Solution (source code)

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= Solution

Let $z_{1-γ/2}$ be the corresponding <standard normal distribution> quantile. <Slutsky theorem> permits replacing $β$ by the consistent estimator $\widehat β$, giving the confidence interval
$$
\left[\widehat β-z_{1-γ/2}{\widehat β\over\sqrt{nα}},\ 
\widehat β+z_{1-γ/2}{\widehat β\over\sqrt{nα}}\right].
$$
After standardization, the preceding convergence in distribution and continuity of the normal distribution show that its coverage tends to $1-γ$.

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