GATE 2026 ST – Question 65

Testing of Hypotheses · Most Powerful and Uniformly Most Powerful Tests · 2 marks · Numerical answer

A single X has density 2x/θ² on 0<x<θ, zero otherwise,θ>0. Test H₀:θ≤1 against H₁:θ>1 at level .05. Let β₁ be the size of a UMP test and β₂ its power at θ=2. Find 10β₁+40β₂ (integer).

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Correct answer: 31 to 31

Explanation

The null CDF is x² at θ=1, so choose upper rejection X>c with c²=.95. For θ≤1 rejection is maximized at θ=1, giving size β₁=.05. For any θ>1, the likelihood ratio is constant on the null support and observations above 1 are null-impossible; this upper region is most powerful and works for all alternatives. At θ=2 its power is β₂=1−c²/4=1−.95/4=.7625. The requested value is .5+40(.7625)=**31**.

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