GATE 2026 ST – Question 34

Testing of Hypotheses · Most Powerful and Uniformly Most Powerful Tests · 1 mark · Numerical answer

X is uniform on (θ,θ+1),θ∈ℝ. For testing H₀:θ≤0 against H₁:θ>1, let β be the power of a level-0.05 UMP test. Find 225β (integer).

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Correct answer: 225 to 225

Explanation

Under every null, X≤1 almost surely; under every alternative θ>1, X>1 almost surely. Rejecting when X>1 has size zero (within level 0.05) and power 1, the largest possible. If exact size is desired, rejection can be added within the null support without reducing power. Thus β=1 and **225β=225**.

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