For iid N(θ,1),n≥2, test H₀:θ∈[1,2] against θ 2. Which statement is true?

A UMP test rejects when c₁≤Σxi≤c₂.

A UMP test rejects whenΣxi>c orΣxi<d.

GATE 2020 ST – Question 50

Testing of Hypotheses · Most Powerful and Uniformly Most Powerful Tests · 2 marks · Multiple choice

For iid N(θ,1),n≥2, test H₀:θ∈[1,2] against θ<1 orθ>2. Which statement is true?

A UMP test rejects when c₁≤Σxi≤c₂.
A UMP test rejects whenΣxi>c orΣxi<d.
No UMP test exists at any level0<α<1.
A UMP test has power belowα.

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Show answer and explanation

Correct answer: (C) No UMP test exists at any level0<α<1.

Explanation

An alternative below1 favours allocating rejection probability to small sample means; an alternative above2 favours large means. Their most powerful rejection directions conflict. No single level-α test is uniformly most powerful over both outside alternatives. Thus C; a sensible two-sided procedure is not automatically UMP.

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