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

At the boundary null ${\pi }_1={\pi }_0=\pi$, $\bar{\pi }=\pi$. Independence of the binomial proportions givesasymptotically, so $T\Rightarrow N(0,1)$. For the composite null $\delta <0$, the mean moves to the nonrejection side, making the boundary the least favorable case.