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

A possibly randomized test is a function $\phi (X)\in [0,1]$, interpreted as the conditional probability of rejecting $H_0$. Its power function isIt has size at most $\alpha$ whenSuch a test is a uniformly most powerful test of size $\alpha$ if, for every other test $\psi$ of size at most $\alpha$,Thus one test maximizes power simultaneously at every parameter value in the alternative.