ASRB Agricultural Research Service (ARS) - Agricultural Statistics scientist · India · Agricultural Scientists Recruitment Board (ICAR)
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 ASRB Agricultural Research Service spread across the units of Statistical Inference II: Testing of Hypotheses & Sequential Analysis. Difficulty: 5 easy, 14 medium, 6 hard.
| Multiple testing: family-wise error rate and false discovery rate | 6 | |
| Power and sample-size calculation for common tests | 5 | |
| Likelihood ratio tests and Wilks' theorem | 3 | |
| Tests based on Z, t, chi-square and F distributions | 3 | |
| Sequential probability ratio test: OC and ASN functions | 3 | |
| Hypotheses, the two types of error, size and power | 2 | |
| Neyman-Pearson lemma and most powerful tests | 1 | |
| Wald and score (Rao) tests | 1 | |
| Relation between tests and confidence intervals | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
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
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
Where is the average sample number (ASN) of an SPRT largest?
D. At parameter values between θ₀ and θ₁, where evidence accumulates slowly
Between the two hypotheses E_θ(Z) is near zero, so the random walk drifts slowly and takes longest to hit a boundary. Far from both values decisions are quick.
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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