MS-DS Master of Data science Statistical Inference Concepts 5 — Questions and Answers
Question 1: The Kolmogorov-Smirnov test is used to:
- Compare means of two groups
- Test whether a sample comes from a specified distribution or compare two samples (Correct answer)
- Test for heteroscedasticity in regression
- Assess multicollinearity
Correct answer: Test whether a sample comes from a specified distribution or compare two samples
The KS test compares the empirical CDF against a theoretical CDF (one-sample) or between two empirical CDFs (two-sample) to detect distributional differences.
Question 2: In maximum likelihood estimation, the score function is defined as:
- The second derivative of the log-likelihood
- The gradient of the log-likelihood with respect to the parameter (Correct answer)
- The ratio of the likelihood to the prior
- The expected value of the log-likelihood
Correct answer: The gradient of the log-likelihood with respect to the parameter
The score is ∂ℓ(θ)/∂θ, the first derivative of the log-likelihood; at the MLE it equals zero.
Question 3: When using the Wilcoxon signed-rank test instead of a paired t-test, the primary reason is:
- The Wilcoxon test has higher power for normally distributed data
- The Wilcoxon test does not assume normality and is more robust to outliers (Correct answer)
- The Wilcoxon test requires larger sample sizes
- The t-test cannot handle paired data
Correct answer: The Wilcoxon test does not assume normality and is more robust to outliers
The Wilcoxon signed-rank test is a non-parametric alternative that only assumes symmetry of differences, making it robust when normality is violated.
Question 4: In the context of regression, what does the term 'heteroscedasticity' refer to?
- Non-linear relationship between predictors and response
- Error variance that changes across levels of the predictors (Correct answer)
- Correlation among the predictor variables
- Errors that follow a non-normal distribution
Correct answer: Error variance that changes across levels of the predictors
Heteroscedasticity means the variance of the regression errors is not constant (σᵢ² varies with X), violating the homoscedasticity assumption of OLS.
Question 5: Which of the following is the correct formula for the standard error of a proportion p̂ estimated from n observations?
- √(p̂/n)
- √(p̂(1−p̂)/n) (Correct answer)
- p̂(1−p̂)/n
- √(p̂(1−p̂)×n)
Correct answer: √(p̂(1−p̂)/n)
For a binomial proportion, SE(p̂) = √[p̂(1−p̂)/n], derived from the variance of the Bernoulli distribution divided by n.
Question 6: A likelihood ratio test statistic −2ln(Λ) follows approximately which distribution under H₀ for large samples?
- Normal distribution with mean 0
- Chi-square distribution with degrees of freedom equal to the number of constrained parameters (Correct answer)
- F-distribution
- t-distribution with n−1 degrees of freedom
Correct answer: Chi-square distribution with degrees of freedom equal to the number of constrained parameters
By Wilks' theorem, −2ln(Λ) is asymptotically χ²(k) where k is the number of parameters constrained under H₀.
Question 7: Which statement best describes the concept of a conjugate prior in Bayesian inference?
- A prior that makes the posterior equal to the likelihood
- A prior from the same distributional family as the posterior, simplifying analytical computation (Correct answer)
- A uniform prior that provides no information
- A prior that always minimizes posterior variance
Correct answer: A prior from the same distributional family as the posterior, simplifying analytical computation
A conjugate prior yields a posterior in the same parametric family, making closed-form Bayesian updating possible (e.g., Beta prior with Binomial likelihood gives a Beta posterior).
MS-DS Master of Data science Statistical Inference Concepts 5
The Kolmogorov-Smirnov test is used to: