GATE - Statistics (ST) · India · IITs and IISc on rotation (2026: IIT Guwahati)
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)
GATE Statistics (ST) and IIT JAM Mathematical Statistics
Most powerful critical regions, size and power computations (including discrete and uniform cases), MLR and UMP tests, LRT statistics and their asymptotic chi-square df, and p-values; multiple-select items on which tests are UMP or unbiased.
How our questions for GATE spread across the units of Statistical Inference II: Testing of Hypotheses & Sequential Analysis. Difficulty: 19 easy, 29 medium, 13 hard.
| Neyman-Pearson lemma and most powerful tests | 9 | |
| Monotone likelihood ratio and uniformly most powerful tests | 7 | |
| Wald and score (Rao) tests | 7 | |
| Likelihood ratio tests and Wilks' theorem | 6 | |
| Power and sample-size calculation for common tests | 6 | |
| Hypotheses, the two types of error, size and power | 5 | |
| Unbiased tests and UMPU tests for exponential families | 5 | |
| Relation between tests and confidence intervals | 5 | |
| Multiple testing: family-wise error rate and false discovery rate | 5 | |
| Sequential probability ratio test: OC and ASN functions | 4 | |
| Tests based on Z, t, chi-square and F distributions | 2 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
Plot yields are N(μ, 16). From 16 plots, H₀: μ = 50 is rejected when x̄ > 51. What is the size of the test?
B. 0.1587
Under H₀, X̄ ~ N(50, 1), so α = P(X̄ > 51) = P(Z > 1) = 0.1587. Using σ = 4 instead of SE = 1 gives z = 0.25; doubling gives a two-sided figure that does not apply to this one-sided rule.
Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics
For H₀: p = 0.5 against H₁: p = 0.8 with 10 Bernoulli trials, the MP test rejects for large X. P(X ≥ 9 | H₀) = 11/1024 and P(X = 8 | H₀) = 45/1024. For an exact size of 0.05, with what probability should H₀ be rejected when X = 8?
B. 0.893
Size = P(X ≥ 9) + γP(X = 8) = 0.05 gives γ = (0.05 − 11/1024)/(45/1024) = 0.893. Dividing 0.05 by P(X = 8) alone ignores the rejections already made at X ≥ 9.
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
Under a monotone likelihood ratio in T, what does the Karlin-Rubin theorem give for H₀: θ ≤ θ₀ against H₁: θ > θ₀?
D. A UMP level-α test that rejects for large T
Karlin-Rubin: rejecting when T > c with P_θ₀(T > c) = α is UMP for the one-sided problem, and its power function is non-decreasing in θ.
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
The generalised likelihood-ratio statistic for H₀: θ ∈ Θ₀ is Λ = sup_Θ₀ L(θ)/sup_Θ L(θ). When is H₀ rejected?
C. When Λ is small, i.e. when −2 log Λ is large
Λ lies in [0, 1]; values near 1 mean the restricted model fits nearly as well as the full one, small values mean H₀ fits poorly. Λ cannot exceed 1 because the full maximum is at least the restricted one.
Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics
A 95% two-sided confidence interval for a mean difference is 0.1 to 0.5 t/ha. What does the corresponding 5% two-sided test conclude?
C. It rejects a difference of 0 and does not reject a difference of 0.3
A two-sided level-α test rejects θ₀ exactly when θ₀ lies outside the 1 − α interval: 0 is outside, 0.3 inside.
Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics
Four ordered p-values are 0.004, 0.011, 0.020 and 0.040. How many hypotheses does Holm's step-down procedure reject at family-wise level 0.05?
D. 4
Holm compares p₍ⱼ₎ with 0.05/(m − j + 1): 0.004 ≤ 0.0125, 0.011 ≤ 0.0167, 0.020 ≤ 0.025, 0.040 ≤ 0.05, so all four are rejected. Bonferroni (all at 0.0125) would reject only two.
Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics
For a one-sided z-test of H₀: μ = μ₀ at level α with known σ and n observations, what is the power at μ₀ + δ (δ > 0)?
C. Φ(δ√n/σ − z_α)
Power = P(X̄ > μ₀ + z_ασ/√n | μ₀ + δ) = P(Z > z_α − δ√n/σ) = Φ(δ√n/σ − z_α). 1 − Φ(...) is β; omitting √n ignores the sample size.
Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics
After a non-significant trial, a scientist computes 'observed power' from the observed effect size. What is wrong with this?
D. Observed power is a function of the p-value and adds no information on the effect
Post-hoc power at the observed effect is a monotone transformation of the p-value; a non-significant result always has low 'observed power'. Power is a planning quantity, computed for a pre-specified effect.
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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