MS-DS Master of Data science Statistical Inference Concepts 3 — Questions and Answers
Question 1: A study finds a statistically significant result with p = 0.03, but the effect size is d = 0.05. What is the most appropriate conclusion?
- The result is practically and statistically meaningful
- Statistical significance guarantees the effect is real
- The effect is statistically significant but likely of negligible practical importance (Correct answer)
- The study should be rerun with fewer observations
Correct answer: The effect is statistically significant but likely of negligible practical importance
With large enough samples, even trivially small effects become statistically significant; effect size (d = 0.05) indicates negligible practical importance.
Question 2: The Fisher information I(θ) quantifies:
- The prior probability of the parameter θ
- The expected curvature of the log-likelihood, reflecting how much data informs about θ (Correct answer)
- The posterior variance of θ
- The number of sufficient statistics for θ
Correct answer: The expected curvature of the log-likelihood, reflecting how much data informs about θ
Fisher information measures how sensitive the log-likelihood is to changes in θ, capturing how much the data can tell us about the parameter.
Question 3: When performing multiple hypothesis tests simultaneously, the Bonferroni correction adjusts α by:
- Multiplying α by the number of tests
- Dividing α by the number of tests (Correct answer)
- Taking the square root of α
- Replacing α with the FDR threshold
Correct answer: Dividing α by the number of tests
Bonferroni sets the per-test significance level to α/m (where m is the number of tests) to control the familywise error rate.
Question 4: Which of the following best describes a sufficient statistic for parameter θ?
- A statistic that equals θ exactly
- A statistic T(X) such that the conditional distribution of X given T does not depend on θ (Correct answer)
- A statistic with minimum variance among all unbiased estimators
- A statistic derived from the moment generating function
Correct answer: A statistic T(X) such that the conditional distribution of X given T does not depend on θ
A sufficient statistic captures all information in the sample about θ; knowing T(X) renders the rest of the data irrelevant for estimating θ.
Question 5: The power of a test is defined as:
- 1 − α (the probability of not making a Type I error)
- The probability of rejecting H₀ when H₀ is false (Correct answer)
- The probability of failing to reject H₀ when H₀ is true
- α × β, the joint error probability
Correct answer: The probability of rejecting H₀ when H₀ is false
Power = 1 − β = P(reject H₀ | H₀ is false), measuring a test's ability to detect a true effect.
Question 6: If X̄ is the sample mean of n i.i.d. observations from a population with mean μ and variance σ², what is Var(X̄)?
- σ²
- σ² / √n
- σ² / n (Correct answer)
- σ / n
Correct answer: σ² / n
The variance of the sample mean is σ²/n because averaging n independent observations reduces variance by a factor of n.
Question 7: Which assumption is NOT required for the classical ordinary least squares (OLS) estimator to be BLUE (Best Linear Unbiased Estimator)?
- Linearity in parameters
- Normality of errors (Correct answer)
- Homoscedasticity (constant error variance)
- No perfect multicollinearity
Correct answer: Normality of errors
The Gauss-Markov theorem guarantees OLS is BLUE under linearity, exogeneity, homoscedasticity, and no perfect multicollinearity — normality is not required.
MS-DS Master of Data science Statistical Inference Concepts 3
A study finds a statistically significant result with p = 0.03, but the effect size is d = 0.05.
What is the most appropriate conclusion?