DSE Statistical Inference and Hypothesis Testing 5 — Questions and Answers
Question 1: A permutation test determines the p-value by:
- Fitting a parametric model to the data and computing residuals
- Repeatedly shuffling group labels and recomputing the test statistic to build a null distribution (Correct answer)
- Resampling with replacement from the observed data
- Comparing the test statistic to a theoretical chi-squared distribution
Correct answer: Repeatedly shuffling group labels and recomputing the test statistic to build a null distribution
Permutation tests approximate the null distribution empirically by randomly reassigning labels many times, making no distributional assumptions.
Question 2: In the context of maximum likelihood estimation, the score function is:
- The second derivative of the log-likelihood with respect to θ
- The first derivative of the log-likelihood with respect to θ (Correct answer)
- The ratio of the likelihood to its maximum value
- The expected value of the log-likelihood under the true parameter
Correct answer: The first derivative of the log-likelihood with respect to θ
The score function S(θ) = ∂ℓ/∂θ equals zero at the MLE and has expectation zero under the true parameter.
Question 3: Which statement about bootstrap confidence intervals is TRUE?
- They require the sampling distribution to be symmetric
- They are exact and always outperform normal-theory intervals
- They approximate the sampling distribution using resampling from observed data (Correct answer)
- They are only valid when the sample size exceeds 1,000
Correct answer: They approximate the sampling distribution using resampling from observed data
Bootstrap intervals use repeated resampling with replacement to estimate the sampling distribution without relying on parametric assumptions.
Question 4: A z-test for a proportion is valid when:
- The sample is drawn without replacement from a finite population
- Both np and n(1−p) are at least 10, ensuring the normal approximation holds (Correct answer)
- The true proportion p is less than 0.5
- The sample size is less than 30
Correct answer: Both np and n(1−p) are at least 10, ensuring the normal approximation holds
The normal approximation to the binomial is reliable when np ≥ 10 and n(1−p) ≥ 10, ensuring enough expected successes and failures.
Question 5: The Cramér-Rao lower bound gives the minimum variance achievable by:
- Any consistent estimator with large n
- Any unbiased estimator of a parameter (Correct answer)
- The maximum likelihood estimator only
- A sufficient statistic under all loss functions
Correct answer: Any unbiased estimator of a parameter
The CRLB states that Var(θ̂) ≥ 1/I(θ) for any unbiased estimator, where I(θ) is the Fisher information.
Question 6: When conducting a one-way ANOVA F-test, rejecting H₀ tells you:
- All group means are significantly different from each other
- At least one group mean differs from the others (Correct answer)
- The group with the highest mean is significantly different from the lowest
- Effect size is practically meaningful across all groups
Correct answer: At least one group mean differs from the others
The ANOVA F-test only indicates that not all means are equal; post-hoc tests (e.g., Tukey HSD) are needed to identify which specific pairs differ.
Question 7: In sequential hypothesis testing, the sequential probability ratio test (SPRT) allows:
- Testing to continue indefinitely with no stopping rule
- Early stopping when accumulated evidence strongly favors H₀ or H₁ (Correct answer)
- Using a fixed sample size determined before data collection
- Controlling only the Type II error rate during data collection
Correct answer: Early stopping when accumulated evidence strongly favors H₀ or H₁
SPRT updates the likelihood ratio after each observation and stops sampling as soon as the ratio crosses predetermined boundaries for H₀ or H₁.
DSE Statistical Inference and Hypothesis Testing 5
A permutation test determines the p-value by: