CHAPTER 11
INFERENCES ABOUT POPULATION VARIANCE
INFERENCES –ABOUT A
POPULATION VARIANCE
Interval Estimate of σsquare
INFERNENCE ABOUT THE VARIANCE
OF TWO POPULATIONS
Hypothesis Testing
Inferences about a population variance
•
Whenever a simple random sample of size n is selected from a normal
population ,the sampling distribution of
( n  1) s

•
2
2
has a chi-square distribution with n-1 degrees of freedom . We can use chi-square
distribution to develop interval estimates and conduct hypothesis tests about a
population variance.
Interval Estimate of σsquare
•
A sample of 20 containers is taken ,and the sample variance for the filling quantities
is found to be =.0025.However, we know we cannot expect the variance of a sample
of 20 containers to provide the exact value of the variance for the population of
containers filled by the production process .Hence, our interest will be in developing
an interval estimate for the population variance.
(n  1)s 2
 2
•
2
based
 2 
(n  1)S 2
2 
1
2
Where the values are
on a chi-square distribution
with n-1 degrees of freedom
and where 1- а is the confidence coefficient.
• Recall that the sample of 20 containers provided a sample
variance of s 2 =.0025,with a sample size of 20 ,we have 19
degrees of freedom. So ,we can determine that.2925 =8.90655.
.2025 =32.8523.Using these values provides the following interval
estimate for the population variance .
•
(19)(. 0025 )
(19)( 0.025 )
32.8523
or
 2 
8.90655
.0014   .0053
2
• Taking the square root of these values provides the following
95% confidence interval for the population standard deviation
.0374    .0728
•
INFERENCE ABOUT THE VARIANCES OF TWO
POPULATIONS
s12
s22
Sampling Distribution of
•
Whenever independent simple random samples of sizes n1 and n2 are selected from
two normal populations with equal variances ,the sampling distribution of
•
s12
s22
When
12   22
•
has an distribution with n1-1 degrees of freedom for the numerator and n2-1 degrees
of freedom for the denominator ;S1*2 is the sample variance for the random sample
of n1 items from population 1, and S2 *2 is the sample variance for the random
sample of n2 items from population 2.
Hypothesis Testing
•
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One-Tailed Test About a Population Variance
H0:  2   02
Ha:  2   02
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•
Test Statistic
2
2
= (n  1)s
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•
•
•
Rejection Rule
2
Using test statistic : Reject H 0 if  2 >  
Using p_ value< α
2
Where  02 is the hypothesized value for the population variance and 
is based on a chi_ square distribution with n_1 degrees of freedom.

 02
• One-Tailed Test About a Population Variance
•
H0:  2   02
•
Ha:  2   02
• Test Statistic
2
(n  1)s 2
•
=
2

0
• Rejection Rule
2
•
Using test statistic : Reject H 0 if  < 12
•
Using p_ value< α
2
• Where  0 is the hypothesized value for the population
variance and
is based on a chi_ square distribution
with n_1 degrees of freedom.
The following is example
•
Assume that a random sample of 10 bus arrivals will be taken at a particular downtown
intersection .If the population of arrival times has a normal probability distribution .we
know from the quantity (n  1) s has a chi-square distribution with n-1 degrees of

freedom .thus ,the test statistical
2
2
•
2 =
( n  1) s 2
2
2
has a chi-square distribution with n-1=9 degrees of freedom .with the null hypothesis 
2
2
=4,a simple size n=10,and
a
sample
variance
provides
the
following
observed
value,

s
2
2
9
s
 2 = 4 with =4.8,we obtain the following  =10.8<  .205
=16.919,so ,we cannot reject H0,.
Hence ,the sample variance of s 2=4.8 is insufficient evidence to conclude that
the arrival –time variance is not meeting the company standard .the p-value
associated with the test statistic  2=10.8 is .29.with a p-value =.29>α=.05,the
null hypothesis cannot be rejected.
• In practice ,one-tailed tests are the most frequently encountered
tests about population variances. That is ,in situations involving
arrival times ,production times ,filling weights ,part
dimensions ,and so on ,low variances are generally
desired ,whereas large variances tend to be unacceptable .With a
statement about the maximum allowance population variance,
we can test the hypothesis that the population is less than or
equal to the maximum allowance value against the alternative
hypothesis that population variance is greater than the
maximum allowance value.
Two –tailed Test About a Population Variance
•
Ho:  2=
•
Ha :  ≠
2
 02
 02
Test Statistic

(n  1)s 2
2
=
 02
Rejection Rule
Using test statistic : Reject H0 if
Using p-value

2
2
2

< 1 or if  >
:Reject H0 if p-value<α
2
 2
2
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•
•
•
•
•
•
Let us demonstrate the use of the chi-square distribution in concluding a two-tailed test about a
population variance by considering a situation faced by a bureau of two motor
vehicles .Historically ,the variance in test scores for individuals applying for driver’s licenses has been
2
 =100.A new examination with new test questions has been developed .Administrators of the bureau
of motor vehicles would like the variances in the test scores for the new examination to remain at the
historical level .To evaluate the variance in new examination test scores , the following two-tailed
hypothesis test has been proposed.
H0:
=100
Ha : 2 ≠100

A sample of 30 applicants for driver’s licenses will be given the new version of the
examination.
The chi-square distribution can be used to conduct this two-tailed test .With a.05 level of
2
2
significance ,the critical values will be .025
=45.7222, .975
=16.0471,with n-1=29degrees of
freedom.
With Ho:  2 =100,the value of the  2 statistic is computed to be

2
2
2
(
n

1
)
s
=
= 29(64)=18.56
 02
100
So ,we are not able to reject H0,There is no statistic evidence that the variance in the new
examination scores differs from the historical variance in examination scores.
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Two –tailed Test About the Variances of Two Populations
H0 :  2   2
1
2
Ha :  12   22
Test Statistic
s12
F=
s22
Rejection Rule
F
Using test statistic :Reject H0 if F >
2
Using p-value : Reject H0 if p-value<a
F
where the value of
is based on an F distribution with n1-1 degrees of freedom for the numerator and
2
n2-1 degrees of freedom for the denominator.
•
one-Tailed Test About the Variance of Two Population
H0:  12   22
Ha:  12   22
•
•
•
•
•
•
Test Statistic
s12
F=
2
Rejection Rule s2
Using test statistic :Reject H0 if F >
F
Using p-value : Reject H0 if p-value<a
where the value of is
Fbased on an F distribution with n1-1 degrees of freedom for the numerator and
•
•
n2-1 degrees of freedom for the denominator.