Math 3339: Statistical Methods for
Engineers - Final Exam
University of Houston
Duration: 3 Hours
Instructions: Answer all questions. Show all your work for full credit.
1. 1. Define the term 'population' in the context of statistics.
Answer: A population is the complete set of items that data can be
collected from, about which information is desired.
2. 2. The probability that a part produced by a machine is defective is 0.05. If 20 parts are
selected, what is the expected number of defective parts?
Answer: Expected value = n * p = 20 * 0.05 = 1.
3. 3. A normal distribution has a mean of 100 and a standard deviation of 15. What is the
probability that a randomly selected value is greater than 130?
Answer: Z = (130 - 100)/15 = 2. P(Z > 2) = 0.0228.
4. 4. What does the Central Limit Theorem state?
Answer: The Central Limit Theorem states that the sampling
distribution of the sample mean approaches a normal distribution
as the sample size increases, regardless of the population's
distribution, provided the sample size is sufficiently large.
5. 5. Construct a 95% confidence interval for a sample mean of 50 with a standard
deviation of 10 and a sample size of 36.
Answer: CI = 50 ± Z*(σ/√n) = 50 ± 1.96*(10/6) = 50 ± 3.27 = (46.73,
53.27)
6. 6. What is the difference between a parameter and a statistic?
Answer: A parameter describes a characteristic of a population; a
statistic describes a characteristic of a sample.
7. 7. Compute the sample variance for the data: 4, 8, 6, 5, 3.
, Answer: Mean = 5.2; Variance = [(4-5.2)^2 + (8-5.2)^2 + (6-5.2)^2 +
(5-5.2)^2 + (3-5.2)^2]/(5-1) = 3.7
8. 8. If two events A and B are independent, what is P(A ∩ B)?
Answer: P(A ∩ B) = P(A) * P(B)
9. 9. A die is rolled twice. What is the probability that the sum is 7?
Answer: There are 6 outcomes that sum to 7: (1,6), (2,5), (3,4), (4,3),
(5,2), (6,1); Total outcomes = 36; Probability = 6/36 = 1/6
10. 10. What is the expected value of a binomial distribution with n = 10 and p = 0.4?
Answer: E(X) = np = 10 * 0.4 = 4
11. 11. Find the standard deviation of a binomial distribution with n = 12 and p = 0.25.
Answer: SD = sqrt(np(1-p)) = sqrt(12*0.25*0.75) ≈ 1.5
12. 12. What is the probability density function of the standard normal distribution?
Answer: f(x) = (1/√(2π)) * e^(-x²/2)
13. 13. What is the standard error of the mean for a population with σ = 20 and sample size
n = 64?
Answer: SE = σ/√n = 20/8 = 2.5
14. 14. Construct a 99% confidence interval for a mean of 70 with σ = 12 and n = 25.
Answer: Z = 2.576; CI = 70 ± 2.576 * (12/5) = 70 ± 6.182 = (63.818,
76.182)
15. 15. State the null and alternative hypotheses for testing whether a mean is different
from 100.
Answer: H0: μ = 100; H1: μ ≠ 100
16. 16. In hypothesis testing, what is a Type I error?
Answer: Rejecting a true null hypothesis.
17. 17. A test yields a p-value of 0.03. What conclusion can you draw at α = 0.05?
Engineers - Final Exam
University of Houston
Duration: 3 Hours
Instructions: Answer all questions. Show all your work for full credit.
1. 1. Define the term 'population' in the context of statistics.
Answer: A population is the complete set of items that data can be
collected from, about which information is desired.
2. 2. The probability that a part produced by a machine is defective is 0.05. If 20 parts are
selected, what is the expected number of defective parts?
Answer: Expected value = n * p = 20 * 0.05 = 1.
3. 3. A normal distribution has a mean of 100 and a standard deviation of 15. What is the
probability that a randomly selected value is greater than 130?
Answer: Z = (130 - 100)/15 = 2. P(Z > 2) = 0.0228.
4. 4. What does the Central Limit Theorem state?
Answer: The Central Limit Theorem states that the sampling
distribution of the sample mean approaches a normal distribution
as the sample size increases, regardless of the population's
distribution, provided the sample size is sufficiently large.
5. 5. Construct a 95% confidence interval for a sample mean of 50 with a standard
deviation of 10 and a sample size of 36.
Answer: CI = 50 ± Z*(σ/√n) = 50 ± 1.96*(10/6) = 50 ± 3.27 = (46.73,
53.27)
6. 6. What is the difference between a parameter and a statistic?
Answer: A parameter describes a characteristic of a population; a
statistic describes a characteristic of a sample.
7. 7. Compute the sample variance for the data: 4, 8, 6, 5, 3.
, Answer: Mean = 5.2; Variance = [(4-5.2)^2 + (8-5.2)^2 + (6-5.2)^2 +
(5-5.2)^2 + (3-5.2)^2]/(5-1) = 3.7
8. 8. If two events A and B are independent, what is P(A ∩ B)?
Answer: P(A ∩ B) = P(A) * P(B)
9. 9. A die is rolled twice. What is the probability that the sum is 7?
Answer: There are 6 outcomes that sum to 7: (1,6), (2,5), (3,4), (4,3),
(5,2), (6,1); Total outcomes = 36; Probability = 6/36 = 1/6
10. 10. What is the expected value of a binomial distribution with n = 10 and p = 0.4?
Answer: E(X) = np = 10 * 0.4 = 4
11. 11. Find the standard deviation of a binomial distribution with n = 12 and p = 0.25.
Answer: SD = sqrt(np(1-p)) = sqrt(12*0.25*0.75) ≈ 1.5
12. 12. What is the probability density function of the standard normal distribution?
Answer: f(x) = (1/√(2π)) * e^(-x²/2)
13. 13. What is the standard error of the mean for a population with σ = 20 and sample size
n = 64?
Answer: SE = σ/√n = 20/8 = 2.5
14. 14. Construct a 99% confidence interval for a mean of 70 with σ = 12 and n = 25.
Answer: Z = 2.576; CI = 70 ± 2.576 * (12/5) = 70 ± 6.182 = (63.818,
76.182)
15. 15. State the null and alternative hypotheses for testing whether a mean is different
from 100.
Answer: H0: μ = 100; H1: μ ≠ 100
16. 16. In hypothesis testing, what is a Type I error?
Answer: Rejecting a true null hypothesis.
17. 17. A test yields a p-value of 0.03. What conclusion can you draw at α = 0.05?
