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.

Question

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Answer 1

Question

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

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^1≥5,n1(1-p^1)≥5,n2p^2≥5,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)=0.26086−0.68627=−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(1−0.26086)46+0.68627(1−0.68627)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=−0.42541−0.15087=−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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Suppose you are gambling on a roulette wheel. Each time the wheel is spun, the result is one of the outcomes 0, 1, and so on through 36. Of these outcomes, 18 are red, 18 are black, and 1 is green. On each spin you bet $5 that a red outcome will occur and $1 that the green outcome will occur. If red occurs, you win a net $4. (You win $10 from red and nothing from green.) If green occurs, you win a net $24. (You win $30 from green and nothing from red.) If black occurs, you lose everything you bet for a loss of $6. a. Use simulation to generate 1,000 plays from this strategy. Each play should indicate the net amount won or lost. Then, based on these outcomes, calculate a 95% confidence interval for the total net amount won or lost from 1,000 plays of the game. (Round your answers to two decimal places and if your answer is negative value, enter "minus" sign.) I worked out the Upper Limit, but I can't seem to arrive at the correct answer for the Lower Limit. What is the Lower Limit?…

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Answer 2

Question

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

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^1≥5,n1(1-p^1)≥5,n2p^2≥5,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)=0.26086−0.68627=−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(1−0.26086)46+0.68627(1−0.68627)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=−0.42541−0.15087=−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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Answer 3

Question

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

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^1≥5,n1(1-p^1)≥5,n2p^2≥5,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)=0.26086−0.68627=−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(1−0.26086)46+0.68627(1−0.68627)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=−0.42541−0.15087=−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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Answer 4

Question

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

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^1≥5,n1(1-p^1)≥5,n2p^2≥5,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)=0.26086−0.68627=−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(1−0.26086)46+0.68627(1−0.68627)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=−0.42541−0.15087=−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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Answer 5

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

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^1≥5,n1(1-p^1)≥5,n2p^2≥5,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)=0.26086−0.68627=−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(1−0.26086)46+0.68627(1−0.68627)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=−0.42541−0.15087=−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.

Answer 6

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

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^1≥5,n1(1-p^1)≥5,n2p^2≥5,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)=0.26086−0.68627=−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(1−0.26086)46+0.68627(1−0.68627)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=−0.42541−0.15087=−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.

Answer 7

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.

Answer 8

Expert Solution & Answer

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.

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

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^1≥5,n1(1-p^1)≥5,n2p^2≥5,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)=0.26086−0.68627=−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(1−0.26086)46+0.68627(1−0.68627)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=−0.42541−0.15087=−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.

Answer 12

Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E Ch. 9.1 - Prob. 7E Ch. 9.1 - Prob. 8E Ch. 9.1 - Prob. 9E Ch. 9.1 - Prob. 10E

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.

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.

Answer to Problem 17E

Explanation of Solution

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Chapter 9 Solutions

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.