Essentials of Statistics for Business and Economics
Essentials of Statistics for Business and Economics
9th Edition
ISBN: 9780357118191
Author: David R. Anderson, Dennis J. Sweeney, Thomas A. Williams
Publisher: Cengage Learning US
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Chapter 11.1, Problem 2E

a.

To determine

Compute the 90% confidence interval estimate of the population variance.

a.

Expert Solution
Check Mark

Answer to Problem 2E

The 90% confidence interval for σ2 is (15.76,46.95).

Explanation of Solution

Calculation:

The given information is that there is a sample of 20 items with a sample standard deviation of 5.

The confidence interval for population variance σ2 is given by:

(n1)×s2χ(α2)2σ2(n1)×s2χ(1α2)2

Here, the significance level is α=0.10.

Degrees of freedom:

n1=201=19

Critical value for χ(1α2)2:

χ1(α2)2=χ1(0.102)2=χ0.952

Procedure:

Step by step procedure to obtain χ20.95 value using Table 11.1 is given below:

  • Locate the value 19 in the left column of the table.
  • Go through the row corresponding to the value 19 and column corresponding to the value χ20.95 of the table.

Thus, the value of χ20.95 with 19 degrees of freedom is 10.117. That is, χ20.95=10.117_.

Critical value for χα22:

χ(α2)2=χ(0.102)2=χ0.052

Procedure:

Step by step procedure to obtain χ20.05 value using Table 11.1 is given below:

  • Locate the value 19 in the left column of the table.
  • Go through the row corresponding to the value 19 and column corresponding to the value χ20.05 of the table.

Thus, the value of χ0.052 with 19 degrees of freedom is 30.144. That is, χ20.05=30.144_.

Substitute n=20,s2=25, χ(α2)2=30.144 and χ(1α2)2=10.117 in confidence interval formula.

((201)×2530.144,(201)×2510.117)=(19×2530.144,19×2510.117)=(47530.144,47510.117)=(15.76,46.95)

Thus, the 90% confidence interval for population variance is (15.76,46.95).

b.

To determine

Compute the 95% confidence interval estimate of the population variance.

b.

Expert Solution
Check Mark

Answer to Problem 2E

The 95% confidence interval for σ2 is (14.46,53.33).

Explanation of Solution

Calculation:

For a 95% confidence interval, the confidence level is 0.95 and α=0.05.

Here, the degrees of freedom is,

n1=201=19

Critical value for χ(1α2)2:

χ(1α2)2=χ(10.052)2=χ0.9752

Procedure:

Step by step procedure to obtain χ0.9752 value using Table 11.1 is given below:

  • Locate the value 19 in the left column of the table.
  • Go through the row corresponding to the value 19 and column corresponding to the value χ20.95 of the table.

Thus, the value of χ20.975 with 19 degrees of freedom is 8.907. That is, χ20.975=8.907_.

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 19 in the left column of the table.
  • Go through the row corresponding to the value 19 and column corresponding to the value χ20.025 of the table.

Thus, the value of χ20.025 with 19 degrees of freedom is 32.852. That is, χ20.025=38.852_.

Substitute n=20,s2=25, χ(α2)2=32.852 and χ(1α2)2=8.907 in confidence interval formula.

((201)×2532.852,(201)×258.907)=(19×2532.852,19×258.907)=(47532.852,4758.907)=(14.46,53.33)

Thus, the 95% confidence interval for population variance is (14.46,53.33).

c.

To determine

Compute the 95% confidence interval estimate of the population standard deviation.

c.

Expert Solution
Check Mark

Answer to Problem 2E

The 95% confidence interval for population standard deviation is (3.8,7.3).

Explanation of Solution

Calculation:

The confidence interval for population standard deviation σ is given by:

(n1)×s2χ(α2)2σ(n1)×s2χ1(α2)2

From part (b), it is clear that 95% confidence interval for population variance is (14.46,53.33).

The confidence interval for population standard deviation is,

(14.46,53.33)=(3.8,7.3).

Therefore, the 95% confidence interval for population standard deviation is (3.8,7.3).

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Essentials of Statistics for Business and Economics

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