×
Get Full Access to Statistics For Engineers And Scientists - 4 Edition - Chapter 5.10 - Problem 8e
Get Full Access to Statistics For Engineers And Scientists - 4 Edition - Chapter 5.10 - Problem 8e

×

# This exercise continues the study of the robustness of the ISBN: 9780073401331 38

## Solution for problem 8E Chapter 5.10

Statistics for Engineers and Scientists | 4th Edition

• Textbook Solutions
• 2901 Step-by-step solutions solved by professors and subject experts
• Get 24/7 help from StudySoup virtual teaching assistants Statistics for Engineers and Scientists | 4th Edition

4 5 1 299 Reviews
23
0
Problem 8E

Problem 8E

This exercise continues the study of the robustness of the Student's t method for constructing confidence intervals. The following figure shows graphs of probability density functions for the N (0, 1) distribution, the lognormal distribution with µ = 1 and σ2 = 0.25, and the gamma distribution with r = 0.5 and λ = 0.5 (this is also known as the chi-square distribution with one degree of freedom). For each of these distributions, generate 10,000 samples of size 5, and for each sample compute the upper and lower limits of a 95% confidence interval using the Student's t method. [If necessary, it is possible to compute the lognormal and gamma random values from normal random values. Specifically, to compute a value X from a lognormal distribution with µ = 1 and σ2 = 0.25, generate Y ~ N (l, 0.25) and compute X = eY. To generate a value X from a gamma distribution with r = 0.5 and λ = 0.5, generate Y ~ N (0, 1) and compute X = Y2.]   a. The true mean of the N (0, 1) distribution is 0. Based on the simulation results, estimate the coverage probability (proportion of samples for which the confidence interval covers the true mean) for samples of size 5 from the N (0, 1) distribution. (Since the assumptions underlying the Student's t method are satisfied here, your answer should be very close to 95%.)

b. The true mean of the lognormal distribution with µ = 1 and σ2 = 0.25 is 3.0802. Based on the simulation results, estimate the coverage probability (proportion of samples for which the confidence interval covers the true mean) for samples of size 5 from the lognormal distribution with µ = 1 and σ2 = 0.25.

c. The true mean of the gamma distribution with r = 0.5 and λ = 0.5 is 1. Based on the simulation results, estimate the coverage probability (proportion of samples for which the confidence interval covers the true mean) for samples of size 5 from the gamma distribution with r = 0.5 and λ = 0.5.

Step-by-Step Solution:
Step 1 of 3

Step 2 of 3

Step 3 of 3

##### ISBN: 9780073401331

Unlock Textbook Solution