There are situations in which we may wish to compare the variances of two populations with independent samples. In that case, the test statistic is the ratio of the sample variances, . Statistical theory implies that if the populations are approximately normally distributed or the sample sizes are large, then under the assumption the population variances are equal, the sampling distribution of this ratio is an F-distribution. This distribution has two parameters, called numerator and denominator degrees of freedom, respectively, which are given by . This implies that a test of the hypotheses,

can be constructed based on the ratio of sample variances,

Strong evidence for this one-sided alternative would be an F-ratio that is much greater than 1. The p-value therefore would be

Note that we could have used

in which case strong evidence for the alternative would be a value for this F-ratio that is much smaller than 1. The p-value then would be

This follows from the fact that

If the hypotheses had been two-sided,

then in practice we could divide the larger sample variance by the smaller sample variance. The corresponding p-value would be two times the area to the right of this ratio under the corresponding F-distribution.

For example, the data given above for the comparison of male and female financial analysts reported
sample sd's , based on sample sizes of *25,18*. Suppose we wish to test
the two-sided hypotheses,

Then the test statistic is

The p-value is taken from the F-distribution with

pvalue = 2*(1 - pf(2.25,17,24))which gives

2017-11-16