Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 28J / b / Solution

Let $z_{1-\gamma /2}$ be the corresponding standard normal distribution quantile. Slutsky theorem permits replacing $\beta$ by the consistent estimator $\hat{\beta }$, giving the confidence intervalAfter standardization, the preceding convergence in distribution and continuity of the normal distribution show that its coverage tends to $1-\gamma$.