Let $Y_{1}, \ldots, Y_{n}$ be observations satisfying

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

where $\epsilon_{1}, \ldots, \epsilon_{n}$ are independent random variables each with the $N\left(0, \sigma^{2}\right)$ distribution. Here $x_{1}, \ldots, x_{n}$ are known but $\beta$ and $\sigma^{2}$ are unknown.

(i) Determine the maximum-likelihood estimates $\left(\widehat{\beta}, \widehat{\sigma}^{2}\right)$ of $\left(\beta, \sigma^{2}\right)$.

(ii) Find the distribution of $\widehat{\beta}$.

(iii) By showing that $Y_{i}-\widehat{\beta} x_{i}$ and $\widehat{\beta}$ are independent, or otherwise, determine the joint distribution of $\widehat{\beta}$ and $\widehat{\sigma}^{2}$.

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