Indian Statistical Service Examination (with Indian Economic Service) · India · Union Public Service Commission
Original practice questions written to the published syllabus · not past papersThe logic and optimality of tests: errors and power, the Neyman-Pearson lemma, uniformly most powerful and unbiased tests, likelihood-ratio tests and their large-sample forms, and Wald's sequential probability ratio test.
In the exams: AICE-JRF Ag Stat Units 1-2 incl. SPRT (heavy); GATE ST 'Testing of Hypotheses' (heavy); IIT JAM MS (heavy); UPSC ISS Statistics I-II (heavy)
How our questions for Indian Statistical Service Examination spread across the units of Statistical Inference II: Testing of Hypotheses & Sequential Analysis. Difficulty: 6 easy, 15 medium, 6 hard.
| Monotone likelihood ratio and uniformly most powerful tests | 4 | |
| Tests based on Z, t, chi-square and F distributions | 4 | |
| Hypotheses, the two types of error, size and power | 3 | |
| Unbiased tests and UMPU tests for exponential families | 3 | |
| Neyman-Pearson lemma and most powerful tests | 2 | |
| Likelihood ratio tests and Wilks' theorem | 2 | |
| Wald and score (Rao) tests | 2 | |
| Relation between tests and confidence intervals | 2 | |
| Sequential probability ratio test: OC and ASN functions | 2 | |
| Power and sample-size calculation for common tests | 2 | |
| Multiple testing: family-wise error rate and false discovery rate | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
What is the power of a test at a particular alternative θ₁?
B. The probability of rejecting H₀ when θ₁ is the true value
Power = 1 − β(θ₁) = P_θ₁(reject H₀). Accepting H₀ under θ₁ is the Type II error probability β; rejecting a true H₀ has probability α. The posterior probability of H₁ is a Bayesian quantity, not power.
Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics
For a fixed sample size, the critical region of a test is made smaller. What happens to the error probabilities?
D. α falls and β rises
A smaller critical region rejects H₀ less often under every θ, so the Type I error probability falls and the Type II error probability rises. Lowering both needs more data or less noise.
Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics
X₁, ..., Xₙ are iid N(μ, σ²) with μ known. For H₀: σ² = σ₀² against H₁: σ² = σ₁² with σ₁² > σ₀², the most powerful test rejects when:
A. Σ(Xᵢ − μ)² is large
log Λ = −(n/2)log(σ₁²/σ₀²) + (1/2)(1/σ₀² − 1/σ₁²)Σ(Xᵢ − μ)², increasing in Σ(Xᵢ − μ)² when σ₁² > σ₀². The test uses χ²ₙ critical values since μ is known.
Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics
Why is there no UMP test of a one-sided hypothesis about the location θ of a Cauchy distribution based on one observation?
C. The likelihood ratio f(x; θ₁)/f(x; θ₀) is not monotone in x, so the MP region depends on θ₁
f(x; θ₁)/f(x; θ₀) rises and then falls back to 1 as x → ±∞, so the MP region against θ₁ is a bounded interval whose location depends on θ₁; no single region is best for all θ₁ > θ₀. Tests do exist; the support is the whole line; information is positive.
Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics
The one-sided test 'reject H₀: μ = μ₀ when Z > z_α' is used against the two-sided alternative μ ≠ μ₀. Why is this test biased?
D. For μ < μ₀ its power falls below α
As μ decreases below μ₀, P(Z > z_α) drops below α: the test is less likely to reject when μ < μ₀ than when H₀ is true.
Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics
A 2 × 2 table of adoption (yes/no) by training (yes/no) has cells a = 30, b = 20, c = 10, d = 40. What is Pearson's χ² without continuity correction?
A. 16.67
χ² = n(ad − bc)²/[(a + b)(c + d)(a + c)(b + d)] = 100 × 1000²/(50 × 50 × 40 × 60) = 16.67, with 1 df. With Yates' correction it is 15.04; omitting n gives 0.17.
Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics
In Wald's SPRT with boundaries B < 1 < A on the likelihood ratio Λₙ = L₁/L₀, what is the rule after n observations?
C. Accept H₀ if Λₙ ≤ B, reject H₀ if Λₙ ≥ A, otherwise take another observation
Λₙ large favours H₁ (reject H₀); small favours H₀. Sampling continues while the ratio stays between the boundaries.
Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics
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