In a normal linear model with design matrix $X \in \mathbb{R}^{n \times p}$, output variables $y \in \mathbb{R}^{n}$ and parameters $\beta \in \mathbb{R}^{p}$ and $\sigma^{2}>0$, define a $(1-\alpha)$-level prediction interval for a new observation with input variables $x^{*} \in \mathbb{R}^{p}$. Derive an explicit formula for the interval, proving that it satisfies the properties required by the definition. [You may assume that the maximum likelihood estimator $\hat{\beta}$ is independent of $\sigma^{-2}\|y-X \hat{\beta}\|_{2}^{2}$, which has a $\chi_{n-p}^{2}$ distribution.]