90;1008,1 ==iiy

ÌQ�SULYLQ D�RPRJHQLW LL�FHORU�GRX �VHULL��VH�SRDWH�DILUPD�

a) VHULD�DOF WXLW �GXS �YDULDELOD�;�HVWH�PDL�RPRJHQ �GHFkW�FHD�GXS �<� b) VHULD�IRUPDW �GXS �<�HVWH�PDL�RPRJHQ �GHFkW�FHD�GXS �;� c) VHULD�DOF WXLW �GXS �YDULDELOD�;�HVWH�PDL�RPRJHQ ��GHRDUHFH�YDORULOH�VXQW�PDL�PLFL� d) QX�VH�SRDWH�FRPSDUD�RPRJHQLWDWHD�FHORU�GRX �VHULL��ILLQG�YRUED�GH�YDriabile diferite; e) ILLQG�YRUED�GH�DFHHDúL�FROHFWLYLWDWH��RPRJHQLWDWHD�FHORU�GRX �VHULL�HVWH�DFHHDúL�

21) $QDOL]D L�XUP WRDUHOH�FDWHJRULL�GH�SHUVRDQH�GH��DQL�úL�SHVWH��GLQWU-R�SHULRDG �GH�UHIHULQ � ��SHUVRDQH�FDUH�QX�OXFUHD] ��QHDYkQG�XQ�ORF�GH�PXQF ��SHUVRDQH�FDUH�VXQW� vQ�F XWDUHD�

XQXL� ORF� GH�PXQF �� SHUVRDQH� FRQFHGLDWH�� SHUVRDQH�FDUH� VXQW� GLVSRQLELOH� V � vQFHDS � LPHGLDW�F XWDUHD�XQXL�ORF�GH�PXQF ��SHUVRDQH�vQ�F XWDUHD�SULPXOXL�ORF�GH�PXQF ��SHUVRDQH�FDUH��GXS �R�vQWUHUXSHUH�YROXQWDU �D�DFWLYLW LL��VROLFLW �UHOXDUHD�DFHVWHLD��SHUVRDQH�VH]RQLHU�RFXSDWH�DIODWH�vQ�RFXSDUHD�XQXL�SHUPDQHQW�ORF�GH�PXQF � ÌQ� GHWHUPLQDUHD� QXP UXOXL� GH� úRPHUL� vQ� VHQVXO� GHILQL LHL� GDWH� GH� %LURXO� ,QWHUQD LRQDO� DO�0XQFLL��GHILQL LD�VWDQGDUG��VH�FXSULQGH�

a) ��úL�� b) 1,��úL�� c) ��úL�� d) ��úL�� e) ��úL��

�� /X P� vQ� FRQVLGHUDUH� WDEHOXO� GH� FRQWLQJHQ � �FRUHOD LH�� ( )[ ]{ }x y n j J k Kj k jk, ; ,∈ ∈ úL�

GLVWULEX LLOH� FRQGL LRQDWH� DOH� OXL� <� GH� ;� ( ){ }y n k Kk jk, , ,j fixat ∈ �� 3HQWUX� DFHVWHD� GLQ� XUP � VH�

FXQRVF�PHGLLOH�úL�GLVSHUVLLOH�

y xn

n yjj

jk kk

( ).

= ∑1úL� [ ]σ y j

jk j jk

k

xn

y y x n2 21( ) ( )

.= −∑ , cu n nj jk

k. = ∑ efectivul

�IUHFYHQ D��PDUJLQDO� ��DVRFLDW �YDULDQWHL�xj.

Dispersia variabilei Y (σ y2 ��vQ�IXQF LH�GH�HOHPHQWH�SUH]HQWDWH�vQ�LSRWH] �VH�SUH]LQW �DVWIHO�

a) [ ]σ σy jj

y j jj

jnn x

nn y x y2 2 21 1= + −∑ ∑. .( ) ( ) ;

b) [ ]σ y jj

jn

n y x y2 21< −∑ . ( ) ;

c) σ σy jj

y jnn x2 21< ∑ . ( ) ;

d) [ ]σ σy jj

y j jj

jn

n xn

n y x y2 2 21 1= − −∑ ∑. .( ) ( ) ;

e) [ ]σ σy jj

y j jj

jnn x

nn y x y2 2 21 1> + −∑ ∑. .( ) ( ) �FX�FRYDULDQ D�GLQWUH�<�úL�FHLODO L�IDFWRUL�

GH�LQIOXHQ �FX�H[FHS LD�IDFWRUXOXL�;� 23) Luându-se în considerare elementele prezentate în ipoteza pUREOHPHL� �� SURSRU LD� YDULD LHL�

PHGLLORU�FRQGL LRQDWH�vQ�GLVSHUVLD�JHQHUDO �D�YDULDELOHL�<�HVWH�QXPLW �JUDG�GH�GHWHUPLQD LH�úL�HVWH�

QRWDW �FX�Ry x.2 ��&DUH�GLQ�XUP WRDUHOH�DILUPD LL�QX�HVWH�DGHY UDW �

a) 0 12≤ ≤Ry x. ;

b) Ry x.2 0= �GDF �WRDWH�PHGLLOH�FRQGL LRQDWH�DOH�OXL�\�vQ�IXQF LH�GH�[�VXQW�HJDOH�vQWUH�HOH�

c) Ry x.2 1= GDF �ILHF UHL�YDORUL�[j�vL�FRUHVSXQGH�R�XQLF �YDORDUH�D�YDUDELOHL�\�

d) Ry x.2 0= � GDF � PHGLLOH� FRQGL LRQDWH� DOH� OXL� \� vQ� IXQF LH� GH� [� VXQW� GLVWLQFWH� GRX � FkWH�

GRX �

e) Ry x.2 �H[SULP �P VXUD�vQ�FDUH�IDFWRUXO�GH�LQIOXHQ �H[SOLF �YDULD LD�YDULDELOHL�\�

24) (URULOH�DOHDWRDUH�GH�UHSUH]HQWDWLYLWDWH�vQWkOQLWH�vQ�FD]XO�RUJDQL] ULL�RULF UXL�WLS�GH�VRQGDM�QX pot

fi evitate ci diminuate:

a) GHRDUHFH� XQLW LORU� GLQ� SRSXOD LLOH� VWDWLVWLFH� OL� VH� DFRUG � DFHHDúL� úDQV � GH� DSDUL LH� vQ�HúDQWLRDQH�

b) GHRDUHFH� vQ� HúDQWLRDQH� VH�FXSULQG�QXPDL�S U L�GLQ�SRSXOD LL� VWDWLVWLFH�úL�SULQ�XUPDUH�QX�pot reproduce identic, decât întâmpO WRU��VWUXFWXULOH�SRSXOD LLORU�GLQ�FDUH�DFHVWHD�DX�IRVW�extrase;

c) DWXQFL�FkQG�FRQWUROXO�GDWHORU�FXOHVH�VH�UHDOL]HD] �SULQ�FHOH�PDL�DGHFYDWH�PHWRGH d) DWXQFL�FkQG�VH�XWLOL]HD] �ED]H�GH�VRQGDM��DFWXDOL]DWH�FRUHVSXQ] WRU� e) DWXQFL�FkQG�VH�IRUPHD] �GLQ�DFHHDúL�SRSXOD LH�WRDWH�HúDQWLRDQHOH�SRVLELOH��FKLDU�GDF �HOH�

DX�DFHODúL�YROXP� 25) 3HQWUX� FRPSDUDUHD� SUH XULORU� SURGXVHORU� GLQWU-R� DU � FX� FHOH� GLQWU-R� DOW � DU � VH� XWLOL]HD] � XQ�

LQGLFH�GH�JUXS�GH�SUH �

a) de tip Laspeyres; b) de tip Paasche; c) de tip Fischer; d) IRUPDW�GXS �PHWRGD�VXEVWLWXLULL�vQO Q XLWH� e) IRUPDW�GXS �PHWRGD�UHVWXOXL�QHGHVFRPSXV�

26) Estimatorul unui parametru θ�VH�QRWHD] � �θ ��$FHVW�HVWLPDWRU�HVWH�QHGHSODVDW�GDF �

a) GLIHUHQ D�GLQWUH�PHGLD�VD�úL�YDORDUHD�SDUDPHWUXOXL�HVWH�GLIHULW �GH�]HUR�( )M ( �θ θ− ≠ 0 ;

b) GDF �PHGLD�VD�HVWH�HJDO �FX�YDORDUHD�SDUDPHWUXOXL�( )M ( � )θ θ= ;

c) GLVSHUVLD� VD� HVWH� PLQLP � �D( � )θ → 0 FkQG� YROXPXO� HúDQWLRQXOXL� WLQGH� F WUH� YROXPXO�SRSXOD LHL�JHQHUDOH��n N→ );

d) dispersia sa este miniP �SHQWUX�XQ�YROXP�IL[DW�DO�HúDQWLRQXOXL� e) GLVSHUVLD�HVWLPDWRUXOXL�HVWH�PD[LPDO �

27) ÌQ�FDOFXOXO�SURGXFWLYLW LL�VRFLDOH�D�PXQFLL�VH�LD�vQ�FRQVLGHUDUH�

a) SRSXOD LD�RFXSDW �WRWDO � b) SRSXOD LD�DFWLY � c) SRSXOD LD�RFXSDW �WRWDO �GLQ�FDUH�VH�VFDGH�SRSXOD LD�GLQ�DUPDW � d) SRSXOD LD� RFXSDW � WRWDO � GLPLQXDW � FX� SRSXOD LD� RFXSDW � vQ� DUPDW � úL� FX� VDODULD LL�

RUJDQL]D LLORU�SROLWLFH�úL�REúWHúWL� e) SRSXOD LD�RFXSDW �WRWDO �GLPLQXDW �FX�VDODULD LL�RUJDQL]D LLORU�SROLWLFH�úL�REúWHúWL�

28) 3HQWUX� D� VWXGLD� HYHQWXDOD� LQIOXHQ a culorii DPEDODMXOXL� DVXSUD� XQXL� QRX� WLS� GH� V SXQ� VH�VWXGLD] �XQ�HúDQWLRQ�GH��PHQDMH��6H�WULPLW�ILHF UXL�PHQDM�GLQ�HúDQWLRQ�SDWUX�EXF L�GH�V SXQ�GH�DFHHDúL�FRPSR]L LH��GLQ�WLSXO�DQDOL]DW��GDU�DPEDODWH�vQ�FXORUL�GLIHULWH��URúX��DOE��EOHX�úL�YHUGH��Pentru a testD�SUHIHULQ D�PHQDMHORU�R�OXQ �PDL�WkU]LX�ILUPD�SURGXF WRDUH�RIHU �JUDWXLW��EXF L�FX�FXORDUHD� GRULW � D� DPEDODMXOXL�� ÌQ� XUPD� VLVWHPDWL] ULL� GDWHORU� REVHUYDWH� V-D� RE LQXW� XUP WRDUHD�serie:

Culoarea 5RúX Alb Bleu Verde TOTAL Nr. menaje 51 74 30 45 200

Pentru o probabilitate 0,1%:

a) VH� UHVSLQJH� LSRWH]D� SRWULYLW� F UHLD� FXORDUHD� DPEDODMXOXL� QX� DU� LQIOXHQ D� YROXPXO�

YkQ] ULORU�GHRDUHFH� χ χcalc. ,, ,20 1%;3220 04 16 3= > = ;

b) VH�DFFHSW �LSRWH]D�SRWULYLW�F UHLD�FXORDUHD�DPEDODMXOXL�QX�DU�LQIOXHQ D�YROXPXO�YkQ] ULORU�deoarece 3,1604,20 2

3%;1,02

. =>= χχ calc ;

c) nu se poate lua nici o decizie deoarece 3,1623%;1,0

2. == χχ calc ;

d) VH� UHVSLQJH� LSRWH]D� DEVHQ HL� LQIOXHQ HL� FXORULL� DPEDODMXOXL� DVXSUD� YROXPXOXL� YkQ] ULORU�GLQ�WLSXO�GH�V SXQ�DQDOL]DW�GHRDUHFH� 3,1610 2

3%;1,02

. =<= χχcalc ;

e) nu se poate lua o decizie deoarece testul χ2 nu poate fi utilizat. 29) 1XP UXO�PD[LP� GH� HúDQWLRDQH� GH� DFHODúL� YROXP� Q� FDUH� SRW� IL� IRUPDWH� SULQ� DFHODúL� SURFHGHX�

DOHDWRU�vQ�YDULDQW �QHUHSHWDW ��I U �UHYHQLUH��GLQWU-R�SRSXOD LH�VWDWLVWLF �GH�YROXP�1�HVWH�

a) nN; b) Nn;

c) ANn ;

d) CNn ;

e) n!.

30) 6H�QRWHD] � HVWLPDWRUXO� SDUDPHWUXOXL�θ cu �θ � LDU�PHGLD� úL� GLVSHUVLD� HVWLPDWRUXOXL� FX� [ ]M �θ � úL�

respectiv cu [ ]D �θ . Estimatorul �θ este abVROXW�FRUHFW�GDF �

a) este nedeplasat ( )M ( � )θ θ= �úL�FRQYHUJHQW��D( � )θ → 0 �GDF � n N→ );

b) M ( � )θ θ→ �úL�D( � )θ → 0 când n N→ ;

c) [ ]D �θ este minim , dar nenul ��SHQWUX�XQ�YROXP�IL[DW�DO�HúDQWLRQXOXL�

d) ( )M ( � )θ θ− ≠ 0 úL��D( � )θ → 0 ��GDF �n N→ ;

e) ( )M ( � )θ θ= , indiferent de valoarea dispersiei estimatorului (pentru orice [ ]D �θ ).

31) Statisticianul enJOH]� <XOH� �� SUHFL]HD] � FRQGL LLOH� SH� FDUH� WUHEXLH� V � OH� vQGHSOLQHDVF �

LQGLFDWRULL�VLQWHWLFL�DL�UHSDUWL LLORU�VWDWLVWLFH�úL�DQXPH�

1) V �ILH�GHILQLW�vQ�PRG�RELHFWLY� 2) V �GHSLQG �GH�WRDWH�REVHUYD LLOH�GLQ�VHULH� 3) V �DLE �R�VHPQLILFD LH�FRQFUHW � 4) V �ILH�VLPSlu de calculat; 5) V �ILH�SX LQ�VHQVLELO�OD�IOXFWXD LLOH�GH�VHOHF LH� 6) V �VH�SUHWH]H�XúRU�OD�FDOFXOH�DOJHEULFH�

$FHVWH�FRQGL LL�vQ�WRWDOLWDWH�

a) sunt respectate de medie; b) sunt respectate de dispersie; c) VXQW�UHVSHFWDWH�GH�FRYDULDQ � d) sunt respectate de valoarea �PRGDO � e) QX�VXQW�UHVSHFWDWH�GH�QLFL�XQ�LQGLFDWRU�DO�WHQGLQ HL�FHQWUDOH�VDX�DO�YDULD LHL�

32) Media ( x �� D� XQXL� HúDQWLRQ�GH�PDUH�YROXP� � �n�� SRDWH� IL� FRQVLGHUDW � F �XUPHD] �R� GLVWULEX LH�

QRUPDO �GH�PHGLH� x 0 �úL�DEDWHUH�PHGLH�S WUDWLF ��HURDUH�PHGLH��σn

, în cazul unui sondaj simplu

cu revenire. Pentru o probabilitate 1-α intervalul de estimare (sau de încredere) a parametrului

0x este:

x x zn

− ≤0 ασ

Pentru ca preciziD� HVWLPD LHL� V � ILH�PDL�PLF � VDX� HJDO � FX� N�� SUHFL]LH� IL[DW � vQ� YDORDUH�

UHODWLY ��GLQ� 0x �YROXPXO�HúDQWLRQXOXL�HVWH�

a) nz

k x≤ ⋅α σ2

2

2

02

;

b) ( )nz

kCV≥ ⋅α

2

22�XQGH�&9� �FRHILFLHQWXO�GH�YDULD LH�

c) nk x

=σ 2

20

;

d) nz k

x= α σ2 2 2

0

;

e) PDL�PDUH�GH��GH�XQLW L� 33) 8Q� FRQWURORU� GH� FDOLWDWH� úWLH� F � SUREDELOLWDWHD� FD� SH� XQ� QRX� DXWRPRELO� V � J VHDVF � GHIHFWH�

YL]LELOH�GH�VXSUDID ��]JkULHWXUL��HVWH�GH��‰. Probabilitatea ca dintr-un lot de 2500 de automobile V �J VHDVF �H[DFW��2 automobile cu defecte vizibile este:

a) 7,5%; b) 3,66%; c) 92,5%; d) 7,32%; e) 15%.

34) Într-un proces de verificare a ipotezelor statistice nivelul de încredere reprezint probabilitatea:

a) α; b) 1-α; c) β; d) 1-β; e) α+β.

35) &RQWXO�³�3URGXF LH´�DO�VHFWRUXOXL�ILUPH�VH�SUH]LQW �VLQWHWLc astfel: 1. Cheltuieli pentru produse LQWHUPHGLDUH�úL�GH�LQYHVWL LL�GLQ�LPSRUW 2. Amortizarea capitalului fix 3. Impozite indirecte brute

��ÌQFDV UL�GLQ�YkQ]DUHD�GH�EXQXUL�F WUH�DOWH�VHFWRDUH -�YkQ] UL�GH�EXQXUL�GH�FRQVXP�F WUH�PHQDMH -�YkQ] UL�GH�SURGXF LH�LQWHUPHGLDU �F WUH�VHFWRUXO�SXEOLF -�YkQ] UL�SULQ�H[SRrt

��6XEYHQ LL�GH�H[SORDWDUH 5. Salarii ��'REkQ]L�úL�UHQWH 7. Profitul din DFWLYLWDWHD�GH�SURGXF LH

��,QYHVWL LL�EUXWH��LQFOXVLY�LPSRUWXULOH� -�LQYHVWL LL�EUXWH�vQ�EXQXUL�GH�FDSLWDO -�LQYHVWL LL�EUXWH�vQ�VWRFXUL

3H�ED]D�HOHPHQWHORU�DFHVWXL�FRQW�VH�SRW�FDOFXOD�PDL�PXO L�LQGLFDWRUL��&DUH�UHOD LH�GLQ�FHOH�SUH]HQWDWH�PDL�MRV�HVWH�IDOV �

a) 9DORDUHD�DG XJDW �OD�SUH XO�SLH HL� �9DORDUHD�DG XJDW �QHW �OD�SUH XO�IDFWRULORU��impozitele indirecte nete (3-4) + amortizarea capitalului fix (2);

b) 9DORDUHD� DG XJDW � EUXW � OD� SUH XO� IDFWRULORU� � YDORDUHD� DG XJDW � OD� SUH XO� IDFWRULORU�(5+6+7) + amortizarea (2);

c) 9DORDUHD�DG XJDW �QHW �OD�SUH XO�SLH HL� �9DORDUHD�DG XJDW �EUXW �OD�SUH XO�SLH HL�- amortizarea; d) Soldul contului: (1+2+3+4+5+6+7)-(8+9)=0; e) Soldul contului: (1+2+3+4+5+6+7)-(8+9)≠ 0.

36) 6 � VH� SUHFL]H]H� FDUH� GLQ� VHULLOH� GH� UHSDUWL LH� FDUDFWHUL]DWH� SULQ� XUP WRDUHOH� VHWXUL� GH� YDORUL�SUH]LQW �R�DVLPHWULH�SR]LWLY �

a) x = 31 2, ; Me=31,2; Mo=31,2;

b) x = 2850; Me=3150; Mo=3300;

c) x = 151 25, ; Me=138,75; Mo=112,58;

d) x = 180 ��0H ��0R ��úL�� e) x = 0; Me=0; Mo=0.

��,QGLFLL�DQXDOL�DL�SUH XULORU�GH�FRQVXP�OD�P UIXrile alimentare în perioada 1991-1994 au fost:

Anul 1991 1992 1993 1994 Indice (an precedent = 100) 286,2 336,6 348,9 236,2

Sursa: Anuarul Statistic al României, 1995, pag. 411 ,QGLFHOH�PHGLX�DO�SUH XULORU�GH�FRQVXP�OD�P UIXULOH�DOLPHQWDUH�vQ�DFHDVW �SHULRDG �D�IRVW� a) 1,985; b) 298,50%; c) 301,975; d) 261,2%; e) 2,612.

��'DF � vQWU-R� VHULH�GH� UHSDUWL LH� UHOD LD�RELHFWLY �GLQWUH�GDWH� �[i�� FRQGXFH� OD�GHFL]LD� FD� WHQGLQ D�

FHQWUDO �V-R�FDUDFWHUL] P�FX�PHGLD�JHRPHWULF �� xg ) atXQFL�DEDWHUHD�PHGLH�S WUDWLF �ID �GH�WHQGLQ D�

FHQWUDO �HVWH�DFHD�YDORDUH��gx

σ pentru care avem:

a) log logσ xi

gig n

x

x=

∑1

2

;

b) ( )σ x i gi

g nx x= −∑1 2

;

c) σ xi gi

g n x x= −

∑1 1 1

2

;

d) ( )σ x i gi

g nx x= −∑1

;

e) σ xi

gig n

x

x=

∑1

.

��&LUFXLWXO�HFRQRPLF�UHIOHFW �vQ�HVHQ �WUDQ]DF LLOH�GLQWUH�VXELHFWHOH�HFRQRPLFH��ÌQ�FDGUXO�DFHVWRUD�XQ�ORF�LPSRUWDQW�vO�RFXS �WUDQ]DF LLOH�GH�SLD ��$FHVWHD�VXQW� a) transferuri curente (impozite, pensii, ajutoare sociale, etc.); E��VXEYHQ LL� c) transferuri de patrimoniu; d) transferuri unilaterale; e) transferuri bilaterale.

��*UDILFHOH�VWDWLVWLFH�vQ�FRRUGRQDWH�SRODUH�VH�XWLOL]HD] �vQ�PRG�FXUHQW�SHQWUX�YL]XDOL]DUHD� D��RULF UHL�VHULL�GH�GDWH�VWDWLVWLFH� E��VHULLORU�GH�UHSDUWL LH� c) eYROX LHL�WUHQGXOXL�GLQ�HYROX LD�XQXL�IHQRPHQ� G��HYROX LHL�XQXL�IHQRPHQ�DIHFWDW�GH�RVFLOD LL�VH]RQLHUH� H��WHQGLQ HL�OHJ WXULL�GLQWUH�YDULDELOH�úL�DOHJHUHD�PRGHOXOXL�GH�UHJUHVLH�

TESTUL 5

��5DWD�úRPDMXOXL�VH�FDOFXOHD] �FX�UHOD LD�

a) neocupatiocupati

ocupati

+;

b) munca de forta

neocupati;

c) ocupati

neocupati;

d) ani 15 de mare mai in varsta populatia

neocupati;

H��QLFL�XQD�GLQWUH�UHOD LLOH�SUHFHGHQWH� ��6H�FXQRVF�XUP WRDUHOH�GDWH�SULYLQG��QXP UXO�GH�IDFWXUL��vQWRFPLWH�GH�6�&��$QRQLPXV��65/�vQ�luna noiembrie 1997:

Data Nr. facturi Data Nr. facturi Data Nr. facturi 1 6 11 10 21 14 2 10 12 14 22 12 3 12 13 6 23 11 4 10 14 18 24 10 5 8 15 13 25 8 6 9 16 9 26 4 7 10 17 14 27 12 8 11 18 12 28 9 9 12 19 17 29 12

10 9 20 12 30 16

I II III Grupe de zile�GXS �

nr. de facturi

Nr. zile Grupe de ]LOH�GXS �

nr. de facturi

Nr. zile Intervale de timp

Nr. facturi

0 - 5 1 0 -5 1 0 - 6 55 6 - 11 13 6 - 10 16 7 - 14 90

11 - 15 13 11 - 15 10 15 - 22 103 15 - 20 3 16 - 20 3 23 - 30 82

A. IV B. V

Intervale ale QXP UXOXL�

zilnic de facturi

Nr. de zile

Intervale ale QXP UXOXL�

de facturi

Nr. de zile

( 0 - 5 ] 1 [ 0 - 5 ) 1 ( 5 - 10 ] 13 [ 5 - 10 ) 9 (10 - 15 ] 13 [10 - 15 ) 17 (15 – 20] 3 [15 – 20) 3

3RSXOD LD�VWDWLVWLF �VWXGLDW �HVWH�VWUXFWXUDW �úL�SUH]HQWDW �vQ�

a) Tabelul I; b) Tabelul II; c) Tabelul III; d) Tabelul IV; e) Tabelul V.

3) Cadranul II al Tabelului Input- Output cuprinde:

D��YDORDUHD�DG XJDW �EUXW �

E��SURGXF LD�ILQDO �

F��SURGXF LD�LQWHUPHGLDU �

G��FRHILFLHQ LL�Fheltuielilor totale;

H��FRHILFLHQ LL�FKHOWXLHOLORU�GLUHFWH�

4) 3HQWUX��GH�VDODULD L�DL�XQHL�VRFLHW L�FRPHUFLDOH��IRQGXO�GH�VDODUL]DUH�D�IRVW��vQWU-R�OXQ ��GH��PLOLRDQH� OHL�� ùWLLQG� F � FHL� PDL� PXO L� GLQWUH� VDODULD L� DX� DYXW� XQ� VDODULX� GH� �� PLL� OHL�� Lar FRHILFLHQWXO�GH�DVLPHWULH�DO�UHSDUWL LHL�GXS �VDODULX�D�IRVW�GH�-��FRHILFLHQWXO�GH�YDULD LH�D�IRVW��

a) 24,5%; b) 200%; c) -25,52%; d) 37,5%; e) 26,67%.

5) 'LVWULEX LD�XQXL�ORW�GH�DXWRPRELOH�GXS �FRQVXPXO�GH�FDUEXUDQW�OD��NP�SDUFXUúL�HVWH�

IQWHUYDOH�GH�YDULD LH�D�FRQVXPXOXL��O� Nr. automobile 6,2 - 6,6 4 6,6 - 7,0 12 7,0 - 7,4 44 7,4 - 7,8 90 7,8 - 8,2 107 8,2 - 8,6 86 8,6 - 9,0 36 9,0 - 9,4 15 9,4 - 9,8 6

Total 400 Utilizând testul χ2� SHQWUX� YHULILFDUHD� FDUDFWHUXOXL� UHSDUWL LHL� Hmpirice, pentru

χ α = = =0 05 62 12 5916, ; ,l se poate afirma:

a) χ χcalc tab2 2> ;

b) QX� VH� DFFHSW � LSRWH]D� FRQIRUP� F UHLD� vQWUH� GLVWULEX LD� HPSLULF � úL� FHD� WHRUHWLF � H[LVW �FRQFRUGDQ �

c) χ calc2 3 44= , ;

d) χ calc2 8 12= , ;

e) χ calc2 12 5916= , .

6) 2�FRPSDQLH�FDUH�SURGXFH�XQ�QRX�WLS�GH�vQJU ú PkQW�DJULFRO�HVWH�LQWHUHVDW �vQ�VWXGLHUHD�UHOD LHL�GLQWUH�SURGXF LD�GH�URúLL��<��úL�FDQWLWDWHD�GH�vQJU ú PkQW�DSOLFDW��;��2�VXSUDID �HVWH�GLYL]DW �vQ�RSW�SDUFHOH�GH�P ULPL�HJDOH�SH�FDUH�VXQW�DSOLFDWH�FDQWLW L�GLIHULWH�GH�vQJU ú PkQW��3URGXF LLOH�GH�URúLL��NJ��úL�FDQWLW LOH�GH�vQJU ú PkQW�DSOLFDW��NJ��vQUHJLVWUDWH�SHQWUX�ILHFDUH�SDUFHO �VXQW�

xi 1 1,5 2 2,5 3 3,5 4 4,5 yi 25 31 27 28 36 35 32 34

(FXD LD�GH�UHJUHVLH�SHWQUX�GHVFULHUHD�GHSHQGHQ HL�GLQWUH�FHOH�GRX �YDULDELOH�HVWH� a) � , ,y x= −24 45 2 38 ; b) � , ,y x= +2 38 24 45 ; c) � , ,y x= +24 45 2 38 ; d) � ,y x= +32 2 01 ; e) � ,y x= −32 2 01 ;

��'LPLQXHD] �DYX LD�QD LRQDO �

a) rezervele de devize; b) rezervele de aur; F��UH]HUYHOH�GH�PRQHG �QD LRQDO �GH LQXWH�GH�DOWH� UL� G��EXQXULOH�QD LRQDOH�DIODWH�SH�WHULWRULXO�DOWRU� UL� H��FUHDQ HOH�DVXSUD�DOWRU� UL�

8) Într-R�UHSDUWL LH�GH�IUHFYHQ H��IUHFYHQ D�UHODWLY �FXPXODW �FUHVF WRU�D�JUXSHL i�DUDW � a) QXP UXO�XQLW LORU�VWDWLVWLFH�FDUH�DX�YDORULOH�FDUDFWHULVWLFLL�PDL�PLFL�VDX�HJDOH�FX�xi;

E��SRQGHUHD�XQLW LORU�VWDWLVWLFH�GLQ�JUXSD�i�vQ�WRWDOXO�FROHFWLYLW LL�VWXGLDWH� c) QXP UXO�XQLW LORU�VWDWLVWLFH�FDUH�DX�YDORULOH�FDUDFWHULVWLFLL�PDL�PDUL�VDX�FHO�SX LQ�HJDle cu xi; d) ponderea unit LORU� VWDWLVWLFH� FDUH� DX� QLYHOXO� FDUDFWHULVWLFLL�PDL�PLF� VDX� HJDO� FX� xi, în

WRWDOXO�FROHFWLYLW LL�VWXGLDWH� e) SRQGHUHD� XQLW LORU� VWDWLVWLFH� FDUH� DX� QLYHOXO� FDUDFWHULVWLFLL�PDL�PDUH� VDX� HJDO� FX� xi, în

WRWDOXO�FROHFWLYLW LL�VWXGLDWH.

9) 8Q� DQDOLVW� ILQDQFLDU� HVWH� LQWHUHVDW� vQ� D� GHWHUPLQD� VDODULXO�PHGLX� DFRUGDW� DQJDMD LORU� ILUPHL�� (O�FXOHJH� GDWH� SULYLQG� VDODULXO� PHGLX� SH� ILHFDUH� GLQ� FHOH� SDWUX� ILOLDOH� DOH� ILUPHL� úL� IRQGXO� GH�salarizare:

Filiala Salariul mediu

(mii lei) Fondul de salarizare

(mil. lei) 1 540 45,90 2 620 33,48 3 480 16,80 4 700 19,60

6DODULXO�PHGLX�SH�vQWUHDJD�ILUP �HVWH� a) 573,168 mii lei; b) 585,5 mii lei; c) 825,325 mii lei; d) 460,5 mii lei; e) 632,375 mii lei.

��5HSDUWL LD�YHQLWXULORU�HVWH�HYLGHQ LDW �vQ�FRQWXO� D��SURGXF LH� E��ILQDQ DUH� c) modificarea patrimoniului; d) redistribuirea venitului; e) contul sintetic de bunuri. 11) 8Q�IDEULFDQW�GH�DXWRPRELOH�GHWHUPLQ �F ��GLQ�UH]HUYRDUHOH�GH�EHQ]LQ �PRQWDWH�SH�PRGHOXO�

�� VXQW� GHIHFWH�� 'DF � �� PDúLQL� VRVHVF� SHQWUX� UHYL]LH�� SUREDELOLWDWHD� FD� �� GLQ� �� PDúLQL� V �necesite înlocuirea rezervorului este:

a) 22,69%; b) 9,72%; c) 12%; d) 30%; e) 70%. ��$QDOL]D L�XUP WRDUHOH�PRGDOLW L�GH�FRQWURO� 1) controlul de volum; 2) controlul cantitativ-numeric; 3) controlul financiar-contabil; 4) FRQWUROXO� ORJLF�� FRQWUROXO� &XU LL� GH� FRQWXUL�� WHVWH� GH� YHULILFDUH� D� VHPQLILFD LHL� LQGLFDWRULORU�FDOFXOD L� ÌQ� WRDWH� HWDSHOH� FHUFHW ULL� VWDWLVWLFH� HVWH� QHFHVDU � YHULILFDUHD� DXWHQWLFLW LL� GDWHORU� úL�eliminarea�HURULORU�GH�REVHUYDUH��FDOFXO�úL�HVWLPDUH��ÌQ�DFHVW�VHQV�HVWH�QHFHVDU� D��úL�� E��úL�� F��úL�� G��úL�� H��úL�� )DF�SDUWH�GLQ�IRU D�GH�PXQF � a) persoanele casnice; b) pensionarii; F��úRPHULL� G��SHUVRDQHOH�I U �XQ�ORF�GH�PXQF �FH�UHQXQ �V �PDL�FDXWH�ORF�GH�PXQF � H��QLFL�XQD�GLQ�FDWHJRULLOH�PHQ LRQDWH� 14) 3HQWUX� GLVWULEX LD� XQHL� YDULDELOH� DOHDWRDUH�� ;�� VH� FXQRDúWH� F � ( )P xi ≤ =151 33 0 5, , , valoarea

quartilei întâi este 105,86 iar�YDORDUHD�TXDUWLOHL�WUHL�HVWH��6HULD�GH�GLVWULEX LH� D��HVWH�SURIXQG�DVLPHWULF �VSUH�YDORULOH�PLFL�DOH�YDULDELOHL� E��DUH�R�DVLPHWULH�PRGHUDW �SR]LWLY � F��DUH�R�DVLPHWULH�PRGHUDW �QHJDWLY � d) nu se poate preciza felul asimetriei; e) este profXQG�DVLPHWULF �VSUH�YDORULOH�PDUL�DOH�YDULDELOHL�

15) 8Q�DQDOLVW�GH�PDUNHWLQJ�XUP UHúWH�vQ��OXQL�HIHFWXO�UHFODPHORU�DVXSUD�YHQLWXULORU�GLQ�YkQ] UL��(O�vQUHJLVWUHD] � FKHOWXLHOLOH� FX� UHFODPD� �;�� PLOLRDQH� OHL�� úL� YHQLWXULOH� GLQ� YkQ] UL� �<�� VXWH�milioane lei)��ÌQ�XUPD�SUHOXFU ULL�GDWHORU�RE LQH�

x y x x yii

ii

ii

i ii

= = = =∑ ∑ ∑ ∑15 10 55 372; ; ; ; ( )i = 1 5, �� ÌQ� LSRWH]D� XQHL� GHSHQGHQ H� � OLQLDUH��

HFXD LD�GUHSWHL�GH�UHJUHVLH�HVWH� a) � , ,y x= −0 7 0 1 ; b) � , ,y x= − +0 1 0 7 ; c) � ,y x= −2 0 7 ; d) � , ,y x= +3 7 5 5 ; e) � ,y x= 0 7 . ��$QDOL]D L�XUP WRDUHOH�QR LXQL� �� DXWRQRPLH�� FRQILGHQ LDOLWDWH�� WUDQVSDUHQ �� VSHFLDOL]DUH�� SURSRU LRQDOLWDWH�� GHRQWRORJLH� SURIHVLRQDO �� LQGHSHQGHQ � SROLWLF �� HODERUDUHD� GHFL]LLlor; 9) argumentarea DF LXQLORU�GXS �GHFODQúDUHD�ORU��SUHOXFUDUHD�GDWHORU��REVHUYDUH� $FWLYLWDWHD�VWDWLVWLFLL�SXEOLFH�WUHEXLH�V �VH�GHVI úRDUH�UHVSHFWkQG�XUP WRDUHOH�SULQFLSLL� D��úL�� E��úL�� c) 3, 4, 5, 6,��úL�� G��úL�� H��úL�� 5HFHQV PkQWXO�FD�PHWRG �GH�REVHUYDUH�VWDWLVWLF �

a) QX�SUHVXSXQH�FXOHJHUHD�GDWHORU�GH�OD�WRDWH�XQLW LOH�SRSXOD LHL�VWDWLVWLFH�ELQH�GHOLPLWDW � b) are exclusiv un caracter demografic; F��VH�vQFDGUHD] �vQ�VIHUD�REVHUY ULORU�FX�FDUDFWHU�SHUPDQHQW� G��VH�RUJDQL]HD] �FX�R�DQXPLW �SHULRGLFLWDWH�

e) VH�RUJDQL]HD] �RUL�GH�FkWH�RUL�6WDWLVWLFD�218�GHFODQúHD] �UHFHQV PLQWH�vQ� ULOH�PHPEUH� 18) ÌQ�PRG�VLQWHWLF�RIHUWD�QD LRQDO �GH�EXQXUL�úL�VHUYLFLL�HVWH�H[SULPDW �GH� a) VN; b) PNNpp; c) PNBpp; d) PNBpf; e) PIBpf.

��)LH�;�úL�<�GRL�YHFWRUL�FX�Q�FRPSRQHQWH� ( )X xi= ’�úL� ( )Y yi= ’cu i n= 1, . 'LVWDQ D�OXL�0LQNRYVNL�HVWH�GDW �GH�UHOD LD�

( )d X Y x yi ip

i

n p, = −

=

∑1

1

cu p N∈ * .

'DF �S� ��VH�RE LQH�GLVWDQ D�DEVROXW ��LDU�GDF �S� ��VH�RE LQH�GLVWDQ D�HXFOLGLDQ � 3ULQ�H[WHQVLH�V �FRQVLGHU P�R�SRSXOD LH�VWDWLVWLF �FX�Q�XQLW L�OD�FDUH�VH�REVHUY �YDULDELOHOH�x1, x2, ... , xr��0DWULFHD�GLVWDQ HORU�GLQWUH�XQLW L��FDOFXODWH�SHQWUX�ILHFDUH�SHUHFKH��xi, yi), cu elemente

de forma ( )d d x xij i j= , :

D��QX�SUH]LQW �LQWHUHV�SHQWUX�DQDOL]D�VWDWLVWLF �

b) SUH]LQW �LQWHUHV�SHQWUX�DQDOL]D�YDULD LHL�úL�D�GHRVHELULORU��GLVLPLODULW LORU��GLQWUH�XQLW L� c��QX�SUH]LQW �LQWHUHV�SHQWUX�VWUXFWXUD�GDWHORU�FXOHVH� d) are toate elementele nenule;

e) DUH�FRPSRQHQWH�QXOH�SH�DPEHOH�GLDJRQDOH�úL�HOHPHQWHOH�VLPHWULFH�ID �GH�GLDJRQDOH�

20) 2�YDULDELO �DOHDWRDUH�;�XUPHD] �R�UHSDUWL LH�QRUPDO �GH�SDUDPHWULL� x �úL�σ 2 �GDF �DUH�GHQVLWDWHD�GH�UHSDUWL LH�

a) ( )1

2

1

22 2

2

σ π σexp − −

x x ;

b) ( )1 1

2 2

2

σ σexp − −

x x ;

c) ( )1

2

1

2 2

2

σ π σexp − −

x x ;

d) ( )1

2

1

2 2

2

σ π σexp − +

x x ;

e) ( )

−

2

22

1exp

2

1xx

σπσ.

21) 6H� FXQRDúWH� IDSWXO� F � PHGLD� H[SULP � WHQGLQ D� FHQWUDO � LDU� SH� DQVDPEOX� DEDWHULOH� YDORULORU�

LQGLYLGXDOH� ID � GH� HD� VH� FRPSHQVHD] � UHFLSURF�� &DUH� YDULDQW � GH� U VSXQV� QX� IRUPDOL]HD] �DILUPD LD�HIHFWXDW �

a) ( )x xii

n− =

=∑

1

0 ;

b) 1 1

01 x xi hi

n−

=

=∑ ;

c) x xi pi

n2 2

1

0−

=

=∑ ;

d) x

xi

gi

n

=∏ =

1

1 ;

e) x x x xh p g= = = .

unde: x �PHGLD�DULWPHWLF

xh �PHGLD�DUPRQLF

x p �PHGLD�S WUDWLF

xg �PHGLD�JHRPHWULF �

22) Trecerea de la venituO�QD LRQDO�OD�YHQLWXO�GLVSRQLELO�VH�UHDOL]HD] �vQ�FDGUXO�FRQWXOXL�

a) repartizarea veniturilor; b) modificarea patrimoniului; F��ILQDQ DUH� d) redistribuirea veniturilor; H��QX�VH�UHDOL]HD] �vQ�FDGUXO�XQXL�DQXPLW�FRQW�PDFURHFRQRPLF�

23) Cifra de afaceri�D�XQHL�VRFLHW L�FRPHUFLDOH�D�FUHVFXW�vQ�SHULRDGD�LDQ��- dec. 1996 cu 10% GH� OD� R� OXQ � OD� DOWD�� ÌQ� SURJUDPXO� VRFLHW LL� FRPHUFLDOH� HVWH� SUHY ]XW� FD� FLIUD� GH� DIDFHUL� vQ�perioada ian. 1990 -�GHF��V �vQUHJLVWUH]H�XQ�LQGLFH�PHGLX�DQXDO�GH��3HQWru perioada ian. 1996 - dec. 1998 indicele mediu anual care trebuie realizat este:

a) anti loglog , log ,79 1 3 45 11

24

−

;

b) anti loglog , log ,1 3 11

2

−

;

c) 120%; d) 95%; e) 118,1%.

24) 3HQWUX� DSUHFLHUHD� P VXULL� vQ� FDUH� XQ� IDFWRU� VHPQLILFDWLY� GH� LQIOXHQ � GHWHUPLQ � R� YDULDELO �GHSHQGHQW ��VH�FDOFXOHD] �DWkW�FRHILFLHQWXO�GH�FRUHOD LH�FkW�úL�UDSRUWXO�GH�FRUHOD LH��9DORULOH�FHORU�GRL�LQGLFDWRUL�ILLQG�LGHQWLFH�DWXQFL�OHJ WXUD�VWDWLVWLF �GLQWUH�FHOH�GRX �YDULDELOH�HVWH�

D��OLQLDU � E��KLSHUEROLF � F��SDUDEROLF � d) logiVWLF � H��H[SRQHQ LDO �

25) 8Q�úRPHU�FDUH�FDXW �XQ�ORF�GH�PXQF �vQWU-R�DQXPLW �LQGXVWULH�FXOHJH�LQIRUPD LL�GHVSUH�VDODULXO�RIHULW�GH�ILUPHOH�GLQ�DFHDVW �LQGXVWULH��3UHVXSXQHP�F �GLVWULEX LD�VDODULXOXL�RIHULW�DQJDMD LORU�GH�FDOLILFDUH�VLPLODU �SRDWH�IL�DSUR[LPDW �FX�R�GLVWULEX LH�QRUPDO �GH�PHGLH��PLL� OHL�úL�DEDWHUH�PHGLH�S WUDWLF �GH��PLL� OHL�� ÌQ�SOXV��SUHVXSXQHP�F �PXQFLWRUXOXL� L�VH�RIHU ��PLL� OHL�GH�F WUH�SULPD�ILUP �FRQWDFWDW ��3URSRU LD�RIHUWHORU�VDODULDOH�FDUH�DU�SXWHD�V �VH�VLWXH]H�SHVWH�DFHVW nivel este:

a) 4,01%; b) 8,02%; c) 54,01%; d) 91,98%; e) 45,99%.

26) 6H� FXQRDúWH� IDSWXO� F � vQWU-R� UDPXU � GH� DFWLYLWDWH� YDULDELOD� �VDODULXO� PHGLX� EUXW�� <�� HVWH�GHSHQGHQW �SULQWUH�DOWHOH�GH��JUDGXO�PHGLX�GH�vQGHSOLQLUH�DO�QRUPHORU��;1��úL��JUDGXO�PHGLX de vQ]HVWUDUH� WHKQLF � D�PXQFLL�� ;2�� &RQWULEX LD� FHORU� GRL� IDFWRUL�;1 úL�;2� OD� YDULD LD� VDODULXOXL�PHGLX�EUXW��<��HVWH�HYDOXDW �SULQ�

a) R r ry x x y x y x/ , / /1 2 1 2

2 2 2= + ;

b) Rr r r r r

ry x x

y x y x y x y x x x

x x

/ ,/ / / /

1 21 2 1 1 2

1 2

2 22

2

2

1=

+ −

−;

c) R r r r r ry x x y x y x y x y x x x/ , / / / /1 2 1 2 1 1 22 2

22= + − ;

d) rx x1 20= ;

e) Rr r

ry x x

y x y x

x x

/ ,/ /

1 21 2

1 2

2 2

21=

+

−.

��&RQVLGHU P�GDWHOH�

Ramura 9DORDUHD�DG XJDW �EUXW ��POG��OHL� (ILFLHQ D�PLMORDFHORU�IL[H 1 3000 0,5 2 1000 0,8

(ILFLHQ D�PLMORDFHORU�IL[H�SH�WRWDO�D�IRVW�

a) 0,5751; b) 0,6500; c) 0,5517; G��QX�SRDWH�IL�FDOFXODW � e) 1,3.

28) ÌQ� HYROX LD� VD� vQGHOXQJDW �� VWDWLVWLFD� V-a dezvoltat continuu atât sub aspectul obiectului de VWXGLX�� GDU�PDL� DOHV� DO� SHUIHF LRQ ULL� úL� GLYHUVLILF ULL�PHWRGHORU� GH� DERUGDUH� DOH� RELHFWXOXL�� ÌQ�WRDWH�PRPHQWHOH�GH]YROW ULL�VWDWLVWLFD�VH�RFXS �GH�DFHOe fenomene:

D��FDUH�VH�SURGXF�OD�R�DQXPLW �XQLWDWH�GH�REVHUYDUH� b) care se produc în mod identic; c) exclusiv tehnice;

d) care se produc într-XQ� QXP U� PDUH� GH� FD]XUL�� SUH]LQW � vQ� SURGXFHUHD� ORU� DQXPLWH�UHJXODULW L�úL�FDUH�SRW�IL�GHQXPLWH�IHQRPHQH�GH�PDV ;

H��FDUH�DX�OD�RULJLQH�XQ�VLQJXU�IDFWRU�GH�LQIOXHQ �

29) 8Q�HOHPHQW�LPSRUWDQW�DO�XQXL�SURJUDP�GH�REVHUYDUH�HVWH�WLPSXO�REVHUY ULL��$FHVWD� D��HVWH�WLPSXO�OD�FDUH�VH�UHIHU �GDWHOH�VDX�WLPSXO�vQUHJLVWU ULL�GDWHORU�

b) nu este altceva decât timpul la care se reIHU � GDWHOH� FXOHVH�� FDUH� SRDWH� IL� XQ� VLQJXU�PRPHQW��SHQWUX�vQUHJLVWU ULOH�VWDWLFH��VDX�R�SHULRDG ��SHQWUX�vQUHJLVWU ULOH�GLQDPLFH��

F��QX�HVWH�DOWFHYD�GHFkW�SHULRDGD�vQ�FDUH�VH�vQUHJLVWUHD] �GDWHOH�VWDWLVWLFH� d) HVWH�SULYLW�úL�SULQ�XUPDUH�SUHFL]DW�FD�WLPS�OD�FDUH�VH�UHIHU �GDWHOH�FXOHVH�úL�FD�LQWHUYDO�GH�

WLPS�vQ�FDUH�VH�UHDOL]HD] �vQUHJLVWUDUHD�GDWHORU� e) nu este altceva decât data la care operatorii statistici trebuie s prezinte datele culese.

��'LVSHUVLD��FDOFXODW �SULP�PHWRGD�PRPHQWHORU��HVWH�

a) momeQWXO�LQL LDO�GH�RUGLQXO�GRL� b) PRPHQWXO�LQL LDO�GH�RUGLQXO�SDWUX�vPS U LW�OD�PRPHQWXO�LQL LDO�GH�RUGLQXO�GRL�OD�S WUDW�

F��PRPHQWXO�FHQWUDW�GH�RUGLQXO�GRL�OD�S WUDW� d) PRPHQWXO�LQL LDO�GH�RUGLQXO�GRL�PLQXV�PRPHQWXO�LQL LDO�GH�RUGLQXO�vQWkL�OD�S WUDW� e) momentul�FHQWUDW�GH�RUGLQXO�GRL�PLQXV�PRPHQWXO�LQL LDO�GH�RUGLQXO�XQX�OD�S WUDW�

31) Scala de interval:

D��DUH�WRDWH�FDUDFWHULVWLFLOH�VFDOHORU�RUGLQDOH�úL�GH�UDSRUW� b) DUH� WRDWH� FDUDFWHULVWLFLOH� VFDOHL� RUGLQDOH� úL� vQ� SOXV�� GLVWDQ D� VDX� GLIHUHQ D� GLQWUH� GRX �

numere�DOH�VFDOHL�DUH�VHPQLILFD LH�FRQFUHW �

c) HVWH�R�VFDO �QXPHULF �úL��vQ�SOXV��UDSRUWXO�GLQWUH�GRX �SXQFWH�DOH�VFDOHL�HVWH�LQGHSHQGHQW�GH�XQLWDWHD�GH�P VXU �

G��SUH]LQW �PXOWH�GLQ�FDUDFWHULVWLFLOH�VFDOHL�RUGLQDOH� H��PDL�HVWH�QXPLW �úL�VFDO �GH�UDSRUW�VDX�VFDO �GLVFUHW �

32) Într-R�UHSDUWL LH�QRUPDO �YDORDUHD�ID �GH�FDUH��GLQ�YDORULOH� LQGLYLGXDOH�VXQW�PDL�PLFL�� LDU�75% din valorile individuale sunt mai mari este:

a) cuartila a doua; b) cuartila a treia; c) cuartila întâi; G��YDORDUHD�PRGDO � e) valoarea�PHGLDQ �

33) 2�DJHQ LH�GH� vQFKLULDW�DXWRPRELOH�KRW U úWH� V -VL�YkQG �DXWRPRELOHOH�GXS �XWLOL]DUHD�DFHVWRUD�timp de un an. Managerul firmei presupune c �GLVWDQ D�SDUFXUV �GH�PDúLQL�LQIOXHQ HD] �FRVWXO�GH�vQWUH LQHUH�DO�DXWRYHKLFXOHORU�úL�GHFL�SUH XO�GH�YkQ]DUe al acestora. Pentru a verifica presupunerea VD��VH�vQUHJLVWUHD] ��SHQWUX��GH�PDúLQL��GLVWDQ D�SDUFXUV �vQ�XOWLPXO�DQ��PLL�NP��úL�FRVWXO�GH�vQWUH LQHUH��VXWH�PLL�OHL��6H�DOF WXLHVF�DVWIHO��JUXSH�GXS �GLVWDQ D�SDUFXUV ��VXE��PLL�NP��- 60 mii km, 60 -��PLL�NP�úL��VDX�SHVWH��PLL�NP�� ÌQ�XUPD�VLVWHPDWL] ULL�GDWHORU�� VH�RE LQ�XUP WRDUHOH�LQIRUPD LL�

&RVWXO�vQWUH LQHULL��VXWH�PLL�OHL�

*UXSH�GXS �GLVWDQ D SDUFXUV ��PLL�NP�

5 - 7 7 - 9 9 - 11 11 - 13

Total

sub 40 16 32 12 - 60 40 - 60 10% 25% 35% 30% 100%

ÌQ� JUXSD� D� WUHLD� GXS � GLVWDQ D� SDUFXUV � ��-80 mii km) s-au înregistrat date pentru 40 de PDúLQL��FRVWXO�PHGLX�GH�vQWUH LQHUH�ILLQG��OHL��FX�XQ�FRHILFLHQW�GH�YDULD LH�GH��LDU�vQ�JUXSD�D�SDWUD� �� úL�SHVWH��PLL�NP�� V-au îQFDGUDW��GLQ�PDúLQL�� FKHOWXLHOLOH� WRWDOH�DX� IRVW��OHL��LDU�DEDWHUHD�PHGLH�S WUDWLF ��PLL�OHL� 'LVWDQ D�SDUFXUV �LQIOXHQ HD] �YDULD LD�FKHOWXLHOLORU�GH�vQWUH LQHUH�vQ�SURSRU LH�GH� a) 51,04%; b) 71,44%; c) 48,96%; d) 69,97%; e) 28,56%. 34) ÌQ� XWLOL]DUHD� PHWRGHL� WDEHOXOXL� GH� FRUHOD LH� SHQWUX� DQDOL]D� OHJ WXULORU� GLQWUH� IHQRPHQH�� GDF �

IUHFYHQ HOH�VXQW�DSUR[LPDWLY�HJDO�UHSDUWL]DWH�vQ�LQWHULRUXO�WDEHOXOXL��DWXQFL� D��OHJ WXUD�HVWH�QHOLQLDU � E��OHJ WXUD�HVWH�OLQLDU �GLUHFW � F��OHJ WXUD�HVWH�LQYHUV � G��OHJ WXUD�HVWH�GLUHFW � H��OHJ WXUD�HVWH�VODE �VDX�QX�H[LVW �OHJ WXU între variabile. 35) Într-R� VHULH� VWDWLVWLF �� vQWUH� SULQFLSDOHOH� WLSXUL� GH� PHGLL� �PHGLD� DULWPHWLF �� PHGLD� DUPRQLF ��

PHGLD�S WUDWLF ��PHGLD�JHRPHWULF ��H[LVW �XUP WRDUHD�UHOD ie:

a) x x x xh g p≤ ≤ ≤ ;

b) x x x xg h p≤ ≤ ≤ ;

c) x x x xh p g≤ ≤ ≤ ;

d) x x x xh g p≤ ≤ ≤ ;

e) x x x xh g p= = = ��GDF �VHULD�HVWH�SHUIHFW�VLPHWULF � 36) 3HQWUX� R� VHULH� GH� GLVWULEX LH� GH� IUHFYHQ H� FX� WHQGLQ � GH� QRUPDOLWDWH�� DEDWHUHD� PHGLH� OLQLDU �

( d x ��ID �GH�DEDWHUHD�WLS��σ x ) este:

a) d x x> σ ;

b) d x x= σ ;

c) d x x≈ 4

5σ ;

d) d x x≈ 2

3σ ;

e) d x x≈ σ 2 . 37) VHQLWXO�EUXW�GLVSRQLELO�OD�QLYHOXO�HFRQRPLHL�QD LRQDOH�VH�FDOFXOHD] � a) VN+Amortizarea; E��91�6ROGXO�YHQLWXULORU�GLQ�WUDQVIHUXUL�vQ�UDSRUW�FX�VWU LQ WDWHD� c) b) + Amortizarea; d) a) -�6ROGXO�YHQLWXULORU�GLQ�WUDQVIHUXUL�vQ�UDSRUW�FX�VWU LQ WDWHD� e) PIB - b). 38) 8Q�GLVWULEXLWRU�GH�VXFXUL�OLYUHD] �FDUWRDQH�GH�FkWH��VWLFOH�úL�FXQRDúWH�SUREDELOLWDWHD��S ��FD�

R� VWLFO � V � VH� VSDUJ � SH� GXUDWD� WUDQVSRUWXOXL�� 'DF � VH� WUDQVSRUW � �� GH� FDUWRDQH�� QXP UXO�FDUWRDQHORU�vQ�FDUH�VH�YRU�J VL�GRX sau mai multe sticle sparte este de aproximativ:

a) 85; b) 14; c) 3960; d) 41; e) 40. ��&DUH�GLQ�REVHUY ULOH�VWDWLVWLFH�SUHFL]DWH�PDL�MRV�QX�DX�XQ�FDUDFWHU�SHUPDQHQW� D��EXJHWHOH�GH�IDPLOLH��VDX�DQFKHWHOH�LQWHJUDWH�vQ�JRVSRG ULL�� b) rapoartele statistice; c) statisticile fiscale; G��vQUHJLVWU ULOH�F V WRULLORU��QDWDOLW LL��PRUWDOLW LL��HWF�� H��UHFHQV PLQWHOH�VWDWLVWLFH�

TESTUL 6

1) ÌQWUH�3,%�úL�31%�H[LVW �QXPDL�UHOD LD�

a) >; b) <; c) =; d) ≥ ; e) ≤ ≥ .

2) Într-o serie de reparWL LH�SH�LQWHUYDOH�IRUPDW �GXS �R�DQXPLW �YDULDELO ��YDORDUHD�VD�PRGDO �HVWH�

LQIOXHQ DW �GH�

a) OLPLWHOH�LQIHULRDUH�úL�VXSHULRDUH�DOH�LQWHUYDOHORU�H[WUHPH�

b) GH�IUHFYHQ HOH�GH�DSDUL LH�DOH�LQWHUYDOHORU�HJDOH�VDX�QHHJDOH�

c) GH�P ULPHD�HJDO �VDX�QHHJDO �D�LQWHUYDOHORU�GH�YDULD LH�

d) GH�OLPLWD�LQIHULRDU �D�LQWHUYDOXOXL�PRGDO�

e) GH�OLPLWD�LQIHULRDU ��GH�IUHFYHQ HOH�LQWHUYDOXOXL�PRGDO��SUHPRGDO�úL�SRVWPRGDO�

3) Într-XQ�VSLWDO��GLQ�SDFLHQ LL�LQWHUQD L�P QkQF �UHJLP�I U �VDUH��$OF WXLQG�JUXSXUL�GH�FkWH��

SDFLHQ L��SUREDELOLWDWHD�FD�H[DFW��SDFLHQ L�V � LQ �UHJLP�I U �VDUH�HVWH�

a) 0,015625; b) 0,2344; c) 0,9375; d) 0,50; e) 0,0468.

4) 3UHFL]D L�FDUH�QX�HVWH�SULQFLSDO �IXQF LH�D�LQGLFDWRULORU�VWDWLVWLFL�

a) GH�P VXUDUH� b) GH�DQDOL] �VDX�VLQWH] � c) GH�YHULILFDUH�D�LSRWH]HORU�úL�VDX�GH�WHVWDUH�D�VHPQLILFD LHL�SDUDPHWULORU�XWLOL]D L� d) de fundamentare a deciziilor; e) de estimare.

5) 3URGXVXO� JOREDO� EUXW� FUHúWH� vQWU-R� SHULRDG � FX� �� LDU� SRQGHUHD� FRQVXPXOXL� LQWHUPHGLDU� vQ�

produsul global brut scade de la 64% la 60%. Produsul intern brut:

a) FUHúWH�FX�� b) scade cu 28,8%; c) FUHúWH�FX�� d) FUHúWH�FX�� e) FUHúWH�FX��

6) ÌQ�WHVWDUHD�LSRWH]HL�VWDWLVWLFH�SULYLWRDUH�OD�SDUDPHWUXO�³PHGLD�SRSXOD LHL´�

H m m

H m ma

0 0

0

:

:

=<

FkQG�GDWHOH�SURYLQ�GH�OD�XQ�HúDQWLRQ�GH�YROXP�UHGXV��UHJLXQHD�FULWLF �HVWH�GDW �GH�

a) z z< − α ;

b) z z< − α / 2 ;

c) t t n< − α , ;

d) t t n< − α / ,2 ;

e) t t n< − −α , 1 .

7) &DUH�GLQ�XUP WRDUHOH�DILUPD LL�SULYLQG�PHGLD�DULWPHWLF �HVWH�DGHY UDW �

a) VXPD� S WUDWHORU� WXWXURU� DEDWHULORU� LQGLYLGXDOH� DOH� WHUPHQLORr seriei de la media lor DULWPHWLF �HVWH�]HUR�

b) P ULPHD� PHGLHL� DULWPHWLFH� HVWH� FXSULQV � vQWUH� [min� úL� [max� GRDU� GDF � VHULD� SUH]LQW � R�WHQGLQ �FODU �GH�VLPHWULH�

c) SHQWUX�R�VHULH�GH�GLVWULEX LH�GH�IUHFYHQ H��PHGLD�DULWPHWLF ��FDOFXODW �SH�ED]D�IUHFYHQ HORU�reGXVH�GH�F�RUL�HVWH�PDL�PLF �GHFkW�PHGLD�VHULHL�LQL LDOH�GH�F�RUL�

d) PHGLD�SURGXVXOXL�D�GRX �YDULDELOH�DOHDWRDUH��HVWH�HJDO �FX�SURGXVXO�PHGLLORU�FHORU�GRX �variabile;

e) SHQWUX�R� VHULH� GH� GLVWULEX LH� GH� IUHFYHQ H��PHGLD� FDOFXODW � SH� ED]D� IUHFYHQ HORU� UHODWLYH�eVWH�HJDO �FX�PHGLD�DULWPHWLF �GHWHUPLQDW �SH�ED]D�IUHFYHQ HORU�DEVROXWH�

8) $QDOL]D� IHQRPHQXOXL� GH�FRQFHQWUDUH� �� GLYHUVLILFDUH�QX�DUH� VHQV�DWXQFL� FkQG�JUXSDUHD� XQLW LORU�

FROHFWLYLW LL�V-a efectuat:

a) pe variante; b) SH�LQWHUYDOH�HJDOH�GH�YDULD LH� c) GXS �R�FDUDFWHULVWLF �GH�VSD LX� d) GXS �R�FDUDFWHULVWLF �DOWHUQDWLY � e) GXS �R�FDUDFWHULVWLF �QHQXPHULF �

9) 7LPSXO�REVHUY ULL�VDX�DO�FXOHJHULL�GDWHORU�VH�DOHJH�

a) vQWRWGHDXQD�vQ�SHULRDGD�vQ�FDUH�SRSXOD LD�VWDWLVWLF �SUH]LQW �XQ�PDUH�JUDG�GH�VWDELOLWDWH� b) vQ�IXQF LH�GH�VFRSXO�FHUFHW ULL�úL�DO�REVHUY ULL�VWDWLVWLFH��GXS �FD]��FkQG�SRSXOD LD�VWDWLVWLF �

SUH]LQW � VDX� QX� XQ� JUDG� PDUH� GH� VWDELOLWDWH� �GH� H[HPSOX� UHFHQV PkQWXO� SRSXOD LHL��REVHUYDUHD�FRWD LLORU�OD�EXUV �D�XQRU�DF LXQL��HWF��

c) vQ�IXQF LH�GH�SRVLELOLW LOH�LQIRUPDWLFH�GH prelucrare a datelor culese; d) DWXQFL�FkQG�VH�GRUHúWH�V �VH�MXVWLILFH�R�DQXPLW �DF LXQH�GHMD�GHFODQúDW � e) OD�vQWkPSODUH��I U �FULWHULL�SUHFL]DWH�úL�DQDOL]DWH�DSULRULF�

10) Într-R� FROHFWLYLWDWH� VWDWLVWLF �� GDF � GLYHUVLILFDUHD� HVWH� PD[LP � �úL�� HYLGHQW� FRQFHQWUDUHD este PLQLP ��DWXQFL�FRHILFLHQWXO�GH�FRQFHQWUDUH�&RUUDGR-Gini este:

a) G=1;

b) Gn

= 1;

c) Gn

= 1;

d) G=0;

e) G n= . 11) ÌQ�FRQWXO�PDFURHFRQRPLF�³9HQLWXUL´��UHVXUVHOH�VH�IRUPHD] �GLQ�

a) $PRUWL]DUH�úL�3,1pp; b) PIBpp�úL�VXEYHQ iile de exploatare; c) PIBpp�úL�LPSR]LWHOH�LQGLUHFWH� d) Amortizare, PIBpp�úL�VXEYHQ LLOH�GH�H[SORDWDUH� e) $PRUWL]DUHD��LPSR]LWH�LQGLUHFWH�úL�LPSR]LWH�SH�LPSRUWXUL�

12) 6 �VH�SUHFL]H]H�FDUH�GLQ�XUP WRDUHOH�DILUPD LL�QX�HVWH�DGHY UDW �

a) vQ�RULFH�UHSDUWL LH�VLPHWULF ��XQLPRGDO �� x Me Mo= = ; b) într-R�UHSDUWL LH�XQLPRGDO ��XúRU�DVLPHWULF ��PHGLDQD�VH�SODVHD] �vQWUH�PHGLH�úL�PRG� c) într-R�UHSDUWL LH�XQLPRGDO ��XúRU�DVLPHWULF ��PHGLD�VH�SODVHD] �vQWUH�PRG�úL�PHGLDQ � d) într-R�UHSDUWL LH�XQLPRGDO �SR]LWLY�vQFOLQDW �F tre valorile mici x Me Mo< < ; e) într-R� UHSDUWL LH� XQLPRGDO � SR]LWLY� vQFOLQDW � F WUH� YDORULOH� PDUL�� PRGXO� VH� VLWXHD] � OD�

VWkQJD�PHGLDQHL�úL�D�PHGLHL� 13) 8Q� DQDOLVW� HFRQRPLF� VWXGLD] � GLVWULEX LD� ILUPHORU� GXS � SURFHQWXO� GLQ� YHQLWXUL� FKHOWXLW� SHQWUX�

cercetare-GH]YROWDUH��3HQWUX�D�YHULILFD�LSRWH]D�QRUPDOLW LL�HPSLULFH�RE LQXWH��HO�XWLOL]HD] �WHVWXO�

χ2�úL�RE LQH��SHQWUX�XQ�QXP U�GH�JUDGH�GH�OLEHUWDWH��O ��YDORDUHD� χ calc2 18 30= , . Probabilitatea

PD[LP �FX�FDUH�VH�JDUDQWHD] �UH]XOWDWXO�HVWH�

a) 75%; b) 90%; c) 95%; d) 99%; e) 97,5%.

14) &DUH�GLQ�XUP WRDUHOH�DILUPD LL�UHIHULWRDUH�OD�FRHILFLHQWXO�GH�YDULD LH�QX�HVWH�DGHY UDW �

a) HVWH�XQ�LQGLFDWRU�VLQWHWLF�DO�vPSU úWLHULL� b) HVWH�H[SUHVLD�UHODWLY �D�DEDWHULL�PHGLL�S WUDWLFH� c) YDORUL�PLFL�DOH�FRHILFLHQWXOXL�GH�YDULD LH�VHPQLILF �XQ�JUDG�PDUH�GH� UHSUH]HQWDWLYLWDWH�D�

mediei caracteristicii studiate; d) YDORUL�PLFL�DOH�FRHILFLHQWXOXL�GH�YDULD LH�UHIOHFW �R�WHQGLQ �DFFHQWXDW �GH�VLPHWULH�D�GLVWULEX LHL� e) YDORUL�PLFL�DOH�FRHILFLHQWXOXL�GH�YDULD LH� UHIOHFW �RPRJHQLWDWHD�FROHFWLYLW LL�GLQ�SXQFWXO�

de vedere al caracteristicii studiate. 15) 2� FDUDFWHULVWLF � LPSRUWDQW � GH� FDOLWDWH� SHQWUX� VXFXULOH� GH� IUXFWH� vPEXWHOLDWH� HVWH� SURSRU LD� GH�

FRQFHQWUDW� QDWXUDO� SH� FDUH� R� FRQ LQ�� ÌQWU-XQ� SURFHV� GH� IDEULFD LH� D� XQHL� P UFL� GH� VXFXUL��presupunem c �SURFHQWXO�GH�FRQFHQWUDW�QDWXUDO�HVWH�DSUR[LPDWLY�QRUPDO�GLVWULEXLW��FX�PHGLD��úL� FRHILFLHQWXO� GH� YDULD LH� �� 6WLFOHOH� FDUH� FRQ LQ�PDL� SX LQ� GH� �� FRQFHQWUDW� QDWXUDO� VXQW�

UHVSLQVH� FDOLWDWLY�� 'DF � �� GH� VWLFOH� VXQW� YHULILFDWH�� OD� FRQWUROXO� FDOLWDtiv pot fi respinse aproximativ:

a) 2112 sticle; b) 4224 sticle; c) 1056 sticle; d) 124 sticle; e) 248 sticle.

16) 3HQWUX�FDUDFWHUL]DUHD�SURFHVHORU�GH�SURGXF LH��VXELHFWHOH�HFRQRPLFH�VH�DJUHJ �

a) SH�VHFWRDUH�LQVWLWX LRQDOH� b) GXS �IRUPD�MXULGLF � c) pe ramuri; d) QX�HVWH�QHFHVDU �R�DJUHJDUH�VSHFLDO �SHQWUX�DFHVW�VFRS� e) DWkW�SH�VHFWRDUH�LQVWLWX LRQDOH�FkW�úL�SH�UDPXUL�

17) 0HGLD�DULWPHWLF �D�S WUDWHORU�DEDWHULORU�LQGLYLGXDOH�GH�OD�WHQGLQ D�ORU�FHQWUDO �VH�QXPHúWH�

a) DEDWHUH�PHGLH�DEVROXW � b) DEDWHUH�PHGLH�S WUDWLF � c) dispersie; d) PHGLDO � e) moment centrat de ordin trei.

18) Într-R� úFRDO � JHQHUDO � VH� RUJDQL]HD] � XQ� WHVW� GH� YHULILFDUH� D� DSWLWXGLQLORU� HOHYLORU� SHQWUX�

LQIRUPDWLF ��'LVWULEX LD�HOHYLORU�GXS �SXQFWDMXO�RE LQXW�HVWH�DSUR[LPDWLY�QRUPDO ��3ULPLL��GLQ�FHL�PDL�SX LQ� vQFOLQD L� VSUH�DFHDVW �GLVFLSOLQ �RE LQ�SkQ � OD��SXQFWH�� LDU� FHL�PDL�SHUIRUPDQ L��RE LQ�� úL�SHVWH��GH�SXQFWH��'DF �YDORDUHD�PHGLDQ �HVWH�GH��GH�SXQFWH�� LDU�FHL�PDL�PXO L�GLQWUH�HOHYL�RE LQ��GH�SXQFWH��FRHILFLHQWXO�OXL�%RZOH\�HVWH�

a) 0,4; b) 0,53; c) 0,07; d) - 0,4; e) - 0,53.

19) PenWUX�VWXGLXO�FDOLW LL�SURGXF LHL�IDEULFDWH�vQ�VHULH��vQWU-R�XQLWDWH�HFRQRPLF ��VH�REVHUY �SHULRGLF�XQ�ORW�GH��H[HPSODUH�SUHOHYDWH�GLQ�SURGXVHOH�DIODWH�OD�VIkUúLWXO�ODQ XOXL�GH�IDEULFD LH��3RSXOD LD�VWDWLVWLF �LQYHVWLJDW �HVWH�

a) SURGXF LD�WRWDO �UHDOL]DW � b) ansamblul loturilor de produse; c) lotul format prin prelevare; d) SURGXF LD�QHWHUPLQDW � e) SURGXF LD�UHDOL]DW ��IDFWXUDW �úL�OLYUDW �EHQHILFLDULORU�

20) 'DF � VH� XUP UHúWH� FXQRDúWHUHD� FRPSRUWDPHQWXOXL� vQ� GRPHQLXO� YHQLWXULORU�� FKHOWXLHOLORU� úL�

RSHUD LXQLORU�ILQDQFLDUH��VXELHFWHOH�HFRQRPLFH�VH�DJUHJ �GXS �FULWHULXO�

a) DFWLYLW LL�GHSXVH� b) LQVWLWX LRQDO� c) GH�UDPXU �

d) IRUPD�MXULGLF �GH�RUJDQL]DUH� e) a)+b)+c)+d).

21) 6DODULD LL�XQHL�ILUPH�DX�VDODULXO�PHGLX��PLL�OHL��FX�R�DEDWHUH�PHGLH�S WUDWLF �D�VDODULLORU��

mii lei. PatroQXO� ILUPHL� KRW U úWH� V � P UHDVF � ILHFDUH� VDODULX� LQGLYLGXDO� GH� �� RUL�� 'LVSHUVLD�QRLORU�VDODULL�ID �GH�VDODULXO�PHGLX�YD�IL�

a) 150;

b) 22500;

c) 29250;

d) 253,5;

e) 38025.

22) ÌQ� PRGHOXO� GH� DQDOL] � GLVSHUVLRQDO � XQLIDFWRULDO �� GLVSHUVLD� FRUHFWDW � GLQWUH� JUXSH� VH� RE LQH�

UDSRUWkQG�YDULDQ D�VLVWHPDWLF ��GLQWUH�FHOH�U�JUXSH��OD�QXP UXO�JUDGHORU�GH�OLEHUWDWH��QRWDW�FX�³O´��Acest num r este:

a) l=n-1; b) l=n-r; c) l=r-1; d) l=n-r-1; e) l=n-2.

23) ÌQ�FD]XO�LQGLFDWRULORU�GH�WLS�FDOLWDWLY��DJUHJDUHD�VH�UHDOL]HD] �SULQWU-o:

a) RSHUD LH�GH�DGXQDUH� b) oSHUD LH�GH�DGXQDUH�VDX�VF GHUH� c) numai printr-R�PHGLH�DULWPHWLF � d) numai prin calculul unei medii armonice; e) SULQ�FDOFXOXO�XQHL�P ULPL�PHGLL�

24) ,GHQWLILFD L�FDUH�GLQWUH�DILUPD LLOH�XUP WRDUH�UHIHULWRDUH�OD�RELHFWXO�REVHUY ULL�HVWH�IDOV �

a) 2ELHFWXO� REVHUY ULL� HVWH� PXO LPHD� GH� IHQRPHQH� GH� PDV � GHVSUH� FDUH� XUPHD] � V � VH�FXOHDJ �GDWHOH�VWDWLVWLFH�

b) 2ELHFWXO�REVHUY ULL�HVWH�R�PXO LPH�GH�XQLW L�VLPSOH�VDX�FRPSOH[H�GH�REVHUYDUH� c) 3RSXOD LD� VXSXV �REVHUY ULL� WUHEXLH�GHOLPLWDW �FD�YROXP�� vQ� WLPS� úL� vQ�VSD LX�� LQGLIHUHQW�

GDF �HD�HVWH�VWDWLF �VDX�GLQDPLF � d) 1X�vQWRWGHDXQD�RELHFWXO�REVHUY ULL�FRLQFLGH�FX�RELHFWXO�FHUFHW ULL� e) &KLDU�GDF �RELHFWXO�REVHUY ULL�HVWH�ELQH�GHILQLW��GXS �FULWHULL�FODUH��HVHQ LDOH��VWDELOH�QX�VH�

SRDWH�DVLJXUD�VXE�QLFL�R�IRUP �FRPSDUDELOLWDWHD�GDWHORU�Ge la o observare la alta. 25) &RUHF LD�OXL�6KHSSDUG�VH�DSOLF �GLVSHUVLHL�FkQG�

a) seria este pe variante; b) VHULD�HVWH�SH�LQWHUYDOH�QHHJDOH�GH�YDULD LH� c) VHULD�HVWH�SH�LQWHUYDOH�HJDOH�GH�YDULD LH� d) FkQG�GLVWULEX LD�HVWH�DSUR[LPDWLY�QRUPDO �úL�SUH]LQW �XQ�QXP U�PDUH�de intervale egale; e) FkQG�VHULD�HVWH�ELPRGDO �

26) 3UHFL]D L�FDUH�HOHPHQWH�GLQ�FHOH�SUH]HQWDWH�PDL�MRV�QX�HVWH�R�IXQF LH�VSHFLILF �PHWRGHL�JUXS ULL��FODVLILF ULL��vQ�DFWLYLWDWHD�VWDWLVWLF �

a) VLVWHPDWL]DUHD�GDWHORU� LQGLYLGXDOH�SULQ� UHVWUkQJHUHD�QXP UXOXL�GH�variante ale variabilei de grupare;

b) IDFLOLWDUHD�FDOFXOXOXL�WXWXURU�LQGLFDWRULORU�SULPDUL�úL�GHULYD L� c) VWUXFWXUDUHD�SRSXOD LHL�VWDWLVWLFH�vQ�FODVH��JUXSH�RPRJHQH� d) SUH]HQWDUHD�úL�GHVFULHUHD�VWUXFWXULL�SRSXOD LHL�VWDWLVWLFH� e) FDUDFWHUL]DUHD�OHJ WXULORU�VWDWLstice dintre variabilele analizate.

27) Programul propriu-]LV� DO� REVHUY ULL� HVWH� IRUPDW� GLQ� WRWDOLWDWHD� FDUDFWHULVWLFLORU� �YDULDELOHORU��SHQWUX�FDUH�XUPHD] �V �VH�FXOHDJ �GDWHOH�GH�OD�XQLW LOH�SRSXOD LHL�VWDWLVWLFH��(O�VH�FRQFUHWL]HD] �

a) vQ� vQWUHE UL� FDUH� VXnt înscrise în formulare statistice la care se primesc obligatoriu U VSXQVXUL�FX�SULYLUH�OD�QLYHOXO��GH�GH]YROWDUH�VDX�IRUPD�ORU�GH�PDQLIHVWDUH�

b) într-R�OLVW �GH�YDULDELOH�GHVSUH�FDUH�VH�FXOHJ�GDWHOH�GLQ�GRFXPHQWHOH�SH�FDUH�XQLW LOH�GH�REVHUYDUH�GRUHVF�V �OH�SXQ �OD�GLVSR]L LH�

c) vQ�vQWUHE UL�vQVFULVH�vQ�FKHVWLRQDUH�VDX�DOWH�IRUPXODUH�VWDWLVWLFH�DGUHVDWH�OD�WRDWH�VDX�OD�R�SDUWH�GLQ�XQLW LOH�SRSXOD LHL�úL�DOH�F URU�SRVLELOH�U VSXQVXUL�UHIOHFW �IRUPHOH�LQGLYLGXDOH�de manifestare ale fenomenelor;

d) vQWUHE UL�LQFOuse în chestionare sau în alte formulare statistice numai atunci când ele sunt DGUHVDWH� GLUHFW� WXWXURU� XQLW LORU� GH� REVHUYDUH� VDX� OD� R� SDUWH� GLQ� DFHVWHD� úL� DOH� F URU�SRVLELOH�U VSXQVXUL�UHIOHFW �IRUPHOH�LQGLYLGXDOH�DOH�IHQRPHQHORU�GH�PDV �LQYHVWLJDWH�

e) în variantele variabilelor precizate.

28) 3HQWUX�GLVWULEX LD�XQHL�YDULDELOH�DOHDWRDUH�;�VH�FXQRDúWH�YDORDUHD�GLVSHUVLHL��úL�YDORDUHD�momentului centrat de ordinul 4: 20876491.

&RHILFLHQWXO�OXL�)LVFKHU�SHQWUX�P VXUDUHD�EROWLULL�DUH�YDORDUHD�

a) γ 2 0 782= − , ;

b) γ 2 2 218= , ;

c) γ 2 5 218= , ;

d) γ 2 0 45= , ;

e) γ 2 2 218= − , .

29) 6H�H[FOXG�GLQ�FDOFXOXO�LQGLFDWRULORU�PDFURHFRQRPLFL�GH�UH]XOWDWH�WUDQ]DF LLOH�

a) intersectoriale; b) bilaterale; c) unilaterale; d) intrasectoriale; e) de SLD �

30) &RQWXO�PDFURHFRQRPLF�3URGXF LH�DUH�FD�VROG�

a) PINpp; b) PIBpf; c) PIBpp; d) VN; e) PNBpf.

31) 'LVWULEX LD� HOHYLORU� GLQ� FODVD� ,� D� XQHL� úFROL� JHQHUDOH� GXS � QXP UXO�PDWHULLORU� OD� FDUH� DX� DYXW�media zece este:

Nr. materii 0 1 2 3 4 5 Nr. elevi 10 30 80 70 50 20

VaORDUHD�PHGLDQ �D�VHULHL�HVWH�

a) 2 materii; b) 2,5 materii; c) 3 materii; d) 2,7 materii; e) 70 elevi.

32) 'LVWULEX LD�XQHL�YDULDELOH�DOHDWRDUH�;�HVWH�SHUIHFW�VLPHWULF ��GH�PHGLH� x = 20 ��'DF �VH�DGDXJ �GRX �XQLW L�VWDWLVWLFH�FX�YDORULOH�YDULDELOHL�VWXGLDWe x1 ��úL�[2 ��DWXQFL�QRXD�GLVWULEX LH�

a) DUH�DVLPHWULH�SR]LWLY � b) HVWH�SHUIHFW�VLPHWULF � c) DUH�DVLPHWULH�QHJDWLY � d) DUH�ILH�DVLPHWULH�SR]LWLY ��ILH�DVLPHWULH�QHJDWLY � e) QX�SUH]LQW �DVLPHWULH�

33) 1X�VXQW�FXSULQVH�vQ�YDORDUHD�SURGXF LHL�

a) serviciile statului; b) EXQXULOH�GH�FDSLWDO�GLQ�SURGXF LH�SURSULH� c) autoconsumul; d) FKLULD�DXWRLPSXV ��LSRWHWLF �� e) EXQXULOH�SURGXVH�úL�FRQVXPDWH�SHQWUX�SURGXFHUHD�DOWRU�EXQXUL�úL�VHUYLFLL�

34) Atunci când datele provin dintr-XQ� HúDQWLRQ� GH� YROXP� UHGXV� �Q�� úL� VH� WHVWHD] � LSRWH]D� QXO �prLYLQG�SDUDPHWUXO�PHGLD�vQWUHJLL�FROHFWLYLW L�

H m m

H m ma

0 0

0

:

:

=>

UHJLXQHD�FULWLF �HVWH�GDW �GH�

a) z z> α ;

b) z z> α / 2 ;

c) t t n> −α ; 1 ;

d) t t> α / 2 ;

e) t t n> − − −α 2 1; .

35) Aplicând modelul analizei GLVSHULVRQDOH� XQLIDFWRULDOH�� VH� DFFHSW � LSRWH]D� FRQIRUP� F UHLD�IDFWRUXO�GH�JUXSDUH�HVWH�VHPQLILFDWLY�GDF ��OD�XQ�SUDJ�GH�VHPQLILFD LH�α��VH�vQGHSOLQHúWH�FRQGL LD�

a) F Fcalc n r< − −α , ,1 1 ;

b) F Fcalc r n r< − −α , ,1 ;

c) F Fcalc r n< − −α , ,1 1 ;

d) F Fcalc r n> − −α , ,1 1 ;

e) F Fcalc r n r> − −α , ,1 .

36) 2�JUXS �GH��GH�VWXGHQ L�VXV LQH�OD�GRX �GLVFLSOLQH�FkWH�XQ�WHVW�SHQWUX�YHULILFDUHD�FXQRúWLQ HORU��

Testele au punctaje diferite, iar rezultatele sunt:

- la testul A: x Aii

2

1

259000=

=∑ ; x Ai

i

==∑ 450

1

25;

- la testul B: xBii

2

1

25425=

=∑ ; xBi

i

==∑ 100

1

25;

*UXSD�GH�VWXGHQ L�HVWH�PDL�RPRJHQ �GLQ�SXQFWXO�GH�YHGHUH�DO�FXQRúWLQ HORU�DFXPXODWH�

a) la disciplina A; b) la disciplina B; c) OD�DPEHOH�GLVFLSOLQH�SUH]LQW �DFHODúL�JUDG�GH�RPRJHQLWDWH� d) nu sunt sufLFLHQWH�GDWH�SHQWUX�D�IL�VWXGLDW �RPRJHQLWDWH�VHULLORU� e) QX�VH�SRW�FRPSDUD�RPRJHQLW LOH�OD�FHOH�GRX �GLVFLSOLQH��GHRDUHFFH�SXQFWDMHOH�WHVWHORU�DX�

fost diferite. 37) 3UREDELOLWDWHD�FD�R�FRPSDQLH�GH�DVLJXU UL�V �ILH� LQXW �V �SO WHDVF �R�SROL �SHQWUX�R�SUREOHP �

PHGLFDO � PDMRU � HVWH� �� 'DF � XQ� JUXS� GH� �� GH� SHUVRDQH� DVLJXUDWH� VXQW� VHOHFWDWH�vQWkPSO WRU�� SUREDELOLWDWHD� FD� DFHDVW � FRPSDQLH� V � SO WHDVF � FHO� SX LQ� R� SROL � SHUVRDQHORU�selectate este:

a) 0,8196; b) 0,1804; c) 0,1353; d) 0,8647; e) 0,001.

38) Valoarea individual �³]HUR´�D�YDULDELOHL�³P ULPHD�H[WUDVXOXL�GH�FRQW´�DUH�VHPQLILFD LH�FRQFUHW �

pe scala:

a) RUGLQDO � b) de rapoarte; c) de intervale; d) GLVFUHW � e) FRQWLQX �

39) 3HQWUX� SDWUX� ILOLDOH� DOH� XQHL� VRFLHW L� FRPHUFLDOH� VWUXFWXUD�FLIUHL� GH�DIDFHUL� UHDOL]DWH� úL� LQGLFLL�

UHDOL] ULL programului la cifra de afaceri au fost:

Filiala Structura cifrei de afaceri realizate (%)

,QGLFHOH�UHDOL] ULL�

programului cifrei de afaceri A 20 1,1 B 40 1,2 C 30 1,25 D 10 1,5

,QGLFHOH�UHDOL] ULL�SURJUDPXOXL�FLIUHL�GH�DIDFHUL�SH�vQWUHDJD�VRFLHWDWH�FRPHUFLDO �D�IRVW�

a) 1,2625; b) 1,225; c) 1,2542; d) 1,2168; e) 1,2465.

TESTUL 7

1) ([SUHVLD� VLQWHWL] ULL� YDORULORU� LQGLYLGXDOH� DOH� XQHL� YDULDELOH� VWDWLVWLFH� vQWU-un singur nivel

UHSUH]HQWDWLY��D�WRW�FHHD�FH�HVWH�HVHQ LDO��RELHFWLY�úL�VWDELO�vQ�DSDUL LD�úL�PDQLIestarea fenomenelor GH�PDV �HVWH�GDW �GH�

a) YDORDUHD�ORU�PHGLDQ � b) valoarea lor medial ; c) valoarea lor medie; d) FRHILFLHQWXO�ORU�GH�YDULD LH� e) FDOFXOXO�XQHL�P ULPL�UHODWLYH�GH�LQWHQVLWDWH

2) 3HULRDGD�GH�WLPS�FkW�IXQF LRQHD] �XQ�DQXPLW�WLS�GH�EDWHULL�HVWH�QRUPDO�GLVWULEXLW �FX�R�PHGLH�GH�

��GH�RUH�úL�R�GHYLD LH�VWDQGDUG�GH��RUH��3UREDELOLWDWHD�FD�R�EDWHULH�V �IXQF LRQH]H�vQWUH��úL��de ore este:

a) 0,5918; b) 0,4082; c) 0,8164; d) 0,1836; e) 0,7959.

3) 'DF � vQWU-R�VHULH�GH�GLVWULEX LH�YDORULOH� LQGLYLGXDOH�VH�VLPSOLILF �GH�N�RUi, atunci dispersia noii

serii este:

a) de k2�RUL�PDL�PLF �ID �GH�GLVSHUVLD�VHULHL�LQL LDOH� b) GH�N�RUL�PDL�PDUH�GHFkW�GLVSHUVLD�VHULHL�LQL LDOH� c) HJDO �FX�GLVSHUVLD�VHULHL�LQL LDOH� d) de k2�RUL�PDL�PDUH�ID �GH�GLVSHUVLD�VHULHL�LQL LDOH� e) GH�N�RUL�PDL�PLF �GHFkW�GLVSHUVLD�VHULHL�LQL LDOH�

4) ÌQ�PRGHOXO�GH�DQDOL] �GLVSHUVLRQDO �XQLIDFWRULDO ��VH�XWLOL]HD] �WHVWXO�³)´�GDW�GH�UHOD LD�

a)

( ) ( )F

y y n

r

y y

n rcalc

i ii

r

ij ij

m

i

r

=−

−

−

−= ==∑ ∑∑2

1

2

11

1: ;

b)

( )

( )F

y y n

y ycalc

i ii

r

ijj

m

i

r=

−

−

=

==

∑

∑∑

2

1

2

11

;

c)

( ) ( )F

y y n

r

y y

ncalc

i ii

r

jj

m

i

r

=−

−

−

−= ==∑ ∑∑2

1

2

11

1 1: ;

d)

( ) ( )F

y y n

r

y y

ncalc

i ii

r

ij ij

m

i

r

=−

−

−

−= ==∑ ∑∑2

1

2

11

1 1: ;

e)

( ) ( )F

y y n

n r

y y n

rcalc

ij i ij

m

i

r

i ii

r

=

−

−

−

−== =∑∑ ∑

2

11

2

1

1: .

5) În teVWDUHD� LSRWH]HL� VWDWLVWLFH�SULYLQG�PHGLD�SRSXOD LHL��FkQG�GDWHOH�SURYLQ�GH� OD�XQ�HúDQWLRQ�GH�YROXP�UHGXV��Q��úL�

H m m

H m ma

0 0

0

:

:

=≠

UHJLXQHD�FULWLF �HVWH�GDW �GH�

a) z z< − α sau z z> α ;

b) z z< − α / 2 sau z z> α /2 ;

c) t t n< − α / ,2 sau t t n> α / ,2 ;

d) t t n< − −α , 1 sau t t n> −α , 1;

e) t t n< − −α / ,2 1sau t t n> −α / ,2 1

6) ÌQ�FDGUXO�XQXL�VWXGLX�GH�SLD �HIHFWXDW�vQ�OXQD�LDQXDULH��V-D�XUP ULW�SULQWUH altele opinia a 100

GH� FRQVXPDWRUL� DVXSUD� FDOLW LL� XQXL� DQXPLW� SURGXV� XWLOL]DW� vQ� PRG� FXUHQW�� 6H� QRWHD] � FX� ;�

YDULDELOD�RSLQLH�úL�FX�[i opinia consumatorului i ( i = 1100, ) intervievat. ��3RSXOD LD�VWDWLVWLF �QRWDW �FX�P �HVWH�IRUPDW �GLQ:

a) WR L�FRQVXPDWRULL� b) consumatorii români; c) FHL��GH�FRQVXPDWRUL�LQWHUYLHYD L� d) FRQVXPDWRULL�SRWHQ LDOL� e) consumatorii din zona de sud-HVW�D� ULL�

��9DULDELOD�;��GHILQLW �vQ�SUREOHPD��HVWH�

a) FDOLWDWLY �FRQWLQX � b) FDOLWDWLY �GLVFUHW � c) QRPLQDO � d) de timp; e) QXPHULF �

8) &DUH�GLQ�XUP WRDUHOH�DILUPD LL�QX�HVWH�DGHY UDW �SHQWUX�R�VHULH�VWDWLVWLF �

a) vQWUH�TXDUWLOD��úL�TXDUWLOD��VH�J VHVF��GLQ�REVHUYD LL��VLWXDWH�vQ�FHQWUXO�GLVWULEX LHL� b) YDORDUHD�TXDUWLOHL��HVWH�HJDO �FX�PHGLDQD��GRDU�SHQWUX�R�VHULH�VLPHWULF � c) YDORDUHD�PHGLDQHL�HVWH�vQWRWGHDXQD�HJDO �FX�YDORDUHD�TXDUWLOHL�� d) SHQWUX�R�VHULH�VLPHWULF ��DEDWHUHD�LQWHUTXDUWLOLF �HVWH�QXO � e) SHQWUX�R�VHULH�VLPHWULF ��DEDWHUHD�LQWHUTXDUWLOLF �FXSULQGH��GLQ�REVHUYD LL�

9) 3HQWUX�R�GLVWULEX LH�VLPHWULF �

a) abaterile�LQGLYLGXDOH�VH�FRPSHQVHD] �GRDU�OD�QLYHOXO�vQWUHJLL�FROHFWLYLW L� b) abaterile individuale sunt nule; c) DEDWHUHD�PD[LP �SR]LWLY �HVWH�HJDO �FX�DEDWHUHD�PD[LP �QHJDWLY � d) DEDWHULOH�LQGLYLGXDOH�VH�FRPSHQVHD] �DWkW�SH�WRWDO��FkW�úL�OD�QLYHOXO�FHQWUDOL]DW�DO�XQLW LORU� e) VXPD�DEDWHULORU�LQGLYLGXDOH�HVWH�HJDO �FX�DPSOLWXGLQHD�YDULD LHL�

10) *UDGXO�GH�OLEHUWDWH�HVWH�XQ�FRQFHSW�VWDWLVWLF�FH�GHVHPQHD] �

a) QXP UXO�GH�HOHPHQWH�GLQWU-o colectivitate; b) num rul de elemente dintr-o colectivitate minus unul; c) num rul de elemente necesare pentru a defini starea unui ansamblu; d) num rul de elemente independente necesare pentru a defini starea unui ansamblu; e) QXP UXO�GH�UHOD LL�LQGHSHQGHQWH�FDUH�OHDJ �HOHPHQWHOH�XQXL�DQVDPEOX�

11) 'RX � YDULDELOH� VWDWLVWLFH� DX� IRVW� PRGHODWH� vQWU-R� SRSXOD LH� VWDWLVWLF � FX� Q� XQLW L� FX� DMXWRUXO�HFXD LHL�OLQLDUH�GH�IRUPD�� y x= +5 64 ��ùWLLQG�F �

( ) ( )x x y yii

n

ii

n− = − =

= =∑ ∑

2

1

2

1

30 1960; � �� SHQWUX� Q �� XQLW L� VWDWLVWLFH�� DWXQFL� LQWHUYDOXO� GH�

încredere pentru panta liniei de regresie, la o probabilitate de 95% este:

a) (63,53 ; 64,47); b) (4,53 ; 5,47); c) (58,92 ; 66,65); d) (61,35 ; 66,65); e) (58,92 ; 69,08).

12) 'LVSHUVLD�XQHL�FDUDFWHULVWLFL�GH�WLS�DOWHUQDWLY�HVWH�PD[LP �FkQG�

a) QXP UXO�GH�U VSXQVXUL�DILUPDWLYH�HVWH�HJDO�FX�QXP UXO�GH�U VSXQVXUL�QHJDWLYH� b) WRDWH�XQLW LOH�FROHFWLYLW LL�vQUHJLVWUHD] �U VSXQVXUL�DILUPDWLYH� c) WRDWH�XQLW LOH�FROHFWLYLW LL�vQUHJLVWUHD] �U VSXQVXUL�QHJDWLYH� d) WRDWH�XQLW LOH�FROHFWLYLW LL�vQUHJLVWUHD] �ILH�U VSXQVXUL�QHJDWLYH��ILH�U VSXQVXUL�SR]LWLYH� e) nu se poate preciza în ce caz dispersia unei caracteristiFL�DOWHUQDWLYH�HVWH�PD[LP �

13) 'HQVLWDWHD�GH�UHSDUWL LH�D�YDULDELOHL�DOHDWRDUH�QRUPDOH��GH�PHGLH�x = 20 9, úL�GLVSHUVLH��HVWH�GHVFULV �GH�

a) f exi( )

,

= ⋅−1

2

20 9

35

2

π;

b) ( )

f ex

x

i

i

( ),

,

,= ⋅

−⋅

−1

17 5 2

1

2 17 520 9

22

π;

c) ( )

f ex

x

i

i

( ),

,

,= ⋅

−⋅

−1

17 5 2

1

2 17 520 9 2

π;

d) ( )

f ex

x

i

i

( ),

,= ⋅

−⋅

−1

4,18 2

1

2 4 1820 9 2

π;

e) ( )

f ex

x

i

i

( ),5

,

,= ⋅

−⋅

−1

4 18 2

1

2 1720 9 2

π.

14) 0HGLD�DUPRQLF �VH�GHILQHúWH�FD�

a) PHGLD�DULWPHWLF �FDOFXODW �GLQ�LQYHUVHOH�YDORULORU�LQGLYLGXDOH��vQUHJLVWUDWH� b) YDORDUHD�LQYHUV ��D�PHGLHL�DULWPHWLFH�D�WHUPHQLORU�VHULHL� c) YDORDUHD� LQYHUV � D� PHGLHL� DULWPHWLFH� FDOFXODW � GLQ� LQYHUVHOH� S WUDWHORU� YDORULORU�

individuale înregistrate; d) YDORDUHD�LQYHUV �D�PHGLHL�DULWPHWLFH�FDOFXODW �GLQ�LQYHUVHOH�YDORULORU�LQGLYLGXDOH� e) YDORDUHD�FDUH��GDF �DU�vQORFXL�WHUPHQLL�VHULHL��QX�DU�PRGLILFD�VXPD�ORU�

15) ,QGLFDWRUXO� ³HQHUJLD� LQIRUPD LRQDO � 2QLFHVFX´� HVWH� XWLOL]DW� vQ� DQDOL]D� VWDWLVWLF � SHQWUX�caracterizarea:

a) asimetriei; b) dispersiei; c) QRUPDOLW LL�GLVWULEX LHL� d) FRQFHQWU ULL� e) WHQGLQ HL�FHQWUDOH�

16) 'LUHFWRUXO�XQXL�SRVW�GH�UDGLR�FRPDQG �R�FHUFHWDUH�VWDWLVWLF �vQ�VFRSXO�GHWHUPLQ ULL�DXGLHQ HL�GH�FDUH�VH�EXFXU �SRVWXO�V X��&HUFHW WRUXO�FXOHJH�SH�XQ�HúDQWLRQ�GDWH�SULYLQG�QXP UXO�GH�RUH�DXGLDWH�úL�DIO �F ��GLQ�SHUVRDQH�DVFXOW �SRVWXO�GH�UDGLR�PDL�SX LQ�GH�GRX �RUH��vQWUH��úL��RUH��vQWUH��úL��RUH��vQWUH��úL��RUH��LDU�UHVWXO�SHVWH��RUH��1XP UXO�PHGLX�GH�RUH�GH�DXGL LH�SH�R�SHUVRDQ �GLQ�HúDQWLRQ�HVWH�

a) 4,8 ore; b) 4,7 ore; c) 6 ore; d) 4,2 ore; e) 4,6 ore.

17) 6 � VH� SUHFL]H]H� FDUH� GLQ� XUP WRDUHOH� GLVWULEX LL� GH� SUREDELOLWDWH� GHVFULH� R� YDULDELO � DOHDWRDUH�Poisson:

a) ( )px xi

i i

x xi i=⋅ −

−10

100 9 0 110!

! !, , (xi=0, 1, 2, … , 10);

b) ( )

pe

xi

x

i

i

=−0 5 0,

!

,5

(xi=0, 1, 2, 3, … );

c) ii xxi Cp −⋅⋅= 6

6 8,02,0 (xi=0, 1, 2, … , 6);

d) ( )

px e

ii

x

x

i

i=

−!

,0 5 (xi=0, 1, 2, 3, … )

e)

( )f ex

x

i

i

( )

,

,

,= ⋅

−−⋅1

31 2

20 9

2 9 61

2

π.

18) 3HQWUX� R� FROHFWLYLWDWH� GH� Q� XQLW L� VLVWHPDWL]DWH� FRQFRPLWHQW� vQ� U� JUXSH� GXS � YDORULOH�caracteriVWLFLL�;� �FRQVLGHUDW � FDUDFWHULVWLF � HVHQ LDO � GH� LQIOXHQ �� úL� vQ�P� JUXSH� GXS � YDORULOH�FDUDFWHULVWLFLL�DQDOL]DWH�<��LQIOXHQ D�IDFWRUXOXL�;�HVWH�P VXUDW �GH�

a) vPSU úWLHUHD�YDORULORU�LQGLYLGXDOH�GLQ�ILHFDUH�JUXS �vQ�MXUXO�PHGLHL�GH�JUXS � b) vPSU úWLHUHD�YDORULORU�LQGLYLGXDOH�GLQ�ILHFDUH�JUXS �vQ�MXUXO�PHGLHL�JHQHUDOH�D�FROHFWLYLW LL� c) vPSU úWLHUHD�PHGLLORU��GH�JUXS �vQ�MXUXO�PHGLHL�JHQHUDOH�D�FROHFWLYLW LL�

d) vPSU úWLHUHD�YDORULORU�LQGLYLGXDOH�vQ�MXUXO�PHGLHL�PHGLLORU�GH�JUXS � e) vPSU úWLHUHD�PHGLHL�PHGLLORU�GH�JUXS �ID �GH�PHGLD�JHQHUDO �D�FROHFWLYLW LL�

19) ÌQ�PRGHOXO�GH�DQDOL] �GLVSHUVLRQDO �XQLIDFWRULDO �YDULDQ D�IDFWRULDO ��VLVWHPDWLF ��HVWH�

a) ( )y yij ij

m

i

r−

==∑∑

2

11

;

b) ( )y y ni ii

r−

=∑

2

1

;

c)

( )y y n

r

i ii

r−

−=∑

2

1

1;

d)

( )y y

n

ijj

m

i

r−

−==∑∑ 2

11

1;

e) ( )y yijj

m

i

r−

==∑∑

2

11

.

20) În testarea ipotezei statistice

H m

H ma

0 100

100

:

:

≥<

XWLOL]kQG�WHVWXO�]�VH�RE LQH�YDORDUHD�WHVWXOXL�] -1,11. Nivelul de încredere al testului este:

a) 86,65%; b) 36,65%; c) 13,35%; d) 26,70%; e) 73,30%.

21) Într-un proces de testare a ipotezelor statistice, eroarea de genul al doilea este:

a) HURDUHD�SH�FDUH�R�IDFHP�DFFHSWkQG�LSRWH]D�QXO �FkQG�HD�HVWH�DGHY UDW � b) HURDUHD�SH�FDUH�R�IDFHP�DFFHSWkQG�LSRWH]D�DOWHUQDWLY �FkQG�HD�HVWH�IDOV � c) HURDUHD�SH�FDUH�R�IDFHP�DFFHSWkQG�LSRWH]D�QXO �FkQG�HD�HVWH�IDOV � d) eroarea pe FDUH�R�IDFHP�HOLPLQkQG�LSRWH]D�QXO �FkQG�HD�HVWH�DGHY UDW � e) HURDUHD�SH�FDUH�R�IDFHP�HOLPLQkQG�LSRWH]D�DOWHUQDWLY �DWXQFL�FkQG�HD�HVWH�IDOV �

22) ÌQ� XWLOL]DUHD�PHWRGHL� UHJUHVLHL� SHQWUX� VWXGLXO� GHSHQGHQ HL� GLQWUH� YDULDELOH��PRGHOXO� GH� IRUPD��

yx

= + +α β ε1 este un model:

a) liniar; b) H[SRQHQ LDO� c) logaritmic; d) hiperbolic; e) parabolic.

23) 'LVSHUVLD�YDORULORU�XQHL�YDULDELOH�DOHDWRDUH�;��ID �GH�R�FRQVWDQW �D�HVWH�PLQLP �FkQG�

a) a = 0;

b) a x= ; c) pentru orice a;

d) a xi= ∑ ;

e) a = xmax. 24) Un VLVWHP�DO�XQXL�YHKLFXO� VSD LDO� WUHEXLH�V � IXQF LRQH]H�FRQWLQXX�SHQWUX�FD�QDYD�V � UHLQWUH� vQ�

VSD LX��2�FRPSRQHQW �D�VLVWHPXOXL�IXQF LRQHD] �FRUHVSXQ] WRU��GLQ� WLPS��3HQWUX�D�DVLJXUD�EXQD�IXQF LRQDUH�D�VLVWHPXOXL��VXQW�PRQWDWH�SDWUX�FRPSRQHQWH�VLPLODUH��vQ�DúD�IHO�vQFkW�VLVWHPXO�IXQF LRQHD] � GDF � FHO� SX LQ� R� FRPSRQHQW � OXFUHD] �� &RPSRQHQWHOH� RSHUHD] � LQGHSHQGHQW��3UREDELOLWDWHD�FD�VLVWHPXO�V �FDG �HVWH�

a) 0,05%;

b) 5%;

c) 15%;

d) 52,20%;

e) 0,5220%.

25) 3HQWUX�R�VHULH�GH�UHSDUWL LH�GH�IUHFYHQ H��PHGLD�S WUDWLF �VH�FDOFXOHD] �FD�

a) ( )

∑

∑

=

==r

ii

r

iii

p

n

nxx

1

1

2

;

b)

∑

∑

=

==r

ii

r

iii

p

n

nxx

1

1

2

;

c) ( )

∑

∑

=

==r

ii

r

iii

p

n

nxx

1

1

2

;

d)

∑

∑

=

==r

ii

r

iii

p

n

nxx

1

1

2

;

e)

∑

∑

=

==r

ii

r

iii

p

n

nxx

1

1 .

26) 'RX �JUXSH�GH� VWXGHQ L�� FX�HIHFWLYH�GH�� úL� UHVSHFWLY��GH�SHUVRDQH��DX� VXV LQXW�XQ� WHVW�GH�

FXOWXU �JHQHUDO ��3ULPD�JUXS �D�RE LQXW�PHGLD��LDU�D�GRXD��1RWD�PHGLH�SH�DQVDPEOXO�FHORU�GRX �JUXSH�HVWH�

a) 8,14; b) 8,10; c) 8,12; d) 8,00; e) 8,09.

27) Într-R�FROHFWLYLWDWHD�VWDWLVWLF �VLVWHPDWL]DW �vQ�U�JUXSH�GXS �YDULD LD�FDUDFWHULVWLFLL�GH�JUXSDUH�;�úL�vQ�P�JUXSH�GXS �YDORULOH�YDULDELOHL�DQDOL]DWH�<�H[LVW �vQWRWGHDXQD�UHOD LD�

a) 222δσσ += i ;

b) 22δσ ≥ ;

c) r

i∑=2

2 σσ ;

d) 22σσ ≤ ;

e)

( )

∑∑

∑∑

= =

= =−

=r

i

m

jij

r

i

m

jijij

n

nyy

1 1

1 1

2

2σ .

28) Într-R�FROHFWLYLWDWHD�GH��VDODULD L��VLVWHPDWL]DW �GXS �IDFWRUXO�GH�JUXSDUH�;��QXP UXO�RUHORU�OXFUDWH�� vQ��JUXSH� úL�GXS �YDULDELOD�GHSHQGHQW �� VDODULXO�� vQ��JUXSH�� DSOLFkQGX-se modelul de DQDOL] � GLVSHUVLRQDO � V-D� RE LQXW�� YDULDQ D� VLVWHPDWLF � 61 �� úL� YDULDQ D� WRWDO �ST ��9DORDUHD�FDOFXODW �D�UDSRUWXOXL�)�SHQWUX�XWLOL]DUHD�WHVWXOXL�FX�DFHODúL�QXPH�HVWH�

a)Fcalc=81,875;

b) Fcalc=73,23; c) Fcalc=1,25; d) Fcalc=61,09; e) Fcalc=36,94.

29) În testarea ipotezei statistice

H m

H ma

0 100

100

:

:

≥<

VH�RE LQH�YDORDUHD�WHVWXOXL�VWDWLVWLF�] -1,11. 3UDJXO�GH�VHPQLILFD LH�DO�WHVWului este: a)0,3665;

b) 0,1335; c) 0,267 ; d) 0,06675; e) 0,43325.

30) Într-un proces de testare a ipotezelor statistice, eroarea de genul întâi este:

a) HURDUHD�SH�FDUH�R�IDFHP�DFFHSWkQG�LSRWH]D�QXO �FkQG�HD�HVWH�DGHY UDW � b) eroarea pe care o facem acceptând ipoteza nXO �DWXQFL�FkQG�HD�HVWH�IDOV � c) HURDUHD�SH�FDUH�R�IDFFHP�DFFHSWkQG�LSRWH]D�DOWHUQDWLY �FkQG�HD�HVWH�DGHY UDW � d) HURDUHD�SH�FDUH�R�IDFHP�HOLPLQkQG�LSRWH]D�QXO �DWXQFL�FkQG�HD�HVWH�DGHY UDW � e) HURDUHD�SH�FDUH�R�IDFHP�HOLPLQkQG�LSRWH]D�DOWHUQDWLY �DWXQFL�FkQG�HD�HVWH�DGHY UDW �

31) 6 � VH� SUHFL]H]H� FDUH� GLQ� XUP WRDUHOH�PHWRGH� GH� FDUDFWHUL]DUH� D� OHJ WXULORU� GLQWUH� YDULDELOHOH�

VWDWLVWLFH�QX�VH�vQFDGUHD] �vQ�FDWHJRULD�³PHWRGHORU�VLPSOH´�

a) metoda seriilor interdependente; b) PHWRGD�JUXS ULORU� c) PHWRGD�JUDILF � d) PHWRGD�FRUHOD iei; e) PHWRGD�WDEHOXOXL�GH�FRUHOD LH�

TESTUL 8

1) Prin AUDIT-ul financiar efectuat în luna ianuarie 1998 la S.C. “ICS” SRL s-D�FRQVWDWDW�F � vQ�

IDFWXULOH�HODERUDWH� vQ��H[LVW �DQXPLWH�HURUL� vQ�FDOFXOXO�79$-XOXL�FDUH� DX�DYXW�GUHSW� VXUV ��QHFXQRDúWHUHD� PHWRdologiei de calcul a TVA-XOXL� OD� XQHOH� SURGXVH�� DSUR[LP UL� HURQDWH��GHIHF LXQL�DOH�PLMORFXOXL�WHKQLF�GH�FDOFXO��HWF�

3RSXOD LD�VWDWLVWLF �VWXGLDW �HVWH�

a) ansamblul facturilor elaborate de S.C. “ICS” SRL; b) DQVDPEOXO�SURGXVHORU�úL�VHUYLFLLORU�UHDOL]DWH�GH�6�&��³,&6´�65/�SkQ �OD��;,,�� c) DQVDPEOXO� SURGXVHORU� úL� VHUYLFLLORU� UHDOL]DWH� GH� 6�&�� ³,&6´� 65/� GHVI FXWH� SH� SLD � úL�

IDFWXUDWH�SkQ �OD��;,,�� d) DQVDPEOXO�SURGXVHORU�úL�VHUYLFLLORU�UHDOL]DWH�SHQWUX�D�IL�GHVI FXWH�SH�SLD �GH�6�&��³,&6´�

SRL, facturate�SkQ �OD��;,,��úL�OD�FDUH�VH�SHUFHSH�79$� e) DQVDPEOXO�IDFWXULORU�HODERUDWH�GH�6�&��³,&6´�65/�SkQ �OD��;,,��SHQWUX�SURGXVHOH�

úL�VHUYLFLLOH�UHDOL]DWH�úL�GHVI FXWH�SH�SLD �

2) &DOFXOXO�DPSOLWXGLQLL�YDULD LHL�YDORULORU�LQGLYLGXDOH�DUH�VHQV�SHQWUX�

a) serii statistice numerice formate pe variante; b) VHULL�VWDWLVWLFH�IRUPDWH�GXS �R�YDULDELO �DOWHUQDWLY � c) VHULL�VWDWLVWLFH�IRUPDWH�SH�LQWHUYDOH�HJDOH�GH�YDULD LH� d) VHULL�VWDWLVWLFH�IRUPDWH�SH�LQWHUYDOH�QHHJDOH�GH�YDULD LH� e) orice tip de serie.

3) Pentru o colectivitDWH�GH�Q�XQLW L�FDUH�DX�IRVW�VWUXFWXUDWH�FRQFRPLWHQW� vQ�r�JUXSH�GXS �YDORULOH�FDUDFWHULVWLFLL�;�úL�vQ�m�JUXSH�GXS �YDORULOH�FDUDFWHULVWLFLL�<��PHGLD�FDUDFWHULVWLFLL�<�SH�vQWUHDJD�

colectivitate ( y ��QX�VH�FDOFXOHD] �FD�

a)

∑

∑

=

==m

jj

m

jjj

n

ny

y

1.

1.

;

b)

∑

∑

=

==r

ii

r

iii

n

nyy

1.

1..

;

c)

∑∑

∑∑

=

==

=

=r m

jij

r m

jijij

in

iny

y

1

1

1

1 ;

d) n

iny

y

r m

jijij∑∑

== =1 1 ;

e)

∑

∑

=

==r

ii

r

iii

n

nyy

1.

1..

.

4) ÌQ� DQXO� �� LQGLFHOH�PHGLX� OXQDU� DO� SUH XULORU� GH� FRQVXP� OD� VHUYLFLL� D� IRVW� SHQWUX� SHULRDGD�ianuarie-DXJXVW��úL�SHQWUX�SHULRDGD�DXJXVW-decembrie 1,0319. Indicele mediu lunar pe anul 1994 a fost:

a) 1,0505; b) 1,0474; c) 1,0435; d) 1,0436; e) 104,35%.

5) 3HQWUX�GRX �YDULDELOH� VWDWLVWLFH�;�úL�<� vQWUH�FDUH� H[LVW �R�GHSHQGHQ � OLQLDU ��XWLOL]kQG�PHWRGD�

JUDILF ��V-au calculat:

σ x x y y x2 9043 2392 8 37 364= = − = =; cov( , ) , ; ; .

(FXD LD�GH�UHJUHVLH�FDUH�PRGHOHD] �GHSHQGHQ D�GLQWUH�FHOH�GRX �YDULDELOH�HVWH�

a) � , ,y x= +133 3 0 2646 ; b) xy 2646,03,133ˆ −= ; c) xy 3,1332646,0ˆ += ; d) xy 78,392,1412ˆ −= ; e) xy 78,392,1338ˆ −−= .

6) ÌQ� UHSDUWL LD� ELQRPLDO � FX� Q� REVHUYD LL�� S� - probabilitatea succesului, q - probabilitatea

LQVXFFHVXOXL��IUHFYHQ D�D�N�VXFFHVH�vQ�1�VHULL�HVWH�

a) q pn k k− ;

b) C q pnk n k k− ;

c) C p qnk n k k− ;

d) N C q pnk n k k⋅ − ;

e) N q pn k k⋅ − . 7) ÌQ�VWXGLXO�PLúF ULL�QDWXUDOH�D�SRSXOD LHL�VH�DQDOL]HD] �SULQWUH�DOWHOH�úL�YDULDELOD�³UDQJXO�Q VFXWXOXL�

YLX´��ÌQ�LDQXDULH��vQ�5RPkQLD�UHSDUWL LD�GXS �DFHDVW �YDULDELO �VH�SUH]LQW �DVWIHO�

5DQJXO�Q VFXWXOXL�YLX 1XP U�GH�QRX�Q VFX L

Primul 139958

Al doilea 64897

Al treilea 19812

Al patrulea 9657

Al cincilea 5650

$O�úDVHOHD 3966

$O�úDSWHOHD 2322

Al 8-OHD�úL�SHVWH 3722

Sursa: Anuarul statistic al României, CNS, 1994, p.126

Datele prezentate mai sus sunt:

a) individuale; b) grupate; c) de stoc; d) de flux; e) agregate.

8) 3HQWUX�GRX �YDULDELOH�VWDWLVWLFH�GHSHQGHQWH�;�úL�<��FX�\� �I�[��DX�IRVW�FXOHVH�GDWH�GH�OD��GH�

XQLW L�VWDWLVWLFH�úL��vQ�XUPD�SUHOXFU ULL�GDWHORU��V-D�RE LQXW�FRHILFLHQWXO�GH�FRUHOD LH�U -0,82. &RHILFLHQWXO�GH�FRUHOD LH�U�

a) HVWH�VHPQLILFDWLY�VWDWLVWLF�OD�XQ�SUDJ�GH�VHPQLILFD LH�α=0,0005; b) nu este semnificativ statistic; c) este semnificativ statistic doar pentru cel mult un nivel de încredere de 1-α=0,90; d) QX�HVWH�VHPQLILFDWLY�VWDWLVWLF�OD�XQ�SUDJ�GH�VHPQLILFD LH�α=0,010; e) DUH�R�YDORDUH�FDUH�QX�SRDWH�IL�WHVWDW �SHQWUX�QLFL�XQ�SUDJ�GH�VHPQLILFD LH�

9) )UHFYHQ HOH�UHGXVH�OD�XQLWDWH�VH�FDOFXOHD] �

a) vQ�FD]XO�UHSDUWL LLORU�SH�LQWHUYDOH�QHHJDOH��SHQWUX�DVLJXUDUHD�FRPSDUDELOLW LL�IUHFYHQ HORU� b) SHQWUX�D�VWXGLD�SRQGHUHD�XQLW LORU�VWDWLVWLFH�GLQWU-R�JUXS �vQ�WRWDOXO�FROHFWLYLW LL�VWXGLDWH� c) pentru a detHUPLQD� QXP UXO� XQLW LORU� FDUH� DX� YDORDUHD� LQGLYLGXDO � VLWXDW � SkQ � OD� XQ�

anumit nivel al caracteristicii; d) SHQWUX�D�VWXGLD�VWUXFWXUD�XQHL�VHULL�GH�GLVWULEX LH� e) SHQWUX� D� VWXGLD� QXP UXO� XQLW LORU� VWDWLVWLFH� FDU� DX� YDORULOH� LQGLYLGXDOH� VLWXDWH� SHVWH� XQ�

anumit nivel al caracteristicii. 10) 3UHVXSXQHP� F �� vQ� XUPD� DSOLF ULL� PHWRGHL� GH� UHJUHVLH� OLQLDU �� V-a calculat, pentru 12 valori

SHUHFKL�� VXPD� S WUDWHORU� DEDWHULORU� YDlorilor empirice de la valorile ajustate

( )y yi ii

− ==∑ � ,2

1

120 507 . Atunci, se

2 ��HVWLPDWRUXO�GLVSHUVLHL�DEDWHULORU�vQWkPSO WRDUH�ε, este:

a) 0,0507; b) 0,04225; c) 0,04609; d) 0,2252; e) 0,2055.

11) 'LVSHUVLD�FDOFXODW �ID �GH�R�FRQVWDQW �a este, comparatLY�FX�GLVSHUVLD�FDOFXODW �ID �GH�PHGLH�

a) de a�RUL�PDL�PLF � b) mai mare de a2 ori; c) PDL�PLF ��FX�S WUDWXO�GLVWDQ HL�GLQWUH�PHGLH�úL�FRQVWDQWD�a; d) mai mare cu a2; e) PDL�PDUH��FX�S WUDWXO�GLVWDQ HL�GLQWUH�PHGLH�úL�FRQVWDQWD�a.

12) 2�YDULDELO �DOHDWRDUH�;�DUH�R�GLVWULEX LH�QRUPDO ��GH�PHGLH��úL�GLVSHUVLH��9DORUHD�[0, pentru care P x x( ) ,≤ =0 0 8413 este:

a) 54,23; b) 45,77; c) 53; d) 59; e) 46.

13) 3HQWUX� YHULILFDUHD� QRUPDOLW LL� UHSDUWL LLORU� HPSLULFH� XWLOL]kQG� WHVWXO� χ2� �� YDORDUHD� FDOFXODW � D�

variabilei χ2 are forma:

a) ( )

i

r

iii

calc np

npn∑=

−= 1

2

2χ ;

b) ∑=

−=r

i i

iicalc np

npn

1

2

2χ ;

c) ∑=

−=r

i i

iicalc np

npn

1

2χ ;

d) ( )∑=

−=r

i i

iicalc

np

npn

12

2χ ;

e) ( )∑

=

−=r

i i

iicalc np

npn

1

22χ .

14) ÌQ�FDGUXO�PHWRGHORU�JHQHUDOH�GH�DERUGDUH� úWLLQ LILF �D� IHQRPHQHORU�HFRQRPLFR-sociale, un loc

LPSRUWDQW�vO�RFXS �DEVWUDFWL]DUHD�VXFFHVLY ��LQGXF LD�úL�GHGXF LD��$FHVWH�PHWRGH� a) nu pot fi utilizate într-XQ� GHPHUV� VWDWLVWLF� GHRDUHFH� RSHUHD] � FX� GDWH� FRQFUHWH� RE LQXWH�

SULQ�REVHUY UL�WRWDOH�úL�SDU LDOH� b) VXQW�XWLOL]DWH�vQ�VWXGLXO�VWDWLVWLF�DO�IHQRPHQHORU�GH�PDV �vQWUXFkW��SHQWUX�D�VH�UH LQH�QXPDi

FHHD�FH�HVWH�HVHQ LDO�úL�WLSLF�WUHEXLH�V �VH�HOLPLQH�DVSHFWHOH�vQWkPSO WRDUH�úL�QHHVHQ LDOH��GH� OD� SDUWLFXODUXO� RE LQXW� SULQ� REVHUYDUHD� �vQUHJLVWUDUHD�� YDORULORU� HPSLULFH� VH� WUHFH�inductiv spre ceea ce este în general valabil pentru întregul ansamblu, etc.;

c) QX�SRW�IL�XWLOL]DWH�GH�VWDWLVWLF �vQWUXFkW�vQ�RULFH�FHUFHWDUH�QX�VH�SRUQHúWH�GH�OD�LSRWH]H�FDUH�ar trebui verificate;

d) VXQW�XWLOL]DWH�QXPDL�vQ�VWXGLXO�UHSDUWL LLORU�VWDWLVWLFH�XQL-�úL�PXOWLGLPHQVLRQDOH� e) VXQW�XWLOL]DWH�QXPDL�GDF �VXQW�RUJDQL]DWH�VRQGDMH�statistice.

15) Analizate la nivelul unui ansamblu, formele individuale de manifestare ale fenomenelor de

PDV �SDU� DVHP Q WRDUH� vQWUH�HOH�� ILLQG� JHQHUDWH�GH�FDX]H� HVHQ LDOH�FRPXQH�� VH� VXSXQ�DFHOHLDúL�OHJL�GH�DSDUL LH�úL�GH]YROWDUH��$FHDVW �OHJH�DF LRQHD] �

a) stDWLF��FkQG�IHQRPHQHOH�VXQW�FLUFXPVFULVH�OD�DFHODúL�VSD LX�úL�DFHHDúL�IRUP �GH�RUJDQL]DUH��GDU�vQ�FRQGL LL�GH�WLPS�GLIHULWH�

b) GLQDPLF��FkQG�IHQRPHQHOH�VXQW�GHOLPLWDWH�vQ�DFHODúL�VSD LX��VXQW�vQUHJLVWUDWH�vQ�DFHOHDúL�XQLW L�GH�WLPS� c) atât static, când fenomenHOH�GH�PDV �VXQW�FLUFXPVFULVH�vQ�DFHOHDúL�FRQGL LL�GH�WLPS�FkW�úL�

GLQDPLF�� FkQG� IHQRPHQHOH� VXQW� GHOLPLWDWH� vQ� VSD LX� úL� RUJDQL]DWRULF�� GDU� vQUHJLVWUDWH� vQ�XQLW L�GH�WLPS�GLIHULWH�

d) exclusiv dinamic, când se face ipoteza c fenomenele au fost înregistrate lD� DFHOHDúL�PRPHQWH�GH�WLPS��YDULDELOH�ILLQG�VSD LXO�úL�IRUPD�GH�RUJDQL]DUH�

e) VWDWLF�� GHRDUHFH� GLQDPLF� QX� SRDWH� DF LRQD� vQWUXFkW� UHOD LLOH� GH� FDX]DOLWDWH� GLQWUH�fenomenele sociale nu au un caracter dinamic.

16) 2�YDULDELO �DOHDWRDUH�;��GLVWULEXLW �QRUPDO�úL�FX�REOLFLWDWH�PRGHUDW ��DUH��GLQ�YDORUL�VLWXDWH�

vQ�FHQWUXO�GLVWULEX LHL��FXSULQVH�vQWUH��úL��FRHILFLHQWXO�GH�YDULD LH�HVWH��LDU�YDORDUHD�PRGDO ��$VLPHWULD�HVWH�

a) PRGHUDW�SR]LWLY � b) PRGHUDW�QHJDWLY � c) HJDO �FX�]HUR� d) nu se poate preciza felXO�úL�P ULPHD�DVLPHWULHL� e) HJDO �FX�DPSOLWXGLQHD�VHPLLQWHUTXDUWLOLF �

17) $EDWHUHD�PHGLH�LQWHUTXDUWLOLF �VH�FDOFXOHD] �FD�

a) 13 QQ − ;

b) 2

13 QQ −;

c) 2

12 QQ −;

d) ( ) ( )

213 MeQMeQ −+−

;

e) ( )122 QQ − .

18) Pentru dou �YDULDELOH�VWDWLVWLFH��D�F URU�WHQGLQ �D�GHSHQGHQ HL�VWDWLVWLFH�\� �I�[��D�IRVW�PRGHODW �

SULQ�HFXD LD�GH�UHJUHVLH� � ,y x= −25 0 7 , se cunosc dispersiile σ x2 2603 04= , �úL�σ y

2 1896 6= , .

&RHILFLHQWXO�GH�FRUHOD LH�DUH�YDORDUHD�

a) 0,82; b) 0,96; c) -0,96; d) -0,82; e) -0,597.

19) 3UREDELOLWDWHD�FD�YDORULOH�XQHL�YDULDELOH�DOHDWRDUH�QRUPDOH�V �VH�VLWXH]H�OD�R�GLVWDQ �PDL�PDUH�GH��RUL�DEDWHUHD�PHGLH�S WUDWLF �GH�R�SDUWH�úL�GH�DOWD�ID �GH�PHGLH�HVWH�

a) 5%; b) 0,05%; c) 2,5%; d) 47,50%; e) 97,5%.

20) Un grup de��DQJDMD L�HVWH�DOHV�GLQWU-R�FRPSDQLH�vQ�VFRSXO�GH�D�GHWHUPLQD�QXP UXO�FHORU�FDUH�VXV LQ�VLQGLFDWXO�FRPSDQLHL��3UHVXSXQHP�F ��GLQ�WR L�DQJDMD LL�VXV LQ�VLQGLFDWXO�

3UREDELOLWDWHD�FD�PDL�SX LQ�GH��DQJDMD L�GLQ�FHL��V �VXV LQ �VLQGLFDWXO�HVWH�

a) 61,78%; b) 38,22%; c) 83,28%; d) 21,50%; e) 16,72%.

21) 'DF �YDORULOH�XQHL�FDUDFWHULVWLFL�VXQW�PLFúRUDWH�FX�FRQVWDQWD�a, atunci dispersia noii serii este:

a) HJDO �FX�GLVSHUVLD�VHULHL�LQL LDOH� b) PDL�PDUH�GHFkW�GLVSHUVLD�VHULHL�LQL LDOH�FX�S WUDWXO�GLVWDQ HL�GLQWUH�PHGLH�úL�FRQstanta a; c) PDL�PLF �GHFkW�GLVSHUVLD�VHULHL�LQL LDOH�FX�S WUDWXO�GLQWUH�PHGLH�úL�FRQVWDQWD�a; d) PDL�PLF �GHFkW�GLVSHUVLD�VHULHL�LQ LDOH�FX�FRQVWDQWD�a; e) PDL�PDUH�GHFkW�GLVSHUVLD�VHULHL�LQL LDOH�FX�FRQVWDQWD�a.

22) Patronul unei firme ce vinde autoturisme presupune� F � UHOD LD� GLQWUH� QXP UXO� GH� DXWRWXULVPH�

vândute într-R�]L��<��úL�QXP UXO�GH�YkQ] WRUL�FH�OXFUHD] �vQ�VDORQXO�GH�SUH]HQWDUH�úL�vQ�VHFWRUXO�GH�SXEOLFLWDWH�D�SURGXVXOXL��;��SRDWH�IL�PRGHODW �SULQWU-R�OLQLH�GUHDSW ��'DWHOH�FXOHVH�SHQWUX��]LOH�sunt:

xi 6 6 4 2 3 yi 20 18 10 6 11

Atunci:

a) OHJ WXUD� GLQWUH� FHOH� GRX � YDULDELOH� SRDWH� IL� PRGHODW � SULQ� HFXD LD� GH� UHJUHVLH�xy 125,3125,0ˆ += ;

b) FRYDULDQ D�GLQWUH�[�úL�\�DUDW �R�OHJ WXU �SXWHUQLF �LQYHUV �vQWUH�FHOH�GRX �YDULDELOH� c) panta linie drepte de regresiH�HVWH��LDU�SXQFWXO�GH�LQWHUFHS LH�FX�D[D�2\�HVWH�-0,125; d) FRHILFLHQWXO�GH�FRUHOD LH�HVWH�VHPQLILFDWLY�VWDWLVWLF��SHQWUX�R�SUREDELOLWDWH�GH��

e) DSUR[LPDWLY��GLQ�YDULD LD�YDORULORU�FDUDFWHULVWLFLL�<�vQ�MXUXO�PHGLHL� y este explLFDW �

SULQ�OHJ WXUD�OLQLDU �GLQWUH�;�úL�<� 23) 3HQWUX� LGHQWLILFDUHD� úL� GLPLQXDUHD� HURULORU� GH� REVHUYDUH� HVWH� QHFHVDU� FRQWUROXO� GDWHORU� FXOHVH��

Acest control nu presupune:

a) FD�SULQ�VRQGDM�V �VH�UHIDF �DQXPLWH�FDOFXOH�GH�RE LQHUH�D�YDORULORU�XQRU�LQGLFDWRUL�însFULúL�în formulare;

b) FD�OD�FHQWUHOH�GH�SUHOXFUDUH�V �VH�YHULILFH�GDF �DX�VRVLW�WRDWH�IRUPXODUHOH��FX�WRDWH�UXEULFLOH�completate;

c) XWLOL]DUHD�XQRU�PRGHOH�GH�YHULILFDUH�D�LSRWH]HORU�VWDWLVWLFH�úL�DSOLFDUH�GH�WHVWH�GH�YHULILFDUH�D�VHPQLILFD LLORU�YDORULORU�LQGLcatorilor;

d) HIHFWXDUHD�GH�FRPSDU UL� e) VXE�QLFL�R�IRUP �XWLOL]DUHD�XQRU�SURJUDPH�LQIRUPDWLFH�VSHFLDO�HODERUDWH�

24) Pentru calculul mediei aritmetice într-R�VHULH�QXPHULF �GH�GLVWULEX LH�GH�IUHFYHQ H�SH�LQWHUYDOH�VH�

iau în considerare: a) limitele inferioare; b) mijloDFHOH�LQWHUYDOHORU��LQGLIHUHQW�GH�UHSDUWL LLOH�GLQ�LQWHUYDOH� c) limitele superioare; d) OLPLWHOH�VXSHULRDUH�GDF �IUHFYHQ HOH�VXQW�UHODWLYH� e) PLMORDFHOH�LQWHUYDOHORU�vQ�LSRWH]D�DSULRULF �D�QRUPDOLW LL�UHSDUWL LLORU�GLQ�LQWHUYDOH�

25) &RHILFLHQWXO� GH� GHWHUPLQD LH� R2, calculat pe baza regulii de adunare a dispersiilor, are întotdeauna valoarea:

a) R 2 1= ;

b) [ ]R 2 0 1∈ , ;

c) ( )R 2 0 1∈ , ;

d) R 2 35%≤ ;

e) R 2 50%> . 26) Inegalitatea dintre abaterile standard ale unei vaULDELOH� ;� QRUPDO� UHSDUWL]DWH� vQ� SRSXOD LLOH�

VWDWLVWLFH� �� GH� DFHODúL� YROXP� úL� DFHHDúL� PHGLH� x , σ σ σ1 2 3> > � HVWH� YL]XDOL]DW � vQ�varianta:

a) b) c) d) e)

2

1

3

3

2

1

3

1

2

1

2

3

2

3

1

27) 3HQWUX� FHOH� GRX � YDULDELOH� VWDWLVWLFH� vQWUH� FDUH� H[LVW � R� GHSHQGHQ � OLQLDU �� V-au înregistrat o PXO LPH�GH�GDWH�VWDWLVWLFH��ÌQ�XUPD�SUHOXFU ULL�DFHVWRUD�V-DX�RE LQXW�LQGLFDWRULL�

11,1 ;9,19485

;8,4099 ;6,96299 ;91,183 ;968 22

==

====

∑∑∑∑∑

iyx

xyxy

ii

iiii

'HVSUH�GHSHQGHQ D�GLQWUH�FHOH�GRX �YDULDELOH�VH�SRDWH�DILUPD�

a) GHSHQGHQ D�HVWH�LQYHUV � b) HVWLPDUHD�FRYDULDQ HL�HVWH�cov(x,y)=130,82; c) FRHILFLHQWXO�GH�FRUHOD LH�HVWH�� d) HFXD LD�GH�UHJUHVLH�HVWH� xy 2213,388ˆ += ;

e) HFXD LD�GH�UHJUHVLH�HVWH� xy 2213,314,34ˆ += . 28) 3HQWUX�XUP WRDUHOH�YDORUL�DOH�XQHL�YDULDELOH�DOHDWRDUH�;�

{ }7 111414 12 20, , , , ��FDUH�GLQ�XUP WRDUHOH�DILUPD LL�HVWH�DGHY UDW �

a) Me = 14; b) 0R� ��úL�0H� ��

c) 13=x �úL�0H� ��

d) 14=x �úL�0R� ��;

e) 14=== MeMox .

29) &RQVLGHUkQG�XUP WRDUHOH�YDORUL�DOH�XQHL�YDULDELOH�DOHDWRDUH�;��{ }5 7 4 5 20 6 4, , , , , , , indicatorul cel

PDL�SRWULYLW�SHQWUX�D�FDUDFWHUL]D�WHQGLQ D�FHQWUDO �HVWH�

a) PHGLD�DUPRQLF � b) media; c) modul; d) PHGLD�S WUDWLF � e) mediana.

30) Un meteoURORJ� GRUHúWH� V � úWLH� vQ� FH� P VXU � SUHYL]LXQLOH� VDOH� VXQW� IRQGDWH�� ÌQ� DFHVW� VHQV�� HO�

DQDOL]HD] �YDULDELOD�QXPLW �SUHYL]LXQH��$FHDVW �YDULDELO �LD�GRX �YDORUL

x =

0

1

,

,

daca previziunea a fost incorecta

daca previziunea este corecta

0HWHRURORJXO�UHVSHFWLY�D�XUP ULW�SUHYL]LXQLOH�HIHFWXDWH�vQ�FHOH��GH�]LOH�FRQVHFXWLYH�úL�D�FRQVWDWDW�F ��DX�IRVW�FRUHFWH�LDU��DX�IRVW�LQFRUHFWH��3H�ED]D�DFHVWRU�GDWH�HO�WHVWHD] �LSRWH]D� H0��SURSRU LD�SUHYL]LXQLORU�HIHFWXDWH�úL�FRUHFWH�HVWH�GH��úL H1��SURSRU LD�HVWH�GLIHULW �GH��úL cu testul χ2:

a)

( )

( ) ( )

χ

χ

22

2 2

1 0 95232 25 18 25

25203 9 3 84

= =

=+ + +

= > =

∑frecventelor observate + frecventa teoretica

frecventa teoretica

, ,; ,

⇒ se respinge ipoteza H0;

b)

( )

( ) ( )

χ

χ

22

2 2

1 0 95232 25 18 25

253 92 3 84

= =

=− + −

= > =

∑frecventelor observate - frecventa teoretica

frecventa teoretica

, ,; ,

⇒ se respinge ipoteza H0;

c) VH�FDOFXOHD] �χ2�vQ�YDULDQWD�E��úL�VH�DOHJH�DWXQFL�LSRWH]D�+0; d) VH�FDOFXOHD] �χ2�vQ�YDULDQWD�D��úL�VH�DOHJH�DWXQci ipoteza H1; e)

( )χ

χ

22

2 2

1 0 95232 18

25 25269 6 3 84

=

=

= ++

= > =

∑∑frecventelor observate

frecventelor teoretice

, ,; ,

⇒ se respinge ipoteza H0. 31) 1RWHOH� RE LQXWH� GH� �� VWXGHQ L� GLQ� DFHHDúL� JUXS �� vQ� VHVLXQHD� GH� YDU �� OD� SDWUX� GLVFLSOLQH� VH�

SUH]LQW �DVWIHO�

1XPH�úL� DISCIPLINA prenume NR. 1 NR. 2 NR. 3 NR. 4

A.B. 9 6 7 8 C.D. 6 7 9 9 E.F. 7 8 8 6 G.H. 10 9 10 8 M.V. 5 7 4 5

3HQWUX�FDUDFWHUL]DUHD�GLVLPLODULW LL�VWXGHQ LORU�GXS �QRWHOH�RE LQXWH��VH�FDOFXOHD] �PDWULFHD�GLVWDQ HORU�DEVROXWH��$FHDVWD�HVWH�

a) AB CD EF GH MV b) AB CD EF GH MV AB 0 7 7 7 11 AB 0 7 7 7 0 CD 7 0 6 8 10 CD 7 0 6 0 8 EF 7 6 0 8 8 EF 7 6 0 8 10 GH 7 8 8 0 15 GH 7 0 8 0 15 MV 11 10 8 15 0 MV 0 8 10 15 0

c) NR1 NR2 NR3 NR4 d) AB CD EF GH MV NR1 0 7 7 11 NR1 0 7 7 7 11 NR2 7 0 6 8 NR2 7 0 6 8 10 NR3 7 6 0 15 NR3 7 6 0 8 8 NR4 11 8 15 0 NR4 11 10 8 0 15

e) AB CD EF GH MV

AB 0 -7 -7 11 7 CD -7 0 -6 8 -10 EF -7 -6 0 8 8 GH 11 8 8 0 15 MV 7 -10 8 15 0

TESTUL 9

1) ÌQ�VWXGLXO� UHSDUWL LHL�QRUPDOH�VH� IRORVHúWH�úL�QR LXQHD��-α) - punct percentilic [ ]α ∈ 0 1, al

UHSDUWL LHL�N(0,1). Acesta:

a. este valoarea Cα pentru care Φ( )Cα α= −1 ;

b. este cuantila de ordin 1-α; c. este cuantila de ordin α;

d. este cuantila de ordin 1

α;

e. este valoarea pentru care Φ( )Cα α= +1 .

2) 'LVSHUVLD�XQHL�VXPH�GH�YDULDELOH��QRWDW �σ x x xn1 2

2+ + +.. :

a) HVWH� HJDO � FX� VXPD� GLVSHUVLLORU� YDULDELOHORU� σ xi

n

i

2

1=∑ GDF � vQWUH� YDULDELOHOH� x1, x2, …, xn

H[LVW �R�GHSHQGHQ �OLQLDU � b) HVWH�HJDO �FX�VXPD�GLVSHUVLLORU�YDULDELOHORU�PDMRUDW �FX�GXEOXO�VXPHL�FRYDULDQ HORU�GLQWUH�

YDULDELOH�OXDWH�VXFFHVLY�GRX �FkWH�GRX � σ xi

n

i jj ii

n

iCov x x2

1 1

2= >=∑ ∑∑+ ( , ) ;

c) HVWH�HJDO �FX�VXPD�FRYDULDQ HORU�GLQWUH�YDULDELOH�OXDWH�VXFFHVLY�GRX �FkWH�GRX � d) nu se poate calcula; e) HVWH�OLSVLW �GH�VHQV�GHRDUHFH�vQ�SUDFWLF �QX�VH�SRW�vQWkOQL�DVWIHO�GH�FD]XUL�

3) 'DF �SH�ED]D�GDWHORU�XQHL� VHULL� VWDWLVWLFH�YDORDUHD�PHGLH�D�FDUDFWHULVWLFLL� V-a calculat ca medie

DUPRQLF ��DWXQFL�vQ�FDOFXOXO�GLVSHUVLHL��SHQWUX�FDUDFWHUL]DUHD�vPSU úWLHULL��VH�XWLOL]HD] �

a) PHGLD�DULWPHWLF � b) media�JHRPHWULF � c) PHGLD�DUPRQLF � d) PHGLD�FURQRORJLF � e) mediana; f) PHGLD�S WUDWLF � g) cuartila a treia.

4) ÌQ�UHSUH]HQW ULOH�JUDILFH�DOH�GDWHORU�VWDWLVWLFH�VH�XWLOL]HD] �

a) vQ�H[FOXVLYLWDWH�VFDUD�DULWPHWLF � b) vQ� IXQF LH� GH� RUGLQXO� GH� P ULPH� DO� GDWHORU� VH� XWLOL]HD] � GXS � FD]� VFDUD� DULWPHWLF ��

ORJDULWPLF ��VHPLORJDULWPLF ��GXEOX�ORJDULWPLF ��HWF�� c) vQ� IXQF LH� GH� RUGLQXO� GH� P ULPH� DO� GDWHORU� VFDOD� QRPLQDO �� RUGLQDO �� GH� LQWHUYDOH�� GH�

UDSRDUWH��FRQWLQX �VDX�GLVFUHW � d) RULFH�VFDU ��QXPDL�JUDILFXO�V �ILH�WUDVDW�vQWU-un sistem de axe rectangulare; e) RULFH�VFDU ��QXPDL�JUDILFXO�V �ILH�WUDVDW�vQWU-un sistem de axe polare.

5) 5HSDUWL LD�PXQFLWRULORU� GLQWU-R�XQLWDWH� LQGXVWULDO � GXS � VDODULXO� ORU�PHGLX� OXQDU�QHW� VH�SUH]LQW �astfel:

Clase salariale (u.m.) Nr. persoane

[800; 1000] 26 [1000; 1100] 33 [1100; 1200] 64 [1200; 1300] 7 [1300; 1500] 10

Procentul muncitorilor cu salariile medii lunare mai mici sau egale cu 1200 u.m. este:

a) 64%; b) 45,8%; c) 87,9%; d) 12,1%; e) 23,5%.

6) 6FRSXO�XQHL�REVHUY UL�VWDWLVWLFH�

a) vQ�PRG�REOLJDWRULX�WUHEXLH�V �FRUHVSXQG �FX�VFRSXO�FHUFHW ULL�VWDWLVWLFH�HIHFWXDWH� b) VH�SUHFL]HD] �vQ�IXQF LH�GH�QHFHVLW LOH�VSHFLILFH�GH�LQIRUPD LL�SHQWUX�GRPHQLXO�vQ�FDUH�VH�

RUJDQL]HD] �FHUFHWDUHD�úL�FDUH�VH�vQFDGUHD] �vQ�VFRSXO�JHQHUDO�DO�DFHVWHLD� c) QX� WUHEXLH� SUHFL]DW� GHRDUHFH� QX� LQIOXHQ HD] � YROXPXO� úL� FDOLWDWHD� GDWHORU� QHFHVDUH�

GHPHUVXOXL�úWLLQ LILF� d) QX� WUHEXLH� SUHFL]DW� GHRDUHFH� QX� SUH]LQW � LPSRUWDQ � SHQWUX� UH]ROYDUHD� SUDFWLF � D� XQRU�

SUREOHPH�FXP�DU�IL��GHOLPLWDUHD�RELHFWXOXL�GH�REVHUYDUH��GHILQLUHD�XQLW LORU�GH�REVHUYDUH��stabilirea programului propriu-]LV�DO�REVHUY ULL�

e) WUHEXLH� FODU� SUHFL]DW� GHRDUHFH� vQ� IXQF LH� GH� DFHVWD� VH� UHDOL]HD] � QXPDL� LQVWUXLUHD�personalului care culege datele.

ÌQ� FRQFHS LD� GH� DVW ]L�� VWDWLVWLFD� IDFH� SDUWH� GLQ� FDGUXO� úWLLQ HORU� FDUH� VWXGLD] �� VXE� DVSHFW�cantitatiY��IHQRPHQHOH�úL�SURFHVHOH�vQWU-R�YL]LXQH�VLVWHPLF ��OD�QLYHO�PLFUR�úL�PDFURHFRQRPLF�� LQkQG�VHDPD�GH�GLQDPLVPXO�VWUXFWXULORU�H[LVWHQWH�úL�GH�IDFWRULL�FDUH�DF LRQHD] �YDULDELO�vQ�WLPS�úL�VSD LX�� ÌQ�FRQVHFLQ ��VFRSXO�XQXL�GHPHUV�VWDWLVWLF�HVWH�

a) excluVLY�GH�FXQRDúWHUH�D�IHQRPHQHORU�úL�SURFHVHORU�LQYHVWLJDWH� b) GH�FXQRDúWHUH�úL�IXQGDPHQWDUH�D�YDULDQWHORU�GHFL]LLORU�GH�DF LXQH� c) GH�FXQRDúWHUH��GH�GHVSULQGHUH�D�OHJLW LORU�VXE�FDUH�VH�PDQLIHVW �IHQRPHQHOH�úL�SURFHVHOH�

LQYHVWLJDWH�úL�GH�IXQGDPHQWDUH�D�DFWLYLW LL�PDQDJHULDOH� d) GH�FRQVHPQDUH�D�PDQLIHVW ULL�IHQRPHQHORU�LQYHVWLJDWH��OD�XQ�PRPHQW�GDW� e) VLPLODU�FX�FHO�DO�XQHL�DFWLYLW L�FRQWDELOH�

8) 3HQWUX�D�vQ HOHJH�FDUDFWHUXO�DFHVWRU�IHQRPHQH�WUHEXLH�SRUQLW�GH�OD�

a) natura raporturilor de cauzalitate a acestora; b) scopXO�FHUFHW ULL� c) VFRSXO�REVHUY ULL� d) prelucrarea datelor culese; e) identitatea formelor lor individuale de manifestare.

9) 8QLW LOH�FRPSOH[H�GH�REVHUYDUH�

a) VXQW�UH]XOWDWH�DOH�RUJDQL] ULL�VRFLDOH�úL�HFRQRPLFH�D�SRSXOD LHL��FROHFWLYLW LL��VWDWLVWLFH� b) pot fi de exemplu: angajatul într-R� vQWUHSULQGHUH�� MXF WRUXO� vQWU-R� HFKLS � GH� IRWEDO��

persoana unei familii; c) VXQW�QXPDL�XQLW LOH�UDSRUWRDUH�FDUH��SRWULYLW�OHJLVOD LHL�vQ�YLJRDUH��LQIRUPHD] �VLVWHPDWLF�

DVXSUD�DFWLYLW LL�XQLW LORU�DIODWH�vQ�VWUXFWXUD�ORU�RJDQL]DWRULF � d) VXQW�HOHPHQWHOH�VLPSOH�FRQVWLWXWLYH�DOH�SRSXOD LHL�VWDWLVWLFH�FDUH�VXQW�GLVSXVH�V �U VSXQG �

OD�vQWUHE ULOH�GLQ�FKHVWLRQDU��GH�H[HPSOX��VWXGHQWXO��SHUVRDQD�FX�GUHSW�GH�YRW��HWF�� e) VXQW�GH�WLSXO��PXO LPHD�vQWUHSULQGHULORU�PLFL�úL�PLMORFLL��PXO LPHD�F V toriilor,etc.

10) 3UHFL]D L�úL�DUJXPHQWD L�GDF �DILUPD LD��³RULFH�QXP U�HVWH�R�GDW �VWDWLVWLF ´�HVWH�

a) DGHY UDW � b) IDOV � c) HVWH�DGHY UDW �GDF �QXP UXO�HVWH�QDWXUDO� d) HVWH�DGHY UDW ��GHRDUHFH�VWDWLVWLFD�VWXGLD] �IHQRPHQHOH�GH�PDV �GLQ�SXQFW�GH�YHGHUH�FDQWLWDWLY� e) QX� HVWH� DGHY UDW �� GHRDUHFH� VWDWLVWLFD� VWXGLD] � úL� YDULDELOH� DOH� F URU� YDULDQWH� VXQW�

nenumerice.

11) /D�H[DPHQXO�GH�%D]HOH�6WDWLVWLFLL�GLQ��SULPLL��VWXGHQ L�GLQ�FDWDORJXO�JUXSHL��DX�RE LQXW�XUP WRDUHOH�QRWH�

Nr. crt 1 2 3 4 5 6 7 8 9 10 Nota 10 7 4 8 9 9 7 3 6 2

Datele prezentat în tabel sunt:

a) individuale; b) agregate; c) de stoc; d) de flux; e) grupate.

12) 0HGLD�DULWPHWLF �HVWH�DFHD�YDORDUHD�QXPHULF �D�YDULDELOHL�VWXGLDWH�FDUH�PLQLPL]HD] �

a) ( )x xii

n−

=∑

2

1

;

b) max x xi − ;

c) x x x xi ji j

i j

n− −

=≠

∑, 1

;

d) ( )x xii

n−

=∑

3

1

;

e)

( )( )x x x xi ji j

n

x

− −=

∑, 1

σ.

13) 3HQWUX� GRX � VHULL� GH� GLVWULEX LH� GH� IUHFYHQ H�� LQGLFDWRUXO� ³DPSOLWXGLQHD� vPSU úWLHULL´� HVWH�FRPSDUDELO�GDF �

a) FHOH�GRX �VHULL�VH�UHIHU �OD�DFHHDúL�FDUDFWHULVWLF � b) FHOH�GRX �VHULL�VH�UHIHU �OD�DFHHDúL�FROHFWLYLWDWH� c) FHOH�GRX �VHULL�VH�UHIHU �OD�FDUDFWHULVWLFL�GLIHULWH��GDU�FX�DFHHDúL�XQLWDWH�GH�P VXU � d) FHOH�GRX �VHULL�DX�DFHODúL�JUDG�GH�DVLPHWULH� e) FHOH�GRX �VHULL�VH�UHIHU �OD�FROHFWLYLW L�GH�DFFHODúL�YROXP�

14) 3XQFWXO�$�VHPQLILF �

a) ( )P X xx x< = −

’

’Φσ

;

1,0

0,5

x − 3σ x − 2σ x − σ x x’ x + σ x + 2σ x + 3σ

-3 -2 -1 0 1 2 3

b) ( )P X xx x> = −

’

’Φσ

;

c) ( )P X xx x= = −

’

’Φσ

;

d) ( )P X xx x< > −

’

’Φσ

;

e) ( )P X xx x< < −

’

’Φσ

.

15) ��GH� HOHYL� GLQ� GRX �RUDúH� SDUWLFLS � OD� XQ� FRQFXUV� GH� FXOWXU � JHQHUDO ��&HL� �� GH� HOHYL� GLQ�

SULPXO�RUDú�RE LQ�XQ�SXQFWDM�PHGLX�GH��SXQFWH��FX�XQ�FRHILFLHQW�GH�YDULD LH�GH��LDU�FHL�GLQ�DO�GRLOHD�RUDú�RE LQ�XQ�SXQFWDM�PHGLX�GH��SXQFWH�FX�R�DEDWHUH�PHGLH�S WUDWLF �GH��SXQFWH��)DFWRUXO�GH�JUXSDUH��RUDúXO��FRQWULEXLH�OD�YDULD LD�SXQFWDMHORU�RE LQXWH�GH�HOHYL�vQ�SURSRU LH�GH�

a) 23,46%; b) 10,28%; c) 76,54%;

A

d) 48,44%; e) 24,48%.

16) 3ULQFLSDOD� SURSULHWDWH� D� IHQRPHQHORU� GH� PDV � HVWH� YDULDELOLWDWHD� vQ� WLPS� úL� vQ� VSD LX� úL�

OHJHD� GH� DSDUL LH� D� DFHVWRUD� VH� PDQLIHVW � FD� WHQGLQ � úL� QX� vQ� ILHFDUH� FD]� vQ� SDUWH�� 'LQ�DFHDVW �FDX] ��IHQRPHQHOH�GH�PDV �VXQW�QXPLWH�úL�IHQRPHQH�QHGHWHUPLQLVWH�VDX�VWRFKDVWLFH��LDU�LQWHUSUHWDUHD�úL�DQDOL]D�ORU�WUHEXLH�V �FDG �VXE�LQFLGHQ D�QXPHUHORU�PDUL��3RWULYLW�DFHVWHL�legi:

a) YDULD LLOH�vQWkPSO WRDUH�GH�OD�WHQGLQ D�JHQHUDO �SURYRFDWH�GH�IDFWRULL�RELHFWLYL�úL�HVHQ LDOL�

VH�FRPSHQVHD] �UHFLSURF�GDF �H[LVW �XQ�QXP U�PDUH�GH�FD]XUL�LQGLYLGXDOH� b) QXPDL�vQWkPSO WRU�YDULD LLOH�GH�OD�WHQGLQ D�JHQHUDO �VH�FRPSHQVHD] � c) QX�HVWH�SRVLELO �WUHFHUHD�GH�OD�QXPHURDVHOH�GDWH�LQGLYLGXDOH�OD�LQGLFDWRUL�VLQWHWLFL�VSHFLILFL�

ansamblului investigat; d) YDULD LLOH�vQWkPSO WRDUH�GH�OD�WHQGLQ D�JHQHUDO �SURYRFDWH�GH�IDFWRULL�DOHDWRUL�GH�LQIOXHQ �

se compeQVHD] � UHFLSURF� GDF � H[LVW � XQ� QXP U�PDUH� GH� FD]XUL� LQGLYLGXDOH� GH� DFHHDúL�HVHQ ��OXDWH�vQ�VWXGLX�

e) WHQGLQ D�GH�PDQLIHVWDUH�WUHEXLH�V �VH�YHULILFH�OD�QLYHOXO�ILHF UHL�XQLW L�GH�REVHUYDUH� 17) 3URFHVXO�GH�FXQRDúWHUH�VWDWLVWLF �VH�vQFKHLH�FX�

a) observarea total �VDX�SDU LDO �D�IHQRPHQHORU�GH�PDV �GHOLPLWDWH� b) prelucrarea datelor înregistrate; c) interpretarea rezultatelor; d) prezentarea datelor în tabele, serii sau grafice; e) DQDOL]D� úL� LQWHUSUHWDUHD� UH]XOWDWHORU� úL� FX� IRUPXODUHD� FRQFOX]LLORU� VWDWLVWLFH� FDUH�

obligatoriu� WUHEXLH�V �FXSULQG �YDULDQWH�SUREDELOH�GH�DSDUL LH�DOH�DFHORUDúL� IHQRPHQH�în viitor.

18) 3HQWUX�VWDELOLUHD�JUHXW LL�VSHFLILFH�D�SURGXF LHL�UHEXWDWH�vQWU-un lot de volum N=3000 produse

s-DX�SUHOHYDW�vQWkPSO WRU�úL�QHUHSHWDW��GH�SURGXVH��ÌQ�XUPD�HIHFWX ULL controlului acestora 9 produse au fost depistate rebuturi. Pentru o probabilitate de 0,975 (zα=1,96) procentul de rebut estimat pentru întregul lot este:

a) (0,015; 0,06);

b) (0,012; 0,048;

c) (0,15; 0,20);

d) (0,02; 0,05);

e) (0,18; 0,25).

19) 2�PHGLH�FDOFXODW �GLQWU-un úLU�GH�YDORUL�LQGLYLGXDOH�HVWH�UHSUH]HQWDWLY �GDF �

a) s-DX�XWLOL]DW�IUHFYHQ HOH�DEVROXWH�GH�DSDUL LH�D�YDORULORU�LQGLYLGXDOH� b) úLUXO�GH�YDORUL�LQGLYLGXDOH�HVWH�RPRJHQ� c) XQXO�GLQ�FRHILFLHQ LL�GH�DVLPHWULH�LD�YDORUL�vQ�LQWHUYDOXO�>-1, 1]; d) úLUXO�GH�YDORUL�HVWH�Vtructurat pe intervale de grupare egale; e) s-D�XWLOL]DW�PHGLD�DULWPHWLF �

20) &DUH�HVWH�UHOD LD�FRUHFW �GH�FDOFXO�D�FRHILFLHQWXOXL�GH�FRUHOD LH�OLQLDU �VLPSO �

a) rxy x y

yxx y

=− ⋅

σ σ;

b) ( )( )

rx x y y

nyxx y

=− −∑σ σ

;

c) r byxx

y=

σσ

;

d)

( ) ( )r

n xy x y

n x x n y yyx =

−

−

−

∑∑∑∑∑ ∑∑2 2 2 2

;

e) ( )( )

ry Y

y yyx

i

i

= −−

−

∑∑

12

2.

21) 2� FRPSDQLH� FDUH� SURGXFH� XQ� QRX� WLS� GH� vQJU ú PkQW� DJULFRO� HVWH� LQWHUHVDW � GH� VWXGLHUHD�FRUHOD LHL�GLQWUH�SURGXF LD�GH�URúLL��<��úL�FDQWLWDWHD�GH�vQJU ú PkQW�DSOLFDW��;��2�VXSUDID �HVWH�GLYL]DW � vQ� RSW� SDUFHOH� GH�P ULPL�HJDOH� SH� FDUH� VXQW� DSOLFDWH� FDQWLW L� GLIHULWH� GH� vQJU ú PkQW��3URGXF LLOH� GH� URúLL� �NJ�� úL� FDQWLW LOH� GH� vQJU ú PkQW� DSOLFDW� �NJ�� vQUHJLVWUDWH� SHQWUX� ILHFDUH�SDUFHO �VXQW�

xi 1 1,5 2 2,5 3 3,5 4 4,5 yi 25 31 27 28 36 35 32 34

&RHILFLHQWXO�GH�FRUHOD LH�DUH�valoarea:

a) 0,37; b) 0,98; c) 0,73; d) -0,84; e) 1,00.

22) Într-XQ� úLU� GH� YDORUL� LQGLYLGXDOH� UHIHULWRDUH� OD� DFHHDúL� YDULDELO � úL� REVHUYDW � vQWU-R� SRSXOD LH�VWDWLVWLF �GHOLPLWDW �vQ�WLPS�úL�vQ�VSD LX��YDORDUHD�PRGDO �HVWH�

a) YDORDUHD�LQGLYLGXDO �SR]LWLY �FHD�PDL�PDUH� b) valoaUHD�LQGLYLGXDO �QHJDWLY �PD[LP �vQ�YDORDUH�DEVROXW � c) YDORDUHD�LQGLYLGXDO �SR]LWLY �VDX�QHJDWLY ��FX�FHD�PDL�PDUH�IUHFYHQ �GH�DSDUL LH� d) DFHD�YDORDUH�FDUH�vQUHJLVWUHD] �FHD�PDL�PDUH�DEDWHUH�DEVROXW �ID �GH�PHGLH� e) YDORDUHD�FDUH�PLQLPL]HD] �GLVSHUVLD�

23) PentrX�DQDOL]D�GHSHQGHQ HORU�VWDWLVWLFH�GLQWUH�YDULDELOH�PHWRGD�JUDILF �SHUPLWH�

a) LQWHUSUHWDUHD�LQWHQVLW LL�OHJ WXULORU�GLQWUH�YDULDELOH� b) FRQVWDWDUHD�H[LVWHQ HL�OHJ WXULL�VWDWLVWLFH� c) LGHQWLILFDUHD�H[LVWHQ HL��GLUHF LHL�úL�IRUPHL�OHJ WXULL�GLQWUH�GRX �YDULDELOH statistice; d) HVWLPDUHD�SDUDPHWULORU�IXQF LHL�GH�UHJUHVLH� e) HVWLPDUHD�UDSRUWXOXL�GH�FRUHOD LH�GDU�QX�úL�D�FRHILFLHQWXOXL�GH�FRUHOD LH�VDX�GH�DVRFLHUH�

24) )LH�R�VHULH�VWDWLVWLF ��IRUPDW �GLQ�Q� ��S��YDORUL�LQGLYLGXDOH�[1, x2, …, xn�úL��51, R2, …, Rn rangurile ordonate ale celor n valori. Mediana (Me) este:

a) Men

xii

n=

=∑1

1

;

b) F Me( ) = 1

2;

c) Me Rp= +1 ;

d) Me xn= +

1

2;

e) Me

xn

xn

=

+ +

2

1

2

2.

25) Într-R�VHULH�GH�UHSDUWL LH�GH�LQWHUYDOH�LQWHUYDOXO�PHGLDQ�HVWH�

a) acela care vPSDUWH�vQ�GRX �S U L�HJDOH�QXP UXO�GH�LQWHUYDOH�DOH�VHULHL�

b) FRUHVSXQ] WRU�SULPHL�IUHFYHQ H�FXPXODWH�FDUH�GHS úHúWH�

nii

r

=∑

1

2;

c) FRUHVSXQ] WRU�SULPHL�IUHFYHQ H�FXPXODWH�FDUH�GHS úHúWH�

nii

r+

=∑ 1

21 ;

d) FRUHVSXQ] WRU�LQWHUYDOXOXL�FX�IUHFYHQ D�FHD�PDL�PDUH� e) FRUHVSXQ] WRU�LQWHUYDOXOXL�vQ�FDUH�HVWH�SODVDW �PHGLD�

26) 'DF �vQWU-R�VHULH�GH�UHSDUWL LH�GH�IUHFYHQ H�x

ni

i i r

=1,

, n nii

r

=∑ =

1

��VH�FDOFXOHD] �IUHFYHQ HOH�

FXPXODWH�FUHVF WRU� ( )Fai i r=1,�úL�IUHFYHQ HOH�FXPXODWH�GHVFUHVF WRU� ( )Fd i i r=1, , atunci între ele

H[LVW �UHOD LD�

a) nFdFa ii =+ ;

b) nFdFa ii += ;

c) nFdFa ii =+ + )1( ;

d) 100=+ ii FdFa ;

e) 100)1( =+ +ii FdFa .

27) )UHFYHQ D�DEVROXW �FXPXODW �FUHVF WRU�D�XOWLPHL�JUXSH�HVWH�vQWRWGHXQD�HJDO �FX�

a) QXP UXO�GH�XQLW L�GLQ�FROHFWLYLWDWH� b) QXP UXO�GH�XQLW L�GLQ�XOWLPD�JUXS � c) 100%; d) IUHFYHQ D�DEVROXW �FXPXODW �GHVFUHVF WRU�D�XOWLPHL�JUXSH� e) IUHFYHQ D�DEVROXW �D�XOWLPHL�JUXSH�SOXV�YROXPXO�FROHFWLYLW LL�

28) În general, în cadrul VRFLHW LL�� GDU� úL� vQ� QDWXU �� IHQRPHQHOH� SRW� V � DSDU � FD� UH]XOWDW� DO� XQHL�

VLQJXUH�FDX]H�VDX�FD�UH]XOWDW�DO�PDL�PXOWRU�FDX]H�FDUH�VH�PDQLIHVW �L]RODW�VDX�vQ�LQWHUGHSHQGHQ �vQWUH� HOH�� ÌQ� SULPXO� FD]�� VXQW� IHQRPHQH� XQLYRF� GHWHUPLQDWH� úL� GH� UHJXO � HOH� VH� SUH]LQW � FD�IHQRPHQH�VLPSOH��LGHQWLFH�vQWUH�HOH��GHQXPLWH�úL�IHQRPHQH�WLSLFH��ÌQ�FHO�GH-al doilea caz, apar ca IHQRPHQH� PXOWLFDX]DOH� DO� F URU� SURFHV� GH� IRUPDUH� SRDWH� V � SUH]LQWH� JUDGH� GLIHULWH� GH�FRPSOH[LWDWH�� FX� UHOD LL� PXOWLSOH� GH� LQWHUGHSHQGHQ �� IRUPkQG� � vPSUHXQ � XQ� DQVDPEOX� D� F UXL�GLPHQVLXQH� úL� VWUXFWXU � SRW� IL� GHOLPLWDWH� vQ� WLPS�� vQ� VSD LX� úL� RUJDQL]DWRULF�� 6WDWLVWLFD� VWXGLD] �fenomenele din:

a) primul caz; b) cazul al doilea; c) ambele cazuri; d) DOW �FDWHJRULH�GHFkW�FHOH�SUHFL]DWH� e) cazul al doilea, dar numai cele cuantificabile.

29) 0HGLD�DULWPHWLF �D�VXPHL�GLQWUH�GRX �YDULDELOH�HVWH�HJDO �FX�VXPD�PHGLLORU�FHORU�GRX �YDULDELOH��FkQG�

a) FHOH�GRX �YDULDELOH�VH�DIO �vQWU-R�UHOD LH�GH�LQWHUGHSHQGHQ � b) FHO�GRX �YDULDELOH�VH�DIO �vQWU-R�UHOD LH�GH�LQYHUV �SURSRU LRQDOLWDWH� c) FHOH�GRX �YDULDELOH�VH�UHIHU �OD�DFHHDúL�FROHFWLYLWDWH� d) FHOH�GRX �YDULDELOH�VXQW�LQGHSHQGHQWH� e) FHOH�GRX �YDULDELOH�VXQW�GLUHFW�SURSRU LRQDOH�

30) 6WXGLD L�XUP WRDUHOH�DILUPD LL��)HQRPHQHOH�GH�PDV ��vQ�JHQHUDO��DSDU�FD�R�PXO LPH�GH�IRUPH�

individuale diferiWH�� DSDUHQW� I U � QLFL� R� OHJ WXU � GH� OD� R� IRUP � OD� DOWD�� GDU� FDUH�� DQDOL]DWH�FRPSDUDWLY��VH�FRQVWDW �F �DX�DFHHDúL�HVHQ ��$FHDVWD�VH�H[SOLF �vQ�SULQFLSDO�SULQ�IDSWXO�F �HOH�VXQW�JHQHUDWH�GH�R�VHULH�GH�FDX]H�FRPXQH�FDUH�VH�PDQLIHVW �GH�UHJXO �vQ�FRQGL LL diferite; 3) La IHQRPHQHOH� GH� PDV � GLQ� VRFLHWDWH�� UHOD LLOH� GH� PXOWLFDX]DOLWDWH� GLUHFW � VDX� LQGLUHFW � FDUH� OH�GHWHUPLQ �IDF�LPSRVLELO �FXQRDúWHUHD�OHJLORU�FDUH�OH�SURGXF�úL�JXYHUQHD] ��GDF �VH�LDX�vQ�VWXGLX�izolat doar câteva din formele lor de manifestare, ignorând ansamblul din care fac parte; 4) În SUDFWLF �IRUPHOH�LQGLYLGXDOH�GH�PDQLIHVWDUH�GLIHU �GH�OD�R�XQLWDWH�OD�DOWD�vQ�IXQF LH�GH�PRGXO�FXP�VH�DVRFLD] �úL�VH�FRPELQ �IDFWRULL�GH�LQIOXHQ �GH�QDWXU �GLIHULW ��O VkQG�LPSUHVLD�F �ILHFDUH�IRUP �de manifHVWDUH� GLQ� FDGUXO� DQVDPEOXOXL� I U � V � H[LVWH� R� FDX]DOLWDWH� FHUW �� $QDOL]DWH�� IRUPHOH�LQGLYLGXDOH�GH�PDQLIHVWDUH�DOH�IHQRPHQHORU�GH�PDV �SDU�DVHP Q WRDUH�vQWUH�HOH��ILLQG�JHQHUDWH�GH�FDX]H�HVHQ LDOH�FRPXQH��VXSXQkQGX-VH�DFHOHLDúL�OHJL�GH�DSDUL LH�úL�GH]Yoltare.

&DUH�GLQ�DILUPD LLOH�GH�PDL�VXV�VXQW�IDOVH�

a) ��úL�� b) ��úL�� c) ��úL�� d) WRDWH��FX�H[FHS LD�� e) nici una.

TESTUL 10

1) 3HQWUX�YHULILFDUHD�QRUPDOLW LL�GLVWULEX LHL�HPSLULFH��vQWU-R�FROHFWLYLWDWH�GH�Q�XQLW L��VLVWHPDWL]DWH�

pe r intHUYDOH� GH� JUXSDUH�� GDF � VH� XWLOL]HD] � WHVWXO� χ2�� YDORDUHD� FDOFXODW � HVWH� FRPSDUDW � FX�valoarea χ2�WDEHODU�FRUHVSXQ] WRDUH�XQXL�QXP U�GH�JUDGH�GH�OLEHUWDWH�HJDO�FX�

a) n-r; b) n-1; c) r-1; d) r-3; e) n-3.

2) 'LVSHUVLD�XQHL�UHSDUWL L�ELQRPLDOH�FX�S�- probabilitatea succesului, q - probabilitatea insuccesului

úL�Q��QXP UXO�GH�REVHUYD LL��HVWH�

a) np; b) nq;

c) npq ; d) npq(q-p); e) npq.

3) &RQVLGHU P� XUP WRDUHOH� SHUHFKL� GH� REVHUYD LL� SHQWUX� YDULDELOHOH� ;� úL� <�� vQWUH� FDUH� H[LVW � R�

GHSHQGHQ �GH�IRUPD�\� �I�[��

xi 1 4 3 2 5 6 0 yi 1 3 3 1 4 7 2

6 �VH�DUDWH�FDUH�GLQ�XUP WRDUHOH�DILUPD LL�HVWH�DGHY UDW �

a) HFXD LD�GH�UHJUHVLH�OLQLDU �HVWH� xy 821,0536,0ˆ −= ;

b) SHQWUX� XQ� SUDJ� GH� VHPQLILFD LH� α=0,05, coeficientul de regresie nu este semnificativ statistic;

c) intervalul de încredere pentru coeficientul de regresie este (0,243, 1,399); d) HFXD LD�GH�UHJUHVLH�OLQLDU �HVWH�

xy 821,0536,0ˆ += ; e) FRHILFLHQWXO�GH�FRUHOD LH�HVWH�U -0,92.

4) 3HQWUX�R�VHULH�GH�UHSDUWL LH�GH�IUHFYHQ H��FX�WHQGLQ �GH�QRUPDOLWDWH��DEDWHUHD�PHGLH�OLQLDU �

a) HVWH�DSUR[LPDWLY��GLQ�DEDWHUHD�LQWHUTXDUWLOLF � b) HVWH�PDL�PLF �VDX�HJDO �FX�� c) HVWH�DSUR[LPDWLY��GLQ�YDORDUHD�DEDWHULL�PHGLL�S WUDWLFH� d) HVWH�HJDO �FX�]HUR� e) FXSULQGH��GLQ�XQLW LOH�VWDWLVWLFH�GLQ�FHQWUXO�GLVWULEX LHL�

5) Pentru verificarea norPDOLW LL� UHSDUWL LLORU� HPSLULFH� XWLOL]kQG� WHVWXO� χ2, valoarea χ calculat2 se

FRPSDU �FX�YDORDUHD� χ tabelar2 �FRUHVSXQ] WRU�XQXL�QXP U�GH�JUDGH�GH�OLEHUWDWH�HJDO�FX�

a) QXP UXO�XQLW LORU�VWDWLVWLFH�GLQ�FROHFWLYLWDWH�PLQXV�XQX� b) QXP UXO�XQLW LORU�VWDWLVWLFH�GLQ�FROHFWLYLWDWH�PLQXV�GRL� c) QXP UXO�LQWHUYDOHORU�GH�YDULD LH�PLQXV�XQX� d) QXP UXO�LQWHUYDOHORU�GH�YDULD LH�PLQXV�QXP UXO�GH�SDUDPHWUL�DL�OHJLL�GH�UHSDUWL LH�χ2 ; e) QXP UXO� LQWHUYDOHORU� GH� YDULD LH�PLQXV� QXP UXO� GH� SDUDPHWUL� DL� OHJLL� GH� UHSDUWL LH�χ2 ,

minus unu. 6) &DUH�GLQ�XUP WRDUHOH�DILUPD LL�HVWH�DGHY UDW �

a) PHGLD�JHRPHWULF �VH�FDOFXOHD] �FD�R�PHGLH�DULWPHWLF �D�ORJDULWPLORU�YDORULORU�LQGLYLGXDOH� b) SURGXVXO�DEDWHULORU�WHUPHQLORU�VHULHL�GH�OD�PHGLD�ORU�JHRPHWULF �HVWH�� c) suma abaterilor WHUPHQLORU�VHULHL�GH�OD�PHGLD�ORU�JHRPHWULF �HVWH�� d) SURGXVXO�DEDWHULORU�WHUPHQLORU�VHULHL�GH�OD�PHGLD�ORU�JHRPHWULF �HVWH�� e) PHGLD�JHRPHWULF �HVWH�DFHD�YDORDUH�FDUH��vQORFXLQG�WHUPHQLL�VHULHL��QX�PRGLILF �SURGXVXO�

S WUDWHORU�ORU� 7) Într-o colectivitate staWLVWLF ��IHQRPHQXO�GH�FRQFHQWUDUH�vQVHDPQ �

a) R�YDULD LH�VF ]XW �D�YDORULORU�FDUDFWHULVWLFLL�vQ�MXUXO�PHGLHL� b) R�GLVWULEX LH�vQ�IRUP �GH�³-´� c) R�DVLPHWULH�VF ]XW �D�GLVWULEX LHL� d) R�GLVWULEX LH�vQ�IRUP �GH�³8´� e) cumularea valorilor caracteristicii în cadrul uneL�JUXSH�FODVH�DOH�FROHFWLYLW LL�

8) 'DF � vQWUH� FXDUWLOHOH� FDOFXODWH�SHQWUX�R�VHULH�GH� UHSDUWL LH� H[LVW � UHOD LD� QQ Q

21 3

2=

+ atunci

UHSDUWL LD�HVWH�

a) DVLPHWULF �OD�GUHDSWD� b) DVLPHWULF �OD�VWkQJD� c) QRUPDO � d) KLSHUEROLF � e) ELGLPHQVLRQDO � f) ELPRGDO �

9) PriQ��FH�WHVW�VH�YHULILF �FRUHVSRQGHQ D�GLQWUH�UHSDUWL LLOH�WHRUHWLFH�úL�FHOH�HPSLULFH"

a) χ2 ; b) F (R.A.Fisher); c) t (Student); d) Kolmogorov-Smirnov; e) Wald.

10) )LH� VHULD� VWDWLVWLF � { }x x xn1 2, , . . . , � RE LQXW � SULQ� REVHUYDUHD� XQHL� YDULDELOH� QXPHULFH� ;� úL�

pentru care s-au calculat media x �úL�GLVSHUVLD��σx2 . Controlându-se calitatea datelor culese, se

FRQVWDW �F �ILHFDUH�GLQ�DFHVWHD�DX�IRVW�PDMRUDWH�OD�vQUHJLVWUDUH�FX��GH�XQLW L��6HULD�FRUHFW �DU�

fi fost: x x xn1 2

100 100 100, , . . . ,

. Calculându-se din nou dispersia pentru valorile corecte se

FRQVWDW �F �DFHDVWD�HVWH�

a) HJDO �FX�GLVSHUVLD�LQL LDO "

b) PDL�PLF �GHFkW�σx2 de 10.000 ori?

c) PDL�PLF �GHFkW�σx2 cu

1

100?

d) mai mare decât σx2 cu

1

100?

e) mai mare decât σx2 de 100 de ori?

11) 3HQWUX�GRX �YDULDELOH�VWDWLVWLFH�GHSHQGHQWH�[�úL�\�FX�\ I�[��V-DX�FXOHV�GDWH�GH�OD��XQLW L�úL�V-au calculat indicatorii:

y y x x x yii

ii

ii

ii

i ii

∑ ∑ ∑ ∑ ∑= = = = =1133 92703 3658 0957118 2967342 2; ; , ; , ; ,

ÌQ� LSRWH]D�XQHL� GHSHQGHQ H� OLQLDUH�HFXD LD� GH� UHJUHVLH� FDUH�PRGHOHD] � OHJ WXUD� GLQWUH� FHOH�GRX �YDULDELOH�HVWH�

a) xy 35,52255,55ˆ += ; b) xy 55,5535,522ˆ +−= ; c) xy 55,5535,522ˆ −= ; d) 55,5535,522ˆ +−= xy ; e) xy 35,52255,55ˆ +−= ;

12) Se precizeaz �XUP WRDUHOH�SRSXOD LL�VWDWLVWLFH� ��SRSXOD LD�%XFXUHúWLXOXL�OD��RFWRPEULH�� SRSXOD LD�GLQ�5RPkQLD�GLQ�PHGLXO�XUEDQ�OD��LXOLH�� PXO LPHD�IDFWXULORU�HODERUDWH�GH�6�&��³,&6´�65/�vQ�WULP��,�� 4) PXO LPHD�F V WRULLORU�vQFKHLDWH�vQ�5RPkQLD�vn anul 1997; 5) VWRFXULOH�GH�P UIXUL�DOH�vQWUHSULQGHULORU�PLFL�úL�PLMORFLL�OD�VIkUúLWXO�DQXOXL��

3RSXOD LL�VWDWLVWLFH�GLQDPLFH�VXQW�

a) ��úL��

b) ��úL��

c) ��úL��

d) ��úL��

e) ��úL��

13) ÌQ�FD]XO�XQHL�REVHUY UL�VWDWLVWLFH�DO�F UHL�VFRS�HVWH�³IRWRJUDILHUHD�]LOHL�GH�Oucru a unui muncitor”:

a) WLPSXO�REVHUY ULL�FRLQFLGH�FX�WLPSXO�OD�FDUH�VH�vQUHJLVWUHD] �GDWHOH� b) WLPSXO�REVHUY ULL�QX�FRLQFLGH�FX�WLPSXO�OD�FDUH�VH�vQUHJLVWUHD] �GDWHOH� c) WLPSXO�REVHUY ULL�VH�UHIHU �OD�GXUDWD�V SW PkQLL�GH�OXFUX�D�PXQFLWRUXOXL� d) precizarea timpXOXL�REVHUY ULL�HVWH�OLSVLW �GH�VHQV� e) WLPSXO�REVHUY ULL�HVWH�PRPHQWXO�FULWLF��³RUD�]HUR´�D�]LOHL�vQ�FDUH�PXQFLWRUXO�VH�SUH]LQW �OD�OXFUX�

14) Coeficientul de concentrare al lui Corrado-*LQL�VH�FDOFXOHD] �FD�

a) ∑=

=n

iixG

1

2 ;

b) ∑∑=

=

=n

in

ii

i

x

xG

1

1

;

c) 1

2

1

1

−

=

=

∑∑

=n

in

ii

i

x

xnG ;

d)

2

1

1

∑∑=

=

=n

in

ii

i

x

xG ;

e) n

ix

x

G

n

in

i

i∑∑=

==

1

2

1 .

15) 0HGLD�DULWPHWLF �D�XQHL�YDULDELOH�DOHDWRDUH�;��UHSUH]LQW �

a) momentul simplu de ordinul întâi;

b) momentul centrat de ordinul întâi;

c) momentul centrat de ordinul al doilea;

d) momentul simplu de ordinul al doilea;

e) PRPHQWXO�VLPSOX�GH�RUGLQXO�DO�GRLOHD�PLQXV�PRPHQWXO�VLPSOX�GH�RUGLQXO�vQWkL�OD�S WUDW�

16) /D�WHVWDUHD�QRUPDOLW LL�UHSDUWL LLORU�HPSLULFH�XWLOL]kQG�WHVWXO�χ2��GDF � χ χcalc tab2 2< , atunci:

a) VH� DFFHSW � LSRWH]D� F � vQWUH� GLVWULEX LD� HPSLULF � úL� FHD� QRUPDO � �WHRUHWLF �� QX� H[LVW �

FRQFRUGDQ � b) VH� DFFHSW � LSRWH]D� F � vQWUH� GLVWULEX LD� HPSLULF � úL� FHD� QRUPDO � �WHRUHWLF �� H[LVW �

FRQFRUGDQ � c) VH� UHVSLQJH� LSRWH]D� FRQIRUP� F UHLD� vQWUH� GLVWULEX LD� HPSLULF � úL� FHD� WHRUHWLF � H[LVW �

concordDQ � d) VH�DFFHSW �LSRWH]D�FRQIRUP�F UHLD�GLVWULEX LD�HPSLULF �GLIHU �VHPQLILFDWLY�GH�FHD�QRUPDO �

�WHRUHWLF �� e) QX�H[LVW �PRWLYH�VXILFLHQWH�SHQWUX�D�DFFHSWD�LSRWH]D�QRUPDOLW LL�GLVWULEX LHL�HPSLULFH�

17) $PSOLWXGLQHD�UHODWLY �D�YDULD LHL�VH�FDOFXOHD] �FD�

a) x xmax min−

100;

b) x x

xmax min

min

−100 ;

c) x x

xmax min

max

−100 ;

d) x x

xmax min−

100 ;

e) x x

xmax −

100 .

18) 5HSDUWL LD�ELQRPLDO ��WLQGH�F WUH�UHSDUWL LD�QRUPDO �

a) GDF �QLFL�S��SUREDELOLWDWHD�VXFFHVXOXL��úL�QLFL�T��SUREDELOLWDWHD�LQVXFFHVXOXi) nu sunt nule; b) GDF �S T �� c) GDF � XQD� GLQWUH� SUREDELOLW L� GHYLQH� IRDUWH� PLF � úL� Q� �QXP UXO� GH� REVHUYD LL�� FUHúWH�

VXILFLHQW�SHQWUX�D�S VWUD�QS�ILQLW� d) GDF �R�SUREDELOLWDWH��S�VDX�T��GHYLQH�QXO � e) GDF �QLFL�S�úL�QLFL�T�QX�VXQW�PLFL��LDU�Q�DUH�YDORUL�PDUL�

19) ÌQ�PRGHOXO�GH�DQDOL] �GLVSHUVLRQDO �XQLIDFWRULDO ��GLVSHUVLD�FRUHFWDW �GLQ�LQWHULRUXO�JUXSHORU�VH�

RE LQH� UDSRUWkQG� VXPD�S WUDWHORU� DEDWHULORU� GLQ� LQWHULRUXO� FHORU� U� JUXSH� OD�QXP UXO� JUDGHORU� GH�OLEHUWDWH��O��$FHVW�QXP U�

a) l=n; b) l=n-1; c) l=r-1; d) l=n-2; e) l=n-r.

20) 3HQWUX� XQ� VWXGLX� VRFLRORJLF� FX� WHPD� ³,PSOLFD LL� DOH� IHQRPHQXOXL� GH� úRPDM� vQ� 5RPkQLD´� V-a

IRUPDW� XQ� HúDQWLRQ� GH� �� GH� SHUVRDQH� LQFOXVH� vQ� ³VIHUD� IRU HL� GH� PXQF ´�� ÌQ� FKHVWLRQDUXO�special elaborat s-DX� FXSULQV� úL� vQWUHE UL� UHIHULWRDUH� OD� VWDUHD� FLYLO � úL� categoria socio-SURIHVLRQDO ��&HOH�GRX �YDULDELOH�VXQW�GH�DFHHDúL�QDWXU �FX�YDORUL�VLWXDWH�SH�R�VFDO �

a) FRQWLQX � b) GLVFUHW � c) RUGLQDO � d) QRPLQDO � e) de intervale.

21) Într-R� SRSXOD LH� VWDWLVWLF � VH� XUP UHVF� YDULDELOHOH� QXPHULFH� [�� \� úL� ]�� vQWUH� FDUH� H[LVW � UHOD LD:

\ [�]��0HGLD�DULWPHWLF �D�YDULDELOHL�[�HVWH�HJDO �FX�SURGXVXO�PHGLLORU�YDULDELOHORU�\�úL�]�DWXQFL�când:

a) \�úL�]�VH�DIO �vQWU-R�DQXPLW �GHSHQGHQ �

b) \�úL�]�VXQW�LQGHSHQGHQWH�vQWUH�HOH� c) \�úL�]�VXQW�YDULDELOH�QRUPDOL]DWH�GDU�GHSHQGHQWH� d) \�úL�]�VXQW�OLQLDU�Gependente; e) \�úL�]�DX�FRYDULDQ D�GLIHULW �GH�]HUR�

22) 'HVSUH�VWRFXO�GH�P UIXUL�H[LVWHQW�OD�R�VRFLHWDWH�FRPHUFLDO �VH�FXQRVF�XUP WRDUHOH�GDWH�RE LQXWH�în urma inventarierii:

Data inventarierii 1.01.1997 1.02.1997 15.03.1997 10.05.1997 1.07.1997

Stocul existent (mil. lei) 50 58 68 70 75 6WRFXO�PHGLX�GH�P UIXUL�DO�VRFLHW LL�FRPHUFLDOH�GLQ�SHULRDGD��- 1.07.197 a fost de:

a) 66 mil. lei; b) 60 mil. lei; c) 64,2 mil. lei; d) 50 mil. lei; e) 75 mil. lei.

23) 2�PDúLQ �XWLOL]DW �SHQWUX�D�GR]D�FDQWLWDWHD�GH�FXORDUH�vQ�YRSVHD��GR]HD] �vQ�PHGLH�m ml pe cutie GH�YRSVHD��&DQWLWDWHD�GH�FXORDUH�GR]DW �HVWH�FXQRVFXW �D�DYHD�R�GLVWULEX LH�QRUPDO �FX�GLVSHUVLD�HJDO �FX��'DF �PDL�PXOW�GH��PO�GH�FXORDUH�HVWH�DPHVWHFDW �vQ�RE LQHUHD�XQHL�QXDQ H�GH�albastru, vopseaua este respins ��9DORDUHD�OXL�m��DVWIHO�vQFkW�V �QX�ILH�UHVSLQVH�PDL�PXOW�GH��din cutii este:

a) 5,072; b) 5,6272; c) 5,536; d) 5,068; e) 5,6288.

24) 3HQWUX� FRQVWUXLUHD� LQGLFLORU� GH� JUXS� VH� XWLOL]HD] � PDL� PXOWH� VLVWHPH� GH� SRQGHUDUH�� ,QIOXHQ D�sistemului de ponderare utilizat poate fi SXV �vQ�HYLGHQ �XWLOL]kQG�

a) H[FOXVLY� FRHILFLHQWXO� GH� FRUHOD LH� DO� LQGLFLORU� LQGLYLGXDOL� DL� IDFWRUXOXL� FDQWLWDWLY� úL� DL�factorului calitativ ( r

i ix f );

b) DQDOL]kQG�FRHILFLHQ LL�GH�YDULD LH�DL�FHORU�GRX �FDWHJRULL�GH�LQGLFL�LQGLYLGXDOL�� fx iiVV ; );

c) produsul )(0/1

)(0/1

01 : fxfx

iiiIIrVV

fixfx =⋅⋅ ;

d) UHOD LD� fxfix iii

fxfx VVrII ⋅⋅+= 1: )()( 01 ;

e) UHOD LD� fxfix iii

fxfx VVrII ⋅⋅+=⋅ 1)()( 01 .

25) &DUH�HVWH�PXO LPHD�YDORULORU�FRHILFLHQWXOXL�GH�FRUHOD LH�vQ�FDGUXO�OHJ WXULORU�GLUHFWH�

a) [ -1, 1 ]; b) [ -1, 0 ]; c) ( 0, 1 ]; d) PXO LPHD�QXPHUelor reale; e) [ -3, 3 ].

26) 3HQWUX�R�FROHFWLYLWDWH�VWUXFWXUDW �vQ�U�JUXSH�GXS �YDORULOH�FDUDFWHULVWLFLL�;��IDFWRU�GH�JUXSDUH��úL�vQ�P�JUXSH�GXS �YDORULOH�YDULDELOHL�DQDOL]DWH�<��UHJXOD�GH�DGXQDUH�D�GLVSHUVLLORU�VH�SRDWH�DSOLFD��XWLOL]D�úL�GDF �DYHP�OD�GLVSR]L LH�IUHFYHQ HOH�UHODWLYH�vQ�ORFXO�IUHFYHQ HORU�DEVROXWH��FX�FRQGL LD�

ca:

a) f jj

m

.=∑ =

1

1 �úL� f ii

r

.=∑ =

11;

b) f ijj

m

i

r=

==∑∑ 1

11

;

c) f ii

r

.=∑ =

11;

d) f ijj

m

=∑ =

11 �úL� f i

i

r

.=∑ =

1

1;

e) f ijj

m

=∑ =

1

1 .

27) Coeficientul lui Pearson βµµ

24

22

= �VH�XWLOL]HD] �SHQWUX�DQDOL]D�VWDWLVWLF �D�

a) asimetriei; b) YDULD LHL� c) boltiri; d) WHQGLQ HL�FHQWUDOH� e) LQGLFDWRULORU�PHGLL�GH�SR]L LH�

28) ÌQ� WHVWDUHD� LSRWH]HL� VWDWLVWLFH�SULYLWRDUH� OD�SDUDPHWUXO�³PHGLD�SRSXOD LHL´�� UHJLXQHD�FULWLF �HVWH�GDW �GH�� t t n< − −α ; 1 când:

a) datele provin dintr-XQ�HúDQWLRQ�GH�YROXP�UHGXV�úL�VH�HIHFWXHD] �WHVW�XQLODWHUDO�VWkQJD� b) datele provin dintr-XQ�HúDQWLRQ�GH�YROXP�UHGXV�úL�VH�HIHFWXHD] �WHVW�XQLODWHUDO�GUHDSWD� c) datele provin dintr-XQ�HúDQWLRQ�GH�YROXP�QRUPDO��VH�FXQRDúWH�GLVSHUVLD�SRSXOD LHL�úL�V-a

efectuat test unilateral stânga; d) datele provin dintr-XQ�HúDQWLRQ�GH�YROXP�UHGXV�úL�V-a efectuat test bilateral; e) datele provin dintr-XQ�HúDQWLRQ�GH�YROXP�QRUPDO�úL�V-a efectuat test bilateral.

29) Într-R� VHULH� GH� GLWULEX LH� QRUPDO � GXS �R� DQXPLW � YDULDELO � QXPHULF �� vQWUH� YDORDUHD�PHGLDQ ��PRGDO �úL�PHGLH�HVWH�DGHY UDW �UHOD LD�

a) x Mo Me< < ;

b) Me x Mo< < ;

c) x Mo Me= = ;

d) x Me Mo< < ;

e) Mo Me x< < .

Unde x �PHGLD�DULWPHWLF ��Me = mediana; Mo� �YDORDUHD�PRGDO 30) 3HQWUX�R�UHSDUWL LH�GH�IUHFYHQ H�FX�WHQGLQ �GH�QRUPDOLWDWH��V �VH�SUHFL]H]H�FDUH�GLQ�XUP WRDUHOH�

DILUPD LL�SULYLWRDUH�OD�PHGLD�VD�QX�HVWH�DGHY UDW � a) UHSUH]LQW �SXQFWXO�IRFDO�GH�SH�VFDO ��vQ�MXUXO�F UXLD�YDORULOH�VH�EDODQVHD] �SHUIHFW�

b) PLQLPL]HD] �IXQF LD�GH�GLVWDQ ��GH�WLSXO� ( )a xii

n−

=∑ 2

1

;

c) HVWH�SX LQ�VHQVLELO �OD�IOXFWXD LLOH�GH�VHOHF LH� d) vQ�SURFHVXO�LQIHUHQ HL�VWDWLVWLFH��HVWH�GH�RELFHL�SUHIHUDW �DOWRU�LQGLFDWRUL�DL�WHQGLQ HL�FHQWUDOH� e) HVWH�PDL�SX LQ�DIHFWDW �GH�YDORULOH�H[WUHPH��GHFkW�PHGLDQD�úL�PRGXO�

31) 6H�FXQRVF�XUP WRDUHOH�GDWH�SULYLQG��QXP UXO�GH�IDFWRUL��vQWRFPLWH�GH�6�&��$QRQLPXV��65/�vQ�OXQD�noiembrie 1997:

3RSXOD LD�VWDWLVWLF �VWXGLDW �HVWH�VWUXFWXUDW �úL�SUH]HQWDW �vQ�

a) Tabelul I; b) Tabelul II; c)Tabelul III; d) Tabelul IV; e) Tabelul V.

Data Nr. facturi Data Nr. facturi Data Nr. facturi 1 6 11 10 21 14 2 10 12 14 22 12 3 12 13 6 23 11 4 10 14 18 24 10 5 8 15 13 25 8 6 9 16 9 26 4 7 10 17 14 27 12 8 11 18 12 28 9 9 12 19 17 29 12 10 9 20 12 30 16

I II III

Grupe de ]LOH�GXS �QU��

de facturi

Nr. zile

Grupe de zile GXS �QU��GH�

facturi

Nr. zile

Intervale de timp

Nr. facturi

0 - 5 1 0 -5 1 0 - 6 55 6 - 11 13 6 - 10 16 7 - 14 90

11 - 15 13 11 - 15 10 15 - 22 103 16 - 20 3 16 - 20 3 23 - 30 82

IV V

Intervale ale QXP UXOXL�]LOQLF�GH�

facturi

Nr. de zile Intervale ale QXP UXOXL�GH�

facturi zilnice

Nr. de zile

( 0 - 5 ] 1 [ 0 - 5 ) 1 ( 5 - 10 ] 13 [ 5 - 10 ) 9 (10 - 15 ] 13 [10 - 15 ) 17 ( 15 - 20 ] 3 [ 15 - 20 ) 3

TESTUL 11

1) 1RW P�FX�σ REV2 �úL�FX�σ FREV

2 �GLVSHUVLLOH�PHGLHL�GH�VHOHF LH�vQ�FD]XULOH�IRUP ULL�HúDQWLRDQHORU�

vQ�YDULDQWHOH�FX�UHYHQLUH�úL�I U �UHYHQLUH��&RPSDUkQG�FHOH�GRX �GLVSHUVLL�V �VH�DSUHFLH]H�FDUH�GLQ�variantele de mai jos sunt corecte:

a) 22

FREVREV σσ < ;

b) 22FREVREV σσ > ;

c) 22REVFREV σσ = ;

d) 12

2

−−>

N

nN

REV

FREV

σσ

;

e) 12

2

−−=

N

nN

REV

FREV

σσ

.

2) 6H�FRQVLGHU �R�SRSXOD LH�VWDWLVWLF �GH�YROXP�1 cu N n k= ⋅ ; unde n -�YROXPXO�HúDQWLRQXOXL�úL�N�-

XQ� QXP U� QDWXUDO� QHQXO�� /D� XQLW LOH� GH� REVHUYDUH� V-D� XUP ULW� YDULDELOD� ;�� DOH� F UHL� YDORUL�individuale sunt x ii = (cu i=1,2,…,N).

'LVSHUVLD�PHGLHL�GH�VHOHF LH�vQ�FD]XO�IRUP ULL�HúDQWLRQXOXL�vQ�YDULDQWD�I U �UHYHQLUH�HVWH�

a) N + 1

2;

b) ( )N N + 1

12;

c) ( )kN− 12

;

d) ( )( )k N− +1 1

12;

e) ( )k −1

12.

3) 6H�FRQVLGHU �R�SRSXOD LH�VWDWLVWLF �GH�YROXP�1�FX� N n k= ⋅ ; unde n -�YROXPXO�HúDQWLRQXOXL�úL�N�-

XQ� QXP U� QDWXUDO� QHQXO�� /D� XQLW LOH� GH� REVHUYDUH� V-D� XUP ULW� YDULDELOD� ;�� DOH� F UHL� YDORUL�individuale sunt x ii = (cu i=1,2,…,N�� 3UHVXSXQHP�F �VH�H[WUDJH�OD�vQWkPSODUH�XQ�QXP U�³a”

vQWUH� �� úL� k ( ( )a k∈ 1, �� (IHFWX P� R� H[WUDJHUH� PHFDQLF � D� HúDQWLRQXOXL� FHHD� FH� vQVHDPQ � F �

HúDQWLRQXO�HVWH�IRUPDW�GLQ�XQLW LOH�a a k a n k, , .. , ( )+ + − 1 . 'LVSHUVLD�PHGLHL�GH�VHOHF LH�HVWH�

a) ( )( )k N− +1 1

12;

b) k 2 1

12

−;

c) ( )k N2 1

12

−;

d) ( )k N− 1

12;

e) ( )N N + 1

12.

6 �FRQVLGHU P�FD]XO�XQXL�VRQGDM�FX�UHYHQLUH�úL�SUREDELOLW LOH�LQHJDOH��$FHDVWD�vQVHDPQ �F �

în fiecare din cele n� SUHOHY UL� XQLW LOH�Ui ( i N= 1, �� VH� UHJ VHVF� vQ� HúDQWLRQ� FX�SUREDELOLW ile Ai

( i N= 1, �� /D� XQLW LOH� GH� REVHUYDUH� V-D� XUP ULW� YDULDELOD� ;�� DOH� F UHL� YDORUL� VXQW� xi ( i N= 1, ). �3UHFL] UL�SHQWUX�SUREOHPHOH��úL��

4) Suma T X ii

N=

=∑

1

HVWH�HVWLPDW �

a) nedeplasat de �Tn

x

Ai

ii

n=

=∑1

1 deoarece [ ] TTM =ˆ , unde cu [ ]M ⋅ s-a notat media;

b) deplasat de �Tn

x

Ai

ii

n=

=∑1

1

deoarece [ ]M T T� − ≠ 0 ;

c) nedeplasat de �Tn

x

Ai

ii

n=

=∑1

1 deoarece T T= � ;

d) deplasat de �Tn

x

Ai

ii

n=

=∑1

1

deoarece [ ]M T T2 2 0� − ≠ ;

e) nedeplasat de �Tn

x

Ai

ii

n=

=∑1

1 deoarece [ ]M T T2 2 0� − = .

5) 1RW P� FX� ( )D x A Ax

ATi i i

i

ii

N/ = −

=∑

2

1dispersia valorilor xi /Ai cu i N= 1, . Dispersia

estimatorului �T este:

a) ( )D Tn

Ax

AT

n

x

ATi

i

ii

Ni

ii

� = −

≠ −

=

∑ ∑1 12

1

22 ;

b) ( )D Tn

Ax

AT

n

x

ATi

i

ii

Ni

ii

� = −

= −

=

∑ ∑1 12

1

22 ;

c) ( )D Tn

A Ax

A

x

Aii j

n

x

ATi j

i

i

j

jj

i

ii

� = −

=

<

−∑∑ ∑1 22

2 22 ;

d) ( )D Tn

x

ATi

ii

� = +∑22

22 ;

e) ( )D Tn

x

AT

n

x

ATi

ii

Ni

ii

N� = −

= −

= =

∑ ∑1 12

1

22

1

.

6) $QDOL]kQG�FRQGL LLOH�IRUPXODWH�OD��úL��VH�FRQVWDW �F �HVWH�DGHY UDW �UHOD LD�

a) ( )Dx

A

n

ND Ti

i

= � ;

b) ( )Dx

A

n N

ND Ti

i

= −

− 1� ;

c) ( )Dx

AnD Ti

i

= � ;

d) ( )Dx

AD Ti

i

= � ;

e) ( )Dx

A nD Ti

i

= 1

� .

7) Un estimator nedeplasat pentru ( )D T� este:

a) ( )� �

( )�D T

n n

x

ATi

ii

n=

−−

=∑1

1

2

1;

b) ( )� � �D Tn

x

ATi

ii

n=

−−

=∑1

1

2

1;

c) ( )� �

( )�D T

n n

x

ATi

ii

N=

−−

=∑1

1

2

1;

d) SUH]HQWDW�vQ�YDULDQWD�D��úL�vQ�SOXV� ( )� �

( )D T

n n

x

A

x

Ai

i

j

jj

n

ii j

n=

−−

∑∑

<

1

12

2

;

e) SUH]HQWDW�vQ�YDULDQWD�F��úL�vQ�SOXV� ( )� �D Tx

A

x

Ai

i

j

jj

N

ii j

N= −

∑∑

<

2

.

8) &RPSDUkQG� GLVSHUVLD� HVWLPDWRUXOXL� GLQ� FD]XO� VHOHF LHL� FX� SUREDELOLW L� LQHJDOH� 'I cu cel cu SUREDELOLW L�HJDOH�'E�FRQVWDW P�F �UHOD LD�DGHY UDW �GLQ�XUP WRDUHOH�YDULDQWH�HVWH :

a) DI > DE�vQ�RULFH�VLWXD LH�

b) DI < DE i i

⇔ >∑ ∑Nxx

Aii

i

22

;

c) DI < DE A >1

Ni⇔ �GDF � xi2 HVWH�PDUH��DGLF �xi este corelat pozitiv cu Ai);

d) DI < DE A <1

Ni⇔ �GDF � xi2 HVWH�PLF��DGLF �xi este corelat pozitiv cu Ai);

e) DI = DE. 6 � FRQVLGHU P� FD]XO� vQ� FDUH� ILHFDUH� XQLWDWH� VHOHFWDW � vQ� HúDQWLRQ� PRGLILF � FRQGL LLOH�

XUP WRDUHORU� H[WUDJHUL� DOH� XQLW LORU� vQ� HúDQWLRQ��'HFL� Ai1HVWH� GDF � SUREDELOLWDWHD� GH� H[WUDJHUH� D�

primei unit L� vQ� HúDQWLRQ� WUHEXLH� V �FXQRDúWHP�DWXQFL� Ai j( )2 SUREDELOLWDWHD�GH� LHúLUH� vQ� HúDQWLRQ�D�

XQLW LORU�8i�vQ�D�GRXD�H[WUDJHUH�GXS �FH�vQ�SULPD�H[WUDJHUH�D�DS UXW�XQLWDWHD�8j, etc. De exemplu,

GDF �8j�HVWH�VHOHF LRQDW�OD�SULPD�H[WUDJHUH�DWXQFL ( )AA

A

A

Ai j

i

j

i

j( )2

1

1

1

11= =

−≠∑ α

α

. Problema aceasta,

GHVWXO�GH�FRPSOLFDW ��D�IRVW�VWXGLDW �WHKQLF�úL�SUDFWLF�vQ��GH�+RURYLW]�úL�7KRPSVRQ��$VWIHO��DX�IL[DW�GRX �JUXSH�GH�QXPHUH��Si - probabilitatea ca Uj�V �ILJXUH]H�vQ�HúDQWLRQ�úL�Sij - probabilitatea ca Uj� úL� 8j� V � ILJXUH]H� VLPXOWDQ� vQ� HúDQWLRQ�� $FHVWH� SUREDELOLW L� VXQW� QXPHUH� VXEXQLWDUH� úL� vQ� SOXV�UHVSHFW �FRQGL LLOH�XUP WRDUH�

p nii

=∑ úL� pij n n

jii j

= −

≠

∑∑ ( )1 �� ÌQ� DFHVWH� FRQGL LL�� HVWLPDWRUXO� XQXL� 7RWDO� T xii

N=

=∑

1

este

�Tx

pHTi

ii

N=

=∑

1��QXPLW�úL�HVWLPDWRUXO� OXL�+RURYLW]� - Thompson. �3UHFL] UL�SHQWUX�SUREOHPHOH�� úL�

10)). 9) ÌQ�FRQGL LLOH�SUH]HQWDWH�SUHFL]D L�YDORDUHD�GH�DGHY U�D�DILUPD LLORU�XUP WRDUH�

a) �THT este un estimator nedeplasat;

b) �THT este un estimator deplasat cu 1

2THT ;

c) [ ]M T THT HT� = ;

d) [ ]M T T pHT HT i� − = ;

e) [ ]M T T pHT HT ij� − = .

10)�ÌQ�FRQGL LLOH�SUH]HQWDWH�GLVSHUVLD�HVWLPDWRUXOXL� �THT este:

a) [ ]D Tp p

pX

p p p

p pX XHT

i i

ii

N

iij i j

i jjii j

i j�

( )=

−−

−

=≠

∑ ∑∑12

1

2 ;

b) [ ]D Tp p

pX

p p p

p pX XHT

i i

ii

N

iij i j

i jjii j

i j�

( )=

−+

−

=≠

∑ ∑∑12

1

2 ;

c) [ ] ( )D T p p px

p

x

pHT i j ijji

i j

i

i

j

j

� = − −

∑∑

≠

1

2

2

;

d) [ ]D Tp p

pX

p p p

p pX XHT

i i

ii

N

iij i j

i jjii j

i j�

( ):=

− −

=≠

∑ ∑∑12

1

2 ;

e) [ ]D Tp p p

p

x

p

x

pHTi j ij

ijj

n

ii j

ni

i

j

j

� =−

−

∑∑

≠

1

2

2

.

11)�1RW P� FX� σ S2 � úL� FX� σ STRAT

2 � GLVSHUVLD�PHGLLORU� GH� VHOHF LH� SHQWUX� FD]XO� VHOHF LHL� VLPSOH� úL�

UHVSHFWLY�SHQWUX�FD]XO�VHOHF LHL�VWUDWLILFDWH��&RPSDUkQG�FHL�GRL�LQGLFDWRUL��FRQVWDW P�YDORDUHD�GH�DGHY U�D�DILUPD LLORU�XUP WRDUH�

a) σ σSTRAT S2 2< , deoarece media dispersiilor straturilor este mai mare decât dispersia

JHQHUDO �

b) σ σSTRAT S2 2< , deoarece media dispersiilor straWXULORU� vQ� RULFH� VLWXD LH� HVWH�PDL�PLF �

GHFkW�GLVSHUVLD�JHQHUDO ��FX�FRQGL LD�FD�GLVSHUVLD�GLQWUH�VWUDWXUL�V �ILH�QHQXO �

c) σ σSTRAT S2 2= � �� GHRDUHFH� PHGLD� GLVSHUVLLORU� VWUDWXULORU� HVWH� HJDO � FX� GLVSHUVLD� GLQWUH�

straturi;

d) σ σSTRAT S2 2− V �ILH�PDL�PLF �GHFkW�GLVSHUVLD�GLQWUH�VWUDWXUL�

e) FRPSDUDUHD�FHORU�GRX �GLVSHUVLL�HVWH�OLSVLW �GH�VHQV�

12)�6H� FRQVLGHU � F � σ STRAT2 � HVWH� GLVSHUVLD� PHGLHL� GH� VHOHF LH� vQ� FD]XO� VHOHF LHL� VWUDWLILFDWH� SH�

straturile “h” (cu mh ,1= ). SH�QRWHD] �FX�nh volumul stratului h ⇒ ==∑n nhh

m

1

��ÌQ�DFHDVW �

VLWXD LH��GLVSHUVLD�HVWH�PD[LP ��h ��GDF �

a) n n n nh m1 2= = = = =. . . . . . ;

b) n n n nh m1 2≠ ≠ ≠ ≠ ≠... ... ;

c) nN

Nh

h

hh

m=

=∑

1

pentru ( )∀ =h m1, ;

d) n n1 2≠ �úL��n n n nh m3 4= = = = =... ... ;

e) N N N Nh m1 2= = = = =... ... .

TESTUL 12*)

1) Estimatorul unui parametru θ VH�QRWHD] �FX� �θ ��$FHVW�HVWLPDWRU�HVWH�QHGHSODVDW�GDF �

a) GLIHUHQ D� GLQWUH� PHGLD� VD� úL� YDORDUHD� SDUDPHWUXOXL� HVWH� GLIHULW � GH� ]HUR�� GDU� DFHDVW �GLIHUHQ �QX�GHS úHúWH��GLQ�YDORDUHD�SDUDPHWUXOXL�

b) GLVSHUVLD� VD� HVWH� PLQLP � � ( )D θ → 0FkQG� YROXPXO� HúDQWLRQXOXL� WLQGH� F WUH� YROXPXO�

SURGXF LHL�n N→ ); c) GLVSHUVLD�VD�HVWH�PLQLP �SHQWUX�RULFH�YROXP�IL[DW�DO�HúDQWLRQXOXL�

d) mHGLD�VD�HVWH�HJDO �FX�YDORDUHD�SDUDPHWUXOXL�� ( )M θ θ→ );

e) GLVSHUVLD�VD�HVWH�PD[LP �

2) 6 � VH� SUHFL]H]H� FDUH� GLQ� VHULLOH� GH� UHSDUWL LH�� FDUDFWHUL]DWH� SULQ� XUP WRDUHOH� VHWXUL� GH� YDORUL��SUH]LQW �R�DVLPHWULH�SR]LWLY �

a) u.m 40=x ; Me = 40 u.m.; Mo = 40 u.m.;

b) u.m 2500=x ; Me = 3000 u.m.; Mo = 3300 u.m.;

c) x = 151 25, u.m ; Me = 138,75 u.m.; Mo = 112,58 u.m.;

d) u.m 180=x ��0H� ��X�P��0R� ��X�P��úL��X�P��

e) x Me Mo= = = 0

3) Scala de interval:

a) DUH�WRDWH�FDUDFWHULVWLFLOH�VFDOHORU�RUGLQDOH�úL�GH�UDSRUW� b) DUH�WRDWH�FDUDFWHULVWLFLOH�VFDOHL�RUGLQDOH�úL��vQ�SOXV��GLVWDQ D�GLQWUH�GRX �QXPHUH�DOH�VFDOHL�

DUH�R�VHPQLILFD LH�FRQFUHW � c) HVWH�R�VFDO �QXPHULF �úL��vQ�SOXV��UDSRUWXO�GLQWUH�GRX �SXQFWH�DOH�scalei este independent

GH�XQLWDWHD�GH�P VXU � d) SUH]LQW �PXOWH�GLQ�FDUDFWHULVWLFLOH�VFDOHL�RUGLQDOH� e) PDL�HVWH�GHQXPLW �úL�VFDO �GH�UDSRUW�VDX�VFDO �GLVFUHW �

4) 3HQWUX� FRQVWUXLUHD� LQGLFLORU� GH� JUXS� VH� XWLOL]HD] � PDL� PXOWH� VLVWHPH� GH� SRQGHUDUH�� ,QIOXHQ D�sistePXOXL�GH�SRQGHUDUH�XWLOL]DW�SRDWH�IL�SXV�vQ�HYLGHQ �

a) XWLOL]kQG�H[FOXVLY�FRHILFLHQWXO�GH�FRUHOD LH�DO�LQGLFLORU�LQGLYLGXDOL�DL�IDFWRULORU�[�úL�I� b) DQDOL]kQG�FRHILFLHQ LL�GH�YDULD LH�DL�FHORU�GRX �FDWHJRULL�GH�LQGLFL�LQGLYLGXDOL��V

ix ; Vi f );

c) utilizând valorile raportului I I V V rx f x f

i i ix f xi f1 0 1 0

1 0/( )

/( ): = , unde r

i xi f este coeficientul

GH�FRUHOD LH�

d) XWLOL]kQG�UHOD LD� I I r V Vx f x f

i i ixi f x f1 0 1 0

1 0 1/( )

/( ): = + ;

e) XWLOL]kQG�UHOD LD� I I r V Vx f x f

i i ixi f x f1 0 1 0

1 0 1/( )

/( )+ = + .

*)�6XELHFW� OD� H[DPHQXO�GH� OLFHQ � OD�GLVFLSOLQD�³7HRULD� VWDWLVWLFLL´� OD� IDFXOWDWHD�GH�&LEHUQDWLF ��6WDWLVWLF � úL�,QIRUPDWLF �(FRQRPLF �-�VSHFLDOL]DUHD�6WDWLVWLF ��6HVLXQHD�0DL��

5) 6H�FRQVLGHU �XQ�WDEHO�GH�FRQWLQJHQ ��FRUHOD LH��vQ�IRUPD�VD�JHQHUDO �

( )[ ]{ }x y n j J k Kj k jk, ; ,∈ ∈ �� 'DF � SH� ED]D� WDEHOXOXL� FRQVLGHUDW� FDOFXO P� YDORULOH� FHQWUDWH� úL�

reduse Ux x

jj

x=

−σ

�úL�Vy y

kk

y=

−σ

( cu x úL� y mediile variabLOHORU�;�úL�<��LDU�σ x úL�σ y abaterile

VWDQGDUG�DOH�YDULDELOHORU�;�úL�<��DWXQFL�FRYDULDQ D�GLQWUH�YDULDELOHOH�;�úL�<��QRWDW �FX�&RY�;�<��este:

a) Cov X Y Cov U Vx y( , ) ( , )≠ σ σ ;

b) Cov X YCov U V

x y( , )

( , )=σ σ

;

c) Cov X Y Cov U Vx y( , ) ( , )= σ σ ;

d) Cov X Y Cov U V x y( , ) ( , )− = σ σ ;

e) Cov X Y Cov U VU V( , ) ( , )= σ σ . 6) 5HIHULWRU�OD�DFWLYLWDWHD�HFRQRPLF �D�XQHL�VRFLHW L�FRPHUFLDOH�VH�FXQRVF�XUP WRDUHOH�GDWH�

9DORDUHD�SURGXF LHL� �PLO��OHL�SUH XUL�FXUHQWH�

Unitatea

Perioada ED]

Perioada FXUHQW

Modificarea valorii SH�VHDPD�PRGLILF ULL SUH XULORU (mil. lei)

A 1200 1850 350 B 875 1600 275

0RGLILFDUHD�UHODWLY �D�YDORULL�SURGXF LHL�OD�QLYHOXO�vQWUHJLL�VRFLHW L��FDX]DW �GH�PRGLILFDUHD�YROXPXOXL�IL]LF�DO�SURGXF LHL�vQWUH�FHOH�GRX �SHULRDGH�D�IRVW�HJDO �FX�

a) -12,5%; b) +100,0%; c) +36,1%; d) +136,1%; e) -112,5%.

7) ÌQ�FD]XO�XQHL�REVHUY UL�VWDWLVWLFH�DO�F UXL�VFRS�HVWH�³IRWRJUDILHUHD´�]LOHL�GH�OXFUX�D�XQXL�OXFU WRU�

a) WLPSXO�REVHUY ULL�FRLQFLGH�FX�WLPSXO�OD�FDUH�VH�vQUHJLVWUHD] �GDWHOH� b) WLPSXO�REVHUY ULL�QX�FRLQFLGH�FX�WLPSXO�OD�FDUH�VH�vQUHJLVWUHD] �GDWHOH� c) WLPSXO�REVHUY ULL�VH�UHIHU �OD�GXUDWD�V SW PkQLL�GH�OXFUX�D�OXFU WRUXOXL� d) SUHFL]DUHD�WLPSXOXL�REVHUY ULL�QX�DUH�VHQV� e) WLPSXO�REVHUY ULL�HVWH�PRPHQWXO�FULWLF�³RUD�]HUR´�D�]LOHL�vQ�FDUH�OXFU WRUXO�VH�SUH]LQW �OD�

lucru. 8) $QDOL]D L�XUP WRDUHOH�SRSXOD LL�VWDWLVWLFH��SRSXOD LD�%XFXUHúWLXOXL�OD��LXOLH��SRSXOD LD�

GLQ�5RPkQLD�GLQ�PHGLXO�XUEDQ�OD��LXOLH��PXO LPHD�IDFWXULORU�HODERUDWH�GH�6�&��³,&6´�65/�vQ�WULP��,��PXO LPHD�F V WRULLORU�vQFKHLDWH�vQ�5RPkQLD�vQ�anul 1997; 5) stocurile de P UIXUL�DO�vQWUHSULQGHULORU�PLFL�úL�PLMORFLL�OD�VIkUúLWXO�DQXOXL��

3RSXOD LL�VWDWLVWLFH�GLQDPLFH�VXQW�

a) ��úL�� b) ��úL�� c) ��úL�� d) ��úL�� e) ��úL��

9) 6H� úWLH� F � ( ) ( )σ σy ii

y x i iir

y yn

y Y2 2 2 21 1= − = −∑ ∑; .� úL� ( )σ Y i

inY y2 21= −∑ � unde

�Y b b xi i= +0 1 ��'DF �U�HVWH�FRHILFLHQWXO�GH�FRUHOD LH�GLQWUH�YDULDELOHOH�<�úL�;��DWXQFL�

a) ( )σ σy x y r.2 2 21= − �úL�σ σ σy y x Y

2 2 2= +. ;

b) ( )σ σy x y r.2 2 21= − �úL�σ σ σy y x Y

2 2 2< +. ;

c) ( )σ σy x y r.2 2 21> − �úL�σ σ σy y x Y

2 2 2= +. ;

d) ( )σ σy x y r.2 2 21> − �úL�σ σ σy y x Y

2 2 2< +. ;

e) ( )σ σy x y r.2 2 21< − �úL�σ σ σy y x Y

2 2 2> +. .

10) 2�YDULDELO � FRPSOH[ �<� HVWH� H[SULPDW � vQ� IXQF LH� GH� IDFWRULL� [�� ]�� X�� Y�� J�� K��Z�� S�� DOH� F URU�

LQIOXHQ H�WUHEXLH�L]RODWH��5HOD LD�RELHFWLY �GLQWUH�<�úL�IDFWRULL�V L�ILLQG�PXOWLSOLFDWLY ��SHUPLWH�R�descompunere în trepte de forma:

cu y x z u v z u v g p u v g w h= ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅

ÌQWUH� LQGLFHOH� YDULDELOHL� <� úL� LQGLFLL� YDULDELOHORU� IDFWRULDOH�� GDF � VH� XWLOL]HD] � WHKQLFD�VXEVWLWXLULL�vQO Q XLWH��QX�H[LVW �UHOD LD�

a) I I Iy y x y z= ⋅( ) ( ) ;

b) I I I Iy y z y u y v= ⋅ ⋅( ) ( ) ( ) ;

c) I I I I I Iy y u y v y g y w y h= ⋅ ⋅ ⋅ ⋅( ) ( ) ( ) ( ) ( ) ;

d) I I I I Iy y x y g y w y h= ⋅ ⋅ ⋅( ) ( ) ( ) ( ) ;

e) I I I I I Iy y u y v y g y w y h− ⋅ ⋅ ⋅ ⋅ ≠( ) ( ) ( ) ( ) ( ) 0 .

Y = x Â��]

p g v u

w h

11) (YROX LD�3URGXVXOXL�,QWHUQ�%UXW�SH�ORFXLWRU�vQ�SHULRDGD��-��VH�SUH]LQW �DVWIHO�

Anii 1994 1995 1996 1997 ��ID �GH�DQXO�SUHFHGHQW 14,0 +7,3 +4,4 -6,3

0HGLD�DQXDO �D�HYROX LHL�SURFHQWXDOH�HVWH�GH�

a) -0,09%; b) 102,0%; c) 201,3%; d) +2,21%; e) +4,35%.

12) 1RW P� FX� σ R2 � úL� FX� σ F

2 � GLVSHUVLLOH� PHGLLORU� GH� VHOHF LH� SHQWUX� SURFHGHHOH� GH� IRUPDUH� D�

HúDQWLRDQHORU� vQ� YDULDQWHOH� FX� UHYHQLUH� úL� I U � UHYHQLUH�� &RPSDUkQG� FHL� GRL� LQGLFDWRUL�� UHOD LD�FRUHFW �HVWH�

a) σ σR F2 2> ;

b) σ σF R2 2> ;

c) σ σR F2 2= ;

d) σ σF R2 21

2= ;

e) σ σF R2 2= �YROXPXO�HúDQWLRQXOXL�

13) Inegalitatea dintre aEDWHULOH� VWDQGDUG� DOH� XQHL� YDULDELOH� ;� QRUPDO� UHSDUWL]DWH� vQ� SRSXOD LLOH�

VWDWLVWLFH� �� GH� DFHODúL� YROXP� úL� DFHHHDúL� PHGLH� x , σ σ σ1 2 3> > � HVWH� YL]XDOL]DW � vQ�varianta:

a) b) c) d)

3

2

1

3

1

2

1

2

3

2

3

1

e)

14) 6H�FXQRDúWH�IDSWXO�F �6�&��³,&6´�65/�D�UHDOL]DW�vQ�OXQD�,$1�¶��R�FLIU �GH�DIDFHUL�GH��POG��lei, iar în perioada IAN ’97 - APR ’98 a cunoscut o modificare�PHGLH�OXQDU �GH��POG��OHL��ÌQ�ipoteza în care în perioada MAI ’98 -�,81�¶��vQUHJLVWUHD] �DFHHDúL�HYROX LH�FD�DFHHD�GLQ�,$1�’97 -�$35�¶��YDORULOH�HVWLPDWH�DOH�FLIUHL�GH�DIDFHUL�vQ�OXQLOH�PDL��LXQLH�úL�LXOLH�VXQW�

a) ��POG��OHL��POG��OHL�úL�� mld. lei; b) ��POG��OHL��POG��OHL�úL��POG��OHL� c) ��POG��OHL��POG��OHL�úL��POG��OHL� d) ��POG��OHL��POG��OHL�úL��POG��OHL� e) ��POG��OHL��POG��OHL�úL��POG��OHL�

15) 'DF � vQWUH� GRX � YDULDELOH� <� úL� ;� QX� H[LVW � QLFL� R� OHJ WXU � �YDULDELlele sunt independente) FRHILFLHQWXO�GH�FRUHOD LH�HVWH�HJDO�FX�]HUR��5HFLSURFD�DFHVWHL�DILUPD LL�

a) HVWH�IDOV � b) HVWH�DGHY UDW � c) HVWH�DGHY UDW �GRDU�GDF �GLVSHUVLLOH�FHORU�GRX �YDULDELOH�VXQW�HJDOH� d) HVWH�DGHY UDW �GRDU�GDF �FHOH�GRX �GLVWULEX LL�IRUPDOH�GXS �;�úL�<�VXQW�QRUPDOH� e) HVWH� DGHY UDW � GRDU� GDF � GDWHOH� FHORU� GRX � YDULDELOH� VXQW� VLVWHPDWL]DWH� vQWU-un tabel de

FRQWLQJHQ ��FRUHOD LH�� 16) La un punct comercial din vânzarea unui produs s-a realizat în luna APR ’98 o încasare de 8

mil. lei. Ca urmare a major ULL�SUH XOXL� ID �GH� OXQD�$35�¶�� vQFDV ULOH�GLQ� OXQD�$35�¶��DX�FXQRVFXW�R�FUHúWHUH�GH��OHL��&DUH�DU�IL�IRVW�vQFDV ULOH�OXQLL�$35�¶��GDF �SUH XO�QX�V-ar fi modificat?

a) 7,7 mil. lei; b) 7,3 mil. lei; c) 5 mil. lei; d) 6 mil. lei; e) 7 mil. lei.

17) ÌQ�HYROX LD�VD�vQGHOXQJDW �VWDWLVWLFD�V-a dezvoltat continuu atât sub aspectul obiectului de studiu

GDU�PDL� DOHV�DO� SHUIHF LRQ ULL� úL� GLYHUVLILF ULL�PHWRGHORU�GH�DERUGDUH�D� DFHVWXL� RELHFW�� ÌQ� WRDWH�PRPHQWHOH�GH]YROW ULL�VDOH�VWDWLVWLFD�VH�RFXS �GH�

a) fenomene care se proGXF�OD�R�DQXPLW �XQLWDWH�GH�REVHUYDUH� b) IHQRPHQH�FDUH�VH�UHSURGXF�vQ�PRG�LGHQWLF�OD�XQLW LOH�GH�REVHUYDUH� c) fenomene care se produc într-XQ� QXP U�PDUH� GH� FD]XUL� úL� DOH� F URU� IRUPH� LQGLYLGXDOH�

PDQLIHVW �DQXPLWH�UHJXODULW L� d) fenomene exclusiv economice; e) fenoPHQH�FDUH�DX�OD�RULJLQHD�SURGXFHULL�ORU�XQ�VLQJXU�IDFWRU�GH�LQIOXHQ �

2

1

3

18) $QDOL]D L� XUP WRDUHOH� QR LXQL�� $XWRQRPLH�� FRQILGHQ LDOLWDWH�� WUDQVSDUHQ �� VSHFLDOL]DUH�� SURSRU LRQDOLWDWH�� GHRQWRORJLH� SURIHVLRQDO �� LQGHSHQGHQ � SROLWLF �� elDERUDUHD�GHFL]LLORU��DUJXPHQWDUHD�DF LXQLORU�GXS �GHFODQúDUHD�ORU��SUHOXFUDUHD�GDWHORU��11) observare.

$FWLYLWDWHD�VWDWLVWLFLL�SXEOLFH�WUHEXLH�V �VH�GHVI úRDUH�UHVSHFWkQG�XUP WRDUHOH�SULQFLSLL�

a) ��úL�� b) ��úL�� c) 3, ��úL�� d) ��úL�� e) ��úL��

19) 6WDELOL L� YDULDQWD� GH� DGHY U� D� DILUPD LHL� XUP WRDUH�� 5HFHQV PkQWXO� FD� PHWRG � GH� REVHUYDUH�

VWDWLVWLF �

a) QX�SUHVXSXQH�FXOHJHUHD�GDWHORU�GH�OD�WRDWH�XQLW LOH�SRSXOD LHL�VWDWLVWLFH�ELQH�GHOLPLWDW � b) are exclusiv un caracter demografic; c) VH�vQFDGUHD] �vQ�VIHUD�REVHUY ULORU�FX�FDUDFWHU�SHUPDQHQW� d) VH�RUJDQL]HD] �FX�R�DQXPLW �SHULRGLFLWDWH� e) VH� RUJDQL]HD] � RUL� GH� FkWH� RUL� 6WDWLVWLFD� 218� GHFODQúHD] � DF LXQHD� GH� HIHFWXDUH� D�

UHFHQV PLQWHORU�vQ� ULOH�membre. 20) *UDILFHOH�VWDWLVWLFH�WUDVDWH�vQ�FRRUGRQDWH�SRODUH�VH�XWLOL]HD] �vQ�PRG�FXUHQW�SHQWUX�YL]XDOL]DUHD�

a) RULF UHL�VHULL�GH�GDWH�VWDWLVWLFH� b) VHULLORU�GH�UHSDUWL LH� c) WUHQGXOXL��WHQGLQ HL��GLQ�HYROX LD�vQ�WLPS�D�XQXL�IHQRPHQ� d) HYROX LD�vQ�WLPS�D�XQXL�IHQRPHQ�DIHFWDW �GH�RVFLOD LL�VH]RQLHUH� e) WHQGLQ HL�OHJ WXULL�GLQWUH�YDULDELOH�úL�SHQWUX�DOHJHUHD�PRGHOHORU�GH�UHJUHVLH�

21) 6H� FXQRDúWH� IDSWXO� F � vQWU-R� UDPXU � GH� DFWLYLWDWH� YDULDELOD� ³VDODULXO� PHGLX� EUXW´� �<�� HVWH�

GHSHQGHQW �SULQWUH�DOWHOH�GH�³JUDGXO�PHGLX�DO� vQGHSOLQLULL�QRUPHORU´��;��úL�³JUDGXO�PHGLX�GH�vQ]HVWUDUH� WHKQLF �D�PXQFLL´� �;��&RQWULEX LD�FHORU�GRL� IDFWRUL�;��úL�;�� OD�YDULD LD�VDODULXOXL�PHGLX�EUXW��\��HVWH�HVWLPDW �SULQ�

a) R r ry x x y x y x/ / /1 2 1 2

2 2 2= + ;

b) Rr r r r r

ry x x

y x y x y x y x x x

x x/

/ / / /

1 2

1 2 1 2 1 2

1 2

2 2

2

2

1=

+ −

−;

c) R r r r r ry x x y x y x y x y x x x/ / / / /1 2 1 2 1 2 1 22 2 2= + − ;

d) rx x1 20= ;

e) Rr r

ry x x

y x y x

x x/

/ /

1 2

1 2

1 2

2 2

21=

+

−.

127 ��r -�FRHILFLHQWXO�GH�FRUHOD LH�OLQLDU �

22) Prin AUDIT-ul financiar efectuat în luna ianuarie 1998 la S.C. “ICS” SRL s-D�FRQVWDWDW�F � vQ�IDFWXULOH�HODERUDWH� vQ��H[LVW �DQXPLWH�HURUL�vQ�FDOFXOXO�79$-ului�FDUH�DX�DYXW�GUHSW�VXUV ��QHFXQRDúWHUHD� PHWRGRORJLHL� GH� FDOFXO� D� 79$-XOXL� OD� XQHOH� SURGXVH�� DSUR[LP UL� HURQDWH��GHIHF LXQL�DOH�PLMORFXOXL�WHKQLF�GH�FDOFXO��HWF�

3RSXOD LD�VWDWLVWLF �VWXGLDW �HVWH�

a) ansamblul facturilor elaborate de S.C. “ICS” SRL; b) ansambOXO�SURGXVHORU�úL�VHUYLFLLORU�UHDOL]DWH�GH�6�&��³,&6´�65/�SkQ �OD��;,,�� c) DQVDPEOXO� SURGXVHORU� úL� VHUYLFLLORU� UHDOL]DWH� GH� 6�&�� ³,&6´� 65/� GHVI FXWH� SH� SLD � úL�

IDFWXUDWH�SkQ �OD��;,,�� d) DQVDPEOXO�SURGXVHORU�úL�VHUYLFLLORU�UHDOL]DWH�SHQWUX�D�IL�GHVI FXWH�SH�SLD �GH�6�&��³,&6´�

65/��IDFWXUDWH�SkQ �OD��;,,��úL�OD�FDUH�VH�SHUFHSH�79$� e) DQVDPEOXO�IDFWXULORU�HODERUDWH�GH�6�&��³,&6´�65/�SkQ �OD��;,,��SHQWUX�SURGXVHOH�

úL�VHUYLFLLOH�UHDOL]DWH�úL�GHVI FXWH�SH�SLD �úL�OD�FDUH�VH�SHUFHSH�79$� 23) G.U�<XOH� �� SUHFL]HD] � F � LQGLFDWRULL� UHSDUWL LLORU� VWDWLVWLFH� WUHEXLH� V � vQGHSOLQHDVF �

XUP WRDUHOH�FRQGL LL��V �ILH�GHILQL L�vQ�PRG�RELHFWLY��V �GHSLQG �GH�WRDWH�GDWHOH�GLQ�VHUL��V �DLE �VHPQLILFD LL�FRQFUHWH��V �ILH�VLPSOX�GH�FDOFXODW��V �ILH�SX LQ�VHQVLELOL�OD�IOXFWXD LLOH�GH�VHOHF LH��V �VH�SUHWH]H�XúRU�OD�FDOFXOH�DOJHEULFH�

$FHVWH�FRQGL LL��vQ�WRWDOLWDWH�

a) sunt respectate de medie; b) sunt respectate de dispersie; c) VXQW�UHVSHFWDWH�GH�FRYDULDQ � d) VXQW�UHVSHFWDWH�GH�YDORDUHD�PRGDO � e) nu sunt rHVSHFWDWH�GH�QLFL�XQ�LQGLFDWRU�DO�WHQGLQ HL�FHQWUDOH�VDX�DO�YDULD LHL�

24) /X P�vQ�FRQVLGHUDUH�XQ�WDEHO�GH�FRQWLQJHQ ��FRUHOD LH�� ( )[ ]{ }KkJjnyx jkkj ∈∈ ,,, úL�GLVWULEX LLOH�

OXL�<�FRQGL LRQDWH�GH�;�� ( ){ }Kk , ,, ∈fixatjny jkk ��3HQWUX�DFHVWHD�GLQ�XUP �VH�FXQRVF�PHGLLOH�úL�

dispersiile:

( ) ∑=k

jkkj

j nyn

xy.

1úL� ( ) ( )[ ]σ y j

jk j jk

k

xn

y y x n2 21= −∑.

unde ∑=k

jkj nn . � HVWH� HIHFWLYXO� �IUHFYHQ D�� PDUJLQDO� �� DVRFLDW � YDULDQWHL� [M� D� YDULDELOHL� ;�� ÌQ�

IXQF LH� GH� HOHPHQWHOH� SUH]HQWDWH� vQ� LSRWH] � GLVSHUVLD� YDULDELOHL� <� �σ y2 �� VH� SUH]LQW � FRUHFW� vQ�

varianta:

a) ( ) ( )[ ]σ σy jj

y j jj

jnn x

nn y x y2 2 21 1= + −∑ ∑. . ;

b) ( )[ ]σ y jj

jnn y x y2 21< −∑ . ;

c) ( )σ σy jj

y jnn x2 21< ∑ . ;

d) ( ) ( )[ ]σ σy jj

y j jj

jnn x

nn y x y2 2 21 1= − −∑ ∑. . ;

e) ( ) ( )[ ]σ σy jj

y j jj

jnn x

nn y x y2 2 21 1> + −∑ ∑. . .

FX�FRYDULDQ D�GLQWUH�<�úL�FHLODO L�IDFWRUL�GH�LQIOXHQ �FX�H[FHS LD�IDFWRUXOXL�;� 25) 3HQWUX� GRX � YDULDELOH� VWDWLVWLFH� D� F URU� WHQGLQ � D� GHSHQGHQ HL� VWDWLVWLFH� D� IRVW�PRGHODW � SULQ�

HFXD LD� GH� UHJUHVLH� � ,y xi i= −25 0 7 � úL� VH� PDL� FXQRVF� GLVSHUVLLOH�� σ x2 2603 04= , úL�

σ y2 1896 6= , ��&RHILFLHQWXO�GH�FRUHOD LH�DUH�YDOoarea:

a) 0,82; b) 0,96; c) -0,96; d) -0,59; e) -0,82.

26) 'DF � vQWUH� FXDUWLOHOH� FDOFXODWH� SHQWUX� R� VHULH� GH� UHSDUWL LH� H[LVW � UHOD LD� ( )Q Q Q2 1 3 2= + /

DWXQFL�UHSDUWL LD�HVWH�

a) DVLPHWULF �VSUH�YDORULOH�PDUL� b) DVLPHWULF �VSUH�YDORULOH�PLFL� c) QRUPDO � d) KLSHUEROLF � e) bidimHQVLRQDO �

27) 9HULILFDUHD�FRQFRUGDQ HL�GLQWUH�UHSDUWL LD�HPSLULF �RE LQXW �vQWU-R�FHUFHWDUH�FRQFUHW �úL�UHSDUWL LD�

WHRUHWLF � SUHVXSXV � VH� HIHFWXHD] � FX� DMXWRUXO� PDL� PXOWRU� WHVWH�� ([SUHVLD�( )n np

npi i

ii

k −

=∑

2

1

UHSUH]LQW �VWDWLVWLFD�WHVWXOXL�

a) Shapiro-Wilk; b) Lilliefors;

c) Hi-S WUDW�� χ 2 );

d) Kolmogorov - Smirnov; e) Gnedenko.

28) 2�JUXS �GH��GH�VWXGHQ L�VXV LQH�OD�GRX �GLVFLSOLQH�FkWH�XQ�WHVW�SHQWUX�YHULILFDUHD�FXQRúWLQ HORU��

7HVWHOH�DX�SXQFWDMH�GLIHULWH��LDU�SH�ED]D�ORU�VH�FXQRDúWH�F �

- la testul A: 900025

1

2 =∑=i

iAx 45025

1

=∑=i

iAx

- la testul B: 42525

1

2 =∑=i

iBx 10025

1

=∑=i

iBx

*UXSD�GH�VWXGHQ L�HVWH�PDL�RPRJHQ �GLQ�SXQFWXO�GH�YHGHUH�DO�FXQRúWLQ HORU�DFXPXODWH� a) la disciplina A; b) la disciplina B; c) la ambele disciplLQH�H[LVW �DFHODúL�JUDG�GH�RPRJHQLWDWH�

d) QX�VXQW�VXILFLHQWH�GDWH�SHQWUX�D�IL�VWXGLDW �RPRJHQLWDWHD� e) QX� VH� SRW� FRPSDUD� RPRJHQLW LOH� FXQRúWLQ HORU� OD� FHOH� GRX � GLVFLSOLQH�� GHRDUHFH�

punctajele testelor au fost diferite.

29) (YROX LD�QXP UXOXL�GH�ERYLQH�úL�YROXPXO�SURGXF LHL�GH�FDUQH�DX�HYROXDW�DVWIHO�

Anii 1990 1991 1992 1993 1994 1995 1996 1997 Nr. bovine (mii) 5381 4355 3683 3597 3481 3496 3435 3431

Prod. carne (mii tone) 2232 2023 1895 1935 1852 1846 1868 1871

'HSHQGHQ D�GLQWUH�FHOH�GRX �YDULDELOH�VH�P VRDU �FX�DMXWRUXO�FRHILFLHQWXOXL�6SHDUPDQ��DOH�F UXL�YDORUL�VXQW�

a) 0,738; b) 0,901; c) -0,675; d) 0,991; e) 0,307.

30) Media ( x �� XQXL� HúDQWLRQ� GH� PDUH� YROXP� �Q�� SRDWH� IL� FRQVLGHUDW � F � XUPHD] � R� GLVWULEX LH�

QRUPDO �GH�PHGLH� x 0 úL�DEDWHUH�PHGLH�S WUDWLF ��HURDUH�PHGLH��σn

în cazul unui sondaj simplu

cu revenire. Pentru o probabilitate 1− α intervalul de estimare (de încredere) a parametrului

x 0 este x x zn

− ≤0 ασ

��3HQWUX�FD�SUHFL]LD�HVWLPD LHL�V �ILH�PDL�PLF �FX�k% din valoarea lui

x 0 �YROXPXO�HúDQWLRQXOXL�� ( )k ∈ 0 1; ) este:

a) nz

k x≤ ⋅α σ2

2

2

02

;

b) ( )nz

kCV≥ α

2

22 �XQGH�&9�HVWH�FRHILFLHQWXO�GH�YDULD LH�

c) nk x

= σ 2

20

;

d) nz k

x= α σ2 2 2

0;

e) Q!��XQLW L�vQ�RULFH�VLWXD LH�

TESTUL 13

��5HFHQV PkQWXO�SRSXOD LHL�HVWH� D��R�REVHUYDUH�WRWDO �XQLF � E��R�REVHUYDUH�WRWDO ��LQWHJUDO ��SHULRGLF � F��R�REVHUYDUH�FXUHQW �SHULRGLF � d) o observare curHQW �GH�R�VLQJXU �GDW � H��R�REVHUYDUH�SHULRGLF �SDU LDO � 2) (URULOH�GH�UHSUH]HQWDWLYLWDWH�VXQW�VSHFLILFH�FHUFHW ULORU� a) prin sistemul rapoartelor statistice; b) prin sondaj; F��SULQ�UHFHQV PkQW� G��SULQ�REVHUYDUHD�S U LL�SULQFLSDOH� e) prin monografii. 3) 3URSULHWDWHD�GHWHUPLQDQW �D�YDULDELOHL�[�vQ�ED]D�F UHLD�VH�GHWHUPLQ �PHGLD�DULWPHWLF �HVWH�

)x ,... ,x ,x( f = )x ,... ,x ,x( f a) 2n

22

21

222

;

)x ,... ,x ,x( f = )x ,... ,x ,x( f b) n21 ;

)x ,... ,x ,x( f = )x ,... ,x ,x( f c) nn

221 ;

)x ,... ,x ,x( f = )x ,... ,x ,x( f d) n21mn

m2

m1 ;

)x ,... ,x ,x( f = )x ,... ,x ,x( f e) n21n21 . 4) 6H�FXQRVF�GDWH�FRQYHQ LRQale referitoare la valorile unei caracteristici:

Grupe )UHFYHQ H

9,0 - 9,5 3

9,5 - 10,0 1

10,0 - 10,5 42

10,5 - 11,0 23

11,0 - 11,5 9

11,5 - 12,0 1

12,0 - 12,5 1

Total n = 80

&XQRVFkQG� F � PHGLD� DULWPHWLF � D� FDUDFWHULVWLFLL� [� HVWH� � � �� XQLW L�� FRHILFLHQWXO� GH�RPRJHQLWDWH��YDULD LH��HVWH�GH�

a) 4,62 %;

b) 13,32 %; c) 22,19 %;

d) 23,32 %; e) 14,62 %.

5) 6WDELOLUHD�LQGLFHOXL�SUH XULORU��EXQXULORU��GH�FRQVXP�VH�IDFH�GXS �UHOD LD�

a) p q

p q

01

11

∑∑

;

b) )q + q( p

)q + q( p

010

011

∑∑

;

c) q p

q p

q p

q p

10

11

00

01

∑∑

⋅∑∑

;

d) p q

p q i

00

00p

∑∑

;

e) p q

i

1

p q

11p

11

∑

∑

.

6) )DFWRUXO�GH�FRUHF LH�LQWURGXV�vQ�FDOFXOXO�HURULL��DEDWHULL��PHGLHL�GH�VRQGDM��vQ�YDULDQWD�SUHOHY ULL�QHUHSHWDWH�D�XQLW LORU��HVWH�HJDO�FX�

a)

2

nσ

;

b) 1 - (n / N) ;

c)

1

N - 1 ;

d)

1

N ;

e)

1

n/

1

N

.

7) &XQRVFkQG�UHSDUWL LD�YDULDELOHL�[�

Grupe )UHFYHQ H 1,1 - 1,3 6 1,3 - 1,5 9 1,5 - 1,7 4 1,7 - 1,9 1

Total ni =∑ 20

valoarea quantilei doi (Q2��HVWH�HJDO �FX� a) 1,375; b) 1,51; c) 1,40;

d) 1,45; e) 1,55.

8) 6H�FXQRVF�XUP WRDUHOH�GDWH��FRQYHQ LRQDOH��DVXSUD�UHSDUWL LHL�XQHL�FDUDFWHULVWLFL�

Grupe )UHFYHQ H 11,0 - 13,0 12 13,0 - 15,0 18 15,0 - 17,0 8 17,0 - 19,0 2

Total n = 40 'DWH� ILLQG��YDORDUHD�PRGDO �0R��[mo) = 13,75; valoarea medie = 14,0, coeficientul de asimetrie (Pearson) are valoarea:

a) 0,377; b) 0,677; c) 1,377; d) 0,963; e) 0,09.

9) ÌQ� OHJ WXU � FX� H[HUFL LXO� DQWHULRU�� VH� FHUH� V � VH� GHWHUPLQH� GLVSHUVLD� FDUDFWHULVWLFLL� DOWHUQDWLYH��ELQDUH��QXP U�XQLW L�FX�YDORUL�PDL�PDUL�FD��

'LVSHUVLD�DFHVWHL�FDUDFWHULVWLFL�HVWH�HJDO �FX� a) 0,1875; b) 0,2875; c) 0,0912; d) 1,1725; e) 0,5000. 10) Se cunosc�GDWH��FRQYHQ LRQDOH��DVXSUD�YROXPXOXL�YkQ] ULORU�vQ�OXQD�PDUWLH��OD�6�&��*,*,�

,03(;�V�U�O��SHQWUX�WUHL�SURGXVH�GLQ�QRPHQFODWRU��FkW�úL�PRGLILFDUHD�SUH XULORU�

Produsul Valoarea vânz rilor în SHULRDGD�FXUHQW ��PLL�OHL�

Modificarea procenWXDO �D�SUH XULORU��

1 13.860 + 5,0 2 51.300 - 10,0 3 57.200 + 10,0

TOTAL Σ= 122.360 X 0RGLILFDUHD�DEVROXW �D�YROXPXOXL�YkQ] ULORU��FDX]DW �GH�PRGLILFDUHD�SUH XULORU� OD�FHOH� WUHL�SURGXVH��D�IRVW�HJDO �FX� a) 0 (zero) mii lei; b) - 80,0 mii lei; c) + 160,0 mii lei; d) + 180,0 mii lei; e) - 102,0 mii lei. 11) &XQRVFkQG� UH]XOWDWHOH� FHUFHW ULL� FDUDFWHULVWLFLL� ;� GLQWU-XQ� VRQGDM� Q� � �� XQLW L�� SUHOHYDWH�

DOHDWRU�úL�QHUHSHWDW�GLQWU-R�SRSXOD LH�1� ��XQLW L�úL�DQXPH = 17,45 u.m., respectiv S2 = 0,61 u.m., pentru o probabilitate P = 95 % (Za� ��LQWHUYDOXO�GH�vQFUHGHUH�SHQWUX�PHGLD�SRSXOD LHL�JHQHUDOH�HVWH�HJDO�FX��YDORUL�URWXQMLWH�FX�GRX �]HFLPDOH��

a) (17,01 ; 17,93); b) (16,25 ; 19,73); c) (10,63 ; 22,75); d) (14,29 ; 21,33); e) (17,29 ; 17,61). 12) 6H� FXQRVF� XUP WRDUHOH� GDWH� � �FRQYHQ LRQDOH�� UHIHULWRDUH� OD� YDORDUHD� YkQ] ULORU� SHQWUX� GRX �

SURGXVH��vQ�SHULRDGD�GH�ED] ��UHVSHFWLY�LQGLFLL�SUH XULORU�úL�PRGLILFDUHD�SURFHQWXDO �D�YROXPXOXL�fizic.

Produsul Valoarea vânz ULORU�vQ�SHULRDGD�GH�ED] ��POG��OHL�

,QGLFHOH�SUH XULORU (%)

0RGLILFDUHD�SURFHQWXDO �D�volumului fizic

(%)

1 10 130 0

2 12 100 0

Total S = 22 X X

0RGLILFDUHD� UHODWLY � D� YROXPXOXL� YDORULF� DO� YkQ] ULORU� OD� QLYHOXO� DQVDPEOXOXL� FHORU� GRX �produse este eJDO �FX� a) + 100 %; b) + 13,64 %; c) 0 %; d) - 25,0 %;

e) + 117,5 %.

13) ÌQ� FDGUXO� DQDOL]HL� GLVSHUVLRQDOH� ELIDFWRULDOH� XWLOL]DW � SHQWUX� D� WHVWD� VHPQLILFD LD� IDFWRULORU� GH�

LQIOXHQ �� GDF � � YDORULOH� � IDFWRUXOXL� $� �i = 1,r ), respectiv B ( j = 1,p �� DWXQFL� QXP UXO�gradelor de libertate al dispersiei reziduale este:

a) rp - 1; b) r(r - 1); c) (r - 1)(p - 1); d) r(p - 1)p; e) r2(p - 1).

14) În cadrul analizei dispersionale unifactoriale, i = 1,r �QXP UXO�JUXSHORU��j = 1,ni -�QXP UXO�

XQLW LORU� vQ� JUXS �� VXPD�S WUDWHORU� DEDWHULORU� SHQWUX�FDOFXOXO� GLVSHUVLHL� UH]LGXDOH� �vQ� LQWHULRUXO�JUXSHL��VH�VWDELOHúWH�FX�UHOD LD�

a)

n

y

- y i

ijj

2

i

2ij

ij

∑

∑∑;

b)

c) n

y

- n

y

ij

ji

2

i

ijj

2

i

∑∑∑

∑;

d) y -

n

y 2ij

iji

ijji

2

∑∑∑∑

;

e) n

y

- y ij

ji

2

2ij

ij

∑∑

∑;

n

y

- n

y

- y i

ijj

2

i

ijji

2

2ij

ij

∑

∑∑∑

∑.

15) ÌQWUH�GRX �YDULDELOH�OHJDWH��XQD�UH]XOWDWLY �\�úL�DOWD�IDFWRULDO �[��SHQWUX�Q� ��SHUHFKL�YDORUL�GH�

REVHUYD LH�� VH� FXQRVF� DJUHJDWHOH� GH� FDOFXO�� 1

100

i x = 2105∑; 1

100

i2 x = 49.665∑

;

1

100

i y = 4160∑; 1

100

i i x y = 96.640∑.

ÌQ�LSRWH]D�XQHL�OHJ WXUL�OLQLDUH�\x� �D��E[��HFXD LD�HVWH� a) yx = 0,963 + 12,735 x; b) yx = 1,259 + 0,799 x; c) yx = 22,933 + 12,811 x; d) yx = 5,937 + 1,694 x; e) yx = 7,339 + 2,437 x. 16) %XQXULOH� GH� IRORVLQ � vQGHOXQJDW � úL� FX� GXUDW �PHGLH� GH� IRORVLQ � LQFOXVH� vQ� DYX LD� QD LRQDO �

DFXPXODW �FXSULQG� a) Bunurile destinate s �VDWLVIDF �SH�R�GXUDW �GH�PD[LP�XQ�DQ��GLIHULWH�FHULQ H�PDWHULDOH�úL�

VSLULWXDOH�DOH�SRSXOD LHL��FD�GH�H[HPSOX��DXWRWXULVPH��WHOHYL]RDUH��DSDUDWH�GH�UDGLR��IULgidere, PRELO �HWF��

b) Bunurile destinate s � VDWLVIDF � SH� R� GXUDW � GH� SHVWH� XQ� DQ�� GLIHULWH� FHULQ H�PDWHULDOH� úL�VSLULWXDOH� DOH� SRSXOD LHL�� FD� GH� H[HPSOX�� IRQGXO� IXQFLDU�� FO GLULOH� GH� ORFXLW�� DXWRWXULVPHOH��televizoarele, aparatele de radio, frigiderele, mobila etc.;

c) Bunurile destinate s �VDWLVIDF �SH�R�GXUDW �PDL�PDUH�GH�XQ�DQ��GLIHULWH�FHULQ H�PDWHULDOH�úL�VSLULWXDOH� DOH� SRSXOD LHL�� FD� GH� H[HPSOX�� FO GLULOH� GH� ORFXLW�� DXWRWXULVPHOH�� WHOHYL]RDUHOH��aparatele de radio, frigiderele, mobila etc.;

d) Bunurile destinate s �VDWLVIDF �SH�R�GXUDW �GH�PD[LP�XQ�DQ��GLIHULWH�FHULQ H�PDWHULDOH�úL�spirituale, ca de exemplu: autoturisme, mijloace financiar-valutare, televizoare, aparate de UDGLR��IULJLGHUH��PRELO �HWF��

e) Bunurile materiale destinate s satisfac � SH� R� GXUDW � GH� SHVWH� XQ� DQ�� GLIHULWH� FHULQ H�PDWHULDOH� úL� VSLULWXDOH� DOH� SRSXOD LHL�� FD� GH� H[HPSOX�� FO GLULOH� GH� ORFXLW�� DXWRWXULVPHOH��PLMORDFHOH�ILQDQFLDU�YDOXWDUH��WHOHYL]RDUH��DSDUDWH�GH�UDGLR��IULJLGHUH��PRELO �HWF�

17) (ILFLHQ D�IRQGXULORU�IL[H�QRL�VH�GHWHUPLQ �FX�UHOD LD�

a) FN

i i

i - 1e =

PIB

FN

∆ / −1

;

b) FN

i

i - 1e =

PIB

FN

∆ /0

;

c) FN

ie =

PIB

FN ;

d) FN

ie =

PIB

Ff;

e) FN

i i

ie =

PIB

Ff

∆ / −1

. unde: eFN -�HILFLHQ D�IRQGXULORU�IL[H�QRL� PIB - produsul intern brut;

i = 0,n - LQGH[XO�SHULRDGHORU�GH�UHIHULQ � Ffi - fondurile fixe în anul i; FNi - fondurile fixe noi; D -�PRGLILFDUHD�DEVROXW �D�LQGLFDWRULORU� 18) 5HVXUVHOH�GH�PXQF �OD�XQ�PRPHQW�GDW�VH�GHWHUPLQ � D�� 5HVXUVH� GLVSRQLELOH� � �7RWDO� SRSXOD LH�� -� �3RSXOD LD� vQ� YkUVW � GH� PXQF � GDU� vQ�

LQFDSDFLWDWH��3RSXOD LD�vQ�DIDUD�YkUVWHL�GH�PXQF �GDU�FDUH�OXFUHD] �� E��5HVXUVH�GLVSRQLELOH� ��7RWDO�SRSXOD LH�DSW �GH�PXQF ��3RSXOD LD� vQ�YkUVW �GH�PXQF �

dar în incapacitate) -��3RSXOD LD�vQ�DIDUD�YkUVWHL�GH�PXQF �GDU�FDUH�OXFUHD] ); F��5HVXUVH�GLVSRQLELOH� ��7RWDO�SRSXOD LH�DSW �GH�PXQF ��-��3RSXOD LD�vQ�YkUVW �GH�PXQF �GDU�

DIODW �vQ�LQFDSDFLWDWH�GH�PXQF ��3RSXOD LD�vQ�DIDUD�YkUVWHL�GH�PXQF �GDU�FDUH�OXFUHD] �� G��5HVXUVH�GLVSRQLELOH� � �7RWDO�SRSXOD LH�� 3RSXOD LD� vQ�DIDUD�YkUVWHL�GH�PXQF �GDU�FDUH�

OXFUHD] ��3RSXOD LD�DIODW �vQ�LQFDSDFLWDWH�GH�PXQF �� H�� 5HVXUVH� GLVSRQLELOH� � �7RWDO� SRSXOD LH� DSW � GH� PXQF �� -� �(OHYL�� VWXGHQ L�� PLOLWDUL� vQ�

WHUPHQ��úRPHUL��FDVQLFH��3RSXOD LD�DIODW �vQ�LQFDSDFLWDWH�GH�PXQF ��

19) Durata mediH�D�]LOHL�GH�OXFUX�VH�FDOFXOHD] ��DZ ):

a) DZ =

durata lunii de lucru

durata zilei de lucru ;

b) DZ =

totalul orelor - om lucrate

totalul om - zilelor lucrate ;

c) DZ =

totalul om - zilelor lucrate

durata lunii de lucru ;

d) DZ =


totalul om - orelor lucrate ;

e) DZ =


numarul mediu al salariatilor . 20) 6H�FXQRVF�XUP WRDUHOH�GDWH�

- mld. lei - Indicatori 1993 1994 Modificarea (‘94/’93) %

��&RQVXPXO�ILQDO�DO�JRVSRG ULLORU 12.670 31.442 + 2,6

��&RQVXPXO�DGPLQLVWUD LHL 2.566 7.011 + 3,1

��)RUPDUHD�EUXW �D�FDSLWDOXOXL�IL[ 3.584 10.096 + 20,7

4) Modificarea stocurilor 2.212 2.253 - 57,0

5) Exportul net - 996 - 1.028 - 72,0

5DWD�LQIOD LHL�vQ�LQWHUYDOXO��-��D�IRVW�HJDO �FX��FDOFXOH�FX�YDORUL�URWXQMLWH�� a) 73 %; b) 173 %; c) 313 %; d) 141 %; e) 256 %.

21) 'DF �VH�QRWHD] � L - durata medie a lunii de lucru; S -�GXUDWD�PHGLH�D�V SW PkQLL�GH�OXFUX� Z - durata medie a zilei de lucru; TH - volumul om-orelor lucrate; TZ - volumul om-zilelor lucrate; WH, WZ -�SURGXFWLYLWDWHD�RUDU �úL�]LOQLF � $WXQFL��SLHUGHUHD�GH�SURGXF LH�FD�XUPDUH�D�QHIRORVLULL�LQWHJUDOH�D�]LOHL�GH�OXFUX�VH�GHWHUPLQ �

WH normata) durata - LD( = Q a) ∆ ;

b) Q = ( DZ - durata normata) TH WH∆ • ; c) Q = ( DL - DS ) TH WZ∆ • ;

d) Q = ( DZ - DS ) TH WH∆ • ;

e) Q = ( DZ - 8) DS WZ∆ • .

22) Dac �VH�QRWHD] �

wi - productivitatea muncii pe subdiviziuni organizatorice (i = 1,n ) iw -�LQGLFLL�LQGLYLGXDOL�DL�SURGXFWLYLW LL�PXQFLL T -�QXP UXO�PHGLX�DO�VDODULD LORU g(T) -�VWUXFWXUD�VDODULD LORU�SH�VXEGLYL]LXQL�RUJDQL]DWRULFH 0,1 -�SHULRDGHOH�FRPSDUDWH��ED] ��HIHFWLY � $WXQFL�HVWH�YDODELO �UHOD LD�

a) w T

T =

w g

w g0 1

1

1 1(T)

0 0(T)

∑∑

∑

∑;

b) w T

T : w T

T =

w g

w g0 1

1

0 0

0

0 1(T)

1 1(T)

∑∑

∑∑

∑

∑;

c) w T

T : w T

T = w T

w T = w T

1

i w T

1 1

1

0 1

1

1 1

0 1

1 1

w 1 1

∑∑

∑∑

∑∑

∑

∑;

d) w T

T : w T

T = w T

w T0 0

0

1 1

1

1 1

0 0

∑∑

∑∑

∑∑ ;

e) w g

w g = w T

T : w T

T1 0

(T)

0 0(T)

1 1

1

0 1

1

∑

∑∑

∑∑

∑.

23) &RQWXO�VLQWHWLF�GH�EXQXUL��FRQWXO��DUH�vQ�FRPSRQHQ �XUP WRULL�LQGLFDWRUL� a)

R U

î�9DORDUHD�SURGXF LHL�LQWHUQH��SURGXF LH�EUXW � - firme - stat -�JRVSRG ULL��PHQDMH�

× Consum intermediar - firme - stat -�JRVSRG ULL��PHQDMH�

× Import de bunuri -�79$�QHGHGXFWLELO

× Consum final - privat - guvernamental

× Impozite pe import î�,QYHVWL LL�EUXWH - firme - guvern

× Export de bunuri Total resurse 7RWDO�XWLOL] UL

b)

R U × Valoarea ad ugat �QHW × Consum intermediar

- firme - stat -�JRVSRG ULL




× Export de bunuri Total resurse 7RWDO�XWLOL] UL

c)

R U î�9DORDUHD�SURGXF LHL�LQWHUQH��SURGXF LH�EUXW � - firme - guvern (stat) -�PHQDMH��JRVSRG ULL�

× Amortizarea capitalului fix - firme - guvern (stat) -�JRVSRG ULL




Total resurse 7RWDO�XWLOL] UL d)

R U

î�9DORDUHD�SURGXF LHL�LQWHUQH��SURGXF LH�EUXW ) - firme - guvern (stat) -�PHQDMH��JRVSRG ULL�

× Consum intermediar - firme - guvern (stat) -�JRVSRG ULL

× Export de bunuri -�79$�QHGHGXFWLELO


× Impozite pe export î�,QYHVWL LL�EUXWH - firme - stat (guvern)

× Import de bunuri

Total resurse 7RWDO�XWLOL] UL

e)

R U

î�9DORDUHD�SURGXF LHL�LQWHUQH��SURGXF LH�EUXW � - firme - guvern (stat) -�PHQDMH��JRVSRG ULL�

× Consum intermediar - firme - guvern (stat) -�JRVSRG ULL��PHQDMH�

× Export de bunuri - TVA nedeductLELO


× Amortismente - firme - guvern

Total resurse 7RWDO�XWLOL] UL

24) 'DF �

ramurilor productiei vectorul reprezinta

X

.

.

.

.

X

X = X

n

2

1

finale productiei vectorul -

Y

.

.

.

.

Y

Y = Y

n

2

1

directelorcheltuieliilorcoeficientmatricea

aaa

aaa

aaa

A

nnnnn

n

n

...

............

...

...

1

22221

11211

=

I - matricea unitate; T - transpus Corect �HVWH�UHOD LD� a) Y = (I - A)-1 X; b) Y = (I - A) X; c) X = (1 - A) Y; d) X = [AT]-1 X + Y; e) Y = (1 - A) XT. 25) 5DWD�DQXDO �D�LQIOD LHL�UHSUH]LQW � D�� &UHúWHUHD� SUH XULORU� GH� FRQVXP�� P VXUDW � SULQ� LQGLFHOH� SUH XULORU� GH� FRQVXP�� vQ� OXQD�

decembrie a anului curent ID �GH�OXQD�GHFHPEULH�D�DQXOXL�SUHFHGHQW� b) &UHúWHUHD�SUH XULORU�GH�FRQVXP��P VXUDW �SULQ�LQGLFHOH�SUH XULORU�GH�FRQVXP��vQ�OXQD�GHFHPEULH�

D�DQXOXL�FXUHQW�ID �GH�OXQD�RFWRPEULH��OXQD�SUHPHUJ WRDUH��OLEHUDOL] ULL��SUH XULORU� F�� &UHúWHUHD� SURFHQWXDO � D� SUH XULORU� GH� FRQVXP�� P VXUDW � SULQ� LQGLFHOH� SUH XULORU� GH�

FRQVXP��vQ�DQXO�FXUHQW�ID �GH�DQXO�GH�ED] � G�� 0HGLD� FUHúWHULORU� OXQDUH� DOH� SUH XULORU�� P VXUDW � SULQ� LQGLFHOH� SUH XULORU� GH� FRQVXP��

FDOFXODW � FD� PHGLH� JHRPHWULF � D� LQGLFLORU� OXQDUL� DL� SUH XULORU� GH� FRQVXP� FX� ED] � PRELO ��ODQ ��GLQ�DQXO�GH�FDOFXO�

H��6F GHUHD�SURFHQWXDO �D�SUH XULORU�GH�FRQVXP��P VXUDW �SULQ�LQGLFHOH�SUH XULORU�GH�FRQVXP��vQ�DQXO�FXUHQW�ID �GH�DQXO�GH�ED] �

26) 6H�FXQRVF�XUP WRDUHOH�GDWH�UHIHULWRDUH�OD�UHVXUVHOH�3,%�SHQWUX�DQXl 1997 (miliarde lei): - industrie: 88.944,7 -�DJULFXOWXU ��VLOYLFXOWXU ��H[��IRUHVWLHU �� -�FRQVWUXF LL�� - servicii: 84.689,3 -�DMXVWDUH�SHQWUX�SURGXF LD�VHUYLFLLORU�EDQFDUH��- 4.415,0 -�LPSR]LWH�úL�VXEYHQ LL�SH�SURGXV��,6 -�YDORDUH�DG XJDW �EUXW �� Produsul intern brut pentru anul 1997 este egal cu: a) 479.484,8; b) 249.750,2; c) 254,165,2; d) 229.734,6; e) 459.469,2. 27) 'DF �vQ�DQXO��FkúWLJXO�VDODULDO�PHGLX�QRPLQDO�QHW�D�IRVW��FRPSDUDWLY cu anul 1996,

LDU�SUH XULOH�GH�FRQVXP�DX�HYROXDW�DVWIHO�� -�P UIXUL�DOLPHQWDUH�� -�P UIXUL�QHDOLPHQWDUH�� - servicii: 276,5 în structura pe cele trei categorii: -�P UIXUL�DOLPHQWDUH�� -�P UIXUL�QHDOLPHQWDUH�� - servicii: 11,8 % 'LQDPLFD�VDODULXOXL�PHGLX�UHDO�D�IRVW�vQ�DQXO��FRPSDUDWLY�FX��HJDO �FX� a) 112,5; b) 88,9; c) 53,9; d) 250,1; e) 77,6. 28) 8WLOL]DUHD�EXQXULORU�úL�VHUYLFLLORU�vQ�HFRQRPLD�QD LRQDO �VH�VWDELOHúWH�FRQIRUP�UHOD LHL� a) Consumul priYDW� �� FRQVXPXO� JXYHUQDPHQWDO� � FRQVXPXO� WRWDO� �� IRUPDUHD� EUXW � D�

FDSLWDOXOXL�IL[��PRGLILFDUHD�VWRFXULORU� �XWLOL]DUHD�ILQDO �D�EXQXULORU�úL�VHUYLFLLORU� E�� &RQVXPXO� SULYDW� �� FRQVXPXO� JXYHUQDPHQWDO� � FRQVXPXO� ILQDO� �� LQYHVWL LLOH� EUXWH� vQ�

capitalul fix + PRGLILFDUHD� VWRFXULORU� � XWLOL]DUHD� ILQDO � vQ� LQWHULRUXO� ULL� �� H[SRUWXO� GH�EXQXUL�úL�VHUYLFLL� �XWLOL]DUHD�ILQDO �D�EXQXULORU�-�LPSRUWXO�GH�EXQXUL�úL�VHUYLFLL� �XWLOL]DUHD�bunurilor ce compun PIB;

F��&RQVXPXO�SULYDW��FRQVXPXO�JXYHUQDPHQWDO��LQYHVWL LLle nete + modificarea stocurilor + importul - exportul = utilizarea bunurilor;

G��&RQVXPXO�VWDWXOXL��JXYHUQDPHQWDO��LQYHVWL LLOH�EUXWH�GH�FDSLWDO�-�H[SRUWXO�GH�EXQXUL�úL�servicii -�LPSRUWXO�GH�EXQXUL�úL�VHUYLFLL�-�XWLOL]DUHD�ILQDO �D�EXQXULORU� �XWLOL]DUHD�EXQXULORU�úL�serviciilor ce compun PIB;

H��&RQVXPXO�SULYDW��FRQVXPXO�VWDWXOXL��JXYHUQDPHQWDO��LQYHVWL LLOH�QHWH�GH�FDSLWDO�IL[��PRGLILFDUHD�VWRFXULORU��H[SRUWXO�QHW� �XWLOL]DUHD�EXQXULORU�úL�VHUYLFLLORU�FH�FRPSXQ�3,%�

29) 6H�FXQRVF�XUP WRDUHOH�GDWH� - Produsul Intern Brut (PIB) în anul 1996: 108396 mld. lei pr. curente - PIB 1997: 249750 mld. lei pr. curente - PIB 1998 (previziune): 411.135 mld. lei pr. curente - Modificarea PIB: 1997/1996 : - 6,6 % 1998/1997 : 0 % 5DWD�LQIOD LHL�vQ�LQWervalul 1996 - 1998 va fi de: a) 64,50 %; b) 125,75 %; c) 115,12 %; d) 305,99 %; e) 225,75 %. 30) &DUH�GLQ�HOHPHQWHOH�GH�PDL�MRV�FH�LQWU �vQ�FRQVXPXO�GLUHFW�DO�SRSXOD LHL�VH�LQFOXGH�vQ�FDOFXOXO�

LQGLFHOXL�SUH XULORU�GH�FRQVXP� a) cheltuieli aferenWH� SO LL� PXQFLL� SHQWUX� SURGXF LD� JRVSRG ULHL� �DUDW�� VHP QDW�� SU úLW��

îngrijirea culturilor, cositul fânului, tratamentul medical al animalelor etc.); E��FKHOWXLHOL�FDUH�VH�UHIHU �OD�SO LOH�HIHFWXDWH�OD�FUHGLWH��UDWH�GH�DVLJXUDUH��DPHQ]L��MRFXUL�GH�

noroc, impozite etc.; F��FKHOWXLHOL�SHQWUX�SURFXUDUHD�SURGXVHORU�YkQGXWH�SH�SLD D� U QHDVF � G�� FKHOWXLHOL� FX� FDUDFWHU� GH� LQYHVWL LL� úL� DFXPXODUH� �FXPS UDUHD� GH� ORFXLQ H�� PDWHULDOH� GH�

FRQVWUXF LL�IRORVLWH�SHQWUX�FRQVWUXLUHD�GH�ORFXLQ H�QRL�VDX�HIHFWXDUHD�GH�UHSDUD LL�OD�ORFXLQ HOH�vechi etc.);

H�� FRQVXPXO� GLQ� UHVXUVH� SURSULL� UHSUH]HQWkQG� FRQWUDYDORDUHD� FDQWLW LORU� GH� SURGXVH�FRQVXPDWH�GH�SRSXOD LH��DOWHOH�GHFkW�GLQ�FXPS U WXUL��VWRF��SURGXF LH�SURSULH��FDGRXUL�HWF��

TESTUL 14

��3HQWUX�GLVWULEX LD� DQJDMD LORU�XQHL�FRPSDQLL�GXS �QLYHOXO�GH�SUHJ WLUH�SURIHVLRQDO �QX�VH�SRDWH�calcula:

a) structura; E��WRWDOXO�DQJDMD LORU� c) valoarea medie; G��YDORDUHD�PRGDO � H��IUHFYHQ HOH�QXOH��GDF �H[LVW �

��5DWD�úRPDMXOXL�QX�VH�SRDWH�FDOFXOD�UDSRUWkQG��

a) popula LD�QHRFXSDW �OD�SRSXOD LD�FXUHQW�DFWLY � E��QXP UXO�GH�úRPHUL�OD�SRSXOD LD�DFWLY � F��SRSXOD LD�QHRFXSDW �OD�SRSXOD LD�DFWLY � G��QXP UXO�GH�úRPHUL�OD�SRSXOD LD�WRWDO � H��QXP UXO�GH�úRPHUL�OD�IRU D�GH�PXQF �

3) Pe termen lung poate fi doar atenuat, dar QX�HYLWDW�úRPDMXO�

a) ciclic; b) structural; c) tehnologic; d) nici una din formele de mai sus; H��SRW�IL�HYLWDWH�WRDWH�IRUPHOH�GH�úRPDM�

��7LWOXULOH�UHSUH]HQWDWLYH�DOH�S U LORU�VRFLDOH�GHILQHVF�vQ�&RQWDELOLWDWHD�1D LRQDO ��

D��REOLJD LXQLOH� b) bonurile de tezaur; c) orice hârtie de valoare; G��DF LXQLOH� H��DFHDVW �GHILQL LH�QX�DSDUH�LQ�&RQWDELOLWDWHD�1D LRQDO �

�� ,QWHUYDOXO� GH� vQFUHGHUH� SHQWUX� HVWLPDUHD� PHGLHL� SRSXOD LHL� JHQHUDOH� SRUQLQG� GH� OD� YDORDUHD�FDOFXODW �SHQWUX�HúDQWLRQ��VH�PDL�QXPHúWH�úL��

a) evantaiul valorilor estimate; b) parametru de încredere; c) interval previzionat; G��FODV �HYDOXDW � H��QX�H[LVW �VLQRQLP�SHQWUX�QR LXQHD�GH�LQWHUYDO�GH�vQFUHGHUH�

��,QGLFDWRUXO�WHQGLQ HL�FHQWUDOH�FH�VLQWHWL]HD] �YDORULOH�vQUHJLVWUDWH�DOH�XQHL�FDUDFteristici cantitative este:

a) cuartila a 3-a; b) coeficientul de asimetrie; c) media; d) mediana; H��QX�H[LVW �XQ�DVWIHO�GH�LQGLFDWRU�

�� &RQWXO� VWDWLVWLF� vQ� FDUH� VH� vQUHJLVWUHD] � � WUDQ]DF LLOH� GH� EXQXUL� úL� VHUYLFLL�� vPSUHXQ � FX�transferurile publice sau private dintr-R�SHULRDG ��vQWUH�R� DU �úL�H[WHULRU�UHSrH]LQW ��

D��EDODQ D�FRPHUFLDO �� E��EDODQ D�GH�SO L� F��EDODQ D�RSHUD LXQLORU�FXUHQWH� G��EDODQ D�OHJ WXULORU�GLQWUH�UDPXUL� e) tabelul input-output.

8) Sinonimul pentru ecart statistic este:

a) difHUHQ �VHPQLILFDWLY � b) abatere; c) echilibru; d) clopotul lui Gauss; H��HFXD LH�VWDWLVWLF �

��&DUWRJUDPD�VH�IRORVHúWH�SHQWUX�D�UHSUH]HQWD�JUDILF��

D��XQLW L�VWDWLVWLFH�FX�GRX �GLPHQVLXQL��vQWRWGHDXQD�GXS �R�FDUDFWHULVWLF � b) XQLW L�VWDWLVWLFH�FX�GRX �GLPHQVLXQL��vQWRWGHDXQD�GXS �PDL�PXOWH�FDUDFWHULVWLFL� c) XQLW L�VWDWLVWLFH�FX�GRX �GLPHQVLXQL��QLFLRGDW �GXS �PDL�PXOWH�FDUDFWHULVWLFL� G��XQLW L�VWDWLVWLFH�FX�GRX �GLPHQVLXQL��GXS �XQD�VDX�PDL�PXOWH�FDUDFWHULVWLFL� e) cartograma nu este un grafic utilizat�GH�VWDWLVWLF �

��0XO LPHD�XQLW LORU�VWDWLVWLFH�H[WUDVH�DOHDWRULX�GLQWU-R�FROHFWLYLWDWH�VWDWLVWLF �HVWH�GHQXPLW �úL�

D��SURE � E��HúDQWLRQDUH� F��VFKHP �SUREDELOLVWLF � G��SDV�GH�QXP UDUH� H��ED] �GH�VRQGDM�

��3HQWUX�D�RE LQH�FHUHUHD�LQWHUQ �EUXW �VH�DGXQ �OD�FHUHUHD�LQWHUQ �QHW �

D��LPSR]LWHOH�LQGLUHFWH�I U �VXEYHQ LL� E��YHQLWXO�QD LRQDO� F��LPSR]LWHOH�LQGLUHFWH�FX�VXEYHQ LL� d) amortizarea; e) cei doi indicatori sunt egali.

��'DF �GLQ�311�H[SULPDW�vQ�SUH XULOH�SLH HL�VH�VFDG�LPSR]LWHOH�LQGLUHFWH�I U �VXEYHQ LL�VH�RE LQH��

D��YHQLWXO�QD LRQDO� b) PNN; c) PIB; G��311�H[SULPDW�vQ�SUH XULOH�IDFWRULORU� e) PNB.

�� ,QGLFDWRUXO�FH�FDUDFWHUL]HD] �DFWLYLWDWHD� XQHL� HFRQRPLL�QD LRQDOH� SULQ� vQVXPDUHD� RSHUD LXQLORU�HIHFWXDWH�GH�DJHQ LL�HFRQRPLFL�VH�PDL�QXPHúWH��

a) aglomerat macroeconomic; E��VXP �PDFURHFRQRPLF � c) agregat macroeconomic; d) agent macroeconomic; e) acord macroeconomic.

��$F LXQHD�GH�FRUHFWDUH�D�YDORULORU�vQUHJLVWUDWH�DOH�XQXL�IHQRPHQ�HFRQRPLFR-social la curba sa de WHQGLQ �GHILQHúWH�RSHUD LXQHD�GH��

a) eliminare a valorilor aberante; b) ajustare; c) grupare; G��HúDQWLRQDUH� H��FRUHF LH�D�HURULORU�VLVWHPDWLFH�

15) Prin formula ( ) ( )f y y x xij j

j

n

i

m

i⋅ − ⋅ −==

∑∑11 ��VH�GHILQHúWH�

D��QXP U WRUXO�UDSRUWXOXL�GH�FRUHOD LH� b) numitorul coeficientului GH�FRUHOD LH�OLQLDU � F��QXP U WRUXO�FRHILFLHQWXOXL�GH�FRUHOD LH�OLQLDU � G��QXPLWRUXO�UDSRUWXOXL�GH�FRUHOD LH� H��FRHILFLHQWXO�GH�FRUHOD LH�OLQLDU ��

��&RPSOHWD L�VSD LXO�OLEHU�GLQ�DILUPD LD� ³&RYDULDQ D� UHSUH]LQW � JHQHUDOL]DUHD� QR LXQLL� GH�«««�� DSOLFDELO � XQXL� VHW� GH� SHUHFKL� GH� GDWH�(atribute statistice)”, alegând dintre variantele:

a) variabilitate; E��YDULDQ � F��YDULD LH� G��YDULDQW � H��YDULDELO �

��&RHILFLHQWXO�GH�HODVWLFLWDWH�VH�FDOFXOHD] �DVWIHO�

a) ritmul de modificare a variabilei factoriale raportat la ritmul de modificare a variabilei rezultative;

E��ULWPXO�GH�PRGLILFDUH�D�YDULDELOHL� IDFWRULDOH� vQPXO LW�FX�ULWPXO�GH�PRGLILFDUH�D�YDULDELOHL�rezultative;

c) ritmul de modificare a variabilei rezultative raportat la ritmul de modificare a variabilei factoriale;

G��ULWPXO�GH�PRGLILFDUH�D�YDULDELOHL�UH]XOWDWLYH�vQPXO LW�FX�ULWPXO�GH�PRGLILFDUH�D�YDULDELOHL�factoriale;

H��PRGLILFDUHD�DEVROXW �D�YDULDELOHL�UH]XOWDWLYH�UDSRUWDW�OD�PRGLILFDUHD�DEVROXW �D�YDULDELOHL�factoriale.

18) Extrapolarea UHSUH]LQW �RSHUD LXQHD�VWDWLVWLF �GH�

a) prelungire a valorilor unei serii cronologice dincolo de perioada pe care s-au efectuat REVHUYD LLOH��FRQIRUP�XQHL�OHJL�GH�WHQGLQ �

b) interpolare; c) estimare a unei valori dintr-o serie de date continue, cunoscâQG�YDORULOH�GLQ�GUHDSWD�úL�GLQ�

stânga sa; d) determinare a unui interval de încredere; H��GHWHUPLQDUH�D�WHQGLQ HL�GH�HYROX LH�D�XQXL�IHQRPHQ�

��&XQRVFkQG�GHILQL LLOH�QR LXQLORU�GH�PHGLH�DULWPHWLF � úL� DUPRQLF �� GDF �DFHVWHD� VXQW� FDOFXODWH�SHQWUX�DFHODúL�VHW�GH�GDWH��SUHFL]D L�FH�DILUPD LH�HVWH�DGHY UDW �

D��R�FUHúWHUH�D�YDORULL�PHGLHL�DULWPHWLFH�YD�GHWHUPLQD�R�FUHúWHUH�D�YDORULL�PHGLHL�DUPRQLFH� E��YDULD LD�PHGLHL�DULWPHWLFH�GHWHUPLQ �YDULD LD�vQ�VHQV�FRQWUDU�D�PHGLHL�DUPRQLFH� F�� FUHúWHUHD� PHGLHL� DUPRQLFH� YD� DWUDJH� GXS � VLQH�� vQ� PRG� DXWRPDW�� FUHúWHUHD� PHGLHL�

aritmetice; G��VF GHUHD�YDORULORU�FDUDFWHULVWLFLL�VWDWLVWLFH�vQUHJLVWUDWH�YD�DWUDJH�GXS �VLQH�VF GHUHD�PHGLHL�

DUPRQLFH�úL�DULWPHWLFH� H��PHGLD�DULWPHWLF �HVWH�LQGHSHQGHQW �GH�FHD�DUPRQLF ��úL�DPkQGRX �VXQW�LQGHSHQGHQWH�GH�

HYROX LD�YDORULORU�vQUHJLVWUDWH�DOH�FDUDFWHULVWLFLL� ��9DORULOH�DMXVWDWH�DOH�XQHL�VHULL�GH�WLPS�SRW�IL��ID �GH�FHOH�vQUHJLVWUDWH�

a) <; b) >; c) =; d) <, =, >; e) nu se pot compara.

�� 'DF �� SHQWUX� R� VHULH� FURQRORJLF �� VXPD� YDORULORU� DMXVWDWH� HVWH� DSUR[LPDWLY� HJDO � FX� VXPD�valorilor înregistrate, atunci:

D��PHWRGD�GH�DMXVWDUH�DOHDV �QX�HVWH�SRWULYLW �� E��PHWRGD�GH�DMXVWDUH�DOHDV �HVWH�RSWLP � c) metoda de aMXVWDUH�DOHDV �HVWH�XQLF � G��PHWRGD�GH�DMXVWDUH�DOHDV �HVWH�FRUHFW �� H��PHWRGD�GH�DMXVWDUH�DOHDV �HVWH�H[DFW �

��5DSRUWXO�GLQWUH�R�YDORDUH�vQUHJLVWUDW ��úL�FHD�WHRUHWLF ��DIODW �SH�FXUED�GH�UHJUHVLH�SRDWH�IL�

a) supraunitar; b) subunitar; c) unitar; d) negativ; e) pozitiv.

$OHJH L�FRPELQD LD�SRWULYLW ��GLQWUH�FHOH�QRWDte cu majuscule: A = a,b,d; B = b,d,e; C = a,b,c; D = a,b; E = a

��'DF � vQ�XUPD�JUXS ULL�XQXL�VHW�GH�GDWH�SH� LQWHUYDOH�GH�YDULD LH�HJDOH��VH�RE LQ�IUHFYHQ H�QXOH��DWXQFL�VH�UHFRPDQG �

D��V �VH�UHJUXSH]H�GDWHOH�GXS �R�DOW �FDUDFWHULVWLF � b) se utilizea] �vQ�FRQWLQXDUH�DFHHDúL�GLVWULEX LH�QH LQkQG�FRQW��GH�IUHFYHQ HOH�QXOH� F��VH�XWLOL]HD] �vQ�FRQWLQXDUH�GLVWULEX LD�GDF �HVWH�GRDU�R�VLQJXU �IUHFYHQ �QXO � G��VH�UHJUXSHD] �GDWHOH��P ULQG�QXP UXO�GH�LQWHUYDOH�GH�YDULD LH� H��VH�UHJUXSHD] �GDWHOH�P ULQG�GLmensiunea intervalului.

��8Q�ULWP�GH�PRGLILFDUH�D�XQXL�IHQRPHQ��ULWP�FH�DUH�YDORDUHD�SR]LWLY ��LPSOLF ��vQ�PRG�QHFHVDU� D��YDORDUHD�VXSUDXQLWDU ��D�LQGLFHOXL�GH�GLQDPLF � E��YDORDUHD�QHJDWLY �D�PRGLILF ULL�DEVROXWH� F��YDORDUHD�QHJDWLY �D�VSRUXOXL�FX�ED] �IL[ � G��GHVFUHúWHUHD�IHQRPHQXOXL� e) stagnarea fenomenului .

��3HQWUX�VHULD�GH�GDWH�UHSUH]HQWkQG�GLVWULEX LD�IDPLOLLORU�GXS �QLYHOXO�GH�YHQLW�

Venit Foarte VF ]XW

6F ]XW Mediu Ridicat Foarte ridicat

Nr. familii 10 20 40 20 10 coeficientul de asimetrie propus de Pearson este:

a) zero; b) unitar; c) egal cu -1; d) nu se poate calcula; H��HJDO�FX�FRHILFLHQWXO�GH�YDULD LH�

�� 'DF � GRX � VHULL� GH� GLVWULEX LH� SDUDOHOH� SUH]LQW � R� WHQGLQ � FXUELOLQLH� SURQXQ DW �� DWXQFL�P VXUDUHD�LQWHQVLW LL�OHJ WXULL�GLQWUH�FHOH�GRX �YDULDELOH�VH�DSUHFLD] �vQ�IXQF LH�GH�YDORDUHD�

D��FRHILFLHQWXOXL�GH�FRUHOD LH�OLQLDU � E��FRYDULDQ HL� F��UDSRUWXOXL�GH�FRUHOD LH� d) valorilor ajustate; H�� QX� VH� SRDWH� FDOFXOD� XQ� LQGLFDWRU� FH� LQGLF � LQWHQVLWDWHD� OHJ WXULL�� GDF � DFHDVWD� este

curbilinie. ��9DULD LD�XQHL�FDUDFWHULVWLFL�VWDWLVWLFH�WLQGH�OD�]HUR�GDF �FROHFWLYLWDWHD�DQDOL]DW �HVWH�

a) de volum normal; E��QRUPDO�GLVWULEXLW �GXS �R�FDUDFWHULVWLF � F��HWHURJHQ � G��RPRJHQ � H��DVLPHWULF �

28) 'DF �SHQWUX��VHULH�GH�GLVWULEX LH�GH�IUHFYHQ H�FRHILFLHQWXO�GH�YDULD LH�LQGLF �XQ�QLYHO�ULGLFDW�GH�

HWHURJHQLWDWH��DWXQFL� VH� UHFRPDQG � vPS U LUHD�FROHFWLYLW LL� vQ�JUXSH�GXS �XQ� IDFWRU�GH�JUXSDUH�semnificativ, ceea ce va duce la:

D��FUHúWHUHD�JUDGXOXL�GH�RPRJHQLWDWH�vQ�LQWHULRUXO�JUXSHORU� b��FUHúWHUHD�JUDGXOXL�GH�HWHURJHQLWDWH�vQ�LQWHULRUXO�JUXSHORU� F��VF GHUHD�JUDGXOXL�GH�RPRJHQLWDWH�vQ�LQWHULRUXO�JUXSHORU� G��FUHúWHUHD�QLYHOXOXL�FRHILFLHQWXOXL�GH�YDULD LH�vQ�LQWHULRUXO�JUXSHORU� e) nivelul de omogenitate în interiorul grupei va fi întotdeauna egal cu cel calculat pentru

GLVWULEX LD�PDUJLQDO � ��3HQWUX�R�SLD �FXQRDúWHP�GDWHOH�

Bunuri 3UH XUL��(u.m.)

&DQWLW L��(u.c.)

3UH XUL��(u.m.)

&DQWLW L��(u.c.)

A 45 300 65 400 B 90 500 105 500 C 260 900 320 1100 D 115 300 150 600 E 118 800 128 700

Atunci:

D��ÌQ��FRVW �FX��PDL�PXOW�GHFkW�vQ��SHQWUX�D�DFKL]L LRQD�DFHHDúL�FDQWLWDWH�GH�PDUI �

E��ÌQ��FRVWD�FX��PDL�SX LQ�GHFkW�vQ��SHQWUX�D�DFKL]L LRQD�DFHHDúL�FDQWLWDWH�GH�PDUI �FD�vQ��

F��$FHHDúL�FDQWLWDWH�GH�EXQXUL��FXPS UDWH�vQ��OD�SUH XULOH�GLQ��FRVW �FX��PDL�VFXPS�GHFkW�HYDOXDWH�OD�SUH XULOH�GLQ��

G��'DF �FDQWLW LOH�FXPS UDWH�vQ��V-DU�IL�DFKL]L LRQDW�OD�SUH XULOH�GLQ��DU�IL�FRVWDW�FX�20,41% mai scump;

e) sunt date insuficiente pentru a alege a,b,c, sau d. ��'DF �vQWU-R� DU ��VDODULXO�PHGLX�D�IRVW�vQ�DQXO�W0�GH��PLOLRQ�OHL�úL�,3&� ��,33� ��LDU�în anul t1��VDODULXO�PHGLX�D�DMXQV�OD�QLYHOXO�GH��PLOLRDQH�OHL�úL�,3&� ��,33� ��ÌQ�DFHVWH�FRQGL LL��salariul real:

a) a crescut; E��D�VF ]XW� F��D�U PDV�QHVFKLPEDW� G��QX�VH�SRDWH�SUHFL]D�HYROX LD�VD� e) este egal cu cel nominal din anul t1.

��3 WUDWXO�DEDWHULL�WLS�P VRDU ��

a) amplitudinea dispersiei unui set de date în jurul mediei lor; b) omogenitatea unui set de date; F��DVLPHWULD�XQHL�GLVWULEX LL� G��WHQGLQ D�FHQWUDO �D�XQXL�VHW�GH�GDWH� H��JUDGXO�GH�FRQFHQWUDUH�D�IUHFYHQ HORU�

��2�UDW �GHVFUHVF WRDUH�D�LQIOD LHL�SRDWH�V �GXF �OD�

D��FUHúWHUHD�úRPDMXOXL� E��GHVFUHúWHUHD�úomajului; c) stagnarHD�úRPDMXOXL� G��úRPDMXO�HVWH�XQ�IHQRPHQ�LQGHSHQGHQW�GH�LQIOD LH�

d) VXQW� LQVXILFLHQWH� LQIRUPD LL� SHQWUX� D� SXWHD� SUHFL]D� LQIOXHQ D� � IHQRPHQHORU� GH� FUHúWHUH�JHQHUDOL]DW �D�SUH XULORU�DVXSUD�úRPDMXOXL�

��3XWHUHD�GH�FXPS UDUH�D�PRQHGHL�QD LRQDOH�HVWH�LQYHUV�SURSRU LRQDO �FX�

D��QLYHOXO�SUH XULORU� b) nivelul veniturilor; c) nivelul PIB/locuitor; G��YROXPXO�EXQXULORU�úL�VHUYLFLLORU�RIHULWH�SH�SLD � H��HVWH�R�P ULPH�LQGHSHQGHQW �

��/D�R�EDQF �VH�DQDOL]HD] �GLVWULEX LD�GHELWRULORU�GXS �VLWXD LD�]LOHORU�GH�vQWkU]LHUH�D�UDPEXUV ULL�creditelor, astfel:

,QWHUYDOH�GH�YDULD LH�D�QXP UXOXL�GH�]LOH�GH�vQWkU]LHUH�D�SO LL�

Nr. debitori “mai mult decât OLPLWD�LQIHULRDU ´

10-20 120 20-30 85 30-40 37 40-50 16 50-60 7 60-70 2

9DORDUHD�PHGLDQ �FDOFXODW �XWLOL]kQG�IUHFYHQ H�UHODWLYH�HVWH

a) 25 zile; b) 35 zile; c) 27 zile; d) 23 zile; H��QX�VH�SRDWH�FDOFXOD�PHGLDQD�XWLOL]kQG�IUHFYHQ H�UHODWLYH�

35) Dintr-XQ�VRQGDM�DX�UH]XOWDW�GDWHOH�UHIHULWRDUH�OD�JUDGXO�GH�vQDYX LUH��DVWIHO�

1LYHO�GH�vQDYX LUH Procente ale populD LHL 6WUXFWXUD�DYX LHL $��1LYHO�IRDUWH�VF ]XW 50 10 %��1LYHO�VF ]XW 25 20 C. Nivel minim acceptabil de avere 10 10 '��1LYHO�PHGLX�GH�vQDYX LUH 10 15 (��1LYHO�ULGLFDW�GH�vQDYX LUH 3 25 F. Nivel extrem de ridicat ... ...

'HFLGH L�GDF �

D��DYX LD�HVWH�HJDO�GLVWULEXLW � b) curba lui Lorentz coincide cu linia de egalitate; c) DYX LD� HVWH� GLVWULEXLW � LQHFKLWDELO�� DYDQWDMD L� ILLQG� FHL� FX� QLYHO� vQDOW� úL� IRDUWH� vQDOW� GH�

vQDYX LUH� G��DYX LD�HVWH�GLVWULEXLW � LQHFKLWDELO��DYDQWDMD L� ILLQG�FHL�FX�QLYHO�VF ]XW�úL� IRDUWH�VF ]XW�GH�

vQDYX LUH� e) nu se poate discuta echitatea distribuirii veniturilor.

��ÌQ�IRUPXOD�HURULL�PD[LPH��YROXPXO�HúDQWLRQXOXL�VH�QRWHD] �FX�n�GDF �VRQGDMXO�UHDOL]DW�HVWH�

a) de serii; E��UHDOL]DW�SH�XQ�HúDQWLRQ�IRUPDW�GLQ�XQLW L�FRPSOH[H; F��UHDOL]DW�SH�XQ�HúDQWLRQ�IRUPDW�GLQ�XQLW L�VLPSOH�

G��UHDOL]DW�SH�XQ�HúDQWLRQ�IRUPDW�GLQ�XQLW L�VLPSOH�VDX�FRPSOH[H��LQGLIHUHQW� e) cu n�VH�QRWHD] �GH�RELFHL�YROXPXO�SRSXOD LHL�JHQHUDOH�

��ÌQWUH�GHFLOD�D�WUHLD�úL�D�úDSWHD�VH�DIO �

a) 40% din termenii seriei; b) 60% din termenii seriei; c) 50% din termeni seriei; d) 30% din termenii seriei; e) 70% din termenii seriei.

38) Formula lui Sturges se poate aplica pentru determinarea:

D��DPSOLWXGLQLL�YDULD LHL�XQHL�FDUDFWHULVWLFL� b) valorii maxime dintr-un set de date; c) valorii minime dintr-un set de date; G��QXP UXOXL�GH�JUXSH�� H��QXP UXOXL�GH�FDUDFWHULVWLFL�

��'LVSHUVLD�HVWH�LQYHUV�SURSRU LRQDO �FX��

D��YROXPXO�HúDQWLRQXOXL� b) volumul caracteristicilor studiate; c) abaterea standard; d) coeficientul de asimetrie; H��HVWH�R�P ULPH�LQGHSHQGHQW �

�� 5DSRUWXO� GLQWUH� LQGLFHOH� DJUHJDW� DO� SUH XULORU� FDOFXODW� GXS � SURFHGHXO� /DVSH\UHV�� vQWU-o HFRQRPLH� LQIOD LRQLVW �� úL� LQGLFHOH� DJUHJDW� DO� SUH XULORU� FDOFXODW� GXS � SURFHGHXO� 3DDVFKH� HVWH�� GH�obicei:

a) supraunitar; b) subunitar; c) unitar; d) negativ; e) QX�VH�úWLH�H[DFW�

TESTUL 15

��&RHILFLHQWXO�GH�YDULD LH�DUDW �

D��GH�FkWH�RUL�HVWH�PDL�PDUH�DEDWHUHD�VWDQGDUG��WLS��ID �GH�PHGLD�DULWPHWLF � E��FX�FkWH�SURFHQWH�HVWH�GHS úLW ��OLPLWD�GH�RPRJHQLWDWH�DGPLV � F��FX�FkW�HVWH�PDL�PDUH�DEDWHUHD�VWDQGDUG�ID �GH�PHGLD�DULWPHWLF � d) de câte ori se cuprinde abaterea standard în medie; H��FkWH�SURFHQWH�GLQ�DEDWHUHD�VWDQGDUG�UHSUH]LQW �PHGLD�DULWPHWLF �

��'DF �XQ�IHQRPHQ�HYROXHD] �GXS �IXQF LD��W��VXPD�S WUDWHORU�DEDWHULORU�GLQWUH�R�SDUWH�GLQ�YDORULOH�vQUHJLVWUDWH��úL�FHOH�DMXVWDWH�FRQVLGHUkQG�YDORULOH�FRUHVSXQ] WRDUH�GLQ�vectorul timpului : 2,4,5,6,7:

a) 0,5; b) 130; c) 133; d) 100; H��DOW �YDULDQW ��«

3) Cunoscând dRDU�KLVWRJUDPD�XQHL�GLVWULEX LL�GH�IUHFYHQ H�QX�SXWHP�UHSUH]HQWD�JUDILF�

a) structura; E��RJLYD�IUHFYHQ HORU�FXPXODWH� c) corelograma; G��SROLJRQXO�IUHFYHQ HORU� H��GLVWULEX LD�IUHFYHQ HORU�UHODWLYH�

��'XS �ULWP��LQIOD LD�QX�SRDWH�IL��

a) târâtoare; b) GHVFKLV � F��JDORSDQW � d) deflatoare; H��IRUPHOH�LQIOD LHL�QX�VH�FODVLILF �GXS �ULWPXO�GH�PRGLILFDUH�

5) Pentru o serie de momente nu se pot calcula:

D��VSRUXUL�FX�ED] �vQ�ODQ � E��VSRUXUL�FX�ED] �IL[ � c) indici; d) ritmuri; H��PHGLD�DULWPHWLF �D�WHUPHnilor seriei.

��&RPSOHWD L�VSD LXO�OLEHU�FX�XQD�GLQ�YDULDQWHOH�D�- e: �9DULDELOD� GLVFUHW � ;� XUPHD] � OHJHD� GH� UHSDUWL LH� «««� D� SDUDPHWULORU� n� úL� p , atunci când IUHFYHQ D�VD�HVWH�FRQIRUP�

( )f C p p n n pnx x n x

= ⋅ ⋅ − > ∈ < <−

1 1 0 1, , ,Ν ¨ D��QRUPDO � E��ELQRPLDO �

c) Student; d) Gauss Laplace; H��QX�H[LVW �DFHDVW �IXQF LH�GH�UHSDUWL LH�

��6WDELOL L�FH�UHOD LH�HVH�DGHY UDW �

a) PIB = VA + TVA + taxe vamale - servicii bancare imputate nerepartizate; b) PIB = VA + TVA + taxe vamale; c) PIB = VA + TVA; d) PIB = VA; e) PIB = VA + TVA + taxe vamale + servicii bancare imputate nerepartizate.

��7HUPHQXO�VWDWLVWLF�GH��UDW ��HVWH�VLQRQLP�FX�FHO�GH��LQGLFH��

a) întotdeauna; E��QLFLRGDW � F��GRDU�GDF �UDWD�H[SULP �R�HYROX LH� G��GRDU�GDF �H[SULP �SRQGHUHD�XQHL�S U L�vQ�vQWUHJ� e) doar vQ�FD]XO�H[FHS LLORU�

��/D�UHFHQV PkQW�VH�FXOHJ�LQIRUPD LL�GH�QDWXU �

D��GHPRJUDILF ��HFRQRPLF ��VRFLDO � E��FXOWXUDO � F��GRDU�HFRQRPLF � G��GRDU�VRFLDO � H��GRDU�GHPRJUDILF �

��6LQRQLPXO�SHQWUX�DEDWHUHD�WLS�QX�VH�PDL�QXPHúWH�

a) abatere standard; E��DEDWHUH�PHGLH�S WUDWLF � c) ecart tip; G��YDULDQ � H��YDULDQW �

��&RPSRQHQWD�VH]RQLHU �D�XQHL�VHULL�GH�WLPS�DSDUH�FD�UH]XOWDW�DO�DF LXQLL�

D�� IOXFWXD LLORU� OHJDWH� GH� DQRWLPS� VDX� FH� VH� UHSURGXF� VLPLODU� vQ� WLPS�� vQ� FXUVXO� XQHL� ]LOH��V SW PkQL��OXQL��trimestre;

E��IOXFWXD LLORU�FLFOLFH� c) factorilor aleatori; G��WHQGLQ HL� e) componenta sezonieU �QX�H[LVW �

��&RPSRQHQWD�FLFOLF �DSDUH�FD�XUPDUH��D�DF LXQLL�

a) factorilor sezonieri; E��IDFWRULORU�FH�GHWHUPLQ �ID]HOH�GH�FRQWUDF LH�úL�UHOD[DUH�D�IHQRPHnelor; c) factorilor aleatori; G��WHQGLQ HL� H��QX�H[LVW �DFHDVW �FRPSRQHQW �

��&XQRVFkQG�RJLYD�IUHFYHQ HORU�FXPXODWH��QX�VH�SRDWH�UHSUH]HQWD�JUDILF��vQ�FRQWLQXDUH�

a) structura; b) histograma;

c) poligonul frecYHQ HORU�DEVROXWH� G��SROLJRQXO�IUHFYHQ HOor relative; H��JUDILFXO�GH�HYROX LH�

��'DF � � UDSRUWXO� GLQWUH� LQGLFHOH� DJUHJDW� DO� SUH XULORU� úL� LQGLFHOH� DJUHJDW� DO�YROXPXOXL� IL]LF� HVWH�supraunitar, atunci, indicele valorii poate fi:

a) supraunitar; b) subunitar; c) unitar; d) a,b,c; e) negativ.

15)� 3HQWUX�P VXUDUHD� HYROX LHL� SUH XULORU�� OD� QLYHOXO� DJHQWXOXL� HFRQRPLF�� VH� UHFRPDQG �XWLOL]DUHD�sistemului de ponderare:

a) Laspeyres; b) Paasche; c) Fisher; d) Edgeworth; e) nici unul din cele precizate mai sus.

�� 3HQWUX� P VXUDUHD� HYROX LHL� SUH XULORU�� OD� QLYHO� PDFURHFRQRPLF�� vQ� 5RPkQLD� VH� XWLOL]HD] �sistemul de ponderare:

a) Laspeyres; b) Paasche; c) Fisher; d) Edgeworth; e) nici unul din cele precizate mai sus.

17) În anul t1�ULWPXO�FLIUHL�GH�DIDFHUL�D�XQHL�ILUPH�D�IRVW�GH��ID �GH�W0. În anul XUP WRU��W2, cifra GH�DIDFHUL�D�FUHVFXW�FX��ùWLLQG�F �VSRUXO�DEVROXW�DO�FLIUHL�GH�DIDFHUL�vQ�SHULRDGD�W2 - t0 a fost de 100 u.m., cifra de afaceri din anul t1 a fost de:

a) 60 u.m.; b) 120 u.m.; c) 20 u.m.; d) 27,27 u.m.; e) nu se poate calcula datorLW �LQIRUPD LHL�LQVXILFLHQWH�

18) 'DF �SH�R�SHULRDG �GH�FLQFL�DQL�� LQGLFHOH�PHGLX�GH�FUHúWHUH�D�SURGXF LHL�XQXL�DJHQW�HFRQRPLF�D�IRVW�GH��vQ�FRQGL LLOH�vQ�FDUH�SURGXF LD�HVWH�R�YDULDELO �vQVXPDELO ��úWLLQG�F �vQ�XOWLPXO�DQ�QLYHOXO�SURGXF LHL�D�IRVW�FX��GH�XQLW L�PDL�PDUH�GHFkW�vQ�SULPXO�DQ��YDORDUHD�LQL LDO �D�SURGXF LHL�D�IRVW�GH�

a) 44,21; b) 4,21; c) 2,24; d) 3; e) nu se poate calcula.

�� 'DF � FKHOWXLHOLOH� SHQWUX� FRQVXP� VXQW� �� X�P�� úL� LQYHVWL LLOH� UHSUH]LQW � �� GLQ� YHQLWXO�QD LRQDO�� LPSRUWXO� � UHSUH]LQW � � GLQ� YHQLW� úL� H[SRUWXO� � GLQ� LQYHVWL LL�� DWXQFL� VROGXO� EDODQ HL�comerciale este:

a) 875 u.m.; b) -875 u.m.; c) 1250 u.m.; d) 375 u.m.; e) 500 u.m..

��$JUHJDWHOH�PDFURHFRQRPLFH�VH�FDOFXOHD] �

D��vQ�XQLW L�IL]LFH� E��vQ�XQLW L�YDORULFH; F��H[FOXVLY�vQ�SUH XULOH�IDFWRULORU� G��H[FOXVLY�vQ�SUH XULOH�SLH HL� H��vQ�SUH XULOH�IDFWRULORU�úL�vQ�SUH XULOH�SLH HL�

��$FWLYLWDWHD�GH�UHGLVWULEXLUH�D�YHQLWXULORU�DUH�ORF�GDF �

D��VH�SO WHVF�DMXWRDUHOH�VRFLDOH� E��VH�SO WHVF�VDODULLOH�DQJDMD LORU�statului; F��VH�SO WHVF�SHQVLLOH� G��VH�DFKLW �LPSR]LWHOH� e) a, c.

��6ROGXO�EDODQ HL�FRPHUFLDOH�VH�FDOFXOHD] �

D��FD�UDSRUW�LQWUH�YDORDUHD�H[SRUWXOXL�úL�YDORDUHD�LPSRUWXOXL� E��FD�GLIHUHQ �vQWUH�FDQWLWDWHD�H[SRUWDW �úL�FHD�LPSRUWDW � F��FD�GLIHUHQ �vQWUH�YDORDUHD�LPSRUWXOXL�úL�D�H[SRUWXOXL� G��FD�GLIHUHQ �vQWUH�YDORDUHD�H[SRUWXOXL�úL�D�LPSRUWXOXL� H��FD�GLIHUHQ �vQWUH�FDQWLWDWHD�LPSRUWDW �úL�FHD�H[SRUWDW �

��'DF �GLQDPLFD�FDQWLW LL�RIHULWH�SH�R�SLD �HVWH�GH��úL�D�SUH XOXL�HVWH�GH��FRHficientul GH�HODVWLFLWDWH�D�FHUHULL�OD�HYROX LD�SUH XOXL��HVWH�

a) unitar; b) supraunitar; c) subunitar; d) 3; e) 0,33.

��8Q�DJUHJDW�PDFURHFRQRPLF�FH�FDUDFWHUL]HD] �QLYHOXO�GH�GH]YROWDUH�D�HFRQRPLHL�QD LRQDOH�HVWH�

a) IPC; b) IPP; c) PNN/locuitor; d) PIB real; H��YROXPXO�SURGXF LHL�WRWDOH�

25) Sunt egale cu zero:

D��HURDUHD�PD[LP ��GDF �GLVSHUVLD�HVWH�HJDO �FX�� E��UDSRUWXO�GH�QHGHWHUPLQD LH��GDF �YDULD LD�UH]LGXDO �HVWH�HJDO �FX�]HUR� F��PHGLD��GDF �VXPD�YDORULORU�FDUDFWHULVWLFLL�HVWH�HJDO �FX�]HUo;

G��FRHILFLHQWXO�GH�DVLPHWULH��GDF �PHGLDQD�HVWH�HJDO �FX�YDORDUHD�PRGDO � e) coeficientul de reprezentativitate, dac media este un indicator reprezentativ.

�� 6WDELOL L� QXP UXO� FRHILFLHQ LORU� vQWRWGHDXQD� � VXSUDXQLWDUL� VWXGLD L� GH� 6WDWLVWLFD� 6RFLDO� - (FRQRPLF �

D��WR L�FRHILFLHQ LL�VXQW�VXSUDXQLWDUL� b) nici unul nu este întotdeauna supraunitar; F��WR L�FRHILFLHQ LL�VXQW�VXEXQLWDUL� G��PDMRULWDWHD�FHORU�VWXGLD L� e) indicii.

��&RHILFLHQWXO�GH�DVLPHWULH�SURSXV�GH�3HDUVRQ�VH�DIO �vQ�UHOD LH�GH�LQYHUV �SURSRU LRQDOLWDWH�FX��

a) abaterea standard; E��DEDWHUHD�VWDQGDUG�úL�YDORDUHD�PRGDO � F��GLVSHUVLD�úL�YDORDUHD�PRGDO � G��PHGLD�DULWPHWLF � H��YDORDUHD�PRGDO �

��&RHILFLHQWXO�GH�FRUHOD LH�OLQLDU �VH�DIO �vQ�UHOD LH�GH�GLUHFW �SURSRU LRQDOLWDWH�FX��

a��FRYDULDQ D� E��DEDWHUHD��VWDQGDUG�FDOFXODW �SHQWUX�YDULDELOD�IDFWRULDO � F��DEDWHUHD�VWDQGDUG�FDOFXODW �SHQWUX�YDULDELOD�UH]XOWDWLY � d) volumul datelor studiate; e) este un indicator independent.

�� 'DF � XQ� IHQRPHQ� HFRQRPLF� vQUHJLVWUHD] � VF GHUL� FX� R� UDW � FUHVF WRDUH�� DWXQFL� XOWLPXO� QLYHO�observat va fi, comparativ cu primul:

a) mai mare; b) mai mic; c) egal; d) nu se poate compara; H��LQIRUPD LD�SUH]HQWDW �HVWH�LQVXILFLHQW �

��³([WHULRUXO´�UHSUH]LQW �vQ�&RQWDELOLWDWHD�1D LRQDO �

D��DJHQ LL�HFRQRPLFL�QD LRQDOL�LPSOLFD L�vQ�WUDQ]DF LL�FX�VWU LQ WDWHD� E��DJHQ L�HFRQRPLFL�GLQ�FHOHODOWH�HFRQRPLL�QD LRQDOH� F��DJHQ LL�HFRQRPLFL�FH�GHVI úRDU �DFWLYLW L�GH�FRPHU �H[WHULRU� d) a,b,c; e) b,c.

��,QWHUYDOXO�GH�vQFUHGHUH�SHQWUX�PHGLD�DULWPHWLF �HVWH�FX�DWât mai mic cu cât:

D��YROXPXO�HúDQWLRQXOXL�HVWH�PDL�PLF� E��SUREDELOLWDWHD�FX�FDUH�VH�JDUDQWHD] �UH]XOWDWHOH�HVWH�PDL�PLF � F��UD LD�GH�VRQGDM�HVWH�PDL�PDUH� G��YROXPXO�HúDQWLRQXOXL�VFDGH� H��SUREDELOLWDWHD�GH�JDUDQWDUH�D�UH]XOWDWHORU�HVWH�FRQVWDQW �

32)�,QIOD LD�HVWH�FDX]DW �GH�GHILFLWXO�EXJHWDU�GDF �

D��FUHúWHUHD�FKHOWXLHOLORU�VWDWXOXL�GHS úHúWH�FUHúWHUHD�YHQLWXULORU� E��FKHOWXLHOLOH�VWDWXOXL�GHS úHVF�YHQLWXULOH� F��FUHúWH�GDWRULD�LQWHUQ �D�VWDWXOXL� G��VH�HPLWH�PRQHG �I U �DFRSHULUH�vQ�EXQXUL�úL�VHUYLcii; H��VH�HFKLOLEUHD] �EDODQ D�GH�SO L�

��2SHUD LXQLOH�HFRQRPLFH�GH�FRQVXP�VXQW�UHDOL]DWH�GH�F WUH�

D��GRDU�GH�F WUH��DJHQ LL�HFRQRPLFL�FH�HIHFWXHD] �RSHUD LXQL�DVXSUD�EXQXULORU�úL�VHUYLFLLORU� E��WR L�DJHQ LL�HFRQRPLFL� F��WR L�DJHQ LL�HFRQRPLFL��H[FOXVLY�PHQDMHOH�úL�DGPLQLVWUD LLOH�SXEOLFH� G��DGPLQLVWUD LLOH�SULYDWH� H��VRFLHW LOH�ILQDQFLDUH�QHILQDQFLDUH�

��ÌQ�&RQWDELOLWDWHD�1D LRQDO �RSHUD LXQLOH�GH�UHGLVWULEXLUH�D�YHQLWXULORU�VXQW�HIHFWXDWH�GH�F WUH��

D��DGPLQLVWUD LLOH�SXEOLFH� b) administrD LLOH�SULYDWH� F��VRFLHW LOH�GH�DVLJXUDUH� G��VRFLHW LOH�EDQFDUH� H��VRFLHW LOH�FRPHUFLDOH�QHILQDQFLDUH�

��/XQJLPHD�RSWLP �D�XQHL�YDORUL�HVWLPDWH�SH�ED]D�GDWHORU�GLQ�HúDQWLRQ�HVWH�

D��GLIHULW �GH�FHD�FDOFXODW �SHQWUX�SROXOD LD�WRWDO � E��DSURSLDW �GH�FHD�FDOFXODW �SHQWUX�SROXOD LD�WRWDO � F��HJDO �FX�FHD�FDOFXODW �SHQWUX�SROXOD LD�WRWDO � G��QX�VH�SRDWH�GHWHUPLQD�R�YDORDUH�RSWLP � e) variantele a - d sunt false, corect este......…

��0HWRGD�LQGLFLORU�QX�SRDWH�IL�XWLOL]DW �SHQWUX�DQDOL]D�

a) compaUD LLORU�vQ�SURILO�WHULWRULDO� E��GHVFRPSXQHULL�XQXL�IHQRPHQ�FRPSOH[�SH�IDFWRUL�GH�LQIOXHQ � F��GHWHUPLQ ULL�JUDGXOXL�GH�LQIOXHQ �D�XQXL�IDFWRU�DVXSUD�DOWXL�IDFWRU� G��HYROX LHL�XQXL�IHQRPHQ� e) analiza estimatorilor unui fenomen complex.

37) 9DULD LD�WRWDO �D�XQHL�FDUDFWHULVWLFL�VWXGLDWH�SHQWUX�XQ�HúDQWLRQ�HVWH�LQYHUV�LQIOXHQ DW �GH�

D��YROXPXO�HúDQWLRQXOXL�� E��YDULD LD�UH]LGXDO � F��YDULD LD�VLVWHPDWLF � G��YDULD LD�H[SOLFDW � H��FRHILFLHQWXO�GH�YDULD LH�

38) 'DF �VH�JUXSHD] �XQ�VHW�GH�GDWH�SH�LQWHUYDOH�HJDOH�GH�YDULD LH��SHQWUX�D�RE LQH��R�GLVWULEX LH�D�

IUHFYHQ HORU� PDL� DSURSLDW � GH� IRUPD� GLVWULEX LHL� QRUPDOH�� QXP UXO� GH� JUXSH� úL� GLPHQVLXQHD�ILHF UXL�LQWHUYDO�

D��YRU��FUHúWH� b) vor scade; F��U PkQ�FRQVWDQWH� G��HYROXHD] �vQ�VHQV�FRQWUDU� e) nu se poate preFL]D�HYROX LD�ORU�

39) Cuartila întâi este:

D��PHGLD�DULWPHWLF �GLQWUH�FXDUWLOD�D�GRXD�úL�D�WUHLD� E��GLIHUHQ D�GLQWUH�FXDUWLOD�D�GRXD�úL�D�WUHLD� F��UDSRUWXO�GLQWUH�FXDUWLOD�D�GRXD�úL�D�WUHLD� G��R�YDORDUH�VLWXDW �OD��GLQ�WHUPHQLL�VHULHL� H��HJDO �FX�YDORDUHD�PLQLP �D�FDUDFWHULVWLFLL�

��ÌQWUH�GHFLOD�D�]HFHD�úL�[min�VH�DIO �

a) 90% din termenii seriei analizate; b) 10% din termenii seriei; c) 100% din termenii seriei; G��MXP WDWH�GLQ�WHUPHQLL�VHULHL� H��GHFLOD�D�]HFHD�QX�H[LVW �

TESTUL 16 1) DesprH�SURGXF LD�PDUI �D�ILOLDOHORU�XQHL�VRFLHW L�FRPHUFLDOH�VH�FXQRVF�XUP WRDUHOH�GDWH�

Filiala Structura valorii

SURGXF LHL�PDUI �vQ�perioada T0 (%)

0RGLILFDUHD�DEVROXW �D�YDORULL�SURGXF LHL�PDUI �FD�XUPDUH�D�PRGLILF ULL�SUH XULORU�

în T1�ID �GH�70 (mld.lei)

0RGLILFDUHD�UHODWLY �D�SUH XULORU�vQ�71�ID �GH�

T0 (%) A B C

25 35 …

10 18 6

+20 +25 + 8

6H�FXQRDúWH�GHDVHPHQHD�F �YDORDUHD�SURGXF LHL�PDUI �OD�ILOLDOD�&�vQ�70 a fost 80 mld. lei. Pentru a FDUDFWHUL]D� vQ� GLQDPLF � SURGXF LD� PDUI � D� VRFLHW LL� FRPHUFLDOe s-a determinat indicele valorii, YROXPXOXL� IL]LF� úL� DO� SUH XULORU� SUHFXP� úL� FRHILFLHQWXO� GH� FRUHOD LH� �U�� GLQWUH� LQGLFLL� LQGLYLGXDOL� DL�YROXPXOXL�IL]LF�úL�DL�SUH XULORU��9DORULOH�DFHVWRUD��vQ�RUGLQHD�SUHFL]DW �PDL�VXV�VXQW�

a) ��úL�U �);

b) ��úL�U!��

c) ��úL�U!��

d) ��úL�U��

e) ��úL�U��

2) 8Q� VRQGDM�$� GH� YROXP�Q� vQ� ED]D� F UXLD� VH� HVWLPHD] �PHGLD�P�D� SRSXOD LHL� VWDWLVWLFH� SHQWUX�

YDULDELOD�;�SULQ� HVWLPD LD� ax � HVWH�PDL� HILFLHQW� GHFkW� VRQGDMXO�%� GH� DFHODúL� YROXP�Q� vQ� ED]D�

F UXLD�VH�HVWLPHD] �DFHHDúL�PHGLH�P�SULQ�HVWLPD LD� bx ��GDF �H[LVW �UHOD LD�

a) )()()()( baba xDxDsimxMxM <== ;

b) )()()()( baba xDxDsauxMxM <= ;

c) )()()()( baba xDxDsimxMxM >= ;

d) )()()()( baba xDxDsixMxM << ;

e) )()()()( baba xDxDsauxMxM << .

1RW ��0�[��úL�'�[��UHSUH]LQW �PHGLD�úL�GLVSHUVLD�OXL�;�

3) Dintre conturile macroeconomice, din SCN, nu are sold:

a) FRQWXO�GH�UHSDUWL LH�D�YHQLWXULORU� b) contul de creare a veniturilor; c) FRQWXO�SURGXF LH� d) contul sintetic de bunuri; e) contul de modificare a patrimoniului;

4) 3H�ED]D�YDORULORU�LQGLYLGXDOH�DOH�DFHOHLDúL�YDULDELOH�QXPHULFH�;��vQUHJLVWUDWH�OD�FHOH�Q�XQLW L�DOH�

SRSXOD LHL�VWDWLVWLFH�3�V-D�FDOFXODW�PHGLD�DULWPHWLF �� x ��YDORDUHD�PHGLDQ ��0H��úL�XQLFD�YDORDUH�PRGDO � �0R�� ÌQWUH� FHOH� WUHL� YDORUL� VLQWHWLFH� V-D� VWDELOLW� F � H[LVW � XUP WRDUHD� LQHJDOLWDWH��

MoMex << ��$FHDVW �LQHJDOLWDWH�

a) VXJHUHD] �F �FHOH�PDL�PXOWH�XQLW L�GLQ�3�vQUHJLVWUHD] �YDORUL�LQGLYLGXDOH�GHSODVDWH�F WUH�valorile mari;

b) VXJHUHD] �F �FHOH�PDL�PXOWH�YDORUL�LQGLYLGXDOH�GLQ�FHOH�vQUHJLVWUDWH�VXQW�GHSODVDWH�F WUH�valorile mici;

c) HYLGHQ LHD] �FRQFHQWUDUHD�YDORULORU�LQGLYLGXDOH�F WUH�0R�

d) HYLGHQ LHD] �VLPHWULD�UHSDUWL LHL�VWDWLVWLFH�

e) QX�DUH�QLFL�R�VHPQLILFD LH�HFRQRPLF �

5) Soldul contului macroeconomic „Crearea veniturilor” este:

a) 9DORDUHD�DG XJDW �EUXW �

b) 3URGXVXO�LQWHUQ�EUXW�OD�SUH XULOH�IDFWRULORU�

c) 3URGXVXO�LQWHUQ�EUXW�OD�SUH XULOH�SLH HL�

d) 3URGXVXO�LQWHUQ�QHW�OD�SUH XULOH�IDFWRULORU�

e) 3URGXVXO�QD LRQDO�QHW�OD�SUH XULOH�IDFtorilor.

6) 6H�FRQVLGHU �R�SRSXOD LH�VWDWLVWLF �3�GH�YROXP�1 Q.k: unde, n-�YROXPXO�HúDQWLRQXOXL�úL�N- un QXP U�QDWXUDO�QHQXO�GLIHULW�GH��/D�XQLW LOH�GH�REVHUYDUH�D�HúDQWLRQXOXL�GH�YROXP�Q�– normal –

s-D� XUP ULW� YDULDELOD� QXPHULF � ;� DOH� F UHL� YDORUL� LQGLYLduale au fost ixi = (cu I=1,2,…,N). 'LVSHUVLD�PHGLHL�GH�VHOHF LH��vQ�FD]XO�vQ�FDUH�HúDQWLRQXO�D�IRVW�IRUPDW�DOHDWRU�úL�I U �UHYHQLUH��este:

a). ;

2

1+N

b). 12

)1( +NN

; c). 2)1(

Nk −

; d). 12

)1)(1( +− Nk

; e). 12

1−k

;

7) 3HQWUX�YHULILFDUHD�OHJ WXULL�DXWRUHJUHVLYH��GH�RUGLQXO�vQWkL��GLQWUH�WHUPHQLL�XQHL�VHULL�FURQRORJLFH�VH�XWLOL]HD] �FHO�PDL�IUHFYHQW�

a) WHVWXO�GH�YHULILFDUH�D�QRUPDOLW LL�UH]LGXXULORU�

b) testul Durbin-Watson definit prin ∑

∑

=

=−−

= n

tt

n

ttt

e

eed

1

2

2

21)(

, unde ttt yye ˆ−= este eroarea de PRGHO��UH]LGXDO ��DIHUHQW �YDORULL�W�D�YDULDELOHL�WLPS�

c) testul Durbin-Watson definit prin ∑

∑

=

=−−

= n

tt

n

ttt

e

eed

1

2

3

21)(

;

d) testul Durbin-Watson definit prin ∑

∑

=

=−−

= n

tt

n

ttt

e

eed

1

2

21)(

;

e) testul Student.

8) $QDOL]kQG� OHJ WXUD� GLQWUH� 6&1� úL� WDEHOXO� LQSXW-RXWSXW� �7,2�� FRQVWDW P� F � SULQ� vQVXPDUHD�FRPSRQHQWHORU�FDGUDQXOXL��DO�7,2�VH�RE LQH�

a) 3,%�OD�SUH XULOH�SLH HL�SULQ�PHWRGD�XWLOL] ULL�ILQDOH�

b) 3,%�OD�SUH XULOH�SLH HL�SULQ�PHWRGD�GH�SURGXF LH�

c) P1%�OD�SUH XULOH�SLH HL�SULQ�PHWRGD�XWLOL] ULL�ILQDOH�

d) 3,%�OD�SUH XULOH�SLH HL�SULQ�PHWRGD�YHQLWXULORU�

e) 311�OD�SUH XULOH�IDFWRULORU�

9) 3HQWUX�DFHHDúL�VHULH�FURQRORJLF �WUHQGXO�V-a determinat prin mai multe metode. Pentru fiecare PHWRG �V-a calculat eroarea mHGLH�S WUDWLF �D�DMXVW ULL��6-DX�RE LQXW�XUP WRDUHOH�UH]XOWDWH�

0HWRGD�GH�DMXVWDUH�XWLOL]DW (URDUHD�PHGLH�S WUDWLF �D�DMXVW ULL��X�P��

• 0HWRGD�LQGLFHOXL�PHGLX�GH�GLQDPLF ��0,0� 29.51

• Metoda sporului mediu (MSM) 27.88

• Metoda mediilor mobile (MMM) cu periodicitatea p=5 ani

17.14

• MMM cu p=2 ani 5.97

3UHFL]D L�FDUH�GLQ�DILUPD LLOH�XUP WRDUH�HVWH�IDOV �

a) FHD�PDL��PLF �SUHFL]LH�vQ�DMXVWDUH�HVWH�RIHULW �GH�0,0�

b) FHYD�PDL�SUHFLV ��GDU�QX�VHPQLILFDWLY�PDL�EXQ ��HVWH�DMXVWDUHD�SULQ�06��

c) WHQGLQ D�GHWHUPLQDW �SULQ�000�FX�S ��HVWH�PDL�DSURSLDW �GH�HYROX LD�UHDO �GHFkW�FHD�RE LQXW �SULQ�000�FX�S ��GHRDUHFH�QHWH]LUHD�RVFLOD LLORU�HVWH�PDL�DFFHQWXDW �

d) DMXVWDUHD�SULQ�000�FX�S ��HVWH�PDL�EXQ �GHFkW�FHD�SULQ�000�FX�S ��

e) FHD�PDL�EXQ �DMXVWDUH�FRPSDUDWLY�FX�FHOHODOWH�PHWRGH�XWLOL]DWH�HVWH�FHD�RE LQXW �SULQ�000�FX�p=2.

10) (IHFWXDUHD�FDOFXOHORU�PDFURHFRQRPLFH�vQ�6&1�QX�SRDWH�IDFH�DEVWUDF LH�GH�IOX[XULOH�JHQHUDWH�GH�DFWLYLWDWHD� VHFWRUXOXL� ÄVWDW´�� 5HIHULWRU� OD� DFHVW� VHFWRU� SUHFL]D L� FDUH� GLQ� XUP WRDUHOH� DILUPD LL�HVWH�IDOV �

a) veniturile sale provin din impozite de la menaje private )( StMI , de la firme )( St

FI �úL�SDU LDO�GLQ�SDUWLFLSDUHD�OD�DFWLYLWDWHD�HFRQRPLF �

b) YHQLWXULOH� VDOH� VXQW� IRORVLWH� SHQWUX� UHPXQHUDUHD� DQJDMD LORU� GLQ� VHFWRUXO� SXEOLF� )( TSTV ,

SHQWUX� FXPS UDUHD� GH� EXQXUL� GH� OD� ILUPH� )( STPI �� SHQWUX� WUDQVIHUXUL� F WUH� ILUPH�

�VXEYHQ LL-6��úL�F WUH�PHQDMH�SULYDWH� )( MSTT ;

c) DFWLYLWDWHD� VWDWXOXL� GH� SURGXFHUH� D� EXQXULORU� FROHFWLYH�� GH� UHJXO �� VH� FRQFHQWUHD] � vQ�fluxuri bilaterale;

d) STM

STSTM

STSTF

STM ESTPIVII ++++=+ , unde EST GLIHUHQ D�GLQWUH�YHQLWXUL�úL�FKHOWXLHOL�

e) DFWLYLWDWHD�VWDWXOXL�JHQHUHD] �SUHSRQGHUHQW�IOX[XUL�vQWU-un singur sens.

11) În cazul sondajului simplu, de volum n, analizându-VH� DEDWHUHD�PHGLH� S WUDWLF � D�PHGLHL� GH�VHOHF LH�FD�P VXU WRU�DO�HURULL�PHGLL�GH�UHSUH]HQWDWLYLWDWH��VH�FRQVWDW �

a) FkQG�Q�FUHúWH�SUHFL]LD�HVWLP ULL�SDUDPHWULORU�FUHúWH�DSUR[LPDWLY�GH� n ori;

b) FkQG�Q�FUHúWH�SUHFL]LD�HVWLP ULL�SDUDPHWULORU�FUHúWH�H[DFW�GH� n ori;

c) FkQG�Q�VFDGH�SUHFL]LD�HVWLP ULL�SDUDPHWULORU�FUHúWH�DSUR[LPDWLY�GH� n ori;

d) FkQG�Q�FUHúWH�SUHFL]LD�HVWLP ULL�SDUDPHWULORU�FUHúWH�WRW�GH�Q�RUL�

e) FUHúWHUHD�VDX�UHGXFHUHD�OXL�Q�QX�DIHFWHD] �SUHFL]LD�HVWLP ULL�SDUDPHWULOor.

12) &RQFRUGDQ D�vQWUH�UDQJXULOH�GH�OD��OD�Q�DOH�YDORULORU�YDULDELOHL��IDFWRULDOH�úL�UDQJXULOH�GH�OD��OD�Q� DOH� YDORULORU� YDULDELOHL� HIHFW� VH� SRDWH� H[SULPD� QXPHULF� FX� FRHILFLHQWXO� GH� FRUHOD LH� DO� OXL�Spearman (rs��úL�FX�FHO�DO�OXL�.HQGDOO��Uk). În cazul îQ�FDUH�Q�HVWH�PDUH�vQWUH�FHL�GRL�FRHILFLHQ L�H[LVW �UHOD LD�

a) ks rr

3

2=; b)

sk rr3

2=; c) 132 =− sk rr ; d) 0=+ sk rr ; e) sk rr = .

13) &DOFXOXO�DYX LHL�SURGXFWLYH��VH�UHDOL]HD] �SH�ED]D�XUP WRDUHORU�HOHPHQWH�

a) UHVXUVHOH�QDWXUDOH��VWRFXUL�úL�UH]HUYH�úL�EXQXUL�GH�FDSLWDO�

b) VWRFXUL�úL�UH]HUYH��SUHFXP�úL�XWLODMH��PDúLQL�úL�HFKLSDPHQWH�

c) VWRFXUL��UH]HUYH�úL�PLMORDFH�IL[H�

d) DYX LD�GH�FRQVXP�D�PHQDMHORU�SULYDWH�úL�D�VWDWXOXL��VWRFXUL��UH]HUYH�úL�EXQXri de capital;

e) VWRFXUL�úL�UH]HUYH��XWLODMH��PDúLQL��HFKLSDPHQWH��FRQVWUXF LL�

14) 8Q� DQDOLVW� GH�PDUNHWLQJ� XUP UHúWH� HIHFWXO� FKHOWXLHOLORU� FX� UHFODPD� �[�� DVXSUD� YHQLWXULORU� GLQ�YkQ] UL� �\�� SH� R� SHULRDG � GH� �� GH� OXQL�� ÌQ� XUPD� SUHOXFU ULL� GDWHORU� SULPDUH� V-au oE LQXW�

XUP WRDUHOH� UH]XOWDWH�� 44.3532 =xσ ; 09.222 =yσ

; cov(x,y)= 70.7; 2.75=x � úL� 84.18=y . În urma analizei statistice s-D�DMXQV�OD�FRQFOX]LD�F �GHSHQGHQ D�GLQWUH�FHOH�GRX �YDULDELOH�SRDWH�IL�modelat �SULQWU-R�IXQF LH�GH�JUDGXO�vQWkL��(FXD LD�PRGHOXOXL�GH�UHJUHVLH�úL�YDORDUHD�FDOFXODW �D�OXL�W��XWLOL]DW �SHQWUX�WHVWDUHD�VHPQLILFD LHL�FRHILFLHQWXOXL�GH�FRUHOD LH��vQ�DFHDVW �RUGLQH��HVWH�

a) xy 2.08.3ˆ += úL�W ��

b) xy 2.08.3ˆ −= úL�W ��

c) xy 8.32.0ˆ += úL�W ��

d) xy 8.32.0ˆ += úL�W ��

e) xy 2.08.3ˆ += úL�W ��

15) Despre patru firme dintr-XQ�FRQFHUQ�VH�FXQRVF�XUP WRDUHOH�GDWH� Firma 6WUXFWXUD�SURGXF LH�

în perioada T0 (%) 3URGXF LD�UHDOL]DW �GH�ILUPH�vQ�T1 în raport cu cea a firmei A

'HS úLUHD��QHUHDOL]DUHD��SURJUDPXOXL�GH�SURGXF LH�vQ�71 (%)

A 15 1.0 +20 B 25 1.8 -10 C 20 1.4 0 D 40 2.5 +30

6H� PDL� FXQRDúWH�� GH� DVHPHQHD�� F � SURGXF LD� UHDOL]DW � GH� ILUPD� $� vQ� 71 a fost de 75 mld. lei, GHYDQVkQG�SURGXF LD�UHDOL]DW �GLn T0�FX��ÌQ�XUPD�SUHOXFU ULL�GDWHORU�SHQWUX�vQWUHJXO�FRQFHUQ�V-D�DMXQV�OD�XUP WRDUHD�FRQFOX]LH�

a) SURJUDPXO�GH�SURGXF LH�D�IRVW�GHS úLW�FX��LDU�SURGXF LD�UHDOL]DW �GLQ�71�D�IRVW�HJDO �FX�FHD�din T0;

b) SURJUDPXO�GH�SURGXF LH�QX�V-a realizat cu 8.83%, LDU�SURGXF LD�UHDOL]DW �vQ�71 a fost mai mare decât cea din T0;

c) LQGLFHOH�UHDOL] ULL�SURJUDPXOXL�GH�SURGXF LH�vQ�71�D�IRVW�GH��LDU�SURGXF LD�SURJUDPDW �pentru T1�D�FUHVFXW�ID �GH�70 cu 120%;

d) LQGLFHOH�UHDOL] ULL�SURJUDPXOXL�GH�SURGXF LH�D�IRVW�GH��LDU�SURGXF LD�UHDOL]DW �vQ�71 a fost PDL�PDUH�GHFkW�FHD�UHDOL]DW �vQ�70 cu 117.9 mld. lei;

e) SURJUDPXO�GH�SURGXF LH�SHQWUX��71�D�IRVW�GHS úLW�FX��POG��OHL�LDU�SURGXF LD�UHDOL]DW �vQ�71 a fost mai mare decât cea din T0 cu peste 5%.

62/8 ,,�$/(�7(67(/25�*5,/

TESTUL 1 1) c; 2) d; 3) b; 4) b; 5) a; 6) c; 7) b; 8) b; 9) a; 10) d; 11) a; 12) d; 13) e; 14) b; 15) b; 16)

c; 17) d; 18) e; 19) e; 20) c; 21) c; 22) a; 23) a; 24) a; 25) b; 26) a; 27) b; 28) a; 29) a; 30)

b; 31) b; 32) c; 33) a; 34) c; 35) e; 36) a; 37) c; 38) c; 39) e; 40) b.

TESTUL 2 1) e; 2) a; 3) b; 4) c; 5) b; 6) d; 7) c; 8) d; 9) a; 10) ; 11) a; 12) b; 13) d; 14) a; 15) c; 16) a;

17) b; 18) c; 19) e; 20) a; 21) e; 22) b; 23) b; 24) b; 25) d; 26) a; 27) b; 28) e; 29) b; 30) b;

31) a; 32) c; 33) c; 34) c; 35) a; 36) e; 37) b; 38) a; 39) b; 40) c.

TESTUL 3 1) d; 2) c; 3) d; 4) d; 5) c; 6) c; 7) e; 8) d; 9) c; 10) b; 11) e; 12) d; 13) b; 14) c; 15) e; 16)

a; 17) b; 18) d; 19) d; 20) c; 21) b; 22) b; 23) a; 24) d; 25) c; 26) c; 27) a; 28) e; 29) d; 30)

d; 31) b; 32) a

TESTUL 4

1) a; 2) b; 3) c; 4) c; 5) e; 6) b; 7) b; 8) b; 9) a; 10) d; 11) c; 12) a; 13) c; 14) d; 15) b; 16)

e; 17) b; 18) e; 19) b; 20) b; 21) a; 22) a; 23) d; 24) b, d; 25) c; 26) b; 27) d; 28) a; 29) d;

30) a; 31) a; 32) b; 33) b; 34) b; 35) e; 36) c; 37) b; 38) a; 39) e; 40) d.

TESTUL 5 1) b; 2) d; 3) b; 4) e; 5) c; 6) c; 7) c; 8) d; 9) a; 10) ; 11) b; 12) e; 13) c; 14) c; 15) b; 16) a;

17) d; 18) c; 19) b; 20) c; 21) e; 22) d; 23) b; 24) a; 25) a; 26) b; 27) c; 28) d; 29) d; 30) d;

31) b; 32) c; 33) c; 34) e; 35) a; 36) c; 37) c; 38) d; 39) e.

TESTUL 6

1) e; 2) e; 3) b; 4) d; 5) c; 6) e; 7) e; 8) d; 9) b; 10) c; 11) e; 12) c; 13) b; 14) d; 15) a; 16)

e; 17) c; 18) a; 19) c; 20) ; 21) e; 22) c; 23) e; 24) e; 25) d; 26) e; 27) c; 28) a; 29) d; 30)

c; 31) c; 32) a; 33) c; 34) c; 35) e; 36) b; 37) d; 38) c; 39) d.

TESTUL 7

1) c; 2) c; 3) a; 4) a; 5) e; 6) c; 7) c; 8) b; 9) d; 10) d; 11) e; 12) a; 13) e; 14) d; 15) d; 16)

b; 17) b; 18) c; 19) b; 20) a; 21) c; 22) d; 23) b; 24) a; 25) d; 26) a; 27) d; 28) a; 29) b; 30)

d; 31) d.

TESTUL 8

1) ; 2) a; 3) e; 4) b; 5) b; 6) d; 7) b; 8) a; 9) a; 10) a; 11) e; 12) c; 13) e; 14) a; 15) c; 16) a;

17) b; 18) d; 19) a; 20) a; 21) a; 22) c; 23) c; 24) e; 25) b; 26) a; 27) e; 28) c; 29) e; 30) ;

31) .

TESTUL 9 1) b; 2) b; 3) c; 4) b; 5) c; 6) b,e; 7) c; 8) a; 9) a; 10) b,e; 11) a; 12) a; 13) a; 14) a; 15) a;

16) d; 17) e; 18) b; 19) b; 20) a,b,c,d; 21) c; 22) c; 23) c; 24) d; 25) c; 26) c; 27) a; 28) b;

29) a,d; 30) e.

TESTUL 10

1) d; 2) e; 3) d; 4) c; 5) e; 6) b; 7) e; 8) c; 9) a; 10) b; 11) e; 12) c; 13) a; 14) d; 15) a; 16)

b; 17) d; 18) e; 19) e; 20) d; 21) b; 22) a; 23) d; 24) d; 25) c; 26) d; 27) c; 28) a; 29) c; 30)

e; 31) d.

TESTUL 11 1) b; 2) d; 3) b; 4) a; 5) b; 6) c; 7) d; 8) b,c,d; 9) b; 10) b,c,e; 11) b; 12) a.

TESTUL 12

1) d; 2) c; 3) b; 4) d; 5) c; 6) c; 7) a; 8) c; 9) a; 10) e; 11) e; 12) a; 13) a; 14) a; 15) a; 16) b;

17) c; 18) a; 19) d; 20) d; 21) b; 22) ; 23) a; 24) a; 25) e; 26) c; 27) c; 28) b; 29) a; 30) b.

TESTUL 13

1) b; 2) b; 3) b; 4) a; 5) d; 6) b; 7) c; 8) a; 9) a; 10) c; 11) e; 12) b; 13) c; 14) a; 15) d; 16)

c; 17) a; 18) c; 19) b; 20) d; 21) b; 22) c; 23) a; 24) b; 25) c; 26) b; 27) e; 28) b; 29) d; 30)

c

TESTUL 14 1) c; 2) d; 3) a; 4) d; 5) a; 6) c; 7) c; 8) b; 9) d; 10)a; 11) d; 12) a; 13) c; 14) b; 15) c; 16) b;

17) c; 18) a; 19) b; 20) d; 21) d; 22) C; 23) e; 24) a; 25) d; 26) c; 27) d; 28) a; 29) a; 30) b;

31) a; 32) a; 33) a; 34) a; 35) c; 36) c; 37) a; 38) d; 39) a; 40) b;

TESTUL 15 1) a; 2) a; 3) c; 4) d; 5) e; 6) b; 7) a; 8) c; 9) a; 10) e; 11) a; 12) b; 13) e; 14) d; 15) b; 16) a;

17) a; 18) b; 19) b; 20) e; 21) e; 22) d; 23) a; 24) c; 25) b; 26) b; 27) d; 28) a; 29) b; 30) d;

31) c; 32) d; 33) b; 34) a; 35) c; 36) d; 37) a; 38) d; 39) d; 40) e.

TESTUL 16 1) b; 2) a; 3) d; 4) b; 5) d; 6) d; 7) b; 8) a; 9) c; 10) c; 11) a; 12) b; 13) e; 14) e;

15) e.

Anexa 1

)XQF LD��*DXVV-Laplace

( ) ∫−

=Φz u

duez0

2

2

2

1

π

z 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,0 0,0000 0,0040 0,0080 0,0120 0,0160 0,0199 0,0239 0,0279 0,0319 0,0359 0,1 0,0398 0,0438 0,0478 0,0517 0,0557 0,0596 0,0636 0,0675 0,0714 0,0753 0,2 0,0793 0,0832 0,0871 0,0910 0,0948 0,0987 0,1026 0,1064 0,1103 0,1141 0,3 0,1179 0,1217 0,1255 0,1293 0,1331 0,1368 0,1406 0,1443 0,1480 0,1517 0,4 0,1554 0,1591 0,1628 0,1664 0,1700 0,1736 0,1772 0,1808 0,1844 0,1879

0,5 0,1915 0,1950 0,1985 0,2019 0,2054 0,2088 0,2123 0,2157 0,2190 0,2224 0,6 0,2257 0,2291 0,2324 0,2357 0,2389 0,2422 0,2454 0,2486 0,2517 0,2549 0,7 0,2580 0,2611 0,2642 0,2673 0,2704 0,2734 0,2764 0,2794 0,2823 0,2852 0,8 0,2881 0,2910 0,2939 0,2967 0,2995 0,3023 0,3051 0,3078 0,3106 0,3133 0,9 0,3159 0,3186 0,3212 0,3238 0,3264 0,3289 0,3315 0,3340 0,3365 0,3389

1,0 0,3413 0,3438 0,3461 0,3485 0,3508 0,3531 0,3554 0,3577 0,3599 0,3621 1,1 0,3643 0,3665 0,3686 0,3708 0,3729 0,3749 0,3770 0,3790 0,3810 0,3830 1,2 0,3849 0,3869 0,3888 0,3907 0,3925 0,3944 0,3962 0,3980 0,3997 0,4015 1,3 0,4032 0,4049 0,4066 0,4082 0,4099 0,4115 0,4131 0,4147 0,4162 0,4177 1,4 0,4192 0,4207 0,4222 0,4236 0,4251 0,4265 0,4279 0,4292 0,4306 0,4319

1,5 0,4332 0,4345 0,4357 0,4370 0,4382 0,4394 0,4406 0,4418 0,4429 0,4441 1,6 0,4452 0,4463 0,4474 0,4484 0,4495 0,4505 0,4515 0,4525 0,4535 0,4545 1,7 0,4554 0,4564 0,4573 0,4582 0,4591 0,4599 0,4608 0,4616 0,4625 0,4633 1,8 0,4641 0,4649 0,4656 0,4664 0,4671 0,4678 0,4686 0,4693 0,4699 0,4706 1,9 0,4713 0,4719 0,4726 0,4732 0,4738 0,4744 0,4750 0,4756 0,4761 0,4767

2,0 0,4772 0,4778 0,4783 0,4788 0,4793 0,4798 0,4803 0,4808 0,4812 0,4817 2,1 0,4821 0,4826 0,4830 0,4834 0,4838 0,4842 0,4846 0,4850 0,4854 0,4857 2,2 0,4861 0,4864 0,4868 0,4871 0,4875 0,4878 0,4881 0,4884 0,4887 0,4890 2,3 0,4893 0,4896 0,4898 0,4901 0,4904 0,4906 0,4909 0,4911 0,4913 0,4916 2,4 0,4918 0,4920 0,4922 0,4925 0,4927 0,4929 0,4931 0,4932 0,4934 0,4936

2,5 0,4938 0,4940 0,4941 0,4943 0,4945 0,4946 0,4948 0,4949 0,4951 0,4952 2,6 0,4953 0,4955 0,4956 0,4957 0,4959 0,4960 0,4961 0,4962 0,4963 0,4964 2,7 0,4965 0,4966 0,4967 0,4968 0,4969 0,4970 0,4971 0,4972 0,4973 0,4974 2,8 0,4974 0,4975 0,4976 0,4977 0,4977 0,4978 0,4979 0,4979 0,4980 0,4981 2,9 0,4981 0,4982 0,4982 0,4983 0,4984 0,4984 0,4985 0,4985 0,4986 0,4986

3,0 0,4987 0,4987 0,4987 0,4988 0,4988 0,4989 0,4989 0,4989 0,4990 0,4990

Anexa 2

Tabel cu valorile reparti LHL��6WXGHQW�vQ�IXQF LH�GH�SUREDELOLWDWHD� ( )P t t≤ α úL�QXP UXO�³I´�DO�

gradelor de libertate 1LYHO�GH�VHPQLILFD LH�SHQWUX�WHVWXO�ELODWHUDO

f

0,50

0,20 0,10 0,05 0,02 0,01 0,002 0,001 0,0001

1 1,000 3,078 6,314 12,706 31,821 63,657 318,309 636,618 6366,198 2 0,816 1,886 2,920 4,303 6,965 9,925 22,327 31,598 99,992 3 0,765 1,638 2,353 3,482 4,541 5,841 10,214 12,924 28,000 4 0,741 1,533 2,132 2,776 3,747 4,604 7,173 8,610 15,544 5 0,727 1,476 2,015 2,571 3,365 4,032 5,893 6,869 11,178 6 0,718 1,440 1,943 2,447 3,143 3,707 5,208 5,959 9,082 7 0,711 1,415 1,895 2,365 2,998 3,499 4,785 5,408 7,885 8 0,706 1,397 1,860 2,306 2,896 3.355 4,501 5,041 7,120 9 0,703 1,383 1,833 2,262 2,821 3,250 4,297 4,781 6,594

10 0,700 1,372 1,812 2,228 2,764 3,169 4,144 4,587 6,211 11 0,697 1,363 1,796 2,201 2,718 3,106 4,025 4,437 5,921 12 0,695 1,356 1,782 2,179 2,681 3,055 3,930 4,318 5,694 13 0,694 1,350 1,771 2,160 2,650 3,102 3,852 4,221 5,513 14 0,692 1,345 1,761 2,145 2,624 2,977 3,787 4,140 5,363 15 0,691 1,341 1,753 2,131 2,602 2,947 3,733 4,073 5,239 16 0,690 1,337 1,746 2,120 2,583 2,921 3,686 4,015 5,134 17 0,689 1,333 1,740 2,110 2,567 2,898 3,646 3,965 5,014 18 0,688 1,330 1,734 2,101 2,552 2,878 3,610 3,922 ‘4,966 19 0,688 1,328 1,729 2,093 2,539 2,861 3,579 3,883 4,897 20 0,687 1,325 1,725 2,086 2,528 2,845 3,552 3,850 4,837 21 0,686 1,323 1,721 2,080 2,518 2,831 3,527 3,819 4,784 22 0,686 1,321 1,717 2,074 2,508 2,819 3,505 3,792 4,736 23 0,685 1,319 1,714 2,069 2,500 2,807 3,485 3,767 4,693 24 0,685 1,318 1,711 2,064 2,492 2,797 3,467 3,745 4,654 25 0,684 1,316 1,708 2,060 2,485 2,787 3,450 3,725 4,619 26 0,684 1,315 1,706 2,056 2,479 2,779 3,435 3,707 4,587 27 0,684 1,314 1,703 2,052 2,473 2,771 3,421 3,690 4,558 28 0,683 1,313 1,701 2,048 2,467 2,763 3,408 3,674 4,530 29 0,683 1,312 1,699 2,045 2,462 2,756 3,396 3,659 4,506 30 0,683 1,310 1,697 2,042 2,457 2,750 3,385 3,646 4,482 35 0,682 1,306 1,690 2,030 2,438 2,724 3,340 3,591 4,389 40 0,681 1,303 1,684 2,021 2,423 2,704 3,307 3,551 4,321 45 0,680 1,301 1,679 2,014 2,412 2,690 3,281 3,520 4,269 50 0,679 1,299 1,676 2,009 2,403 2,678 3,261 3,496 4,288 60 0,679 1,296 1,671 2,000 2,390 2,660 3,232 3,460 4,169 70 0,678 1,294 1,667 1,994 2,381 2,648 3,211 3,435 4,127 80 0,678 1,292 1,664 1,990 2,374 2,639 3,195 3,416 4,096 90 0,677 1,291 1,662 1,987 2,368 2,632 3,183 3,402 4,072

100 0,677 1,290 1,660 1,984 2,364 2,626 3,174 3,390 4,053 120 0,677 1,289 1,658 1,980 2,358 2,617 3,160 3,373 4,025 200 0,676 1,286 1,653 1,972 2,345 2,601 3,131 3,340 3,970 500 0,675 1,283 1,648 1,965 2,334 2,586 3,107 3,310 3,922

1000 0,675 1,282 1,646 1,962 2,330 2,581 3,098 3,300 3,906 � 0,675 1,282 1,645 1,960 2,326 2,576 3,090 3,290 3,891

f

0,25 0,10 0,05 0,025 0,01 0,005 0,001 0,0005 0,00005

1LYHO�GH�VHPQLILFD LH�SHQWUX�WHVWXO�XQLODWHUDO

Anexa 3 5HSDUWL LD�)��)LVKHU�6QHGHFRU�

9DORULOH�IXQF LHL�)��22

21 ssF = ) pentru l1, l2�JUDGH�GH�OLEHUWDWH�úL�D = 0,05, D = 0,01 nivel de

VHPQLILFD LH

l1 = 1 l1 = 2 l1 = 3 l1 = 4 l1 = 5 l2 α =

0,05 α = 0,01 α =

0,05 α = 0,01 α =

0,05 α = 0,01 α =

0,05 α = 0,01 α =

0,05 α = 0,01

1 161,40 4052,00 199,50 4999,00 215,70 5403,00 224,60 5625,00 230,20 5764,00 2 18,51 98,49 19,00 99,00 19,16 99,17 19,25 99,25 19,30 99,30 3 10,13 34,12 9,55 30,81 9,28 29,46 9,12 28,71 9,01 28,24 4 7,71 21,20 6,94 18,00 6,59 16,69 6,39 15,98 6,26 15,52 5 6,61 16,26 5,79 13,27 5,41 12,06 5,19 11,39 5,05 10,97 6 5,99 13,74 5,14 10,91 4,76 9,78 4,53 9,15 4,39 8,75 7 5,59 12,25 4,74 9,55 4,35 8,45 4,12 7,85 3,97 7,45 8 5,32 11,26 4,46 8,65 4,07 7,59 3,84 7,01 3,69 6,63 9 5,12 10,56 4,26 8,02 3,86 6,99 3,63 6,42 3,48 6,06

10 4,96 10,04 4,10 7,56 3,71 6,55 3,48 5,99 3,33 5,64 11 4,84 9,65 3,98 7,20 3,59 6,22 3,36 5,67 3,20 5,32 12 4,75 9,33 3,88 6,93 3,49 5,95 3,26 5,41 3,11 5,06 13 4,67 9,07 3,80 6,70 3,41 5,74 3,18 5,20 3,02 4,86 14 4,60 8,86 3,74 6,51 3,34 5,56 3,11 5,03 2,96 4,69 15 4,54 8,68 3,68 6,36 3,29 5,42 3,06 4,89 2,90 4,56 16 4,49 8,53 3,63 6,23 3,24 5,29 3,01 4,77 2,85 4,44 17 4,45 8,40 3,59 6,11 3,20 5,18 2,96 4,67 2,81 4,34 18 4,41 8,28 3,55 6,01 3,16 5,09 2,93 4,58 2,77 4,25 19 4,38 8,18 3,52 5,93 3,13 5,01 2,90 4,50 2,74 4,17 20 4,35 8,10 3,49 5,85 3,10 4,94 2,87 4,43 2,71 4,10 21 4,32 8,02 3,47 5,78 3,07 4,87 2,84 4,37 2,68 4,04 22 4,30 7,94 3,44 5,72 3,05 4,82 2,82 4,31 2,66 3,99 23 4,28 7,88 3,42 5,66 3,03 4,76 2,80 4,26 2,64 3,94 24 4,26 7,82 3,40 5,61 3,01 4,72 2,78 4,22 2,62 3,90 25 4,24 7,77 3,38 5,57 2,99 4,68 2,76 4,18 2,60 3,86 26 4,22 7,72 3,37 5,53 2,98 4,64 2,74 4,14 2,59 3,82 27 4,21 7,68 3,35 5,49 2,96 4,60 2,73 4,11 2,57 3,78 28 4,20 7,64 3,34 5,45 2,95 4,57 2,71 4,07 2,56 3,75 29 4,18 7,60 3,33 5,42 2,93 4,54 2,70 4,04 2,54 3,73 30 4,17 7,56 3,32 5,39 2,92 4,51 2,69 4,02 2,53 3,70 40 4,08 7,31 3,23 5,18 2,84 4,31 2,61 3,83 2,45 3,51 60 4,00 7,08 3,15 4,98 2,76 4,13 2,52 3,65 2,37 3,34

120 3,92 6,85 3,07 4,79 2,68 3,95 2,45 3,48 2,29 3,17 ∞ 3,84 6,64 2,99 4,60 2,60 3,78 2,37 3,32 2,21 3,02

Anexa 4

Valorile variabilei χα2

�vQ�IXQF LH�GH�SUREDELOLWDWHD�α χ χα= >P( )2 2

l

0,990

0,975 0,950 0,900 0,100 0,050 0,025 0,010 0,001

1 0,0002 0,0010 0,0039 0,0158 2,71 3,84 5,02 6,63 10,83 2 0,02 0,05 0,10 0,21 4,61 5,99 7,38 9,21 13,82 3 0,12 0,22 0,35 0,58 6,25 7,81 9,35 11,34 16,27 4 0,30 0,48 0,71 1,06 7,78 9,49 11,14 13,28 18,47 5 0,55 0,83 1,15 1,61 9,24 11,07 12,83 15,09 20,52 6 0,87 1,24 1,64 2,20 190,64 12,59 14,45 16,81 22,46 7 1,24 1,69 2,17 2,83 12,02 14,07 16,01 18,47 24,32 8 1,65 2,18 2,73 3,49 13,36 15,51 17,53 20,09 26,13 9 2,09 2,70 3,33 4,17 14,68 16,92 19,02 21,67 27,88

10

2,56 3,25 3,94 4,87 15,99 18,31 20,48 23,21 29,59

11 3,05 3,82 4,57 5,58 17,27 19,67 21,92 24,72 31,26 12 3,57 4,40 5,23 6,30 18,55 21,03 23,34 26,22 32,91 13 4,11 5,01 5,89 7,04 19,81 22,36 24,74 27,69 34,53 14 4,66 5,63 6,57 7,79 21,06 23,68 26,12 29,14 36,12 15 5,23 6,26 7,26 8,55 22,31 25,00 27,49 30,58 37,70 16 5,81 6,91 7,96 9,31 23,54 26,30 28,84 32,00 39,25 17 6,41 7,56 8,67 10,08 24,77 27,59 30,19 33,41 40,79 18 7,01 8,23 9,39 10,86 25,99 28,87 31,53 34,80 42,31 19 7,63 8,91 10,12 11,65 27,20 30,14 32,85 36,19 43,82 20

8,26 9,59 10,85 12,44 28,41 31,41 34,17 37,57 45,32

21 8,90 10,28 11,59 13,24 29,61 32,67 35,48 38,93 46,80 22 9,54 10,98 12,34 14,04 30,81 33,92 36,78 40,29 48,27 23 10,20 11,69 13,09 14,85 32,01 35,17 38,08 41,64 49,73 24 10,86 12,40 13,85 15,66 33,20 36,41 39,37 42,98 51,18 25 11,52 13,12 14,61 16,47 34,38 37,65 40,65 44,31 52,62 26 12,20 13,84 15,38 17,29 35,56 38,88 41,92 45,64 54,05 27 12,88 14,57 16,15 18,11 36,74 40,11 43,19 46,96 55,48 28 13,57 15,31 16,93 18,94 37,92 41,34 44,46 48,28 56,89 29 14,26 16,05 17,71 19,77 39,09 42,56 45,72 49,59 58,30 30

14,95 16,79 18,49 20,60 40,26 43,77 46,98 50,89 59,70

Anexa 5 Tabel cu numere aleatoare

Coloana Rândul

1 2 3 4 5 6 7 8 9 10 1 87 35 67 44 51 49 18 98 97 84 75 22 53 29 10 52 26 87 54 92 2 25 52 29 67 35 99 48 88 40 08 63 68 82 39 38 47 91 39 11 00 3 87 17 83 31 25 59 87 48 25 80 24 08 81 45 21 32 90 08 44 31 4 05 04 40 35 72 95 48 56 77 57 63 19 80 16 48 52 06 47 64 98 5 81 16 09 21 91 71 29 76 54 01 53 17 30 67 62 95 56 58 10 91 6 54 85 78 88 57 91 11 69 10 22 71 87 24 92 52 64 42 82 78 95 7 44 78 19 18 35 40 27 66 89 72 21 17 71 69 95 17 97 17 62 60 8 97 20 98 97 37 33 93 75 18 88 35 85 46 05 07 20 08 17 66 24 9 98 77 57 51 40 41 76 21 18 54 60 61 79 13 94 57 50 73 89 68

10 78 12 77 30 83 30 59 28 73 33 47 07 60 07 45 38 82 10 73 19 11 41 19 70 62 43 46 06 13 22 38 31 18 61 60 07 14 49 16 28 16 12 70 64 30 55 67 46 95 79 63 66 82 56 67 10 76 77 03 22 42 18 13 06 56 09 89 68 87 79 19 35 94 66 18 17 91 72 81 72 77 92 39 14 29 46 18 28 08 88 48 56 49 44 67 82 72 67 28 83 10 26 58 13 15 42 14 55 51 72 95 29 25 15 18 25 68 48 92 87 16 78 43 17 47 16 33 75 87 15 15 23 13 79 62 73 76 69 09 77 82 65 72 47 59 56 17 09 80 99 61 98 08 34 11 88 79 08 32 46 78 35 58 44 16 12 23 18 98 31 57 50 85 80 53 39 05 92 54 42 29 01 35 23 09 81 96 64 19 51 70 52 55 83 12 95 02 79 11 49 79 87 95 98 48 88 08 64 77 20 27 83 61 07 49 05 46 20 35 78 31 34 42 50 68 11 42 14 29 77 21 78 84 69 15 64 42 92 39 36 08 56 39 35 02 92 78 46 63 82 98 22 22 12 89 66 49 09 99 10 62 53 19 31 81 83 50 43 37 42 10 00 23 69 41 59 54 82 72 44 66 64 03 76 59 12 12 41 56 34 90 26 06 24 54 99 46 54 51 38 59 07 64 21 81 17 88 47 23 05 63 43 08 67 25 99 91 82 79 92 62 44 24 01 34 45 16 33 56 17 78 42 86 70 94 26 54 96 58 70 68 38 48 37 08 25 51 28 42 09 45 96 78 84 23 59 27 88 37 31 01 58 97 65 10 96 97 32 75 65 73 06 01 60 19 68 10 28 49 40 84 90 19 91 75 41 55 63 98 25 01 69 29 57 72 77 75 95 29 21 18 87 99 98 48 38 12 64 18 78 50 43 97 04 07 85 18 98 87 30 06 31 99 51 27 13 43 21 07 55 90 35 29 84 08 70 66 03 18 96 31 58 56 75 14 10 32 35 15 47 05 38 54 92 08 08 72 79 66 65 20 32 83 00 61 74 76 68 58 67 31 58 77 93 92 44 94 16 01 29 32 19 33 51 45 05 35 16 73 80 43 44 64 19 14 52 22 81 62 79 68 13 37 34 90 55 27 93 33 06 76 42 16 08 16 81 86 15 09 79 30 86 74 77 35 68 65 05 79 82 57 78 22 37 41 82 44 07 43 98 89 78 23 96 04 36 51 84 34 98 03 38 87 12 79 88 57 88 46 81 62 56 94 91 06 59 37 85 23 92 01 84 05 83 64 12 95 51 44 60 64 68 86 53 05 26 16 38 31 95 14 27 80 20 36 96 81 84 88 55 50 47 22 39 70 56 66 41 39 91 55 02 31 42 31 54 72 45 32 43 76 89 95 74 04 79 33 46 91 40 46 80 10 25 92 61 83 83 58 58 71 46 29 95 72 38 89 05 26 05 41 07 78 66 31 63 37 30 38 95 57 84 99 85 32 91 28 06 85 67 13 42 48 69 61 99 15 93 98 43 38 84 46 49 84 89 44 51 75 42 45 17 43 70 69 68 47 02 10 60 99 02 84 41 36 81 65 44 08 49 93 49 62 44 98 63 62 58 95 62 61 53 28 49 20 20 21 49 45 35 01 57 45 64 45 76 50 34 97 87 14 65 07 59 45 02 56 89 50 85 03 42 65 07 35 46 23 47 83 35 73 08 83 52 47 19 79 85 26 07 53 80 00 33 18 07 47 18 38 27 88 50 09 58 11 96 83 05 86 49 61 35 03 46 63 88 48 48 93 93 51 63 96 33 82 81 58 02 19 36 55 80 74 84 14 63 79 22 49 65 23 13 22 01 10 20 01 11 15 36 43 49 49 19 24 77 70 95 08 50 77 19 27 34 65 55 62 52 47 73 09 34 58 38 28 28 18 15 82 65

Anexa 5 (continuare) Coloana Rândul

1 2 3 4 5 6 7 8 9 10 51 61 39 57 28 82 11 49 79 26 68 85 55 27 69 41 27 70 74 51 72 52 20 79 22 91 99 39 15 98 39 00 87 41 31 42 37 79 94 48 90 37 53 96 28 37 52 96 74 84 48 85 77 65 63 36 72 11 78 83 00 01 39 54 35 25 98 71 48 26 14 39 73 31 81 15 76 22 17 76 03 72 32 70 55 65 29 37 18 09 40 08 51 59 45 46 94 29 57 71 74 74 19 16 83 56 30 06 86 39 75 15 85 61 83 12 60 77 88 14 67 62 50 00 59 77 57 62 63 22 21 37 17 88 28 54 66 45 52 75 40 99 32 81 19 22 04 58 44 86 78 61 36 08 79 10 56 33 15 56 66 39 74 69 73 49 36 74 59 70 30 42 32 63 10 22 08 32 39 43 53 02 33 45 90 62 92 50 23 60 53 31 94 17 71 65 36 71 69 62 96 13 24 22 91 24 36 89 98 05 61 01 19 00 22 89 97 50 76 89 94 09 25 09 40 22 55 00 11 94 82 62 59 53 30 97 09 82 69 01 51 35 69 28 68 18 68 62 34 37 36 16 63 12 64 27 42 66 42 47 13 10 10 52 61 34 19 91 68 00 50 19 99 64 83 91 19 54 89 49 54 79 30 64 16 09 07 34 69 84 09 30 11 44 65 16 84 22 33 12 54 06 08 44 90 77 47 41 03 68 80 32 94 60 48 66 64 20 55 02 45 72 18 37 53 54 01 07 13 90 41 61 99 14 79 21 67 83 21 49 79 27 82 84 32 91 86 21 35 30 65 29 07 42 09 69 16 68 58 34 75 23 57 30 02 73 03 17 06 18 71 83 98 61 47 77 12 32 69 49 74 56 30 49 30 04 11 31 75 07 02 03 56 36 78 66 59 03 66 70 14 36 53 40 17 30 00 12 32 71 07 35 92 67 32 69 77 14 47 44 71 56 77 08 01 57 35 91 01 34 17 82 26 17 74 73 63 28 82 81 52 72 62 47 85 70 90 70 96 78 94 75 63 83 92 83 15 17 22 98 66 54 73 62 33 40 58 01 57 14 36 29 50 42 99 77 10 90 09 65 93 78 59 74 00 51 35 07 12 40 90 41 69 86 17 61 75 05 63 77 50 23 06 87 75 55 30 12 69 80 94 51 03 79 62 86 41 57 05 71 98 13 08 82 17 76 15 05 72 68 21 08 61 57 62 54 74 83 24 07 86 09 21 14 40 95 77 24 56 81 69 20 97 25 19 59 55 80 77 81 34 56 35 38 42 33 01 78 32 97 31 52 09 41 18 77 88 79 38 43 07 74 33 39 54 26 02 66 79 55 11 78 50 47 47 17 94 44 98 16 93 09 55 53 73 54 00 52 26 80 48 11 03 79 08 84 00 32 26 94 50 67 98 35 98 68 68 04 86 09 81 74 24 53 19 95 66 61 98 45 46 89 64 44 73 56 57 91 52 39 56 82 62 35 99 91 01 37 76 51 83 10 63 92 61 96 48 59 39 75 60 18 83 90 14 34 82 90 53 25 80 96 50 63 41 52 21 41 51 15 10 44 53 84 35 19 70 16 35 40 68 61 42 32 39 52 93 16 18 67 05 37 21 44 85 10 34 98 89 60 50 54 77 33 50 03 89 09 02 83 11 46 82 26 31 86 95 88 45 42 65 85 77 56 59 71 96 40 49 10 70 55 02 98 64 44 87 93 90 00 60 22 99 86 12 00 21 48 74 74 90 99 82 75 08 93 39 88 68 22 99 19 65 37 71 50 69 31 56 45 59 63 50 92 63 52 02 35 89 78 44 32 84 79 93 84 63 86 57 13 70 87 01 26 91 55 91 58 67 90 03 58 35 81 62 40 53 21 63 24 83 66 37 48 94 67 11 66 21 73 91 28 91 01 98 65 56 38 27 09 45 15 13 41 60 38 99 89 78 94 29 92 00 40 05 09 71 05 35 60 25 77 19 17 30 54 38 37 82 14 86 42 93 85 61 58 86 74 13 13 69 23 03 61 36 52 86 27 92 45 59 61 63 94 06 35 53 06 68 30 23 00 65 54 30 13 69 65 61 28 61 28 12 73 95 39 69 61 11 02 81 98 55 93 20 65 13 75 06 27 21 27 21 45 71 96 12 26 23 24 42 18 34 64 08 47 39 64 96 42 49 20 28 10 91 80 97 76 93 55 27 27 17 55 98 17 07 77 29 29 88 32 20 87 18 90 95 98 52 08 40 32 40 33 67 70 25 18 45 79 33 53 29 99 88 35 97 59 99 91 49 00 88 14 76 96 02 50 36 93 96 07 29 97 18 55 79 84 63

100 16 78 05 19 94 15 12 18 25 87 93 58 30 60 49 28 05 57 47 17 101 02 22 85 19 48 74 55 24 89 69 15 53 00 20 88 48 95 08 00 47 102 85 76 34 51 40 44 62 93 65 99 72 64 09 34 01 13 09 74 90 65 103 00 88 96 79 38 24 77 00 70 91 47 43 43 82 71 67 49 90 37 00

$QH[D��VIkUúLW� Coloana Rândul

1 2 3 4 5 6 7 8 9 10 104 64 29 81 85 50 47 36 50 91 19 09 15 98 75 60 58 33 15 51 44 105 94 03 80 04 21 49 54 91 77 85 00 45 68 23 12 94 23 44 36 88 106 42 28 52 73 06 41 37 47 47 31 52 99 89 92 22 81 86 55 99 09 107 09 27 52 72 49 11 30 93 33 29 54 17 54 48 47 42 04 79 18 64 108 54 68 64 07 85 32 05 96 54 79 57 43 96 97 30 72 12 19 41 70 109 25 04 92 29 71 11 64 10 42 23 23 67 01 19 20 58 35 93 39 46 110 28 58 32 91 95 28 42 36 98 59 66 32 15 51 46 63 57 10 83 55 111 64 35 04 62 24 87 44 85 45 68 41 66 19 17 13 09 63 37 15 33 112 61 05 55 88 25 01 15 77 12 90 69 34 36 93 52 39 36 23 59 73 113 98 93 18 93 86 98 99 04 75 28 30 05 12 09 57 35 90 15 98 07 114 61 89 35 47 16 32 20 16 78 52 82 37 26 33 67 42 11 93 35 61 115 94 40 82 18 06 61 54 67 03 66 76 82 90 31 71 90 39 27 97 85 116 54 38 58 65 27 70 93 57 59 00 63 56 18 79 85 52 21 03 03 16 117 63 70 89 23 76 46 97 70 00 62 15 35 97 42 47 54 60 60 78 12 118 61 58 65 62 81 29 69 71 95 53 53 69 20 95 66 60 50 70 22 97 119 51 68 98 15 05 64 43 32 74 03 44 63 52 38 67 59 56 69 11 14 120 59 25 41 48 64 79 62 26 87 86 94 30 43 54 26 98 61 38 53 44 121 85 00 02 24 67 85 88 10 34 01 54 53 23 77 33 11 19 68 13 50 122 01 46 87 56 19 19 19 43 70 25 24 29 48 22 44 81 35 40 33 23 123 42 41 25 10 87 27 77 28 05 90 73 03 95 46 88 82 25 02 05 00 124 03 57 14 03 17 80 47 85 94 49 89 55 10 27 19 50 20 37 02 71 125 18 95 93 40 45 53 04 57 17 03 34 54 83 91 69 02 90 72 98 45 126 74 11 04 66 68 52 70 11 97 01 55 36 63 49 42 68 82 15 48 64 127 31 54 98 82 61 64 40 50 42 48 96 84 82 42 55 15 72 34 90 96 128 85 51 93 55 89 63 47 92 88 42 00 08 21 52 27 28 77 48 02 42 129 19 95 97 55 27 91 15 20 96 25 48 75 49 95 88 68 36 09 66 17 130 75 74 55 98 33 02 36 99 11 84 07 71 40 65 95 54 01 90 44 32

Anexa 6

9DORULOH�IXQF LHL�GH�UHSDUWL LH�D�DPSOLWXGLQLL�vPSU úWLHULL

≤

−= t

xxPP

n

σ)1()(

t n 5 6 7 8 9

4,00 0,9623 0,9469 0,9300 0,9120 0,8929 4,10 0,9693 0,9566 0,9425 0,9273 0,9112 4,20 0,9752 0,9647 0,9530 0,9404 0,9269 4,25 0,9777 0,9682 0,9576 0,9461 0,9338 4,35 0,9821 0,9744 0,9657 0,9562 0,9460 4,45 0,9857 0,9795 0,9724 0,9647 0,9563 4,50 0,9873 0,9817 0,9754 0,9684 0,9608 4,55 0,9887 0,9837 0,9780 0,9917 0,9649 4,60 0,9899 0,9855 0,9804 0,9747 0,9686 4,65 0,9911 0,9871 0,9825 0,9775 0,9719 4,70 0,9921 0,9885 0,9845 0,9799 0,9750 4,75 0,9930 0,9898 0,9862 0,9822 0,9777 4,80 0,9938 0,9910 0,9878 0,9842 0,9802 4,85 0,9945 0,9920 0,9892 0,9860 0,9824 4,90 0,9952 0,9930 0,9904 0,9876 0,9844 4,95 0,9958 0,9938 0,9916 0,9890 0,9862 5,00 0,9963 0,9946 0,9926 0,9903 0,9876 5,05 0,9967 0,9952 0,9935 0,9915 0,9893 5,10 0,9971 0,9958 0,9942 0,9925 0,9906 5,15 0,9975 0,9963 0,9950 0,9934 0,9917 5,20 0,9978 0,9968 0,9956 0,9942 0,9927 5,25 0,9981 0,9972 0,9961 0,9949 0,9936 5,30 0,9983 0,9975 0,9966 0,9956 0,9944 5,35 0,9985 0,9979 0,9971 0,9961 0,9951 5,40 0,9987 0,9980 0,9974 0,9966 0,9957 5,45 0,9989 0,9984 0,9978 0,9971 0,9963 5,50 0,9994 0,9986 0,9981 0,9975 0,9968 5,70 0,9995 0,9992 0,9989 0,9986 0,9989 6,00 0,9998 0,9997 0,9996 0,9994 0,9993

BIBLIOGRAFIE

1. Andrei, T.

Stancu, S. 6WDWLVWLF ��WHRULH�úL�DSOLFD LL��(GLWXUD�$//��%XFXUHúWL��

2. Baron, T. Anghelache, C.

L DQ��(�

6WDWLVWLF ��(GLWXUD�(FRQRPLF ��%XFXUHúWL��

3. Baron T. L DQ�(�

Matache S.

6WDWLVWLF ��/XFU UL�DSOLFDWLYH, A.S.E., %XFXUHúWL��

4. % GL ��0� Baron, T. Korka, M.

Statistica pentru afaceri��(GLWXUD�(ILFLHQW��%XFXUHúWL��

5. Biji, E. Baron, T. (coordonatori):

6WDWLVWLF � WHRUHWLF � úL� HFRQRPLF �� (GLWXUD� 'LGDFWLF � úL� 3HGDJRJLF ��

%XFXUHúWL��

6. Biji, E. Lilea, E. Wagner, P.

6WDWLVWLF ��(GLWXUD�8QLYHUVLWDWHD�³7LWX�0DLRUHVFX´��%XFXUHúWL��

7. Capanu, I. Wagner, P. 6HF UHDQX��&��

6WDWLVWLF �PDFURHFRQRPLF ��(GLWXUD�(FRQRPLF ��%XFXUHúWL��

8. Craiu, V. %kVF ��2��

Teste de omogenitate��(GLWXUD�(FRQRPLF ��%XFXUHúWL� 1998.

9. Isaic-Maniu, Al. *U GLQDUX��$� Voineagu, V. 0LWUX ��&�

6WDWLVWLF �WHRUHWLF �úL�HFRQRPLF ��(GLWXUD�7HKQLF ��&KLúLQ X��


0DFURHFRQRPLH�úL�DQDOL] �PDFURHFRQRPLF , Editura Univ. “Constantin %UkQFRYHDQX´��%XFXUHúWL��


Statistica pentru managementul afacerilor�� (GLWXUD� (FRQRPLF ��

%XFXUHúWL��


Statistica pentru managementul afacerilor�� HGL LD� D� GRXD�� (GLWXUD�

(FRQRPLF ��%XFXUHúWL��

13. Pecican, E. Econometrie��(GLWXUD�$//��%XFXUHúWL��

14. Porojan, D. 6WDWLVWLFD�úL�WHRULD�VRQGDMXOXL��(GLWXUD�ùDQVD�65/��%XFXUHúWL��

15. Voineagu V. 0LWUX �&� Isaic-Maniu Al.

L DQ�(� Baron T. Matache S. Isaic-Maniu I. ùHUEDQ�'� Voineagu M.

6WDWLVWLF �WHRUHWLF �úL�PDFURHFRQRPLF ��7HVWH��OXFU UL�SUDFWLFH��VWXGLL�GH�

caz��(G��(FRQRPLF ��%XFXUHúWL��

16. Biji, E. Wagner, P. Lilea, E. Petcu N. 9 WXL�0��

6WDWLVWLF , Editura Universitatea “Titu 0DLRUHVFX´��%XFXUHúWL��

Date post:	08-Nov-2014
Category:	Documents
Upload:	lupasco-chris
View:	102 times
Download:	25 times

Problemar Statistica

Documents