GATE 2022 ST – Question 61

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

For iid N(μ,2²) observations,test μ=10 against μ=12 by rejecting for large X̄. Given Φ(1.645)=.95 and Φ(2.33)=.99,find the smallest integer sample size with type-I error .05 and type-II error at most .01.

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Correct answer: 16 to 16

Explanation

The size-.05 cutoff is c=10+1.645·2/√n. Under μ=12,the standardized cutoff is (c−12)/(2/√n)=1.645−√n. Requiring its CDF≤.01 gives 1.645−√n≤−2.33,or n≥3.975²=15.800625. Smallest integer **16**.

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