Consider the simple linear regression model $y=\beta_{0}+\beta_{1} x+\varepsilon$, with $E(\varepsilon)=0$, $\operatorname{Var}(\varepsilon)=\sigma^{2}$, and $\varepsilon$ uncorrelated.

a. Show that $\operatorname{Cov}\left(\hat{\beta}_{0}, \hat{\beta}_{1}\right)=-\bar{x} \sigma^{2} / S_{x x}$.

b. Show that $\operatorname{Cov}\left(\bar{y}, \hat{\beta}_{1}\right)=0$.