Solution
= Solution
Averaging the centered model gives
$$
\hat\alpha'=\bar Y
=\alpha'+\bar\epsilon.
$$
Therefore
$$
\boxed{\hat\alpha'\sim N\left(\alpha',\frac{\sigma^2}{n}\right)}.
$$
Writing $z_{0.975}=\Phi^{-1}(0.975)$, an exact 95% confidence interval is
$$
\boxed{\bar Y\pm z_{0.975}\frac\sigma{\sqrt n}}.
$$