For a $n$ sized sample of independent observations from a normal population, we are interested in checking \begin{align} H_{0}: \sigma^{2} = \sigma_{0}^{2} \quad \text{versus} \quad \sigma^{2} \neq \sigma_{0}^{2} \end{align}
Recall from section \begin{align} \frac{(n-1)S^{2}}{\sigma^{2}} \sim \chi_{n-1}^{2} \end{align}
Then if $H_{0}$ is true, our test statistic \begin{align} TS = \frac{(n-1)S^{2}}{\sigma_{0}^{2}} \sim \chi_{n-1}^{2} \end{align}
and from the test simply becomes \begin{alignat}{2} \text{Accept} \quad &H_{0} \quad &&\text{if} \quad \chi_{1-\alpha/2, n-1}^{2} \leq TS \leq \chi_{\alpha/2, n-1}^{2}\newline \text{Reject} \quad &H_{0} \quad &&\text{otherwise} \end{alignat}
One sided test can be done in a similar manner, comparing with $\chi_{1-\alpha, n-1}^{2}$ or $\chi_{\alpha, n-1}^{2}$ based on which side we want to reject $H_{0}$.
We are interested in comparing \begin{align} H_{0}: \sigma_{x}^{2} = \sigma_{y}^{2} \quad \text{versus} \quad \sigma_{x}^{2} \neq \sigma_{y}^{2} \end{align}
Recall that the ratio of sample variance with population variance is $\chi^{2}$-distributed, and the ratio of two $\chi^{2}$-distributed variables has an F-distribution. Hence, when $H_{0}$ is true, \begin{align} TS = \frac{S_{x}^{2}}{S_{y}^{2}} \sim F_{n-1, m-1} \end{align}
Noting that F-distribution is always positive, the region for accepting $H_{0}$ simply become \begin{alignat}{2} \text{Accept}\quad &H_{0} \quad &&\text{if} \quad F_{1-\alpha/2, n-1, m-1} \leq TS \leq F_{\alpha/2, n-1, m-1}\newline \text{Reject}\quad &H_{0} \quad &&\text{otherwise} \end{alignat}