GATE 2026 ST – Question 21

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

For testing H₀ against H₁ at level α∈(0,1), which statement holds?

A UMP test always exists.
If a UMP test exists, its size is α.
A UMP unbiased test is necessarily UMP.
A UMP test, when it exists, is unbiased.

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Correct answer: (D) A UMP test, when it exists, is unbiased.

Explanation

The constant randomized test that rejects with probability α is a level-α competitor and has power α at every alternative. A UMP test must have at least that power, while its rejection probability under any null is at most α. Hence it is unbiased. **D**. UMP tests need not exist, a best unbiased test need not dominate biased tests, and some problems permit full power without exhausting the level.

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