GATE 2025 ST – Question 20

Testing of Hypotheses · Most Powerful and Uniformly Most Powerful Tests · 1 mark · Multiple choice

X has density f₀=1 on [0,1] or f₁=1 on [1,2], zero elsewhere. Let φ be a most powerful level-.05 test of f₀ against f₁. Which is necessary?

E_f₀φ=.05
E_f₁φ=1
P_f(φ=1)=P_f(X>1) for both f
P_f₁(φ=1)<1

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Correct answer: (B) E_f₁φ=1

Explanation

The test rejecting X>1 has null rejection zero and alternative power 1. Therefore any most powerful test must attain power 1, proving **B**. Its size need not exhaust .05, and additional null-support rejection can be added without changing power, so A,C are not necessary.

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