Reject $H_0$ in favor of $H_1$ if $\frac{S_1^2}{S_2^2} > F_{\alpha/2, n_1 - 1, n_2 - 2}$ provided $S_1 > S_2$ # Derivation Derive a test of size $\alpha$ for $\begin{align}H_0: \frac{\sigma_1^2}{\sigma_2^2} = 1 && H_1: \frac{\sigma_1^2}{\sigma_2^2} \ne 1\end{align}$ Under the assumption that $H_0$ is true, $\sigma_1^2 = \sigma_2^2$ and so $\frac{S_1^2}{S_2^2} \sim F(n_1-1, n_2-1)$