GATE 2020 ST – Question 57

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

X takes1,2,3,4,5. Under H₀ probabilities are(.10,.10,.10,.10,.60); under H₁ they are(.05,.06,.08,.09,.72). Find power of the most powerful level0.1 test (two decimals).

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: 0.12 to 0.12

Explanation

Likelihood ratios f₁/f₀ are0.5,0.6,0.8,0.9,1.2. Reject preferentially at X=5, the largest ratio. Since its null probability is0.6, randomize rejection there with probability0.1/0.6=1/6. Power=0.72/6=0.12. Randomization is necessary to obtain the optimal exact level.

More Most Powerful and Uniformly Most Powerful Tests questions

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…
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…
GATE 2026 ST Q57 – From a single observation, test H₀:θ=1 against H₁:θ≠1 at level 0.05. For which density…
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…
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…

All Most Powerful and Uniformly Most Powerful Tests questions →