1. Inference concerning two population variances

2. Introduction
Methods for comparing two population variances are occasionally needed, though not as frequently as inference for means or proportions.
For the case in which the populations under investigation are normal, the procedures are based on the F distribution.

3. Introduction (continued)
We encountered the distribution is Section 7.4 when forming confidence intervals for a population variance
The F distribution arises as the ratio of two chi-square random variables, each divided by their corresponding numbers of degrees of freedom.

4. The F distribution
Let and be independent chi-square rv’s with and d.f., respectively. Then
has an F distribution with
d.f.
Since and both have a chi-squared distribution with d.f., respectively,
has an F distribution ,

5. Using an F table
Let be the value on the horizontal axis such that the area under the F curve with and degrees of freedom is
The distribution is not symmetric, but lower tail probabilities do not need to be tabulated since
Appendix Table A-9 gives for
and various values of and

6. The F test for equality of variances
Null hypothesis: Test statistic value: Alternative Hypothesis Rejection region or

7. P-values for F tests
It is cumbersome to tabulate critical values for the F curve because of the combinations of two different numbers of degrees of freedom.
The p-value for a number between two critical values lies between the corresponding to the two values.