Essentials Of Statistics For Business & Economics
Essentials Of Statistics For Business & Economics
9th Edition
ISBN: 9780357045435
Author: David R. Anderson, Dennis J. Sweeney, Thomas A. Williams, Jeffrey D. Camm, James J. Cochran
Publisher: South-Western College Pub
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Textbook Question
Chapter 11, Problem 30SE

A sample standard deviation for the number of passengers taking a particular airline flight is 8. A 95% confidence interval estimate of the population standard deviation is 5.86 passengers to 12.62 passengers.

  1. a. Was a sample size of 10 or 15 used in the statistical analysis?
  2. b. Suppose the sample standard deviation of s = 8 was based on a sample of 25 flights. What change would you expect in the confidence interval for the population standard deviation? Compute a 95% confidence interval estimate of α with a sample size of 25.

a.

Expert Solution
Check Mark
To determine

Check whether a sample size of 10 or 15 was used in the statistical analysis.

Answer to Problem 30SE

The sample size 15 is used in the statistical analysis.

Explanation of Solution

Calculation:

It is given that the sample standard deviation is 8. The 95% confidence interval for population standard deviation is (5.86, 12.62).

The confidence interval formula for population variance σ2 is given below:

(n1)×s2χ(α2)2σ2(n1)×s2χ1(α2)2

The confidence interval formula for population standard deviation σ is given below:

(n1)×s2χ(α2)2σ(n1)×s2χ1(α2)2

For n=15, the degrees of freedom is calculated as follows:

n1=151=14

Critical value for χ(1α2)2:

χ1(α2)2=χ1(0.052)2=χ0.9752

Procedure:

Step-by-step procedure to obtain χ20.975 value using Table 11.1 is given below:

  • Locate the value 14 in the left column of the table.
  • Find the row corresponding to the value 14 and column corresponding to the value χ20.975 of the table.

Thus, the value of χ20.975 with 14 degrees of freedom is 5.629. That is, χ20.975=5.629_.

Critical value for χα22:

χ(α2)2=χ(0.052)2=χ0.0252

Procedure:

Step-by-step procedure to obtain χ20.025 value using Table 11.1 is given below:

  • Locate the value 14 in the left column of the table.
  • Find the row corresponding to the value 14 and column corresponding to the value χ20.025 of the table.

Thus, the value of χ20.025 with 14 degrees of freedom is 26.119. That is, χ20.025=26.119_.

Substitute n=15, s2=64, χ(α2)2=26.119, and  χ(1α2)2=5.629 in the confidence interval.

((151)×6426.119,(151)×645.629)=(14×6426.119,14×645.629)=(89626.119,8965.629)=(34.3,159.2)

Thus, the 95% confidence interval for population variance is (34.3,159.2).

The 95% confidence interval for population standard deviation is given below:

(34.3,159.2)=(5.86,12.62)

Therefore, the 95% confidence interval for population standard deviation is (5.86,12.62).

From the given information, the 95% confidence interval for population standard deviation is 5.86 passengers to 12.62 passengers. For the sample size n=15, the 95% confidence interval for population standard deviation is (5.86,12.62). Hence, the sample size 15 is used in the statistical analysis.

b.

Expert Solution
Check Mark
To determine

Compute the 95% confidence interval estimate of σ with a sample size of 25. Explain the changes expected in the confidence interval.

Answer to Problem 30SE

The 95% confidence interval estimate of σ with a sample size of 25 is (6.25,11.13).

The change expected in the confidence interval is that the width of the confidence interval is smaller for n=25 compared to n=15.

Explanation of Solution

Calculation:

For n=25, the degrees of freedom is calculated as follows:

n1=251=24

Critical value for χ(1α2)2:

χ1(α2)2=χ1(0.052)2=χ0.9752

Procedure:

Step-by-step procedure to obtain χ20.975 value using Table 11.1 is given below:

  • Locate the value 24 in the left column of the table.
  • Find the row corresponding to the value 24 and column corresponding to the value χ20.975 of the table.

Thus, the value of χ20.975 with 24 degrees of freedom is 12.401. That is, χ20.975=12.401_.

Critical value for χα22:

χ(α2)2=χ(0.052)2=χ0.0252

Procedure:

Step-by-step procedure to obtain χ20.025 value using Table 11.1 is given below:

  • Locate the value 24 in the left column of the table.
  • Find the row corresponding to the value 24 and column corresponding to the value χ20.025 of the table.

Thus, the value of χ20.025 with 24 degrees of freedom is 39.364. That is, χ20.025=39.364_.

Substitute n=25, s2=64, χ(α2)2=39.364, and  χ(1α2)2=12.401 in the confidence interval.

((251)×6439.364,(251)×6412.401)=(24×6439.364,24×6412.401)=(1,53639.364,1,53612.401)=(39.02,126.86)

Thus, the 95% confidence interval for population variance is (39.02,126.86).

The 95% confidence interval for population standard deviation is as follows:

(39.02,126.86)=(6.25,11.13)

Therefore, the 95% confidence interval for population standard deviation is (6.25,11.13).

The 95% confidence interval for population standard deviation when n=25 is (6.25,11.13) and from part (a), the 95% confidence interval for population standard deviation when n=15 is (5.86,12.62).

The interval width is 4.88(=11.36.25) for n=25 and the interval width is 6.76(=12.625.86) for n=15. Hence, the interval width is decreased for n=25.

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Chapter 11 Solutions

Essentials Of Statistics For Business & Economics

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