GATE Testing of Hypotheses: Most Powerful and Uniformly Most Powerful Tests – Previous Year Questions
20 GATE previous year questions on Most Powerful and Uniformly Most Powerful Tests (Testing of Hypotheses, Statistics) with answers and explanations, from every paper.
- GATE 2021 ST Q15 – 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…
- GATE 2021 ST Q62 – A size-100 sample from N(μ,9) has mean 5.608. Given Φ(1.96)=.975 and Φ(1.64)=.95,find the p-value of the UMP unbiased test μ=5.02 against μ≠5.02,to…
- GATE 2021 ST Q63 – X takes 7,8,9,10. Under p₀ the probabilities are .90,.05,.04,.01;under p₁ they are .69,.10,.16,.05. Find the power of the most-powerful size-.05 test…
- GATE 2022 ST Q20 – Six iid Bernoulli(θ) observations are used to test θ=1/2 against θ>1/2 via Y=ΣX i. Observed Y=4. Find the p-value.
- GATE 2022 ST Q60 – Four iid Bernoulli(θ) observations test θ=1/4 against θ>1/4 by rejecting when ΣX i>2. Let α be type-I error and γ(θ) the power. Find 16α+7γ(1/2),as an…
- GATE 2022 ST Q61 – 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…
- GATE 2023 ST Q28 – From one observation x,test standard normal density exp(−y²/2)/√(2π) against Laplace density exp(− y )/2. Which most-powerful rejection region has the…
- GATE 2023 ST Q42 – For (X,Y)∈0,1²,P(0,0)=P(1,1)=θ and P(1,0)=P(0,1)=1/2−θ,0≤θ≤1/2. Test θ=1/4 against θ=1/3 using n iid pairs. If m counts equal pairs,which…
- GATE 2023 ST Q64 – One observation has CDF F(x)=0 for x<0,1−(1−x)θ for 0≤x<1,and 1 for x≥1,θ>0. Test θ=1 against θ=2 by rejecting if x<0.5. If α is the level and β the…
- GATE 2024 ST Q34 – 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…
- GATE 2024 ST Q56 – An iid exponential-scale sample n≥2 has density e(−x/λ)/λ,x≥0,λ>0. Put T₁=ΣX i,T₂=1/T₁. Write χ² (ν,α) for the upper-tail-α quantile. Which test is…
- GATE 2025 ST Q20 – X has density f₀=1 on [0,1] or f₁=1 on [1,2], zero elsewhere. Let φ be a most powerful level-.05 test of f₀ against f₁. Which is necessary?
- GATE 2025 ST Q21 – For simple densities f₀,f₁, let φ be a most powerful level-.05 test based on X. Which is not necessarily correct?
- GATE 2025 ST Q43 – X₁,X₂ are iid exponential of scale θ>0. For H₀:θ≤1 against H₁:θ>1, φ=1 (X₁>1). With S=X₁+X₂, which test has the same power function as φ?
- GATE 2025 ST Q62 – X₁,X₂ are iid with density e(x−θ) for x≤θ, zero otherwise,θ∈ℝ. For H₀:θ≥0 against H₁:θ<0, let β(θ) be the power of a level-.09 UMP test. Find β(ln…
- GATE 2026 ST Q21 – For testing H₀ against H₁ at level α∈(0,1), which statement holds?
- GATE 2026 ST Q34 – X is uniform on (θ,θ+1),θ∈ℝ. For testing H₀:θ≤0 against H₁:θ>1, let β be the power of a level-0.05 UMP test. Find 225β (integer).
- GATE 2026 ST Q45 – An iid sample n>1 has f β(x)=βe(−x)(1−e(−x))(β−1),x>0,β>0. Test H₀:β=1 against H₁:β>1 at level α. Which holds?
- GATE 2026 ST Q57 – From a single observation, test H₀:θ=1 against H₁:θ≠1 at level 0.05. For which density families does a UMP test exist?
- GATE 2026 ST Q65 – A single X has density 2x/θ² on 0<x<θ, zero otherwise,θ>0. Test H₀:θ≤1 against H₁:θ>1 at level .05. Let β₁ be the size of a UMP test and β₂ its power…