GATE 2021 ST – Question 64
X₁,…,X₁₀ are iid with density f_θ(x)=f(x−θ),where f is symmetric about zero. For testing θ=1.2 against θ≠1.2,let T⁺ be the Wilcoxon signed-rank statistic. If η=P(T⁺<50) under H₀,find 32η,to two decimals.
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Correct answer: 31.60 to 31.80
Explanation
Under the continuous symmetric null,each of the 2¹⁰ sign patterns on ranks 1,…,10 is equally likely and the total rank sum is 55. By symmetry,P(T⁺≥50)=P(T⁺≤5). There are 10 subsets of distinct ranks with sum at most 5:1,1,1,2,2,3 for sums 0,1,2,3,4,5. Hence η=1−10/1024=1014/1024 and 32η=31.6875,rounded **31.69**.