F Distribution

Suppose 2 independent populations (of sizes $N_1$ and $N_2$) are given. And samples of sizes $n_1$ and $n_2$ are taken from each population.

F statistic can be computed using:

Properties

Degrees of Freedom

Degrees of freedom are $\nu_1 = n_1 - 1$ and $\nu_2 = n_2 - 1$.

F Reciprocal

Uses

• Variance comparison: test whether 2 populations have equal variance
• ANOVA: test whether means of 3 or more groups are equal (ratio of between-group to within-group variance)
• Regression F-test: test whether the regression model as a whole explains a significant portion of variance in the response

Hypothesis Test for Equal Variance

$H_0$: $\sigma_1^2 = \sigma_2^2$

Test statistic:

Under $H_0$, $F \sim F(\nu_1, \nu_2)$.

Reject $H_0$ if $F > F_{\nu_1, \nu_2, \alpha}$ (right-tail) or compare against both tails for a 2-tailed test. Critical values are read from the F-table for the given $\nu_1$, $\nu_2$, and significance level $\alpha$.