Consider the linear regression model

$$Y_{i}=\alpha+\beta x_{i}+\epsilon_{i}, \quad 1 \leqslant i \leqslant n$$

where $\epsilon_{1}, \ldots, \epsilon_{n}$ are independent, identically distributed $N\left(0, \sigma^{2}\right), x_{1}, \ldots, x_{n}$ are known real numbers with $\sum_{i=1}^{n} x_{i}=0$ and $\alpha, \beta$ and $\sigma^{2}$ are unknown.

(i) Find the least-squares estimates $\widehat{\alpha}$and $\widehat{\beta}$ of $\alpha$ and $\beta$, respectively, and explain why in this case they are the same as the maximum-likelihood estimates.

(ii) Determine the maximum-likelihood estimate $\widehat{\sigma}^{2}$ of $\sigma^{2}$ and find a multiple of it which is an unbiased estimate of $\sigma^{2}$.

(iii) Determine the joint distribution of $\widehat{\alpha}, \widehat{\beta}$ and $\widehat{\sigma}^{2}$.

(iv) Explain carefully how you would test the hypothesis $H_{0}: \alpha=\alpha_{0}$ against the alternative $H_{1}: \alpha \neq \alpha_{0}$.