 Solution

For bootstrapping, repeatedly sample $X_1^{\ast },\dots ,X_n^{\ast }$ with replacement from the empirical distribution of the observations and compute ${\hat{\beta }}^{\ast }=\alpha /{\overline{X}}^{\ast }$. Conditional on the data, estimate the $\gamma /2$ and $1-\gamma /2$ quantiles $q_l,q_u$ of $\sqrt{n}({\hat{\beta }}^{\ast }-\hat{\beta })$. The basic bootstrap interval isAs the number of resamples tends to infinity it estimates the conditional quantiles; bootstrap consistency and the continuous mapping theorem then give asymptotic coverage $1-\gamma$.