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# A general, method for finding a confidence interval for

ISBN: 9780073401331 38

## Solution for problem 10E Chapter 5.10

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition

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Problem 10E

Problem 10E

A general, method for finding a confidence interval for the difference between two means of normal populations is given by expression (5.21) on page 365. A pooled method that can be used when the variances of the populations are known to be equal is given by expression (5.22) on page 367. This exercise is designed to compare the coverage probabilities of these methods under a variety of conditions. A fair amount of coding may be required, depending on the software used.

a. Let nX = 10, nY = 10. σX = 1, and σY = 1. Generate 10.000 pairs of samples: X1*…, XnX * from

a  N(0. σ2) distribution, and Y1*…., Y*nY from a N (0, σ2) distribution. For each pair of samples, compute a 95% confidence interval using the general method, and a 95% confidence interval using the pooled method. Note that each population has mean 0, so the true difference between the means is 0. Estimate the coverage probability for each method by computing the proportion of confidence intervals that cover the true value 0.

b. Repeat part (a), using nX = 10, nY = 10, σX = 1, and σY = 5.

c. Repeat part (a), using nX = 5, nY = 20, σX = 1, and σY = 5.

d. Repeat part (a), using nX = 20, nY = 5, σX = 1, and σY = 5.

e. Does the coverage probability for the general method differ substantially from 95% under any of the conditions in parts (a) through (d)? (It shouldn't.)

f. Based on the results in parts (a) through (d), under which conditions does the pooled method perform most poorly?

i. When the sample sizes are equal and the variances differ.

ii. When both the sample sizes and the variances differ, and the larger sample comes from    the population with the larger variance.

iii. When both the sample sizes and the variances differ, and the smaller sample comes from the population with the larger variance.

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##### ISBN: 9780073401331

This textbook survival guide was created for the textbook: Statistics for Engineers and Scientists , edition: 4. This full solution covers the following key subjects: method, Using, sample, variances, differ. This expansive textbook survival guide covers 153 chapters, and 2440 solutions. Statistics for Engineers and Scientists was written by and is associated to the ISBN: 9780073401331. Since the solution to 10E from 5.10 chapter was answered, more than 245 students have viewed the full step-by-step answer. The answer to “A general, method for finding a confidence interval for the difference between two means of normal populations is given by expression (5.21) on page 365. A pooled method that can be used when the variances of the populations are known to be equal is given by expression (5.22) on page 367. This exercise is designed to compare the coverage probabilities of these methods under a variety of conditions. A fair amount of coding may be required, depending on the software used.a. Let nX = 10, nY = 10. ?X = 1, and ?Y = 1. Generate 10.000 pairs of samples: X1*…, XnX * froma N(0. ?2) distribution, and Y1*…., Y*nY from a N (0, ?2) distribution. For each pair of samples, compute a 95% confidence interval using the general method, and a 95% confidence interval using the pooled method. Note that each population has mean 0, so the true difference between the means is 0. Estimate the coverage probability for each method by computing the proportion of confidence intervals that cover the true value 0.________________b. Repeat part (a), using nX = 10, nY = 10, ?X = 1, and ?Y = 5.________________c. Repeat part (a), using nX = 5, nY = 20, ?X = 1, and ?Y = 5.________________d. Repeat part (a), using nX = 20, nY = 5, ?X = 1, and ?Y = 5.________________e. Does the coverage probability for the general method differ substantially from 95% under any of the conditions in parts (a) through (d)? (It shouldn't.)________________f. Based on the results in parts (a) through (d), under which conditions does the pooled method perform most poorly? i. When the sample sizes are equal and the variances differ.________________ ii. When both the sample sizes and the variances differ, and the larger sample comes from the population with the larger variance.________________ iii. When both the sample sizes and the variances differ, and the smaller sample comes from the population with the larger variance.” is broken down into a number of easy to follow steps, and 324 words. The full step-by-step solution to problem: 10E from chapter: 5.10 was answered by , our top Statistics solution expert on 06/28/17, 11:15AM.

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