Two Normal Variances

Out test statistic is

f=\frac{s_x^2}{s_y^2}

which follows $F_{n_x-1,n_y-1}$ distribution under $H_0$ (larger sample variance is the numerator and smaller is the denominator).

$H_0$	$H_1$	$p$ -value	Decision Rule
$\sigma_x^2=/\le\sigma_y^2$	$\sigma^2_x\gt\sigma_y^2$	$\mathbb{P}(F\gt f)$	$f\gt F_{n_x-1,n_y-1,\alpha}$
$\sigma_x^2=\sigma_y^2$	$\sigma_x^2\ne\sigma_y^2$	$2\cdot\mathbb{P}(F\gt f)$	$f\gt F_{n_x-1,n_y-1,\alpha/2}$

$F$ -distribution

If $X_1$ follows $\chi^2_{d_1}$ distribution and $X_2$ follows $\chi^2_{d_2}$ distribution, $F$ distribution is defined as

\frac{X_1/d_1}{X_2/d_2}\sim F_{d_1,d_2}