statistics for business and economics: bab 11
TRANSCRIPT
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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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$$
Inferences Aout a !opulation "ariance
Chi-S%uare &istriution
Inter'al (stimation of 2
)*pothesis Testing
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Inter'al (stimation of 2
Inter'al (stimate of a !opulation "ariance
#here the 2
'alues are ased on a chi-s%uaredistriution #ith n- 1 degrees of freedom and#here 1 - is the condence coe3cient
( ) ( )
/ ( / )
n s n s
!
!!
!!
!!
( ) ( )
/ ( / )
n s n s
!
!!
!!
!!
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Inter'al (stimation of
Inter'al (stimate of a !opulation Standard
&e'iation
Ta5ing the s%uare root of the upper andlo#er limits of the 'ariance inter'al pro'idesthe condence inter'al for the population
standard de'iation
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Chi-S%uare &istriution With Tail Areas of 02
78 of thepossile 2'alues
78 of thepossile 2'alues
2
2
00
02020202
."#$!
."#$!
.%!$!
.%!$!
Inter'al (stimation of 2
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(:ample;
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(:ample;
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Inter'al (stimation of 2
n- 1 10 - 1 7 degrees of freedom and 0
22
00
0202
260260
22
022
- 1.260
n s
(:ample;
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(:ample;
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Inter'al (stimation of 2
Sample 'ariance s2pro'ides a point estimate of 2
A 78 condence inter'al for the population'ariance is gi'en *;
$$ 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
)*pothesis TestingAout a !opulation "ariance
!!
%!
=
( )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
distriu- tion #ith n- 1 df. or ?eDect H0if p-
'alue
)*pothesis TestingAout a !opulation "ariance
!!
%!
=
( )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