 Solution

Let $v_i=[(X^{T}X{)}^{-1}{]}_{ii}>0$. Part (b) givesThe residual vector and coefficient estimator are independent: they are jointly normal and their cross-covariance matrix is zero, since $X^T(I-H)=0$. Hence $Z_i$ is independent of $U=(n-p)s^2/{\sigma }^2\sim {\chi }_{n-p}^2$. By the definition of Student's t-distribution,Let $t_{n-p,1-\alpha /2}$ be the $1-\alpha /2$ quantile of this distribution. Inverting the central probability statement gives the exact marginal confidence intervalIts coverage is $1-\alpha$ under the stated Gaussian linear model; simultaneous coverage for all coefficients would require a separate adjustment.