F-test

Remarks

Example: in one-way analysis of variance, there are $k$ groups and $N$ observations in total, with independent normal errors of common variance. Let $\mathrm{SSB}$ be the sum of squares between groups and $\mathrm{SSW}$ the sum of squares within groups. Under $H_0$ that all group means are equal, $F = \frac{\mathrm{SSB}/(k - 1)}{\mathrm{SSW}/(N - k)}$ has the F-distribution with $k - 1$ and $N - k$ degrees of freedom. The check: $\mathrm{SSB}/\sigma^2$ and $\mathrm{SSW}/\sigma^2$ are independent chi-squared variables with $k - 1$ and $N - k$ degrees of freedom, by Cochran's theorem. Non-example: for two non-normal samples, the variance ratio $S_1^2/S_2^2$ does not have an F-distribution under equal variances in general. So comparing it with F quantiles is not an F-test in this sense, and its level can be far from $\alpha$.