Suppose that $X_{1}, \ldots, X_{n}$ are i.i.d. $N\left(\mu, \sigma^{2}\right)$. Let

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

(a) Compute the distributions of $\bar{X}$ and $S_{X X}$ and show that $\bar{X}$ and $S_{X X}$ are independent.

(b) Write down the distribution of $\sqrt{n}(\bar{X}-\mu) / \sqrt{S_{X X} /(n-1)}$.

(c) For $\alpha \in(0,1)$, find a $100(1-\alpha) \%$ confidence interval in each of the following situations: (i) for $\mu$ when $\sigma^{2}$ is known; (ii) for $\mu$ when $\sigma^{2}$ is not known; (iii) for $\sigma^{2}$ when $\mu$ is not known.

(d) Suppose that $\widetilde{X}_{1}, \ldots, \widetilde{X}_{\widetilde{n}}$ are i.i.d. $N\left(\widetilde{\mu}, \widetilde{\sigma}^{2}\right)$. Explain how you would use the $F$ test to test the hypothesis $H_{1}: \sigma^{2}>\tilde{\sigma}^{2}$ against the hypothesis $H_{0}: \sigma^{2}=\tilde{\sigma}^{2}$. Does the $F$ test depend on whether $\mu, \widetilde{\mu}$ are known?