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ASRB NET Statistical Inference II: Testing of Hypotheses & Sequential Analysis: practice questions

ASRB NET - Agricultural Statistics (and Computer Application / Bioinformatics) · India · Agricultural Scientists Recruitment Board (ICAR)

Original practice questions written to the published syllabus · not past papers
10worked here, free
41written to this exam in this subject
11syllabus units

What this subject covers

The 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)

What ASRB NET asks from it

ASRB NET / ARS Agricultural Statistics

Choosing and interpreting tests for trial and survey data, multiple comparisons, sequential sampling for pests, and replication from power and CV.

Where the 41 questions fall

How our questions for ASRB NET spread across the units of Statistical Inference II: Testing of Hypotheses & Sequential Analysis. Difficulty: 11 easy, 20 medium, 10 hard.

Sequential probability ratio test: OC and ASN functions9
Multiple testing: family-wise error rate and false discovery rate6
Likelihood ratio tests and Wilks' theorem5
Power and sample-size calculation for common tests5
Tests based on Z, t, chi-square and F distributions4
Hypotheses, the two types of error, size and power3
Neyman-Pearson lemma and most powerful tests2
Monotone likelihood ratio and uniformly most powerful tests2
Unbiased tests and UMPU tests for exponential families2
Relation between tests and confidence intervals2
Wald and score (Rao) tests1

10 practice questions, worked

Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.

1 · Hypotheses, the two types of error, size and power · hard

Plot yields are N(μ, 16). From 16 plots, H₀: μ = 50 is rejected in favour of H₁: μ = 52 when x̄ > 51.645. What is the power of the test?

  1. 0.639
  2. 0.361
  3. 0.050
  4. 0.535
Answer and working

A. 0.639

SE = 4/√16 = 1. Power = P(X̄ > 51.645 | μ = 52) = P(Z > −0.355) = 0.639. 0.050 is the size; 0.361 is β; using σ = 4 instead of the standard error gives 0.535.

Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics

2 · Monotone likelihood ratio and uniformly most powerful tests · medium

Under a monotone likelihood ratio in T, what does the Karlin-Rubin theorem give for H₀: θ ≤ θ₀ against H₁: θ > θ₀?

  1. A UMP level-α test that rejects for small T
  2. An LRT that is UMP only as n becomes large
  3. A UMPU test, though no UMP test exists here
  4. A UMP level-α test that rejects for large T
Answer and working

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

3 · Unbiased tests and UMPU tests for exponential families · hard

In what sense is Fisher's exact test for a 2 × 2 table optimal?

  1. It has exactly size α for every table, with no randomisation needed
  2. Its randomised version is UMPU for the odds ratio, by conditioning on the margins
  3. It is the likelihood-ratio test, so Wilks' theorem gives its exact size
  4. It is UMP among all tests of independence, conditionally and unconditionally, for any table size
Answer and working

B. Its randomised version is UMPU for the odds ratio, by conditioning on the margins

The margins are sufficient for the nuisance parameters; conditioning on them gives a hypergeometric null distribution, and the randomised conditional test is UMPU. The non-randomised version is conservative.

Further reading: Lehmann and Romano, Testing Statistical Hypotheses; Casella and Berger, Statistical Inference; Rohatgi and Saleh, An Introduction to Probability and Statistics

4 · Likelihood ratio tests and Wilks' theorem · easy

The generalised likelihood-ratio statistic for H₀: θ ∈ Θ₀ is Λ = sup_Θ₀ L(θ)/sup_Θ L(θ). When is H₀ rejected?

  1. When Λ exceeds 1, so the null fits better
  2. When Λ equals 0.5, the midpoint of its range
  3. When Λ is small, i.e. when −2 log Λ is large
  4. When Λ is large, close to its maximum of 1
Answer and working

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

5 · Tests based on Z, t, chi-square and F distributions · medium

Germination is 45 of 100 seeds for treated and 30 of 100 for untreated seed. What is the pooled two-proportion z-statistic?

  1. 2.191
  2. 3.098
  3. 2.218
  4. 32.000
Answer and working

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

6 · Tests based on Z, t, chi-square and F distributions · medium

In a 2 × 2 table one expected frequency is 3. What is the recommended analysis?

  1. Fisher's exact test, because the χ² approximation fails with small expected counts
  2. Pooling the table with another experiment until every cell exceeds 3
  3. The z-test for two proportions, because it never needs large expected counts in any cell
  4. Pearson's χ² without correction, because small counts make it more powerful
Answer and working

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

7 · Multiple testing: family-wise error rate and false discovery rate · medium

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?

  1. 3
  2. 1
  3. 2
  4. 4
Answer and working

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

8 · Multiple testing: family-wise error rate and false discovery rate · easy

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?

  1. Duncan's multiple range test
  2. Scheffé's method
  3. Dunnett's test
  4. Tukey's HSD test
Answer and working

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

9 · Sequential probability ratio test: OC and ASN functions · medium

In Wald's SPRT with boundaries B < 1 < A on the likelihood ratio Λₙ = L₁/L₀, what is the rule after n observations?

  1. Reject H₀ if Λₙ ≤ B, accept H₀ if Λₙ ≥ A, otherwise take another observation
  2. Accept H₀ if Λₙ ≤ A, otherwise reject H₀, after a fixed number of steps
  3. Accept H₀ if Λₙ ≤ B, reject H₀ if Λₙ ≥ A, otherwise take another observation
  4. Reject H₀ if B < Λₙ < A, otherwise stop sampling and accept H₀
Answer and working

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

10 · Power and sample-size calculation for common tests · medium

After a non-significant trial, a scientist computes 'observed power' from the observed effect size. What is wrong with this?

  1. Observed power must be computed at α = 0.10, not at the α used in the original trial
  2. Observed power is always exactly 0.5, whatever the trial found
  3. Observed power requires the non-central F, which applies only to ANOVA
  4. Observed power is a function of the p-value and adds no information on the effect
Answer and working

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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Statistical Inference II: Testing of Hypotheses & Sequential AnalysisThe full study manual, free ASRB NETFormat, sections and how it has changed

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