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 Incoming links: Crossing-power obstruction to a uniformly most powerful test

Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 18H / d / Solution Created 2026-09-24 Updated 2026-09-25

The two tests have the same size, but their powers cross. At

θ = 0.1

,

π_{1} (0.1) = 0.15, π_{2} (0.1) = \frac{1}{2} (\frac{1}{10} + 0.2)^{2} < 0.15.

At

θ = 0.5

,

π_{1} (0.5) = 0.55, π_{2} (0.5) = 1 - \frac{1}{2} (1 - \frac{1}{10})^{2} > 0.55.

Thus Test 2 is a same-size test that beats Test 1 at one alternative, while Test 1 beats Test 2 at another. By the crossing-power obstruction to a uniformly most powerful test,

neither Test 1 nor Test 2 is uniformly most powerful .

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