MS-DS Master of Data science Statistical Inference Concepts 4 — Questions and Answers
Question 1: In a chi-square goodness-of-fit test, the test statistic is large when:
- Observed frequencies closely match expected frequencies
- Observed frequencies deviate substantially from expected frequencies (Correct answer)
- The sample size is very small
- The degrees of freedom equal the number of categories
Correct answer: Observed frequencies deviate substantially from expected frequencies
The chi-square statistic Σ(O−E)²/E is large when observed counts differ greatly from what the null hypothesis predicts.
Question 2: The delta method is used to approximate the variance of g(X̄) when:
- g is a non-differentiable transformation
- g is a smooth (differentiable) function and n is large (Correct answer)
- The population is normally distributed
- X̄ is a sufficient statistic
Correct answer: g is a smooth (differentiable) function and n is large
The delta method uses a first-order Taylor expansion of g around μ; it requires g to be differentiable and works well for large n via the CLT.
Question 3: A p-value of 0.001 in a study with n = 10,000 primarily reflects:
- A large effect size
- A very small effect size that is detectable only due to the huge sample (Correct answer)
- High posterior probability that H₀ is false
- An error in the statistical analysis
Correct answer: A very small effect size that is detectable only due to the huge sample
With n = 10,000, even tiny effects produce very small p-values; the p-value conflates effect size with sample size.
Question 4: In hypothesis testing, which error is controlled directly by setting the significance level α?
- Type II error (failing to reject a false H₀)
- Type I error (rejecting a true H₀) (Correct answer)
- Both Type I and Type II errors simultaneously
- The false discovery rate
Correct answer: Type I error (rejecting a true H₀)
The significance level α is defined as the maximum acceptable probability of a Type I error (false positive).
Question 5: The Rao-Blackwell theorem states that if T is a sufficient statistic and U is an unbiased estimator, then E[U|T] is:
- Biased but with lower variance than U
- Unbiased and has variance no greater than that of U (Correct answer)
- Unbiased and always equal to U
- Consistent but not efficient
Correct answer: Unbiased and has variance no greater than that of U
Conditioning an unbiased estimator on a sufficient statistic yields an unbiased estimator with equal or lower variance — the Rao-Blackwell improvement.
Question 6: Which of the following correctly describes the relationship between a two-sided 95% confidence interval and a two-sided hypothesis test at α = 0.05?
- They are unrelated procedures
- H₀: μ = μ₀ is rejected at α = 0.05 if and only if μ₀ falls outside the 95% CI (Correct answer)
- The CI always equals the acceptance region of the test
- Rejecting H₀ guarantees the CI excludes zero
Correct answer: H₀: μ = μ₀ is rejected at α = 0.05 if and only if μ₀ falls outside the 95% CI
There is a duality: a value μ₀ is in the 95% CI if and only if the test of H₀: μ = μ₀ fails to reject at α = 0.05.
Question 7: A Bayesian credible interval differs from a frequentist confidence interval in that:
- Credible intervals are always wider
- A 95% credible interval has a 95% posterior probability of containing the true parameter (Correct answer)
- Credible intervals do not require a prior distribution
- Credible intervals are only valid for large samples
Correct answer: A 95% credible interval has a 95% posterior probability of containing the true parameter
A Bayesian credible interval directly states that P(θ ∈ I | data) = 0.95, a direct probability statement about the parameter given the data.
MS-DS Master of Data science Statistical Inference Concepts 4
In a chi-square goodness-of-fit test, the test statistic is large when: