Beginning Statistics, 2nd Edition
Beginning Statistics, 2nd Edition
2nd Edition
ISBN: 9781932628678
Author: Carolyn Warren; Kimberly Denley; Emily Atchley
Publisher: Hawkes Learning Systems
Question
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Chapter 9.4, Problem 17E
To determine

To Construct:

Construct and interpret a 90% confidence interval for the true differences between the percentages of Northerners and Southerners who attend church on a weekly basis.

Expert Solution & Answer
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Answer to Problem 17E

Solution:

The true differences between the percentages of Northerners and Southerners who attend church on a weekly basis is between -57.6% and -27.5% with 90% confidence. Thus with 90 % confidence a large percentage of Southerners than Northerners attend church on a weekly basis.

Explanation of Solution

Formula Used:

When the samples taken are independent, simple random samples, the conditions for a binomial distribution are met for both samples, and the sample sizes are large enough to ensure that n1p^15,n1(1-p^1)5,n2p^25,n2(1-p^2)5, the margin of error of a confidence interval for the differences between two population proportions is given by

E =zα2p^1(1-p^1)n1+p^2(1-p^2)n2

Where zα2 is the critical value for the level of significance, c =1-α, such that the area under the standard normal distribution to the right of zα2 is equal to α2, p^1 and p^2 are two sample proportions, and n1 and n2 are two sample sizes.

The confidence interval for the difference between two population proportions is given by

(p^1- p^2)- E< p1- p2<(p^1- p^2)+ E or ((p^1- p^2)- E, (p^1- p^2)+ E)

Where p^1 and p^2 are two sample proportions, and n1 and n2 are two sample sizes, (p^1- p^2) is the point estimate for the difference between population proportions, E is the margin of error.

Calculation:

Let population 1 be the sample of Northerners and population 2 be the samples of Southerners. The sample size for the first and second sample is calculated as follows:

n1=34+12=46n2=35+16=51

At 90% level of confidence so zα/2=z0.10/2=1.645

Using these sample sizes to calculate the sample proportions

p^1=x1n1=1246=0.26086

p^2=x2n2=3551=0.68627

Subtracting these two sample proportions produces the point estimate.

(p^1- p^2)=0.260860.68627=0.42541

The margin of error is calculated as follows-:

E =zα2p^1(1-p^1)n1+p^2(1-p^2)n2=1.6450.26086(10.26086)46+0.68627(10.68627)51=0.15087

Subtracting the margin of error from the point estimate and then adding the margin of error to the point estimate gives the following endpoints of the confidence interval

Lower end point:

(p^1- p^2)E=0.425410.15087=0.57628

Upper end point:

(p^1- p^2)+E=0.42541+0.15087=0.27454

Thus, the 90% confidence interval for the difference between the two population ranges from -0.57628 to -0.27454. The confidence interval can be written mathematically using either inequality symbols or interval notation, as follows.

-0.57628 < p1- p2< -0.27454 or (-0.57628, -0.27454)

Convert into percentage:

(-0.57628×100, -0.27454×100)

(-57.6%, -27.5%)

Interpretation:

The true differences between the percentages of Northerners and Southerners who attend church on a weekly basis is between -57.6% and -27.5% with 90% confidence. Thus with 90 % confidence a large percentage of Southerners than Northerners attend church on a weekly basis.

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Chapter 9 Solutions

Beginning Statistics, 2nd Edition

Ch. 9.1 - Prob. 11ECh. 9.1 - Prob. 12ECh. 9.1 - Prob. 13ECh. 9.1 - Prob. 14ECh. 9.1 - Prob. 15ECh. 9.1 - Prob. 16ECh. 9.2 - Prob. 1ECh. 9.2 - Prob. 2ECh. 9.2 - Prob. 3ECh. 9.2 - Prob. 4ECh. 9.2 - Prob. 5ECh. 9.2 - Prob. 6ECh. 9.2 - Prob. 7ECh. 9.2 - Prob. 8ECh. 9.2 - Prob. 9ECh. 9.2 - Prob. 10ECh. 9.2 - Prob. 11ECh. 9.2 - Prob. 12ECh. 9.2 - Prob. 13ECh. 9.2 - Prob. 14ECh. 9.2 - Prob. 15ECh. 9.2 - Prob. 16ECh. 9.2 - Prob. 17ECh. 9.2 - Prob. 18ECh. 9.2 - Prob. 19ECh. 9.2 - Prob. 20ECh. 9.2 - Prob. 21ECh. 9.2 - Prob. 22ECh. 9.2 - Prob. 23ECh. 9.2 - Prob. 24ECh. 9.3 - Prob. 1ECh. 9.3 - Prob. 2ECh. 9.3 - Prob. 3ECh. 9.3 - Prob. 4ECh. 9.3 - Prob. 5ECh. 9.3 - Prob. 6ECh. 9.3 - Prob. 7ECh. 9.3 - Prob. 8ECh. 9.3 - Prob. 9ECh. 9.3 - Prob. 10ECh. 9.3 - Prob. 11ECh. 9.3 - Prob. 12ECh. 9.3 - Prob. 13ECh. 9.3 - Prob. 14ECh. 9.3 - Prob. 15ECh. 9.3 - Prob. 16ECh. 9.3 - Prob. 17ECh. 9.3 - Prob. 18ECh. 9.3 - Prob. 19ECh. 9.3 - Prob. 20ECh. 9.3 - Prob. 21ECh. 9.4 - Prob. 1ECh. 9.4 - Prob. 2ECh. 9.4 - Prob. 3ECh. 9.4 - Prob. 4ECh. 9.4 - Prob. 5ECh. 9.4 - Prob. 6ECh. 9.4 - Prob. 7ECh. 9.4 - Prob. 8ECh. 9.4 - Prob. 9ECh. 9.4 - Prob. 10ECh. 9.4 - Prob. 11ECh. 9.4 - Prob. 12ECh. 9.4 - Prob. 13ECh. 9.4 - Prob. 14ECh. 9.4 - Prob. 15ECh. 9.4 - Prob. 16ECh. 9.4 - Prob. 17ECh. 9.4 - Prob. 18ECh. 9.4 - Prob. 19ECh. 9.4 - Prob. 20ECh. 9.4 - Prob. 21ECh. 9.4 - Prob. 22ECh. 9.4 - Prob. 23ECh. 9.4 - Prob. 24ECh. 9.4 - Prob. 25ECh. 9.5 - Prob. 1ECh. 9.5 - Prob. 2ECh. 9.5 - Prob. 3ECh. 9.5 - Prob. 4ECh. 9.5 - Prob. 5ECh. 9.5 - Prob. 6ECh. 9.5 - Prob. 7ECh. 9.5 - Prob. 8ECh. 9.5 - Prob. 9ECh. 9.5 - Prob. 10ECh. 9.5 - Prob. 11ECh. 9.5 - Prob. 12ECh. 9.5 - Prob. 13ECh. 9.5 - Prob. 14ECh. 9.5 - Prob. 15ECh. 9.5 - Prob. 16ECh. 9.5 - Prob. 17ECh. 9.5 - Prob. 18ECh. 9.5 - Prob. 19ECh. 9.5 - Prob. 20ECh. 9.5 - Prob. 21ECh. 9.5 - Prob. 22ECh. 9.5 - Prob. 23ECh. 9.5 - Prob. 24ECh. 9.5 - Prob. 25ECh. 9.5 - Prob. 26ECh. 9.5 - Prob. 27ECh. 9.5 - Prob. 28ECh. 9.CR - Prob. 1CRCh. 9.CR - Prob. 2CRCh. 9.CR - Prob. 3CRCh. 9.CR - Prob. 4CRCh. 9.CR - Prob. 5CRCh. 9.CR - Prob. 6CRCh. 9.CR - Prob. 7CRCh. 9.CR - Prob. 8CRCh. 9.CR - Prob. 9CRCh. 9.CR - Prob. 10CRCh. 9.CR - Prob. 11CRCh. 9.CR - Prob. 12CRCh. 9.CR - Prob. 13CRCh. 9.CR - Prob. 14CRCh. 9.CR - Prob. 15CRCh. 9.P - Prob. 1P
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