= Solution
For <bootstrapping (statistics)>, repeatedly sample $X_1^*,\ldots,X_n^*$ with replacement from the empirical distribution of the observations and compute $\widehat β^*=α/\overline X^*$. Conditional on the data, estimate the $γ/2$ and $1-γ/2$ quantiles $q_l,q_u$ of $\sqrt n(\widehat β^*-\widehat β)$. The basic bootstrap interval is
$$
[\widehat β-q_u/\sqrt n,\widehat β-q_l/\sqrt n].
$$
As the number of resamples tends to infinity it estimates the conditional quantiles; bootstrap consistency and the continuous mapping theorem then give asymptotic coverage $1-γ$.
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