Consider the linear regression model

$$Y_{i}=\alpha+\beta x_{i}+\varepsilon_{i}$$

for $i=1, \ldots, n$, where the non-zero numbers $x_{1}, \ldots, x_{n}$ are known and are such that $x_{1}+\ldots+x_{n}=0$, the independent random variables $\varepsilon_{1}, \ldots, \varepsilon_{n}$ have the $N\left(0, \sigma^{2}\right)$ distribution, and the parameters $\alpha, \beta$ and $\sigma^{2}$ are unknown.

(a) Let $(\hat{\alpha}, \hat{\beta})$ be the maximum likelihood estimator of $(\alpha, \beta)$. Prove that for each $i$, the random variables $\hat{\alpha}, \hat{\beta}$ and $Y_{i}-\hat{\alpha}-\hat{\beta} x_{i}$ are uncorrelated. Using standard facts about the multivariate normal distribution, prove that $\hat{\alpha}, \hat{\beta}$ and $\sum_{i=1}^{n}\left(Y_{i}-\hat{\alpha}-\hat{\beta} x_{i}\right)^{2}$ are independent.

(b) Find the critical region of the generalised likelihood ratio test of size $5 \%$ for testing $H_{0}: \alpha=0$ versus $H_{1}: \alpha \neq 0$. Prove that the power function of this test is of the form $w\left(\alpha, \beta, \sigma^{2}\right)=g(\alpha / \sigma)$ for some function $g$. [You are not required to find $g$ explicitly.]