MS-DS Master of Data science Master of Data science Statistical and Probabilistic Analysis 1 — Questions and Answers
Question 1: Which of the following is the correct interpretation of a 95% confidence interval for a population mean?
- There is a 95% probability that the true mean lies within this specific interval
- If we repeated the sampling procedure many times, 95% of the constructed intervals would contain the true mean (Correct answer)
- 95% of the sample data points fall within this interval
- The true mean equals the sample mean with 95% accuracy
Correct answer: If we repeated the sampling procedure many times, 95% of the constructed intervals would contain the true mean
A confidence interval is a procedure, not a probability statement about a fixed parameter. The frequentist interpretation is that 95% of intervals constructed via this method across repeated samples will capture the true population mean — not that any one interval has a 95% chance of doing so.
Question 2: A random variable X follows a Poisson distribution with parameter λ. What are its mean and variance?
- Mean = λ, Variance = λ²
- Mean = √λ, Variance = λ
- Mean = λ, Variance = λ (Correct answer)
- Mean = 1/λ, Variance = 1/λ²
Correct answer: Mean = λ, Variance = λ
The Poisson distribution has the unique property that both its mean and variance equal the rate parameter λ. This equidispersion property is one reason the Poisson is used to model rare events and serves as a baseline for detecting overdispersion in count data.
Question 3: In hypothesis testing, what does the p-value represent?
- The probability that the null hypothesis is true
- The probability of observing a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true (Correct answer)
- The probability that the alternative hypothesis is true
- One minus the probability of a Type I error
Correct answer: The probability of observing a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true
The p-value is computed under the assumption that H₀ is true and measures how surprising the observed data would be in that world. It is not the probability that H₀ is true (a Bayesian quantity) nor the significance level α itself.
Question 4: Which theorem justifies approximating the distribution of a sample mean as normal, regardless of the population's distribution, when the sample size is large?
- Bayes' Theorem
- The Law of Large Numbers
- The Central Limit Theorem (Correct answer)
- Chebyshev's Inequality
Correct answer: The Central Limit Theorem
The Central Limit Theorem states that the sampling distribution of the sample mean converges to a normal distribution as n → ∞, regardless of the underlying population distribution (given finite mean and variance). This underpins z-tests, t-tests, and many inferential procedures.
Question 5: For two events A and B, which condition defines statistical independence?
- P(A ∩ B) = P(A) + P(B)
- P(A | B) = P(A) and P(B | A) = P(B) (Correct answer)
- P(A ∪ B) = P(A) × P(B)
- P(A ∩ B) = 0
Correct answer: P(A | B) = P(A) and P(B | A) = P(B)
Events A and B are independent if knowing B occurred provides no information about A, i.e., P(A|B) = P(A), which equivalently implies P(A ∩ B) = P(A)·P(B). Mutual exclusivity (P(A ∩ B) = 0) is a different — and often opposite — concept.
Question 6: What is the primary purpose of Maximum Likelihood Estimation (MLE)?
- To find the parameter values that minimize the squared residuals between observations and predictions
- To find the parameter values that maximize the probability of observing the given data under the assumed model (Correct answer)
- To compute the posterior distribution of parameters given prior beliefs and data
- To find parameter values that minimize the Kullback-Leibler divergence from a reference distribution
Correct answer: To find the parameter values that maximize the probability of observing the given data under the assumed model
MLE selects the parameter values θ that maximize the likelihood function L(θ | data) = P(data | θ), making the observed data most probable under the model. Minimizing squared residuals is OLS (equivalent to MLE only under Gaussian errors), and maximizing the posterior is MAP estimation.
MS-DS Master of Data science Master of Data science Statistical and Probabilistic Analysis 1
Which of the following is the correct interpretation of a 95% confidence interval for a population mean?