MS-DS Master of Data science Statistical and Probabilistic Analysis 4 — Questions and Answers
Question 1: The Kolmogorov-Smirnov test is used to compare which of the following?
- An empirical distribution to a reference distribution, or two empirical distributions (Correct answer)
- Means of two independent groups
- Variances across multiple groups
- Proportions in a contingency table
Correct answer: An empirical distribution to a reference distribution, or two empirical distributions
The KS test measures the maximum difference between cumulative distribution functions, used for goodness-of-fit or two-sample comparisons.
Question 2: If X ~ Binomial(n=10, p=0.5), what is Var(X)?
- 2.5 (Correct answer)
- 5
- 0.25
- 1.25
Correct answer: 2.5
Variance of a Binomial is np(1-p) = 10 × 0.5 × 0.5 = 2.5.
Question 3: In multiple hypothesis testing, the Bonferroni correction adjusts the significance threshold by:
- Dividing α by the number of tests (Correct answer)
- Multiplying each p-value by the number of tests, then using rank-based thresholds
- Dividing each p-value by the sample size
- Using the false discovery rate instead of family-wise error rate
Correct answer: Dividing α by the number of tests
Bonferroni sets the per-test threshold to α/m (where m is the number of tests) to control the family-wise error rate.
Question 4: A researcher notices that residuals from a regression model increase in spread as fitted values increase. This pattern indicates:
- Heteroscedasticity (Correct answer)
- Multicollinearity
- Autocorrelation
- Overfitting
Correct answer: Heteroscedasticity
Heteroscedasticity is non-constant variance in residuals, often visible as a fan shape in residual plots.
Question 5: The Central Limit Theorem guarantees that the sampling distribution of the mean approaches normality as n increases, regardless of population shape. Which condition is essential?
- The samples must be independent and identically distributed with finite variance (Correct answer)
- The population must be symmetric
- The sample size must exceed 100
- The population must be continuous
Correct answer: The samples must be independent and identically distributed with finite variance
CLT requires i.i.d. sampling and finite population variance; sample size of ~30 is often sufficient for most distributions.
Question 6: Maximum Likelihood Estimation (MLE) selects parameters that:
- Maximize the probability of observing the given sample under the assumed model (Correct answer)
- Minimize the sum of squared residuals
- Maximize the posterior probability given a prior
- Minimize the Bayesian Information Criterion
Correct answer: Maximize the probability of observing the given sample under the assumed model
MLE finds parameters θ that maximize L(θ|data), the likelihood of the observed data under the parametric model.
Question 7: Given two events A and B where P(A) = 0.5 and P(B|A) = 0.4, what is P(A ∩ B)?
- 0.2 (Correct answer)
- 0.4
- 0.9
- 0.1
Correct answer: 0.2
By the multiplication rule, P(A ∩ B) = P(B|A) × P(A) = 0.4 × 0.5 = 0.2.
MS-DS Master of Data science Statistical and Probabilistic Analysis 4
The Kolmogorov-Smirnov test is used to compare which of the following?