DSE Statistical Inference and Hypothesis Testing 3 — Questions and Answers
Question 1: What does the power of a hypothesis test measure?
- Probability of rejecting H₀ when H₀ is true
- Probability of rejecting H₀ when H₀ is false (Correct answer)
- Probability of failing to reject H₀ when H₀ is false
- Probability that the confidence interval contains the true parameter
Correct answer: Probability of rejecting H₀ when H₀ is false
Power = 1 − β, where β is the Type II error rate; it is the probability of correctly detecting a true effect.
Question 2: A one-sample chi-squared test is used to test whether observed category frequencies match:
- A specified mean vector
- Expected frequencies from a theoretical distribution (Correct answer)
- Two sample variances are equal
- Regression coefficients are jointly zero
Correct answer: Expected frequencies from a theoretical distribution
The chi-squared goodness-of-fit test compares observed counts to expected counts derived from a hypothesized distribution.
Question 3: Increasing sample size while keeping α fixed will generally:
- Increase β (Type II error)
- Decrease the power of the test
- Narrow the confidence interval and increase power (Correct answer)
- Have no effect on the width of confidence intervals
Correct answer: Narrow the confidence interval and increase power
Larger samples reduce standard error, producing narrower confidence intervals and greater ability to detect true effects (higher power).
Question 4: The Kolmogorov-Smirnov test is primarily used to:
- Compare two population means
- Test whether a sample comes from a specified distribution (Correct answer)
- Assess homogeneity of variances
- Determine optimal sample size
Correct answer: Test whether a sample comes from a specified distribution
The KS test measures the maximum distance between an empirical CDF and a theoretical CDF to assess distributional fit.
Question 5: Which scenario correctly illustrates a Type II error?
- Concluding a drug works when it actually does not
- Failing to detect a real difference between two drug dosages (Correct answer)
- Setting α too high in a clinical trial
- Rejecting a true null hypothesis due to a large sample
Correct answer: Failing to detect a real difference between two drug dosages
A Type II error (false negative) occurs when we fail to reject H₀ even though H₁ is actually true.
Question 6: A likelihood ratio test compares two nested models by:
- Subtracting the AIC values of the two models
- Computing twice the difference in log-likelihoods, which follows a chi-squared distribution (Correct answer)
- Dividing the residual sum of squares of the restricted by the unrestricted model
- Using bootstrap resampling to compare model fits
Correct answer: Computing twice the difference in log-likelihoods, which follows a chi-squared distribution
Under regularity conditions, −2(ℓ_restricted − ℓ_unrestricted) ~ χ²(df) where df equals the number of constraints imposed.
Question 7: Bayesian credible intervals differ from frequentist confidence intervals in that they:
- Always produce wider intervals for the same data
- Express a direct probability statement about where the parameter lies (Correct answer)
- Do not require any distributional assumptions
- Are only valid for large samples
Correct answer: Express a direct probability statement about where the parameter lies
A Bayesian 95% credible interval means there is a 95% posterior probability that the parameter falls in that range, incorporating prior information.
DSE Statistical Inference and Hypothesis Testing 3
What does the power of a hypothesis test measure?