CUET-PG - Statistics · India · National Testing Agency
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)
ICAR AIEEA-PG Group 07, CUET-PG Statistics, SSC JSO and state statistical services
Two types of error, level of significance, and the z, t, chi-square and F tests applied to agricultural data.
How our questions for CUET-PG spread across the units of Statistical Inference II: Testing of Hypotheses & Sequential Analysis. Difficulty: 31 easy, 34 medium, 5 hard.
| Tests based on Z, t, chi-square and F distributions | 12 | |
| Hypotheses, the two types of error, size and power | 9 | |
| Power and sample-size calculation for common tests | 8 | |
| Multiple testing: family-wise error rate and false discovery rate | 7 | |
| Neyman-Pearson lemma and most powerful tests | 6 | |
| Wald and score (Rao) tests | 6 | |
| Monotone likelihood ratio and uniformly most powerful tests | 5 | |
| Unbiased tests and UMPU tests for exponential families | 5 | |
| Likelihood ratio tests and Wilks' theorem | 5 | |
| Relation between tests and confidence intervals | 5 | |
| Sequential probability ratio test: OC and ASN functions | 2 |
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
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
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 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 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
Germination is 45 of 100 seeds for treated and 30 of 100 for untreated seed. What is the pooled two-proportion z-statistic?
A. 2.191
Pooled p̄ = 75/200 = 0.375; SE = √(0.375 × 0.625 × (1/100 + 1/100)) = 0.0685; z = 0.15/0.0685 = 2.191. The unpooled SE gives 2.218; forgetting the two sample sizes gives 3.098.
Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics
In a 2 × 2 table one expected frequency is 3. What is the recommended analysis?
A. Fisher's exact test, because the χ² approximation fails with small expected counts
With expected counts below 5, the χ² approximation can be poor; Fisher's exact test (conditioning on the margins) is the standard alternative. Yates' correction is a cruder fix. The z-test for two proportions is equivalent to the uncorrected χ² and has the same problem.
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
In a variety trial, eight new entries are each to be compared only with one check variety, with control of the family-wise error rate. Which procedure is designed for this?
C. Dunnett's test
Dunnett's procedure handles many-to-one comparisons with a control and is more powerful there than Tukey's all-pairs method or Scheffé's all-contrasts method.
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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