Hypothesis Tests for Two Variances

Suppose we have independent random samples from two normal populations with variances $\sigma_1^2$ and $\sigma_2^2$, and we want to test whether one population has a larger variance than the other.

For the right-tailed test

$$H_0:\ \sigma_1^2 = \sigma_2^2 \qquad H_1:\ \sigma_1^2 > \sigma_2^2,$$

the test statistic is

$$F = \dfrac{s_1^2}{s_2^2},$$

which, under $H_0$, follows an $F$-distribution with $(n_1 - 1,\, n_2 - 1)$ degrees of freedom. We reject $H_0$ at significance level $\alpha$ if

$$F > F_\alpha(n_1 - 1,\, n_2 - 1).$$

For example, suppose $n_1 = 11,\ s_1^2 = 4.5,\ n_2 = 16,\ s_2^2 = 1.5$, and $\alpha = 0.05$. Then

$$F = \dfrac{4.5}{1.5} = 3,$$

with df $= (10,\,15)$. Since $F_{0.05}(10,\,15) = 2.54$ and $3 > 2.54$, we reject $H_0$. There is sufficient evidence that $\sigma_1^2 > \sigma_2^2$.