Past exam of the options I course of the University of Oxford / 2012 / O / O2 / b / i / Solution

The design columns are $1$, $x$ and $z$. Their pairwise inner products are zero by the three supplied sums. Put $S_{xx}={\sum }_i{x}_i^2$ and $S_{zz}={\sum }_i{z}_i^2$, positive under the full-rank assumption. ThenApplying part (a),The off-diagonal covariances vanish because the design columns are orthogonal. With Gaussian errors, the coefficient estimators are also jointly normal and therefore independent.