MS-DS Master of Data science Master of Data science Statistical and Probabilistic Analysis 1

Question 1

Which of the following is the correct interpretation of a 95% confidence interval for a population mean?

Accepted Answer

If we repeated the sampling procedure many times, 95% of the constructed intervals would contain the true mean

Answer

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?

Accepted Answer

Mean = λ, Variance = λ

Answer

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?

Accepted Answer

The probability of observing a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true

Answer

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?

Accepted Answer

The Central Limit Theorem

Answer

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?

Accepted Answer

P(A | B) = P(A) and P(B | A) = P(B)

Answer

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.

MS-DS Master of Data science Practice Test

MS-DS Master of Data science Practice Test

MS-DS Master of Data science Master of Data science Statistical and Probabilistic Analysis 1