 Solution

The normal-model log-likelihood is $-\frac{n}{2}\log {\sigma }^2-\Vert Y-X\beta {\Vert }^2/(2{\sigma }^2)$ up to a constant. Differentiating gives the maximum-likelihood estimatorsLet $P=X(X^{T}X{)}^{-1}{X}^T$. The fitted component $P\epsilon$ and residual $(I-P)\epsilon$ are orthogonal jointly normal projections and hence independent. ThereforeThe prediction error $\tilde{y}-y^{\ast }=x^{\ast T}(\hat{\beta }-\beta )-{\epsilon }^{\ast }$ is normal with mean zero and variance ${\sigma }^2{\tau }^2$, and is independent of the residual variance estimate. Dividing by the independent estimated standard deviation yieldsThus the prediction interval in a normal linear model isThe extra $1$ includes the new observation's independent noise; omitting it would give a confidence interval for its conditional mean instead.