GATE 2021 ST – Question 15

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

X₁,…,X_n are iid N(0,σ²),n≥2. Let T₁=ΣX_i²,T₂=X̄² and let 𝒮 be the joint density family for σ>0. Write χ²_(ν,α) for the (1−α) quantile. To test σ=1 against σ>1 at level α,which holds?

𝒮 has MLR in T₁;reject when T₁>χ²_(n,α).
𝒮 has MLR in T₁;reject when T₁>χ²_(n,1−α).
𝒮 has MLR in T₂;reject when T₂>χ²_(n,α).
𝒮 has MLR in T₂;reject when T₂>χ²_(1,1−α).

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Correct answer: (A) 𝒮 has MLR in T₁;reject when T₁>χ²_(n,α).

Explanation

The log likelihood is −n ln σ−T₁/(2σ²)+constant. For larger σ,the likelihood ratio increases with T₁. Under σ=1,T₁∼χ²_n,so its upper-tail α cutoff is χ²_(n,α). This is **A**.

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