Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 207 / 3 / e / Solution

Let $X_1\sim \mathrm{Binomial}(22,{\pi }_0)$ be interim responses and independently $X_2\sim \mathrm{Binomial}(32,{\pi }_0)$ be second-stage responses. The design rejects immediately when $X_1\ge e_1=10$, continues when $r_1<X_1<e_1$, and after continuation rejects when $X_1+X_2\ge r=20$. Hence at ${\pi }_0=0.3$,This is the type-I error of the specified two-stage clinical trial design.