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,
If the hypotheses had been two-sided,
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,
pvalue = 2*(1 - pf(2.25,17,24))which gives pvalue = 0.0673. Therefore, we would reject the null hypothesis at the 10% level of significance. This conclusion is based on the assumption that the populations are approximately normal, so that assumption should be checked.