Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 2 / a / Solution

Let ${\hat{p}}_k=X_k/n_k$, where $X_k\sim \mathrm{Binomial}(n_k,p_k)$ independently, and putThe Wald statistic is $Z=({\hat{p}}_1-{\hat{p}}_0)/\sqrt{\hat{V}}$. The central limit theorem and Slutsky theorem give $Z\dot{\sim }N(0,1)$ under $H_0$. When $p_1-p_0={\delta }^{\ast }>0$,