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Stat 2263
Section 8-6 Inferences on Two Population Variances
In addition to comparing two means, researchers may be interested in
comparing two population variances. For example, is the variation in
two quality control processes different? Another reason may be in
determining which t test to use when comparing two means: the
pooled variance case, or the case of unequal variances.
For the comparison of two variances or standard deviations, an F test
is used. Note that when comparing means, we look at the difference
between the two means. When comparing variances we look to the
s21
2
ratios of two variances, or 12 . Not surprisingly, the ratio 2 will be
2
s2
used as the test statistic, where s12 is the larger variance. The
s21
sampling distribution of 2 is the F distribution.
s2
Properties of the F distribution
 The values of F cannot be negative, because variances are
always positive or zero
 The distribution is positively skewed
 The F distribution is a family of curves based on the degrees of
freedom of the variance in the numerator and the degrees of
freedom of the variance in the denominator.
 The assumptions here are that the two independent random
samples come from normally distributed populations
Test Statistic:
F
s21
s22
with numerator df 1  n1  1 and denominator
df, 2  n2  1 , where s12 is the larger variance.
Table 5 in Appendix A gives the critical values for the F distribution for
=0.05, and 0.025. It is limited since the requirement of two sets of
degrees of freedom for the numerator and the denominator means a
1
lot of numbers. We may use the relationship F1 , , 
for the
2 1 2
F , ,
2
2
1
2
lower tailed critical value. Choosing the larger of the two sample
variances as the numerator, may save some work.
Example:
Find the critical values for a
a.
right tailed F test when =0.05 and  1  15, 2  21
(2.18)
b,
two tailed F test when =0.05 and  1  20, 2  12
1
(3.07,
)
2.68
When testing the equality of two variances, these hypotheses are
used:
H0 : 21  22
H0 : 21  22
H0 : 21  22
HA : 21  22
HA : 21  22
HA : 21  22
Two-tailed
=
Right-tailed
Left-tailed
Test Statistic:
F
s21
s22
with numerator df 1  n1  1 and denominator
df, 2  n2  1
Rejection Region:
a)
two tailed test:Reject H 0 if F > F
, ,
2 1 2
b)
right tailed test:Reject H 0 if F > F,1 , 2
c)
left tailed test:
or if F < F1
Reject H 0 if F < F1 ,1 ,2 
, ,
2 1 2

1
F
, ,
2 2 1
1
F,2 ,1
Calculations:
No p-value necessary for the F test due to the limitations of the table.
Conclusion:
Exercises 419 - 420