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Indian Statistical Institute admission tests Statistical Inference II: Testing of Hypotheses & Sequential Analysis: practice questions

Indian Statistical Institute admission tests (BStat, MStat, JRF in Statistics) · India · Indian Statistical Institute

Original practice questions written to the published syllabus · not past papers
10worked here, free
50written 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)

Where the 50 questions fall

How our questions for Indian Statistical Institute admission tests spread across the units of Statistical Inference II: Testing of Hypotheses & Sequential Analysis. Difficulty: 18 easy, 27 medium, 5 hard.

Neyman-Pearson lemma and most powerful tests6
Wald and score (Rao) tests6
Power and sample-size calculation for common tests6
Hypotheses, the two types of error, size and power5
Monotone likelihood ratio and uniformly most powerful tests5
Unbiased tests and UMPU tests for exponential families5
Relation between tests and confidence intervals5
Likelihood ratio tests and Wilks' theorem4
Multiple testing: family-wise error rate and false discovery rate4
Tests based on Z, t, chi-square and F distributions2
Sequential probability ratio test: OC and ASN functions2

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 · easy

Plot yields are N(μ, 16). From 16 plots, H₀: μ = 50 is rejected when x̄ > 51. What is the size of the test?

  1. 0.0500
  2. 0.1587
  3. 0.3173
  4. 0.4013
Answer and working

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

2 · Neyman-Pearson lemma and most powerful tests · medium

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:

  1. Σ(Xᵢ − μ)² is large
  2. X̄ is large
  3. |X̄ − μ| is large
  4. Σ(Xᵢ − μ)² is small
Answer and working

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

3 · 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

4 · Unbiased tests and UMPU tests for exponential families · medium

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?

  1. For μ > μ₀ its power falls below α
  2. Its power is exactly α everywhere
  3. Its size exceeds α at μ = μ₀
  4. For μ < μ₀ its power falls below α
Answer and working

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

5 · 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

6 · Wald and score (Rao) tests · hard

Which drawback is specific to the Wald test among the three asymptotic likelihood tests?

  1. It needs the model to be refitted under H₀ every time
  2. It has a different asymptotic null distribution from the others
  3. It cannot be used for more than one parameter at a time
  4. It is not invariant to reparameterisation of θ
Answer and working

D. It is not invariant to reparameterisation of θ

Testing θ = 1 and log θ = 0 by Wald can give different statistics, because the standard error is computed on the chosen scale. The LRT is invariant; all three are asymptotically χ² under H₀ and all extend to several parameters.

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

7 · Relation between tests and confidence intervals · easy

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?

  1. It rejects neither, because the interval does not include 0.6
  2. It rejects both a difference of 0 and a difference of 0.3
  3. It rejects a difference of 0 and does not reject a difference of 0.3
  4. It rejects a difference of 0.3 and does not reject a difference of 0
Answer and working

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

8 · 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

9 · Power and sample-size calculation for common tests · easy

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)?

  1. Φ(δ/σ − z_α)
  2. Φ(δ√n/σ + z_α)
  3. Φ(δ√n/σ − z_α)
  4. 1 − Φ(δ√n/σ − z_α)
Answer and working

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

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 Indian Statistical Institute admission testsFormat, sections and how it has changed

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