e-Statistics

F-Test

In the comparison of population means in two independent groups, say ``Group 1'' and ``Group 2,'' test procedures (pooled t-test and Wilcoxon test) often assume that the two population variances are approximately equal. This assumption itself can be treated as hypothesis test, and F-test is introduced for the plausibility of equal variances. The test is actually concerned with how Group 1 and Group 2 differ in terms of their respective population variance $ \sigma_1^2$ and $ \sigma_2^2$.

$ H_A:\hspace{0.05in}\sigma_1^2$ $ \sigma_2^2$

The test procedure is based upon the sample standard deviations $ S_1 =$ and $ S_2 =$ from Group 1 and 2 with the respective sample size $ n =$ and $ m =$ . The test statistic $ F = \frac{S_1^2}{S_2^2}$ is likely observed around, greater than, or less than unity under the respective null hypothesis `` $ \sigma_1^2 = \sigma_2^2$,'' `` $ \sigma_1^2 \ge \sigma_2^2$,'' or `` $ \sigma_1^2 \le \sigma_2^2$.'' The opposite of such observation is expressed by p-value < $ \alpha$, suggesting an evidence against the null hypothesis $ H_0$. To check the plausibility of equal variances, we reason that the null hypothesis `` $ \sigma_1^2 = \sigma_2^2$'' can be accepted when we fail to reject it. The F-test is known to be very sensitive to the appropriateness of normality assumption.