∑i · testul f (fisher) phqwux yhulilfduhd hjdolw ll glvshuvlloru d grx hœdqwlrdqh de volum...

3ULQFLSDOHOH�QR LXQL�SHUHFKH�GLQ�FROHFWLYLWDWHD�JHQHUDO �úL�GLQ�FROHFWLYLWDWHD�GH�VHOHF LH�

Tabelul 4.1.

&DUDFWHULVWLFD�QHDOWHUQDWLY &DUDFWHULVWLFD�DOWHUQDWLY Denumirea indicatorului &ROHFWLYLWDWHD�JHQHUDO Colectivitatea

GH�VHOHF ie Colectivitatea

JHQHUDO Colectivitatea

GH�VHOHF LH A 1 2 3 4

Media

N

x

x

N

ii∑

== 10

n

x

x

n

ii∑

== 1 N

MP =

n

mw =

Dispersia

N

xxN

ii∑

=−

= 1

20

20

)(

σ n

xxn

ii∑

=−

= 1

2

2

)(

σ

)1(2 ppp −=σ )1(2 www −=σ

Abaterea medie S WUDWLF

200 σσ =

2σσ = 2pp σσ = 2

ww σσ =

Eroarea de reprezentativitate:

0xxEr

−=

&RQGL LD�FD�PHGLD�HúDQWLRQXOXL�V �ILH�UHSUH]HQWDWLY �

&HUFHWDUHD�VHOHFWLY

%51000

0 ≤⋅−x

xx

Eroarea medie de reprezentativitate:

∑

∑

=

=

−

==k

ss

k

sss

xx

n

nxx

1

1

20 )(

σµ ,

în care: k -�UHSUH]LQW �QXP UXO�HúDQWLRDQHORU�SRVLELOH� ns -�IUHFYHQ HOH�PHGLLORU�GH�VHOHF LH�SRVLELOH�

(URDUH�OLPLW �PD[LP �DGPLV )( x∆ :

xx z σα=∆ ,

unde coeficientul αz � UHSUH]LQW �DUJXPHQWXO�IXQF LHL�*DXVV� - Laplace pentru probabilitatea

)( zΦ FX�FDUH�VH�JDUDQWHD] �UH]XOWDWHOH� 6(/(& ,$�$/($72$5(�6,03/

6HOHF LD�DOHDWRDUH�VLPSO �VH�DSOLF �vQ�FD]XO�FROHFWLYLW LORU�RPRJHQH�GLQ�FDUH�VH�H[WUDJ�XQLW L�VLPSOH�IRORVLQG�SURFHGHH�GH�VHOHF LH�DOHDWRDUH�


• SHQWUX�VHOHF LH�UHSHWDW �

nnrepx

220

σσσ ≈=

• SHQWUX�VHOHF LH�QHUHSHWDW �

−≈

−−≈

−−=

N

n

nN

nN

nN

nN

nnerepx 1

11

2220

σσσσ

Pentru cDUDFWHULVWLFD�DOWHUQDWLY ��HURDUHD�PHGLH�GH�UHSUH]HQWDWLYLWDWH�VH�YD�QRWD�FX� wσ , deci: • SHQWUX�VHOHF LD�UHSHWDW �

n

ww

n

pprepw

)1()1( −≈−=σ

• SHQWUX�VHOHF LD�QHUHSHWDW �

)1()1(

)1()1(

N

n

n

ww

N

n

n

ppnerepw −⋅

−=−⋅

−=σ

(URDUHD�OLPLW �PD[LP �DGPLV :

• pentru selHF LH�UHSHWDW

rep xx z σ∆ ⋅= , respectiv rep wx z σ∆ ⋅=

• SHQWUX�VHOHF LH�UHSHWDW

repxx z ne σ⋅=∆ , respectiv repwx z ne σ⋅=∆

Intervalul de încredere al mediei�FROHFWLYLW LL�JHQHUDOH:

• pentru caracterisWLFD�QHDOWHUQDWLY ��

xx xxx ∆+<<∆− 0

• SHQWUX�FDUDFWHULVWLFD�DOWHUQDWLY �

ww wpw ∆+<<∆−

,QWHUYDOXO�GH�YDULD LH�DO�QLYHOXOXL�WRWDOL]DW al caracteristicii:

• SHQWUX�FDUDFWHULVWLFD�QHDOWHUQDWLY ��

( )xxN ∆− < ∑=

N

iix

1

< ( )xxN ∆+

• SHQWUX�FDUDFWHULVWLFD�DOWHUQDWLY ��

( )wwN ∆− < M < ( )wwN ∆+

9ROXPXO�HúDQWLRQXOXL

• pentru sondajul simplu repetat:

nzx

20σ

α=∆ , de unde: 2

20

2

x

zn

∆=

σα

• pentru sondajul simplu nerepetat:

)1(20

N

n

nzx −=∆

σα , de unde:

N

z

zn

x

20

22

20

2

σ

σ

α

α

+∆

=

5HGLPHQVLRQDUHD�HúDQWLRQXOXL prin modificarea erorii limite maxime admise ( ’

x∆ ): • SHQWUX�VHOHF LH�UHSHWDW �

( )2’

22

’x

zn

∆=

σα

• SHQWUX�VHOHF LH�QHUHSHWDW �

( )N

z

zn

x

22’

22’

σσ

α

α

+∆=

6(/(& ,(�7,3,& ��675$7,),&$7 � 6HOHF LD� WLSLF � VH� DSOLF � vQ� FD]XO� FROHFWLYLW LORU� QHRPRJHQH� IRUPDWH� GLQ� XQLW L� VLPSOH�

vPS U LWH�vQ�JUXSH��VWUDWXUL��VXEFROHFWLYLW L��FkW�PDL�RPRJHQH�FX�HYLGHQ LHUHD�WLSXULORU�FDOLWDWLYH�� (úDQWLRQXO� VH� RE LQH� SULQ� H[WUDJHUHD� GH� VXEHúDQWLRDQH� GLQ� VWUDWXULOH� SRSXOD LHL� WRWDOH� SULQ�

SURFHGHH�GH�VHOHF LH�DOHDWRDUH�

ÌQ�YHGHUHD� UHSDUWL] ULL� HúDQWLRQXOXL�SH�VXEHúDQWLRDQH�FRUHVSXQ] WRU� WLSXULORU� FDOLWDWLYH�� VH�SRW�aplica WUHL�PRGDOLW L:

1. RepartizaUHD� vQ� PRG� HJDO� D� HúDQWLRQXOXL� SH� VXEHúDQWLRDQH� LQGLIHUHQW� GH� QXP UXO�

XQLW LORU�FH�FRPSXQ�VWUDWXULOH�SRSXOD LHL�WRWDOH�

k

nni = ,

unde: ni –�GLPHQVLXQHD�ILHF UXL�VXEHúDQWLRQ� k –�QXP UXO�GH�VWUDWXUL�vQ�SRSXOD LD�WRWDO �

Acest tip de seleF LH�PDL�SRDUW �GHQXPLUHD�GH�VHOHF LH�WLSLF �VWUDWLILFDW ��QHSURSRU LRQDO �

2. (úDQWLRQXO� VH� VHSDU � SH� VXEHúDQWLRQH� vQ� IXQF LH� GH� SRQGHUHD� ILHF UXL� VWUDW� vQ�

FROHFWLYLWDWHD�JHQHUDO �

∑∑∑

======ii

i

i

i

N

n

N

n

N

n

N

n

N

n ......

2

2

1

1 ,

de unde:

i

ii

N

Nnn

∑=

Acest tip de VHOHF LH�VH�PDL�QXPHúWH�VHOHF LD�WLSLF ��VWUDWLILFDW ��SURSRU LRQDO .

3. /D� IRUPDUHD� VXEHúDQWLRDQHORU� VH� LD� vQ� FRQVLGHUD LH� DWkW� SRQGHUHD� ILHF UXL� VWUDW� vQ�

FROHFWLYLWDWHD�JHQHUDO �FkW�úL�JUDGXO�GH�RPRJHQLWDWH�DO�VWUDWXULORU��UHSUH]HQWDW�GH�DEDWHUHD�PHGLH�S WUDWLF �

∑=

=k

iii

iii

N

Nnn

1

σ

σ

$FHVW�WLS�GH�VHOHF LH�VH�PDL�QXPHúWH�VHOHF LD�WLSLF ��VWUDWLILFDW ��RSWLP �GHRDUHFH�G �FHOH�PDL�

PLFL�HURUL�vQ�SUDFWLF ��GDU�HVWH�JUHX�GH�DSOLFDW� 5HOD LLOH�GH�FDOFXO�vQ�FD]XO�VRQGDMXOXL�WLSLF�SURSRU LRQDO: • eroarea medie de reprezentativitate:

ny

2σσ = SHQWUX�VHOHF LD�UHSHWDW

−=

N

n

ny 12σ

σ SHQWUX�VHOHF LD�QHUHSHWDW

• HURDUHD�OLPLW �PD[LP �DGPLV

nzy

2σ⋅=∆ SHQWUX�VHOHF LD�UHSHWDW

−⋅=∆

N

n

nzy 1

2σ SHQWUX�VHOHF LD�QHUHSHWDW

• HVWLPDUHD�PHGLHL�OD�QLYHOXO�FROHFWLYLW LL�JHQHUDOH

yy yyy ∆+<<∆− 0 ) ,

unde:

∑∑=

i

ii

n

nyy

• estimarea nivelului totalizat al caracteristicii

)()(1

y

N

iiy yNyyN ∆+<<∆− ∑

=

*

• GLPHQVLRQDUHD�HúDQWLRQXOXL

2

22

x

zn

∆=

σ pentru VHOHF LH�UHSHWDW

N

z

zn

x

222

22

σσ

+∆= SHQWUX�VHOHF LH�QHUHSHWDW

6(/(& ,$�'(�6(5,,

6H� IRORVHúWH� vQ� FD]XO� vQ� FDUH� FROHFWLYLWDWHD� JHQHUDO � HVWH� IRUPDW � GLQ� XQLW L� FRPSOH[H�

�HFKLSH��EULJ ]L��IHUPH�HWF�� Cu cât mediile din serii sunt mai apropiate între ele, vor estima mai corect valoarea medie a

vQWUHJLL�FROHFWLYLW L��UHOL]kQGX-VH�FRQGL LD�GH�UHSUH]HQWDWLYLWDWH�D�HúDQWLRQXOXL� (úDQWLRQDUHD�I FkQGX-VH�SH�ED] �GH�VHULL��HúDQWLRQXO�HVWH�IRUPDW�GLQWU-XQ�QXP U�GH�VHULL�QRWDW�

cu r, iar vQ�FROHFWLYLWDWHD�JHQHUDO �QXP UXO�VHULLORU�VH�YD�QRWD�FX�R.


• pentru caracteristici nealternative

r

xx

2

rep

δσ =

'H�UH LQXW�

*�3HQWUX� D� RE LQH� DFHODúL� JUDG� GH� SUHFL]LH� D� UH]XOWDWHORU�� HúDQWLRQXO� FRQVWLWXLW� SULQ�stratificare este mai mic decât cel pentru sondajul aleator simplu.

*�3HQWUX� DFHODúL� YROXP� DO� HúDQWLRQXOXL�� SUHFL]LD� HVWH� PDL� PDUH� vQ� FD]XO� VRQGDMXOXL�stratificat decât în cazul sondajului aleator simplu.

*�(úDQWLRQXO� WUHEXLH�GLPHQVLRQDW� DVWIHO� vQFkW� ILHFDUH� VXEHúDQWLRQ� V � FRQ LQ � XQ�QXP U�VXILFLHQW�GH�XQLW L�SHQWUX�D�SHUPLWH�FDOFXODUHD�GLVSHUVLLORU�OD�QLYHOXO�VXEHúDQWLRDQHORU�(ni > 35).

1

2

nerep −−=

R

rR

r

xx

δσ

• pentru caracteristici alternative

rp

w

2

rep

δσ =

1

2

nerep −−=

R

rR

r

p

w

δσ

(URDUHD�OLPLW �PD[LP �DGPLV

rzx

2δ⋅=∆ �SHQWUX�VHOHF LD�UHSHWDW

−−⋅=∆

1

2

R

rR

rzx

δ SHQWUX�VHOHF LD�QHUHSHWDW

'LPHQVLRQDUHD�HúDQWLRQXOXL

2

22

x

zn

∆=

δ SHQWUX�VHOHF LH�UHSHWDW

222

22

)1( δδ

zR

Rzn

x +∆−= SHQWUX�VHOHF LH�QHUHSHWDW

7HVWH�GH�VHPQLILFD LH 6H� SRUQHúWH� GH� OD� ´LSRWH]D� GLIHUHQ HL� QXOH´� FDUH� SUHVXSXQH� F � QX� H[LVW � GLIHUHQ H�

VHPQLILFDWLYH� vQWUH� YDORULOH� GH� VRQGDM� úL� SDUDPHWULL� SRSXOD LHL� VDX� vQWUH� YDORULOH� GH� VRQGDM� D� GRX �HúDQWLRDQH�DOHDWRDUH�

&RPSDUDUHD�PHGLHL�GH�VRQGDM�FX�PHGLD�SRSXOD LHL�vQWU-R�UHSDUWL LH�QRUPDO

n

xxt

/0

σ−

=

'DF �SHQWUX�XQ�QLYHO�GH�VHPQLILFD LH�GH�� 96,1>t �VH�UHVSLQJH�LSRWH]D�GLIHUHQ HL�QXOH�

&RPSDUDUHD�PHGLLORU�D�GRX �HúDQWLRDQH�PDUL

2

22

1

21

21

nn

xxt

σσ+

−=

Se DFFHSW � LSRWH]D� úL� VH� FRQVLGHU � GLIHUHQ HOH� QHVHPQLILFDWLYH� GDF � _W_� �� QLYHO� GH�

VHPQLILFD LH�� Se respinge�LSRWH]D�GDF �_�W�_��!��QLYHO�GH�VHPQLILFD LH��

&RPSDUDUHD�D�GRX �SURSRU LL�GH�VRQGDM�SHQWUX�R�FDUDFWHULVWLF �DOWHUQDWLY

Cazul 1 �QX�FXQRDúWHP�SURSRU LD�p�vQ�FHOH�GRX �SRSXOD LL�GLQ�FDUH�V-DX�IRUPDW�HúDQWLRDQHOH�

$GPLWHP�F ��21

2211

nn

wnwnp

++

=

'DF ��

+⋅

−

21

21

11

nnqp

ww > 3 ,

GLIHUHQ D�GLQWUH�FHOH�GRX �SURSRU LL�HVWH�VHPQLILFDWLY � Cazul 2 �SURSRU LLOH�vQ�FHOH�GRX �SRSXOD LL�GLQ�FDUH�V-DX�IRUPDW�HúDQWLRDQHOH�QX�VXQW�HJDOH��

IDSW�SHQWUX�FDUH�VH�IRORVHVF�GLVSHUVLLOH�HúDQWLRDQHORU�

'DF ��

2

22

1

11

21

)1()1(

n

ww

n

ww

ww

−+

−

−>3 ,

GLIHUHQ D�GLQWUH�FHOH�GRX �SURSRU LL�HVWH�VHPQLILFDWLY �

Cazul 3 (FRPSDU P�SURSRU LLOH�HúDQtioanelor cu 21

2211

nn

wnwnp

+

+= )

&DOFXO P��GH�H[HPSOX�

1

2

21

11

n

n

nn

qp

wpt

⋅+⋅−

=

'DF ��W1�!��GLIHUHQ D�HVWH�VHPQLILFDWLY � 3HQWUX�HúDQWLRDQHOH�GH�YROXP�UHGXV�VH�IRORVHúWH�testul lui Student. &RPSDUDUHD�PHGLHL�GH�VRQGDM�FX�PHGLD�SRSXOD LHL�vQWU-o rHSDUWL LH�6WXGHQW

1/

0

−

−=

n

xxt calc

σ

tabelart VH�GHWHUPLQ �SHQWUX�XQ�DQXPLW�QLYHO�GH�VHPQLILFD LH�úL��Q-1) grade de libertate 'DF �Wcalculat<ttabelar�VH�DGPLWH�LSRWH]D�GLIHUHQ HL�QXOH�

&RPSDUDUHD�GLVSHUVLLORU�QHFXQRVXWH�D�GRX �HúDQWLoane de volum redus

21

21

11

nnS

xxt calc

+

−= ,

unde 2

)()(

21

1 1

22

21

1 2

−+

−+−

=∑ ∑= =

nn

xxxx

S

n

i

n

iii

'DF �Wcalculat < ttabelar�VH�DGPLWH�LSRWH]D�GLIHUHQ HL�QXOH��YDORDUHD�WDEHODU �VH�VWDELOHúWH�SHQWUX�

QXP UXO�JUDGHORU�GH�OLEHUWDWH� 221 −+= nnV ). Testul F (Fisher) pHQWUX�YHULILFDUHD�HJDOLW LL�GLVSHUVLLORU�D�GRX �HúDQWLRDQH� de volum redus

22

21

S

SFcalc =

tabelarF � VH� GHWHUPLQ � SHQWUX� XQ� DQXPLW� QLYHO� GH� VHPQLILFD LH� úL� SHQWUX� JUDGHOH� GH� OLEHUWDWH�

111 −= nf ��úL�� 122 −= nf . 'DF � tabelarcalc FF > , rezultatul sondajului trebuie interpretat ca fiind semnificativ, deci

GLIHUHQ D�GLQWUH�GLVSHUVLLOH�FHORU�GRX �HúDQWLRDQH�WUHEXLH�H[SOLFDW �GH�LQIOXHQ D�DOWRU�IDFWRUL�GHFkW�FHL�FDUH�GHWHUPLQ �HURULOH�GH�VRQGDM�

PROBLEME REZOLVATE

1. 3HQWUX�XQ�HúDQWLRQ�GH��GH�DJHQ L�HFRQRPLFL�V-DX�RE LQXW�XUP WRDUHOH�UH]XOWDWH� • profitul mediu: 505 mil. lei / agent economic; • DEDWHUHD�PHGLH�S WUDWLF �SULYLQG�SURILWXO��PLO��OHL� ùWLLQG�F �HúDQWLRQXO�UHSUH]LQW ��GLQ�WRWDOXO�DJHQ LORU�HFRQRPLFL��V-D�IRUPDW�SULQ�VHOHF LH�

QHUHSHUDW �úL�UH]XOWDWHOH�VH�JDUDQWHD] �FX�R�SUREDELOLWDWH�GH��]� ��VH�FHUH� 1. HVWLPDUHD�SURILWXOXL�PHGLX�úL�D�SURILWXOXL�WRWDO�OD�QLYHOXO�FROHFWLYLW LL�JHQHUDOH� 2. GLPHQVLRQDUHD�XQXL�QRX�HúDQWLRQ�GDF �HURDUHD�OLPLW �PD[LP �DGPLV �HVWH�GH��PLO��OHL� Rezolvare 1. 3HQWUX� HVWLPDUHD� SDUDPHWULORU� vQ� SRSXOD LD� WRWDO � HVWH� QHFHVDU� V � VH� FDOFXOH]H� HURDUHD�

OLPLW �PD[LP �DGPLV �

88,129,0200

409631

2

=⋅=

−⋅=∆

N

n

nzx

σ mil. lei

Estimarea profitului mediu:

xx xxx ∆+<<∆− 0 de unde:

505 – 12,88 < 0x < 505 + 12,88 492,12 < 0x < 517,88 mil. lei / agent economic

Estimarea profitului total:

)()(1

x

N

iix xNxxN ∆+<<∆− ∑

=

88,517200012,49220002000

1

⋅<<⋅ ∑=i

ix

10357609842402000

1

<< ∑=i

ix mil. lei

2. Pentru GLPHQVLRQDUHD�QRXOXL�HúDQWLRQ�VH�IRORVHúWH�UHOD LD�

( )N

z

zn

x

222

22

σσ

⋅+∆ ′

⋅=′ �DJHQ L�HFRQRPLFL

( )88

2000

40969400

4096922

2

22

=⋅+

⋅=⋅+∆ ′

⋅=′

N

z

zn

xσ

σ�DJHQ L�HFRQRPLFL

2. 3HQWUX�XQ�HúDQWLRQ�GH��GH�DJHQ L�HFRQRPLFL�VH�FXQRVF�GDWHOH��

'LVWULEX LD�DJHQ LORU�HFRQRPLFL�GXS �QXP UXO�GH�DQJDMD L�úL�GXS �P ULPHD�SURILWXOXL Tabelul 4.2.

*UXSH�GH�DJHQ L�HFRQRPLFL�GXS �P ULPHD�SURILWXOXL

(mil. lei) *UXSH�GH�DJHQ L�

HFRQRPLFL�GXS �

QXP UXO�GH�DQJDMD L sub 400 400-440

440-480

480-520

520-560

560-600

600 úL�SHVWH

Total

A 1 2 3 4 5 6 7 8 Sub 20 5 25 20 10 - - - 60 ��úL�SHVWH - 5 20 40 30 25 20 140 Total 5 30 40 50 30 25 20 200

Se cere: 1. V � VH� HVWLPH]H� OLPLWHOH� vQWUH� FDUH� VH� YD� vQFDGUD� SURILWXO� PHGLX� úL� SURILWXO� WRWDO� vQ�

FROHFWLYLWDWHD� JHQHUDO � úWLLQG� F � HúDQWLRQXO� UHSUH]LQW � �� GLQ� WRWDOXO� DJHQ LORU�economici, s-D� IRUPDW� SULQ� VHOHF LH� QHUHSHWDW � úL� UH]XOWDWHOH� VH� JDUDQWHD] � FX� R�probabilitate de 0,9973 (z = 3);

2. V � VH� HVWLPH]H� SURFHQWXO� PD[LP� DO� DJHQ LORU� HFRQRPLFL� FX� XQ� SURILW� GH� FHO� SX LQ� 500 mil. lei.

Rezolvare

1. Pentru estimarea paramHWULORU� SRSXOD LHL� WRWDOH� VXQW� QHFHVDUH� R� VHULH� GH� FDOFXOH�

prealabile: • calculul mediei generale:

∑

∑

=⋅

=⋅

=m

jj

m

jjj

n

ny

y

1

1

..ecmil.lei/ag505

200

2062025580305405050040460304205380

=

=⋅+⋅+⋅+⋅+⋅+⋅+⋅=y

• FDOFXOXO�PHGLLORU�GH�JUXS :

∑

∑

=

==

m

jij

m

jijj

i

n

ny

y

1

1

333,44360

1050020.4602542053801 =⋅++⋅+⋅=y mil. lei / agent economic

=⋅+⋅+⋅++⋅+⋅=140

20620255803054040.5002046054202y

429,531= mil. lei / agent economic

• FDOFXOXO�GLVSHUVLLORU�GH�JUXS :

( )

∑

∑

=

=−

=m

jij

m

jijij

i

n

nyy

1

1

2

2σ

889,118860

10443,33)-(500

60

20 ) 443,333- (460 52) 443,333- (420 5443,333)- (380

2

22221

=⋅

+

+⋅+⋅+⋅

=σ

245,30122

2 =σ

• FDOFXOXO�PHGLHL�GLVSHUVLLORU�GH�JUXS :

238,2465200

140245,301260889,1188

1

1

2

2 =⋅+⋅

==

∑

∑

=⋅

=⋅

r

ii

r

iii

n

nσσ

• calculul erorii limite maxime admise:

109,0200

238,246531

2

≅⋅=

−⋅=∆

N

n

nzy

σ mil. lei

Estimarea profitului mediu:

yy yyy ∆+<<∆− 0 ,

de unde:

505 – 10< 0y <505+10

495 < 0y < 515 mil. lei / agent economic

Estimarea profitului total:

)()(1

y

N


=

515200049520002000

1

⋅<<⋅ ∑=i

iy

10300009900002000

1

<< ∑=i

iy mil. lei

2. 3RUQLQG�GH�OD�GLVWULEX LD�ELGLPHQVLRQDO �GLQ�WDEHOXO��VH�RE LQH�XUP WRUXO�WDEHO�

'LVWULEX LD�DJHQ LORU�HFRQRPLFL�GXS �QXP UXO�GH�DQJDMD L�úL�P ULPHD�SURILWXOXL

Tabelul 4.3.

*UXSH�GH�DJHQ L�

HFRQRPLFL�GXS �

QXP UXO�GH�

DQJDMD L

$JHQ L�

economici cu profit < 500

mil. lei

$JHQ L�

economici cu profit ≥ 500 mil. lei )( im

Total )( in i

ii n

mw =

)1(2iiw ww

i−=σ

A 1 2 3 4 5 Sub 20 55 5 60 0,0833 0,0764 ��úL�SHVWH 45 95 140 0,6786 0,2181 Total 100 m = 100 n = 200 w = 0,5 0,25

ÌQ�YHGHUHD�HVWLP ULL�SURFHQWXOXL�PD[LP�VXQW�QHFHVDUH�R�VHULH�GH�FDOFXOH� • FDOFXODUHD�PHGLLORU�GH�JUXS �� iw ��úL�D�PHGLHL�SH�WRWDO��w) (vezi tabelul 4.3. coloana 4);

• calcularea dispersiilRU�GH�JUXS �� 2

iwσ ) (vezi tabelul 4.3., coloana 5);

• FDOFXODUHD�PHGLHL�GLVSHUVLLORU�GH�JUXS �

1756,0200

1402181,0600764,02

2 =⋅+⋅=⋅

=∑

∑i

iww n

ni

σσ

• FDOFXODUHD�HURULL�OLPLWH�PD[LPH�DGPLV �

084,02000

2001

200

1756,031

2

=

−⋅=

−⋅=∆

N

n

nz iw

w

σ

Procentul maxim admis:

p = 584,0084,05,0 =+=∆+ ww sau 58,4%

5H]XOW �F �FHO�PXOW��GLQ�WRWDOXO�DJHQ LORU�HFRQRPLFL�GLQ�FROHFWLYLWDWHD�JHQHUDO �DX�XQ�SURILW�GH�FHO�SX LQ��PLO��OHL�

3. 3HQWUX�XQ�HúDQWLRQ�GH��GH�DJHQ L�HFRQRPLFL�VH�FXQRVF��GDWHOH�

'LVWULEX LD�DJHQ LORU�HFRQRPLFL�GXS �QXP UXO�GH�DQJDMD L�úL�GXS �FLIUD�GH�DIDFHUL

Tabelul 4.4.

*UXSH�GH�DJHQ L�HFRQRPLFL�GXS �

QXP UXO�GH�DQJDMD L 1XP U�GH�DJHQ L�

economici Cifra medie de afaceri

(mil. lei / agent economic) A 1 2

Sub 20 40 1440 20-40 100 2140 ��úL�SHVWH 60 2300

Total 200 … ùWLLQG�F ��SH�WRWDO��FRHILFLHQWXO�GH�YDULD LH�D�IRVW�GH��VH�FHUH� 1. V �VH�HVWLPH]H� OLPLWHOH�vQWUH�FDUH�VH�YD� vQFDGUD�FLIUD�PHGLH�GH�DIDFHUL�úL�FLIUD� WRWDO �GH�

DIDFHUL� vQ� FROHFWLYLWDWHD� JHQHUDO � GDF � HúDQWLRQXO� UHSUH]LQW � �� GLQ� WRWDO�� V-a format SULQ�VHOHF LH�QHUHSHWDW �úL�UH]XOWDWHOH�VH�JDUDQWHD] �FX�R�SUREDELOLWDWH�GH��]� ��

2. V � VH� VWDELOHDVF � FDUH� DU� IL� GLPHQVLXQHD� HúDQWLRQXOXL� GDF � HURDUHD� OLPLW � DU� FUHúWH� FX��V �VH�GLVWULEXLH�QRXO�HúDQWLRQ�SH�VXEHúDQWLRDQH�

Rezolvare 1. Pentru estimarea paramHWULORU� SRSXOD LHL� WRWDOH� VXQW� QHFHVDUH� R� VHULH� GH� FDOFXOH�

prealabile: • calculul mediei generale:

∑

∑

=⋅

=⋅

=m

jj

m

jjj

n

ny

y

1

1

2048200

6023001002140401440 =⋅+⋅+⋅=y mil. lei /agent economic

• calculul dispersiei pe total:

100⋅=y

vσ

de unde 292736,255890856,505100

204837,24

1002 =⇒=⋅=⋅= σσ yv

• calculul dispersiei dintre grupe:

( )

97216200

60)20482300(100)20482140(40)20481440( 222

1

1

2

2

=⋅−+⋅−+⋅−

=

=−

=

∑

∑

=

=r

ii

r

iii

n

nyy

δ

• PHGLD�GLVSHUVLLORU�GH�JUXS �VH�GHWHUPLQ �SH�ED]D�UHJXOLL�GH�DGXQDUH�D�GLVSHUVLLORU�

292736,15867497216292736,255890222 =−=−= δσσ

• HURDUHD�OLPLW �DGPLV ( )y∆

802000

2001

200

292736,1586741

2

≅

−=

−⋅=∆

N

n

ny

σ

Estimarea cifrei medii de afaceri:

yy yyy ∆+<<∆− 0 ,

de unde:

2048 – 80< 0y < 2048 + 80 1968 < 0y < 2128 mil. lei / agent economic

Estimarea cifrei totale de afaceri:

)()(1

y

N


=

21282000196820002000

1

⋅<<⋅ ∑=i

iy

425600039360002000

1

<< ∑=i

iy mil. lei

2. &DOFXOXO�YROXPXOXL�HúDQWLRQXOXL

( )N

z

zn

y

222

22

σσ

⋅+∆′

⋅=′ ,

unde

962,1 =∆⋅=∆′ yy mil. lei

153

2000

292736,158674996

292736,1586749

2

=⋅+

⋅=′n �DJHQ L�HFRQRPLFL

5HSDUWL]DUHD�SH�VXEHúDQWLRDQH�OXkQG�vQ�FDOFXO�VRQGDMXO�WLSLF�QHUHSHWDW��Q� ��WXULúWL�� • QHSURSRU LRQDO

513

153 ≅=′

=′r

nni �DJHQ L�HFRQRPLFL

• SURSRU LRQDO &RQVLGHUkQG�F �SULPXO�HúDQWLRQ�UHVSHFW �VWUXFWXUD�SRSXOD LHL��JUXSD�,�� grupa a II-a: 50%, grupa a III-D��RE LQHP�

31153100

201 =⋅=′n �DJHQ L�HFRQRPLFL

76153100

502 =⋅=′n �DJHQ L�HFRQRPLci

46153100

303 =⋅=′n �DJHQ L�HFRQRPLFL

4. 3HQWUX�HVWLPDUHD�FKHOWXLHOLORU�VXSOLPHQWDUH�HIHFWXDWH�GH�WXULúWLL�GLQ�FDGUXO�XQXL�VHMXU�GH��]LOH�

s-a efectuat un sondaj stratificat, datele înregistrate s-DX� SUHOXFUDW� úL� UH]XOWDWHOH� V-au trecut în tabelul 4.5.

&KHOWXLHOLOH�PHGLL�]LOQLFH�úL�FRHILFLHQWXO�GH�YDULD LH�vQ�HúDQWLRQ

Tabelul 4.5.

*UXSH�GH�WXULúWL�

GXS �YkUVW (ani)

1XP UXO�

WXULúWLORU

Cheltuieli suplimentare medii zilnice

(mii lei)

&RHILFLHQWXO�GH�YDULD LH�DO�

cheltuielilor suplimentare (%)

0 1 2 3 Sub 30 30-50 50 si peste

130 180

90

200 300 400

20 25 33

Total 400 … ...

Se cere: 1. FRQVLGHUkQG�F �FHOH��GH�SHUVRDQH�UHSUH]LQW �XQ�HúDQWLRQ�VWUDWLILFDW�GH��VHOHFWDW�vQ�

PRG�DOHDWRU�úL�QHUHSHWDW�GLQ�QXP UXO� WRWDO�DO� WXULúWLORU�GLQWU-o VWD LXQH�� vQWU-un sejur de ��]LOH�GH�VIkUúLW�GH�V SW PkQ ��V �VH�GHWHUPLQH�vQWUH�FH�OLPLWH�VH�YRU�vQFDGUD�FKHOWXLHOLOH�VXSOLPHQWDUH�PHGLL�úL�WRWDOH�]LOQLFH�DOH�WXULúWLORU�GLQ�vQWUHDJD�FROHFWLYLWDWH��UH]XOWDWHOH�VH�vor garanta cu P = 0,9545 (z = 2);

2. V �VH�VWDELOHDVF �FH�YROXP�DO�HúDQWLRQXOXL�DU�IL�IRVW�QHFHVDU�GDF �V-ar fi utilizat sondajul aleatoriu simplu;

3. V �VH�VWDELOHDVF �FH�YROXP�GH�VHOHF LH�YD�IL�QHFHVDU�GDF �VH�RUJDQL]HD] �R�QRX �FHUFHWDUH�VHOHFWLY �úL�HURDUHD�OLPLW �DGPLV �FDOFXODW �OD�SULPXO�SXQFW�VH�YD�P UL�FX��FHOHODOWH�FRQGL LL�U PkQkQG�QHVFKLPEDWH��

Rezolvare

1. &DOFXOXO�LQGLFDWRULORU�GH�VHOHF LH�úL�HVWLPDUHD�SDUDPHWULORU�FROHFWLYLW LL�JHQHUDOH • PHGLD�PHGLLORU�GH�JUXS

290400

90400180300130200

1

1 =⋅+⋅+⋅==∑

∑

=

=r

ii

r

iii

n

nyy mii lei / turist

• GLVSHUVLLOH�GH�JUXS � ( )2

iσ

100⋅=i

ii y

vσ

, de unde: 100

iii

yv ⋅=σ

40100

200201 =⋅=σ mii lei / turist ⇒ 16002

1 =σ

75100

300252 =⋅=σ mii lei / turist ⇒ 56252

2 =σ

132100

400333 =⋅=σ mii lei / turist ⇒ 174242

3 =σ

• PHGLD�GLVSHUVLLORU�GH�JUXS � ( )2σ

65,6971400

901742418056251301600

1

1

2

2 =⋅+⋅+⋅==∑

∑

=⋅

=r

i

r

iii

n

nσσ

• eroarea medie de reprezentativitate ( )yσ

069,4800

4001

400

65,69711

2

=

−=

−⋅=

N

n

ny

σσ mii lei

• HURDUHD�OLPLW �DGPLV � ( )y∆

138,8069,42 =⋅=⋅=∆ yy z σ mii lei

• estimarea cheltuielilor medii zilnice suplimentare pe total colectivitate:

yy yyy ∆+<<∆− 0 ,

de unde: 290 – 8,138< 0y <290 + 8,138 281,862 < 0y < 298,138 mii lei / turist

• estimarea cheltuielilor zilnice suplimentare totale

)()(1

y

N


=

138,2988000862,28180008000

1

⋅<<⋅ ∑=i

iy

238510421548968000

1

<< ∑=i

iy mii lei

2. 'HWHUPLQDUHD�YROXPXOXL�HúDQWLRQXOXL�SHQWUX�VRQGDMXO�DOHDWRULX�VLPSOX�QHUHSHWDW�

N

z

zn

y

20

22

20

2

σσ

⋅+∆

⋅=

'HRDUHFH� QX� FXQRDúWHP� GLVSHUVLD� WRWDO � GLQ� FROHFWLYLWDWHD� JHQHUDO � FL� QXPDL� GDWHOH�

GHVSUH� HúDQWLRQXO� IRUPDW� SULQ� UHVSHFWDUHD� SULQFLSLLORU� XQHL� VFKHPH� probabilistice, putem vQORFXL�DFHVW�LQGLFDWRU�SULQ�GLVSHUVLD�WRWDO �D�GDWHORU�GLQ�HúDQWLRQ�

'LVSHUVLD� WRWDO � D� HúDQWLRQXOXL� VH� GHWHUPLQ � vQ� DFHVW� FD]� GLQ� UHJXOD� GH� DGXQDUH� D�

dispersiilor:

222 σδσ +=

( )5400

400

90)290400(180)290300(130)290200( 222

1

1

2

2 =⋅−+⋅−+⋅−

=−

=

∑

∑

=

=

r

ii

r

iii

n

nyy

δ

65,1237165,697154002 =+=σ

684

8000

65,123712138,8

65,1237122

2

2

=⋅+

⋅=n �WXULúWL

'HFL�� vQ� FD]XO� XWLOL] ULL� XQXL� VRQGDM� DOHDWRU� VLPSOX�� SHQWUX� DFHODúL� JUDG� GH�

UHSUH]HQWDWLYLWDWH��WUHEXLH�OXD L�vQ�VWXGLX��WXULúWL�ID �GH�QXPDL��FkW�VXQW�QHFHVDUL��GDF �SRSXOD LD�VH�VWUDWLILF �SH�FHOH�WUHL�JUXSH�GH�YkUVW �

3. &DOFXOXO�YROXPXOXL�QRXOXL�HúDQWLRQ

( )N

z

zn

y

222

22

σσ

⋅+∆′

⋅=′ ,

unde:

545,805,1 =∆⋅=∆′ yy mii lei / turist

364

8000

65,69714545,8

65,69714

2

=⋅+

⋅=′n �WXULúWL

5HSDUWL]DUHD�SH�VXEHúDQWLRDQH�OXkQG�vQ�FDOFXO�VRQGDMXO�WLSLF�QHUHSHWDW��Q� ��WXULúWL�� • neproSRU LRQDO

1213

364 ≅==r

nni �WXULúWL

• SURSRU LRQDO �

∑=

=r

ii

ii

N

Nnn

1

Rezultatele calculelor sunt prezentate în tabelul 4.6.

5HGLPHQVLRQDUHD�HúDQWLRQXOXL

Tabelul 4.6.

*UXSH�GH�WXULúWL�

GXS �YkUVW (ani)

1XP UXO�

WXULúWLORU ( iN )

% WXULúWLORU

100⋅∑ i

i

N

N

ni

0 1 2 4 Sub 30 30-50 50 si peste

2600 3600 1800

32,5 45,0 22,5

118 164 82

Total 8000 100,0 364

5. /D�R�VRFLHWDWH�FRPHUFLDO �FX�ED] �GH�WUDWDPHQW�úL�RGLKQ �vQ�OXQD�LXOLH��V-au înregistrat

3200 GH�WXULúWL�UHSDUWL]D L�DVWIHO�SH�VH[H� • 1376 masculin; • 1824 feminin. Se cere: 1. V � VH� VWDELOHDVF � YROXPXO� QHFHVDU� DO� HúDQWLRQXOXL� IRORVLQG� GUHSW� FDUDFWHULVWLF � GH�

UHSUH]HQWDWLYLWDWH�UHSDUWL LD�SH�VH[H�3� ��]� ��LDU�HURDUHD�PD[LP �DGPLV �HVWH�de 5%;

2. HIHFWXkQG�XQ�VRQGDM�SLORW�úL�SUHOXFUkQG�GDWHOH�GLQ�HúDQWLRQ�V-D�RE LQXW�R�GXUDW �PHGLH�D�VHMXUXOXL�GH��]LOH� �� WXULVW�FX�R�DEDWHUH�VWDQGDUG�GH��]LOH��V �VH�GLPHQVLRQH]H�XQ�QRX�HúDQWLRQ�JDUDQWkQG�UH]XOWDWHOH�FX�R�SUREDELOLWDWH�3� ��]� ��

Rezolvare 1. )RORVLQG� GUHSW� FDUDFWHULVWLF � GH� DVLJXUDUH� D� UHSUH]HQWDWLYLW LL� R� YDULDELO � DOWHUQDWLY �

�VH[XO��YRP�RE LQH�

337

3200

)43,01(43,096,105,0

)43,01(43,096,1

)1(

)1(2

2

2

22

2

≅−⋅⋅

+

−⋅⋅=

−⋅+∆

−⋅=

N

ppz

ppzn

x

�WXULúWL

În formula de mai sus:

43,03200

1376 ===N

Mp

ÌQ�SUDFWLF ��VH�FRQVLGHU �HURDUHD�OLPLW �GH��úL�vQ�DFHVW caz:

Q� ��WXULúWL 2. 'HRDUHFH� GXUDWD� VHMXUXOXL� HVWH� YDULDELO � QXPHULF �� HURDUHD� OLPLW � VH� SRDWH� GHWHUPLQD�

astfel:

5,010100

5 =⋅=∆ x

9ROXPXO�QRXOXL�HúDQWLRQ�

121

3200

1696,125,0

1696,122

2

22

=⋅

+

⋅=

⋅+∆

⋅=′

N

z

zn

xσ

σ�WXULúWL

6. Pentru verificarea duratei medii de ardere a unui bec s-a organizat un sondaj de 5% dintr-un

ORW�GH��GH�EHFXUL�DPEDODWH�vQ�FXWLL�GH�FkWH��EXF L��ÌQ�XUPD�FURQRPHWU ULL�GXUDWHL�GH�DUGHUH�D�becurilor din fiecare cutie (serie) s-a calculat durata medie de ardere (vezi tabelul 4.7. coloana 1).

Mediile seriilor

Tabelul 4.7.

Nr.crt. al seriei

Durata medie de ardere (în ore)

)( ix

2)( si xx −

A 1 2 1 2100 2809 2 2250 9409 3 2080 5329 4 1950 41209 5 2325 29584 6 2230 5929 7 2170 289 8 2050 10609 9 2175 484 10 2200 2209 Total 21530 107860

ùWLLQG�F � OD�VHOHFWDUHD�FXWLLORU�V-D�IRORVLW�H[WUDJHUHD�QHUHSHWDW � LDU�SUREDELOLWDWHD�FX�FDUH�VH�

JDUDQWHD] �UH]XOWDWHOH�HVWH��]� ��VH�FHUH� 1. V �VH�HVWLPH]H�OLPLWHOH�vQWUH�FDUH�VH�YD�vQFDGUD�GXUDWD�PHGLH�GH�DUGHUH�SH�vQWUHJXO�ORW�GH�

becuri; 2. GDF �VH�UHSHW �VRQGDMXO��FDUH�YD�IL�YROXPXO�QHFHVDU�DO�HúDQWLRQXOXL��GDF �VH�IRORVHúWH�R�

probabilitate p = 0,9973 pentru care z = 3. Rezolvare

1. 'HRDUHFH�FXWLLOH��VHULLOH��VXQW�GH�DFHHDúL�GLPHQVLXQH��SHQWUX�FDOFXOXO�PHGLHL�SH�WRWDO�VH�

IRORVHúWH�UHOD LD�

215310

21530 === ∑r

xx i

s ore / bec

3HQWUX�FDOFXODUHD�HURULL�PHGLL�HVWH�QHFHVDU �FDOFXODUHD�GLVSHUVLHL�GLQWUH�VHULL��YH]L�WDEHOXO�

4.7., coloana 2):

( )10786

10

1078601

2

2 ==−

=∑

=

r

xxr

isi

sδ

Fiind un sondaj de serii unde R = 200, r� ��HURDUHD�OLPLW �VH�FDOFXOHD] �DVWIHO�

9,621200

10200

10

1078696,1

1

2

=−

−⋅=−−⋅=∆

R

rR

rz s

x

δ ore

Limitele între care se va încadra media pe total:

xsxs xxx ∆+<<∆− 0 ,

de unde:

2153 - 62,9 < 0x < 2153 + 62,9 2090,1 < 0x < 2215,9 ore / bec

2. Noul volum de sondaj va fi:

( )( ) ( )

2295,211078639,62199

107863200

1 22

2

222

22

≅=⋅+⋅

⋅⋅=⋅′+∆⋅−

⋅′⋅=′

sx

s

zR

zRr

δδ

cutii

PROBLEME PROPUSE

1. 3HQWUX�R�XQLWDWH�HFRQRPLF �VH�FXQRVF�GDWHOH�

Tabelul 4.8.

*UXSH�GH�PXQFLWRUL�GXS �P ULPHD�

SURGXF LHL��PLL�EXF�� Nr.

muncitori Sub 10 2 10-12 6 12-14 12 14-16 22 16 -18 18 ��úL�SHVWH 6 Total 66

Se cere:

1. V �VH�YHULILFH�GDF �GLVWULEX LD�PXQFLWRULORU�GXS �P ULPHD�SURGXF LHL�HVWH�RPRJHQ � 2. V � VH� HVWLPH]H� OLPLWHOH� vQWUH� FDUH� VH� YD� vQFDGUD� SURGXF LD�PHGLH� úL� SURGXF LD� WRWDO � vQ�

FROHFWLYLWDWHD� JHQHUDO � GDF � HúDQWLRQXO� UHSUH]LQW � ��GLQ�FROHFWLYLWDWHD� JHQHUDO � úL� V-a IRUPDW� SULQ� VHOHF LH� QHUHSHWDW ��SUREDELOLWDWHD�FX� FDUH� VH� JDUDQWHD] � UH]XOWDWHOH�HVWH� GH�0,9545 (z=2);

3. V � VH� GHWHUPLQH� YROXPXO� HúDQWLRQXOXL� GDF � SUREDELOLWDWHD� FX� FDUH� VH� JDUDQWHD] �rezultatele este de 0,9973 (z=3);

4. V � VH� GHWHUPLQH� QXP UXO� PD[LP� DO� PXQFLWRULORU� FDUH� DX� UHDOL]DW�� vQ� PHGLH�� FHO� SX LQ� ��PLL�EXF L�

2. Dintr-XQ�VRQGDM�VWDWLVWLF�GH��VHOHFWDW�vQWkPSO WRU�úL�QHUHSHWDW��V-DX�RE LQXW�GDWHOH�

Tabelul 4.9.

*UXSH�WLSLFH�GH�DJHQ L�

HFRQRPLFL�GXS �P ULPHD�

capitalului (mil. lei)

1U��DJHQ L�

economici

Profit pe agent economic

(mil.lei/ag.ec.) A 1 2

I 20 14 II 75 21 III 55 23

ùWLLQG� F � YDORDUHD� PRGDO � D� SURILWXOXL�� SH� WRWDO� HúDQWLRQ�� D� IRVW� GH� �� PLO�� OHL� FX� XQ�coeficient de asimetrie de - 0,21, se cere:

1. V �VH�VWDELOHDVF �GDF �IDFWRUXO�GH�JUXSDUH��P ULPHD�FDSLWDOXOXL��HVWH�VHPnificativ pentru YDULD LD�SURILWXOXL��

2. V � VH� VWDELOHDVF � OLPLWHOH� vQWUH� FDUH� VH� YRU� vQFDGUD� SURILWXO� PHGLX� úL� SURILWXO� WRWDO� vQ�FROHFWLYLWDWHD�JHQHUDO �GDF �SUREDELOLWDWHD�FX�FDUH�VH�JDUDQWHD] �UH]XOWDWHOH�HVWH�3 ��(z=1,96).

3. Într-o unitate economLF �VH�FXQRVF�GDWHOH�

Tabelul 4.10.

Grupe de muncitori GXS �P ULPHD�

SURGXF LHL��EXF��

Nr. muncitori

37 4 38 12 39 24 40 44 41 36 42 10 Total 130

Se cere:

1. V � VH� FDOFXOH]H� HURDUHD� PHGLH� GH� UHSUH]HQWDWLYLWDWH� úL� HURDUHD� OLPLW � DGPLV � GDF �

HúDQWLRQXO�UHSUH]LQW ��GLQ�QXP UXO�WRWDO�DO�PXQFLWRULORU�LDU�UH]XOWDWHOH�VH�JDUDQWHD] �cu o probabilitate de 0,95 (z = 1,96);

2. V �VH�HVWLPH]H�SDUDPHWULL�FROHFWLYLW LL�JHQHUDOH� 3. V �VH�GHWHUPLQH�QRXO�YROXP�DO�HúDQWLRQXOXL�GDF �HURDUHD�OLPLW �VH�P UHúWH�GH�GRX �RUi.

4. Dintr-XQ�VRQGDM�VWDWLVWLF�GH��SURSRU LRQDO�VWUDWLILFDW�V-DX�RE LQXW�GDWHOH�

Tabelul 4. 11.

*UXSH�GH�DJHQ L�HFRQRPLFL�GXS �P ULPHD�SURILWXOXL��PLO��

lei) *UXSH�GH�DJHQ L�

HFRQRPLFL�GXS �

QXP UXO�GH�VDODULD L sub 12 12-16 16-20 20-24 24-28 2��úL�peste

Total

A 1 2 3 4 5 6 7 I 8 20 12 - - - 40 II - 15 25 30 20 10 100 Total 8 35 37 30 20 10 140

Se cere:

1. V �VH�HVWLPH]H�OLPLWHOH�vQWUH�FDUH�VH�YD�vQFDGUD�SURILWXO�PHGLX�úL�SURILWXO�WRWDO�vQ�DO�GRLOHD�VWUDW� � GLQ� FROHFWLYLWDWHD� JHQHUDO �� GDF � UH]XOWDWHOH� VH� JDUDQWHD] � FX� R� SUREDELOLWDWH� GH�0,9973 (z = 3);

2. V � VH� HVWLPH]H� SDUDPHWULL� FROHFWLYLW LL� JHQHUDOH�� GDF � UH]XOWDWHOH� VH� JDUDQWHD] � FX� R�probabilitate de 0,9973 (z = 3);

3. V � VH� GHWHUPLQH� YROXPXO� XQXL� QRX� HúDQWLRQ� GDF � SUREDELOLWDWH� FX� FDUH� VH� JDUDQWHD] �UH]XOWDWHOH�HVWH��]� ��úL�V �VH�GLVWULEXLH�HúDQWLRQXO�SH�VXEHúDQWLRDQH�

5. Pentru verificarea duratei medii de ardere a unui bec s-a organizat un sondaj de 5% dintr-un

ORW�GH��GH�EHFXUL�DPEDODWH�vQ�FXWLL�GH�FkWH��EXF L��ÌQ�XUPD�FURQRPHWU ULL�GXUDWHL�GH�DUGHUH�D�becurilor din fiecare cutie (serie) s-a calculat durata medie de ardere (vezi tabelul 4.12., coloana 1).

Mediile seriilor

Tabelul 4.12.

Nr. crt. al seriei

Durata medie de ardere (sute ore)

)( ix

A 1 1 8 2 10 3 7,2 4 7 5 10,2 6 8,5 7 9 8 9,6 9 7,5 10 8

ùWLLQG�F � OD�VHOHFWDUHD�FXWLLORU�V-D�IRORVLW�H[WUDJHUHD�QHUHSHWDW � LDU�SUREDELOLWDWHD�FX�FDUH�VH�

JDUDQWHD] �UH]XOWDWHOH�HVWH��]� ��VH�FHUH� 1. V �VH�HVWLPH]H�OLPLWHOH�vQWUH�FDUH�VH�YD�vQFDGUa durata medie de ardere pe întregul lot de

becuri; 2. V �VH�GHWHUPLQH�YROXPXO�XQXL�QRX�HúDQWLRQ��GDF �HURDUHD�OLPLW �VH�P UHúWH�FX��

∑i · testul f (fisher) phqwux yhulilfduhd hjdolw ll glvshuvlloru d grx hœdqwlrdqh de volum...

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