BSIE Probability & Statistics 4 — Questions and Answers
Question 1: Which nonparametric test is the rank-based alternative to the two-sample t-test?
- Chi-square test
- Mann-Whitney U test (Correct answer)
- Kruskal-Wallis test
- Friedman test
Correct answer: Mann-Whitney U test
The Mann-Whitney U test (also called Wilcoxon rank-sum test) compares two independent groups without assuming normality.
Question 2: If a 90% confidence interval is (18, 26), which conclusion is correct?
- 90% of the population falls between 18 and 26
- We are 90% confident the population mean lies between 18 and 26 (Correct answer)
- There is a 90% probability the true mean is 22
- The sample mean is 90% accurate
Correct answer: We are 90% confident the population mean lies between 18 and 26
A confidence interval means that 90% of such intervals constructed from repeated samples would contain the true population mean.
Question 3: The F-distribution is primarily used in which statistical procedure?
- t-test for one mean
- Analysis of Variance (ANOVA) (Correct answer)
- Chi-square goodness of fit
- Linear regression t-test
Correct answer: Analysis of Variance (ANOVA)
ANOVA uses the F-statistic (ratio of between-group to within-group variance) to compare three or more group means.
Question 4: In simple linear regression y = β₀ + β₁x + ε, what does β₁ represent?
- y-intercept when x = 0
- Average change in y for a one-unit increase in x (Correct answer)
- Correlation coefficient between x and y
- Standard error of the estimate
Correct answer: Average change in y for a one-unit increase in x
β₁ is the slope, representing the expected change in the response variable y per unit increase in the predictor x.
Question 5: A quality engineer takes samples of size n = 25 from a process with σ = 10. What is the standard error of the sample mean?
- 0.4
- 2.0 (Correct answer)
- 10
- 25
Correct answer: 2.0
Standard error = σ / √n = 10 / √25 = 10 / 5 = 2.0.
Question 6: Bayes' theorem is used to:
- Compute cumulative probabilities in a normal distribution
- Update prior probabilities based on new evidence (Correct answer)
- Test whether two samples come from the same population
- Determine sample size for a given margin of error
Correct answer: Update prior probabilities based on new evidence
Bayes' theorem updates the prior probability of a hypothesis to a posterior probability after observing new data or evidence.
Question 7: A continuous uniform distribution U(a, b) has variance equal to:
- (b − a) / 2
- (b − a)² / 4
- (b − a)² / 12 (Correct answer)
- (b + a) / 2
Correct answer: (b − a)² / 12
The variance of a continuous uniform distribution is (b − a)² / 12.
BSIE Probability & Statistics 4
Which nonparametric test is the rank-based alternative to the two-sample t-test?