Solution
= Solution
By the definition of $c$, $\sum_i z_i=0$. Put $S=\sum_i z_i^2>0$. Then
$$
X^TX=\begin{pmatrix}n&0\\0&S\end{pmatrix}.
$$
Part (a) therefore gives
$$
\boxed{\operatorname{Cov}\begin{pmatrix}\widehat\alpha\\\widehat\beta\end{pmatrix}
=\sigma^2\begin{pmatrix}1/n&0\\0&1/S\end{pmatrix}.}
$$
In particular the two estimates are uncorrelated, and under the assumed normal-error model they are independent.