BSIE Probability & Statistics 5 — Questions and Answers
Question 1: The power of a statistical test is defined as:
- 1 − α (one minus the significance level)
- 1 − β (one minus the Type II error probability) (Correct answer)
- The probability of a Type I error
- The p-value when H₀ is true
Correct answer: 1 − β (one minus the Type II error probability)
Power = 1 − β, the probability of correctly rejecting a false null hypothesis.
Question 2: In a 2×2 contingency table, the chi-square test of independence assesses:
- Whether two sample means are equal
- Whether two categorical variables are associated (Correct answer)
- Whether a dataset follows a normal distribution
- Whether two variances are equal
Correct answer: Whether two categorical variables are associated
The chi-square test of independence determines whether the distribution of one categorical variable differs across levels of another.
Question 3: A negative covariance between two random variables X and Y indicates:
- X and Y tend to increase together
- When X is above its mean, Y tends to be below its mean (Correct answer)
- X and Y are independent
- The correlation coefficient is zero
Correct answer: When X is above its mean, Y tends to be below its mean
Negative covariance means that high values of X tend to be paired with low values of Y, indicating an inverse linear relationship.
Question 4: Which sampling method divides the population into subgroups and randomly samples from each subgroup proportionally?
- Simple random sampling
- Cluster sampling
- Systematic sampling
- Stratified sampling (Correct answer)
Correct answer: Stratified sampling
Stratified sampling divides the population into strata (subgroups) and draws random samples from each, ensuring representation of all subgroups.
Question 5: The interquartile range (IQR) is calculated as:
- Q3 − Q1 (Correct answer)
- Q3 + Q1
- (Q3 − Q1) / 2
- Q2 − Q1
Correct answer: Q3 − Q1
IQR = Q3 − Q1, representing the spread of the middle 50% of the data.
Question 6: For a Poisson random variable with mean λ = 3, what is the variance?
- √3
- 1.5
- 3 (Correct answer)
- 9
Correct answer: 3
For a Poisson distribution, the mean and variance are both equal to λ, so Var(X) = 3.
Question 7: The coefficient of determination R² in regression analysis measures:
- The slope of the regression line
- The proportion of variance in y explained by x (Correct answer)
- The probability that the model is correct
- The standard error of the residuals
Correct answer: The proportion of variance in y explained by x
R² represents the proportion of total variability in the response variable explained by the regression model.
BSIE Probability & Statistics 5
The power of a statistical test is defined as: