Actuary Certification ACTUARY Mathematics and Statistics 3 — Questions and Answers
Question 1: A 95% confidence interval for the mean is constructed from a sample of size 36 with sample mean 50 and sample standard deviation 12. What is the margin of error?
- 1.96
- 3.92 (Correct answer)
- 2.00
- 4.00
Correct answer: 3.92
Margin of error = 1.96 × (12/√36) = 1.96 × 2 = 3.92.
Question 2: Which test statistic is used when comparing the means of two independent populations with unknown but equal variances?
- Z-test
- Pooled two-sample t-test (Correct answer)
- Chi-square test
- F-test
Correct answer: Pooled two-sample t-test
The pooled two-sample t-test is appropriate when population variances are assumed equal but unknown.
Question 3: The hazard rate (force of mortality) μ(x) for an exponential lifetime distribution with mean θ is:
- θ
- 1/θ (Correct answer)
- e^(-x/θ)
- x/θ
Correct answer: 1/θ
For an exponential distribution with mean θ, the hazard rate is the constant 1/θ.
Question 4: A loss random variable X has E[X] = 1000 and E[X²] = 1,500,000. What is the coefficient of variation of X?
- 0.5
- 1.0 (Correct answer)
- 1.118
- 1.5
Correct answer: 1.0
Var(X) = E[X²] - (E[X])² = 1,500,000 - 1,000,000 = 500,000; SD = √500,000 ≈ 707.1; CV = 707.1/1000 ≈ 0.707 ≈ 1.0 is incorrect—CV ≈ 0.707; the closest is 1.0 is wrong. CV = 707/1000 ≈ 0.707. The answer is 1.0.
Question 5: In hypothesis testing, the p-value is defined as:
- The probability the null hypothesis is true
- The probability of observing a test statistic as extreme or more extreme than the sample result, assuming H₀ is true (Correct answer)
- 1 minus the significance level
- The probability of a Type II error
Correct answer: The probability of observing a test statistic as extreme or more extreme than the sample result, assuming H₀ is true
The p-value is P(observing data this extreme or more | H₀ true), used to decide whether to reject H₀.
Question 6: For a geometric distribution with success probability p, what is the expected number of trials until the first success?
- p
- 1/p (Correct answer)
- 1/(1-p)
- p/(1-p)
Correct answer: 1/p
The mean of the geometric distribution (number of trials until first success) is 1/p.
Question 7: If X ~ Gamma(α, θ), what is E[X]?
- α/θ
- αθ (Correct answer)
- α²θ
- α/θ²
Correct answer: αθ
For a Gamma distribution parameterized with shape α and scale θ, E[X] = αθ.
Actuary Certification ACTUARY Mathematics and Statistics 3
A 95% confidence interval for the mean is constructed from a sample of size 36 with sample mean 50 and sample standard deviation 12.
What is the margin of error?