DSE Statistical Inference and Hypothesis Testing 4 — Questions and Answers
Question 1: Which nonparametric test is the rank-based alternative to the independent samples t-test?
- Wilcoxon signed-rank test
- Mann-Whitney U test (Correct answer)
- Kruskal-Wallis test
- Friedman test
Correct answer: Mann-Whitney U test
The Mann-Whitney U test compares rank sums from two independent groups without assuming normality, serving as an alternative to the two-sample t-test.
Question 2: In a two-sided hypothesis test with α = 0.05, the critical region is placed:
- Entirely in the right tail
- Entirely in the left tail
- Split equally between both tails (Correct answer)
- Determined by the sign of the test statistic
Correct answer: Split equally between both tails
A two-sided test splits α/2 = 0.025 into each tail, rejecting H₀ when the test statistic is extreme in either direction.
Question 3: What is a sufficient statistic?
- A statistic that has minimum variance among all unbiased estimators
- A statistic that captures all the information in the data about the parameter of interest (Correct answer)
- A statistic whose sampling distribution is known exactly
- A statistic computed from the entire population
Correct answer: A statistic that captures all the information in the data about the parameter of interest
A sufficient statistic T(X) for parameter θ means the conditional distribution of the data given T does not depend on θ, so T contains all parameter information.
Question 4: The Neyman-Pearson lemma states that the most powerful test for simple H₀ vs. simple H₁ uses:
- The sample mean as the test statistic
- A likelihood ratio threshold as the rejection region (Correct answer)
- The largest observed value in the sample
- A chi-squared statistic based on observed frequencies
Correct answer: A likelihood ratio threshold as the rejection region
Neyman-Pearson shows that rejecting H₀ when the likelihood ratio L(θ₁)/L(θ₀) exceeds a constant k produces the most powerful level-α test.
Question 5: A researcher tests 20 independent null hypotheses, each at α = 0.05. The expected number of false rejections under all null hypotheses being true is:
- 0
- 1 (Correct answer)
- 5
- 20
Correct answer: 1
Expected false positives = 20 × 0.05 = 1, illustrating the multiple testing problem even when every null is actually true.
Question 6: The false discovery rate (FDR), controlled by the Benjamini-Hochberg procedure, is defined as:
- The probability of at least one false rejection
- The expected proportion of rejected nulls that are false rejections (Correct answer)
- The probability that a specific rejected null is true
- One minus the familywise error rate
Correct answer: The expected proportion of rejected nulls that are false rejections
FDR = E[V/R] where V is false rejections and R is total rejections; BH controls this expected proportion, offering more power than Bonferroni.
Question 7: An estimator is consistent if, as sample size increases:
- Its variance approaches zero regardless of the true parameter
- It converges in probability to the true parameter value (Correct answer)
- Its bias becomes exactly zero for all finite samples
- Its sampling distribution becomes exactly normal
Correct answer: It converges in probability to the true parameter value
Consistency requires that the estimator converges in probability to the true θ as n → ∞, meaning it gets arbitrarily close with high probability.
DSE Statistical Inference and Hypothesis Testing 4
Which nonparametric test is the rank-based alternative to the independent samples t-test?