GATE 2024 ST – Question 64

Testing of Hypotheses · Likelihood Ratio and Large Sample Tests · 2 marks · Numerical answer

Four observations from N(μ,1),μ unknown, have sample mean .16. Given Φ(2.28)=.989,Φ(1.96)=.975,Φ(1.64)=.95, a size-.05 LRT tests μ=0 against μ≠0. Find its power “at the sample mean” to three decimals.

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Correct answer: 0.055 to 0.067

Explanation

The test rejects when |2X̄|>1.96. Interpret the source phrase as evaluating the power function at parameter μ=.16 (a plug-in effect), not as a probability conditional on the observed sample mean. Under this parameter 2X̄~N(.32,1). Power=1−Φ(1.96−.32)+Φ(−1.96−.32)=1−.95+1−.989=**.061**.

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