Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 5J / Solution

Assume $X$ has full column rank. The log-likelihood, up to constants, isThe ordinary least squares normal equations giveand maximizing over the variance gives the maximum-likelihood estimatorBecause $I-H$ is an orthogonal projection of rank $n-p$,Writing ${\chi }_{\nu ,q}^2$ for the $q$-quantile, a confidence interval of level $1-\alpha$ is