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Why Divide by n— 1? Let a population consist of the values

Elementary Statistics | 12th Edition | ISBN: 9780321836960 | Authors: Mario F. Triola ISBN: 9780321836960 18

Solution for problem 45BB Chapter 3.3

Elementary Statistics | 12th Edition

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Elementary Statistics | 12th Edition | ISBN: 9780321836960 | Authors: Mario F. Triola

Elementary Statistics | 12th Edition

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Problem 45BB

Why Divide by n − 1 ? Let a population consist of the values 2 min, 3 min, 8 min. (These are departure delay times taken from American Airlines flights from New York's JFK airport to Los Angeles. See Data Set 15 in Appendix B.) Assume that samples of two values are randomly selected with replacement from this population. (That is, a selected value is replaced before the second selection is made.)

a. Find the variance σ 2 of the population {2 min, 3 min, 8 min}.

b. After listing the nine different possible samples of two values selected with replacement, find the sample variance s 2 (which includes division by n − 1 ) for each of them; then find the mean of the nine sample variances s 2 .

c. For each of the nine different possible samples of two values selected with replacement, find the variance by treating each sample as if it is a population (using the formula for population variance, which includes division by n), then find the mean of those nine population variances. d. Which approach results in values that are better estimates of σ 2 : part (b) or part (c)? Why? When computing variances of samples, should you use division by n or n − 1 ?

e. The preceding parts show that s 2 is an unbiased estimator of σ 2 . Is s an unbiased estimator of σ ? Explain.

Step-by-Step Solution:
Step 1 of 3

Solution 45BB

(a)

We know that the mean of the population by the following formula:

Therefore, the mean of the population is 4.33 minutes.

We know that population variance  found by the following formula:

Therefore, the variance of the given population is.

(b)

We know that, sample of size  taken from the population of size  by with replacement in ways. From the given information,  therefore, the possible number of samples of size 2 from a population of size 3 in  ways.

These 9 samples and their respective means and sample variances are listed in the following table:

Sample

Sample mean

Sample Variance

The mean of the nine sample variances found by

Therefore, the mean of the nine sample variances is.

(c)

By considering each sample as a population and the population variances for nine samples given in the following table:

Population Variance

Step 2 of 3

Chapter 3.3, Problem 45BB is Solved
Step 3 of 3

Textbook: Elementary Statistics
Edition: 12
Author: Mario F. Triola
ISBN: 9780321836960

This full solution covers the following key subjects: population, min, Find, values, variance. This expansive textbook survival guide covers 121 chapters, and 3629 solutions. Elementary Statistics was written by and is associated to the ISBN: 9780321836960. Since the solution to 45BB from 3.3 chapter was answered, more than 464 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Elementary Statistics, edition: 12. The answer to “?Why Divide by n ? 1 ? Let a population consist of the values 2 min, 3 min, 8 min. (These are departure delay times taken from American Airlines flights from New York's JFK airport to Los Angeles. See Data Set 15 in Appendix B.) Assume that samples of two values are randomly selected with replacement from this population. (That is, a selected value is replaced before the second selection is made.) a. Find the variance ? 2 of the population {2 min, 3 min, 8 min}. b. After listing the nine different possible samples of two values selected with replacement, find the sample variance s 2 (which includes division by n ? 1 ) for each of them? then find the mean of the nine sample variances s 2 .c. For each of the nine different possible samples of two values selected with replacement, find the variance by treating each sample as if it is a population (using the formula for population variance, which includes division by n), then find the mean of those nine population variances. d. Which approach results in values that are better estimates of ? 2 : part (b) or part (c)? Why? When computing variances of samples, should you use division by n or n ? 1 ? e. The preceding parts show that s 2 is an unbiased estimator of ? 2 . Is s an unbiased estimator of ? ? Explain.” is broken down into a number of easy to follow steps, and 239 words. The full step-by-step solution to problem: 45BB from chapter: 3.3 was answered by , our top Statistics solution expert on 03/15/17, 10:30PM.

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Why Divide by n— 1? Let a population consist of the values