Consider independent random variables $X_{1}, \ldots, X_{n}$ with the $N\left(\mu_{X}, \sigma_{X}^{2}\right)$ distribution and $Y_{1}, \ldots, Y_{n}$ with the $N\left(\mu_{Y}, \sigma_{Y}^{2}\right)$ distribution, where the means $\mu_{X}, \mu_{Y}$ and variances $\sigma_{X}^{2}, \sigma_{Y}^{2}$ are unknown. Derive the generalised likelihood ratio test of size $\alpha$ of the null hypothesis $H_{0}: \sigma_{X}^{2}=\sigma_{Y}^{2}$ against the alternative $H_{1}: \sigma_{X}^{2} \neq \sigma_{Y}^{2}$. Express the critical region in terms of the statistic $T=\frac{S_{X X}}{S_{X X}+S_{Y Y}}$ and the quantiles of a beta distribution, where

$$S_{X X}=\sum_{i=1}^{n} X_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^{n} X_{i}\right)^{2} \text { and } S_{Y Y}=\sum_{i=1}^{n} Y_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^{n} Y_{i}\right)^{2}$$

[You may use the following fact: if $U \sim \Gamma(a, \lambda)$ and $V \sim \Gamma(b, \lambda)$ are independent, then $\left.\frac{U}{U+V} \sim \operatorname{Beta}(a, b) .\right]$