Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 18H / b / i / Solution

The likelihood ratio of the simple alternative to the simple null isThis is a strictly increasing function of $x$. By the Neyman-Pearson lemma, the most powerful test therefore rejects for $X>c$. The logistic distribution hasThe size condition ${\mathbb{P}}_0(X>c)=\alpha$ givesBecause the distribution is continuous, no boundary randomization is needed. The most powerful size-$\alpha$ test is therefore