GATE 2024 ST – Question 34

Testing of Hypotheses · Most Powerful and Uniformly Most Powerful Tests · 1 mark · Numerical answer

One X~N(0,σ²) tests H₀:σ²=1 against H₁:σ²=2 by the size-.05 Neyman–Pearson rule f₂(X)/f₁(X)>c. Given Φ(1.96)=.975,Φ(1.64)=.95,upper-tail χ²_(1,.05)=3.841,χ²_(2,.05)=5.991, find c to two decimals.

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Correct answer: 1.81 to 1.89

Explanation

The alternative/null density ratio is 2^(−1/2)exp(x²/4), increasing in x². Under the null X²~χ²₁, so reject above 3.841. Consequently c=exp(3.841/4)/√2≈**1.85**.

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