MS-DS Master of Data science Statistical and Probabilistic Analysis 2 — Questions and Answers
Question 1: A dataset has a mean of 50 and a standard deviation of 10. Using Chebyshev's inequality, what is the minimum proportion of data within 3 standard deviations of the mean?
- At least 88.9% (Correct answer)
- At least 75%
- At least 95%
- Exactly 99.7%
Correct answer: At least 88.9%
Chebyshev's inequality states at least 1 - 1/k² of data lies within k standard deviations; for k=3, that's 1 - 1/9 ≈ 88.9%.
Question 2: Which test is most appropriate to assess whether two continuous variables are linearly associated when both are normally distributed?
- Pearson correlation test (Correct answer)
- Spearman rank correlation
- Chi-square test of independence
- Mann-Whitney U test
Correct answer: Pearson correlation test
Pearson correlation tests linear association between two normally distributed continuous variables and is parametric.
Question 3: In a Type II error scenario, a researcher fails to reject a null hypothesis that is actually false. Which factor, if increased, most directly reduces this error?
- Sample size (Correct answer)
- Significance level α
- Confidence level
- Standard deviation
Correct answer: Sample size
Increasing sample size reduces the standard error, increasing statistical power and reducing the probability of a Type II error (β).
Question 4: A random variable X follows a Poisson distribution with λ = 4. What is P(X = 0)?
- e⁻⁴ (Correct answer)
- 4e⁻⁴
- 1 - e⁻⁴
- e⁻⁴/4
Correct answer: e⁻⁴
For Poisson, P(X=k) = (λᵏ e⁻λ)/k!; at k=0, this simplifies to e⁻⁴.
Question 5: The joint probability P(A ∩ B) = 0.12, P(A) = 0.4, and P(B) = 0.3. Are events A and B independent?
- Yes, because P(A)·P(B) = 0.12 (Correct answer)
- No, because P(A ∩ B) ≠ P(A) + P(B)
- No, because the events are mutually exclusive
- Yes, because P(A|B) = P(B|A)
Correct answer: Yes, because P(A)·P(B) = 0.12
Independence holds when P(A ∩ B) = P(A)·P(B); here 0.4 × 0.3 = 0.12, which matches.
Question 6: Which property of estimators means the estimator's expected value equals the true parameter value?
- Unbiasedness (Correct answer)
- Consistency
- Efficiency
- Sufficiency
Correct answer: Unbiasedness
An unbiased estimator has E[θ̂] = θ, meaning on average it hits the true parameter without systematic error.
Question 7: A 95% confidence interval for a population mean is (42, 58). Which interpretation is correct?
- If the procedure were repeated many times, 95% of such intervals would contain the true mean (Correct answer)
- There is a 95% probability the true mean is between 42 and 58
- 95% of the population falls between 42 and 58
- The sample mean has a 95% chance of being correct
Correct answer: If the procedure were repeated many times, 95% of such intervals would contain the true mean
Confidence intervals reflect the long-run frequency of the procedure, not the probability that a specific interval contains the true parameter.
MS-DS Master of Data science Statistical and Probabilistic Analysis 2
A dataset has a mean of 50 and a standard deviation of 10.
Using Chebyshev's inequality, what is the minimum proportion of data within 3 standard deviations of the mean?