Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 6H / Solution

The Neyman–Pearson lemma says that among tests of size at most $\alpha$ for two simple hypotheses, a likelihood-ratio test rejecting where $p(x;{\theta }_1)/p(x;{\theta }_0)>k$ is most powerful, with boundary randomisation if required. If $\phi$ is that test and $\psi$ any competing test, choose $k$ so the sizes agree. Pointwise,Integration and the size inequality yield $E_1\phi \ge E_1\psi$.

Herewhich decreases with $|x|$. Thus reject for $|X|\le c$, where