Statistics for Business and Economics: bab 11

7/24/2019 Statistics for Business and Economics: bab 11

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Slides Prepared by

JOHN S. LOUCKSSt. Edwards University

2002 South-Western/Thomson Learning


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Chapter 11Inferences Aout !opulation "ariances

Inference aout a !opulation "ariance

Inferences aout the "ariances of T#o!opulations


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Inferences Aout a !opulation "ariance

Chi-S%uare &istriution

Inter'al (stimation of 2

)*pothesis Testing


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Inter'al (stimate of a !opulation "ariance

#here the 2

'alues are ased on a chi-s%uaredistriution #ith n- 1 degrees of freedom and#here 1 - is the condence coe3cient

( ) ( )

/ ( / )

n s n s

!

!!

!!

!!

( ) ( )

/ ( / )

n s n s

!

!!

!!

!!


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Inter'al (stimation of

Inter'al (stimate of a !opulation Standard

&e'iation

Ta5ing the s%uare root of the upper andlo#er limits of the 'ariance inter'al pro'idesthe condence inter'al for the population

standard de'iation


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Chi-S%uare &istriution With Tail Areas of 02

78 of thepossile 2'alues

78 of thepossile 2'alues

2

2

00

02020202

."#$!

."#$!

.%!$!

.%!$!



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(:ample;


9/21 77

(:ample;


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n- 1 10 - 1 7 degrees of freedom and 0

22

00

0202

260260

22

022

- 1.260

n s

(:ample;


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(:ample;


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Sample 'ariance s2pro'ides a point estimate of 2

A 78 condence inter'al for the population'ariance is gi'en *;

$$ 2 2$$

s x x

n

i!!

& '

"#%=

= =

( ) ..s

x x

n

i!!

& '

"#%=

= =

( ) ..

( ).

.

( ).

.

% #%

" %!

% #%

! #%

!

( ).

.

( ).

.

% #%

" %!

% #%

! #%

!

(:ample;


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Left-Tailed Test

)*potheses

Test Statistic

?eDection ?ule

?eDect H0if #here is ased ona chi-s%uare distriution #ith n- 1 df. or

?eDect H0if p-'alue

)*pothesis TestingAout a !opulation "ariance

H%!

%! H%

!%!

Ha ! %

!>Ha !

%!

>

!!

%!

=

( )n s

!!

%!

=

( )n s

! !> ! !> 2


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?ight-Tailed Test

)*potheses

Test Statistic

?eDection ?ule

?eDect H0if #here isased on

a chi-s%uare distriution #ith n- 1 df. or

?eDect H0if p-'alue


!!

%!

=

( )n s

!!

%!

=

( )n s

2

-1 .

Ha ! %

!


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T#o-Tailed Test

)*potheses

Test Statistic

?eDection ?ule

?eDect H0if #hereare ased on a chi-s%uare

distriution #ith n- 1 df. or ?eDect H0if p-

'alue


!!

%!

=

( )n s

!!

%!

=

( )n s

Ha ! %

!Ha !

%!

H%!

%! =H%

!%! =

! !!< ( / ) ! !!< ( / ) or ! !!> /or ! !!> /2 2

-1 / 2. / 2and


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Ene-Tailed Test

)*potheses

Test Statistic

?eDection ?ule

?eDect H0if FF F#here the 'alue of F is

ased on an Fdistriution #ith n1- 1

numerator. and n2 - 1 denominator. df

)*pothesis Testing Aout the"ariances of T#o !opulations

F s

s=

!

!!

F s

s=

!

!!

H% !

!! H%

!!!

Ha

!!!

>Ha !

!!>


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T#o-Tailed Test

)*potheses

Test Statistic

?eDection ?ule

?eDect H0if FF F/2 #here the 'alue of F/2 is

ased on an Fdistriution #ith n1- 1

numerator. and n2 - 1 denominator. df

)*pothesis Testing Aout the"ariances of T#o !opulations

H% !

!! =H%

!!! =

Ha

!!!

Ha !

!!

F s

s=

!

!!

F s

s=

!

!!


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as #as

descried earlier> on another 10 thermostats>this time

manufactured * TempGing The temperature

readingsof the ten thermostats are listed elo#

We #ill conduct a h*pothesis test #ith 10to see

if the 'ariances are e%ual for Thermo?ite=sthermostats

and TempGing=s thermostats

Therm 1 2 $ + 4 6 9 7 10

(:ample;


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)*pothesis Testing Aout the "ariances of T#o

!opulations )*potheses

Thermo?ite and TempGingthermo- stats ha'e same

temperature 'ariance. Their 'ariances are not e%ual.

?eDection ?ule

The F distriution tale sho#s that #ith 10>

7 df numerator.> and 7 dfdenominator.>

F0 $19

?eDect H0if FF $19

H% !

!! =H%

!!! =

Ha ! !!Ha ! !!

(:ample;


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)*pothesis Testing Aout the "ariances of T#o

!opulations Test Statistic

Thermo?ite=s sample 'ariance is 60

TempGing=s sample 'ariance is 12

F 12/60 216

Conclusion

We cannot reDect H0 There is insu3cient

e'idence to conclude that the population'ariances diHer for the t#o thermostat

rands

(:ample;


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(nd of Chapter 11

Statistics for Business and Economics: bab 11

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