A statistician is interested in the power of a $t$-test with level $5 \%$ in linear regression; that is, the probability of rejecting the null hypothesis $\beta_{0}=0$ with this test under an alternative with $\beta_{0}>0$.

(a) State the distribution of the least-squares estimator $\hat{\beta}_{0}$, and hence state the form of the $t$-test statistic used.

(b) Prove that the power does not depend on the other coefficients $\beta_{j}$ for $j>0$.