LaunchPad for Moore's Introduction to the Practice of Statistics (12 month access)
LaunchPad for Moore's Introduction to the Practice of Statistics (12 month access)
8th Edition
ISBN: 9781464133404
Author: David S. Moore, George P. McCabe, Bruce A. Craig
Publisher: W. H. Freeman
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Chapter 8.1, Problem 32E

(a)

To determine

To test: Whether an 99% confidence interval would be wider or narrower than the 95% confidence interval obtained in Exercise 8.31.

(a)

Expert Solution
Check Mark

Answer to Problem 32E

Solution: The 99% confidence interval would be wider. An 99% confidence interval is obtained as (0.3962,0.4638), which is wider than the 95% confidence interval (0.4043,0.4557)_ obtained in Exercise 8.31.

Explanation of Solution

Calculation: The 95% confidence interval is obtained as (0.4043,0.4557) in the previous Exercise 8.31. Compute an 99% confidence interval for the same data.

The formula for 99% confidence interval for population proportion p is defined as:

Confidence interval=p^±m

Where, p^ the proportion of sample and m is the margin of error.

The sample proportion is provided as:

p^=43%=43100=0.43

Therefore, the sample proportion p^ is obtained as 0.43.

The formula for margin of error m is defined as:

m=z*×SEp^

In the above formula, z* is the critical value of the standard normal density curve and SEp^ is the standard error.

The formula for standard error SEp^ of sample proportion p^ and sample size n is defined as:

SEp^=p^(1p^)n

The sample proportion p^ is obtained as 0.43 in the previous step. Substitute the obtained sample proportion of 0.43 and sample size of 1430 in the standard error formula. So,

SEp^=p^(1p^)n=0.43×(10.43)1430=0.24511430=0.0131

Therefore, the standard error is obtained as 0.0131. The value of z* for 99% confidence level is z*=2.58 from the standard normal table.

So, the margin of error is obtained as:

m=z*×SEp^=2.58×0.0131=0.033798

Therefore, the margin of error is obtained as 0.033798.

Substitute the obtained values of margin of error and sample proportion in the formula for confidence interval. Therefore, an 99% confidence interval is obtained as:

p^±m=(0.43±0.033798)=(0.430.033798,0.43+0.033798)=(0.3962,0.4638)

Therefore, an 99% confidence interval is obtained as (0.3962,0.4638).

The width of the 99% confidence interval is obtained as:

0.46380.3962=0.0676

The width of the 95% confidence interval is obtained as:

0.45570.4043=0.0514

Conclusion: The obtained widths of the two confidence levels show that the 99% confidence interval is wider than the 95% confidence interval.

(b)

To determine

To test: Whether a 90% confidence interval would be wider or narrower than the 95% confidence interval obtained in Exercise 8.31.

(b)

Expert Solution
Check Mark

Answer to Problem 32E

Solution: A 90% confidence interval would be wider. A 90% confidence interval is obtained as (0.40845,0.45155) is narrower than the 95% confidence interval (0.4043,0.4557)_ obtained in Exercise 8.31.

Explanation of Solution

Calculation: The 95% confidence interval is obtained as (0.4043,0.4557) in the previous exercise 8.31. Compute a 90% confidence interval for the same data.

The formula for 90% confidence interval for p is defined as:

Confidence interval=p^±m

The sample proportion is provided as:

p^=43%=43100=0.43

Therefore, the sample proportion p^ is obtained as 0.43.

The formula for margin of error m is defined as:

m=z*×SEp^

In the above formula, z* is standard normal density curve and SEp^ is the standard error.

The formula for standard error SEp^ of sample proportion p^ and sample size n is defined as:

SEp^=p^(1p^)n

The sample proportion p^ is obtained as 0.43 in the previous step. Substitute the obtained sample proportion of 0.43 and sample size of 1430 in the standard error formula. So,

SEp^=p^(1p^)n=0.43×(10.43)1430=0.24511430=0.0131

Therefore, the standard error is obtained as 0.0131. The value of z* for 90% confidence level is z*=1.645 from the standard normal table.

So, the margin of error is obtained as:

m=z*×SEp^=1.645×0.0131=0.02155

Therefore, the margin of error is obtained as 0.02155.

Substitute the obtained values of margin of error and sample proportion in the formula for confidence interval. Therefore, a 90% confidence interval is obtained as:

p^±m=(0.43±0.02155)=(0.430.02155,0.43+0.02155)=(0.40845,0.45155)

Therefore, a 90% confidence interval is obtained as (0.40845,0.45155).

The width of the 90% confidence interval is obtained as:

0.408450.45155=0.043

The width of the 95% confidence interval is obtained as:

0.45570.4043=0.0514

Conclusion: The obtained widths of the two confidence levels show that 90% confidence interval is narrower than the 95% confidence interval.

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Chapter 8 Solutions

LaunchPad for Moore's Introduction to the Practice of Statistics (12 month access)

Ch. 8.1 - Prob. 11UYKCh. 8.1 - Prob. 12ECh. 8.1 - Prob. 13ECh. 8.1 - Prob. 14ECh. 8.1 - Prob. 15ECh. 8.1 - Prob. 16ECh. 8.1 - Prob. 17ECh. 8.1 - Prob. 18ECh. 8.1 - Prob. 19ECh. 8.1 - Prob. 20ECh. 8.1 - Prob. 21ECh. 8.1 - Prob. 22ECh. 8.1 - Prob. 23ECh. 8.1 - Prob. 24ECh. 8.1 - Prob. 25ECh. 8.1 - Prob. 26ECh. 8.1 - Prob. 27ECh. 8.1 - Prob. 28ECh. 8.1 - Prob. 29ECh. 8.1 - Prob. 30ECh. 8.1 - Prob. 31ECh. 8.1 - Prob. 32ECh. 8.1 - Prob. 33ECh. 8.1 - Prob. 34ECh. 8.1 - Prob. 35ECh. 8.1 - Prob. 37ECh. 8.1 - Prob. 39ECh. 8.1 - Prob. 40ECh. 8.1 - Prob. 41ECh. 8.1 - Prob. 42ECh. 8.1 - Prob. 43ECh. 8.1 - Prob. 44ECh. 8.1 - Prob. 36ECh. 8.1 - Prob. 38ECh. 8.2 - Prob. 45UYKCh. 8.2 - Prob. 46UYKCh. 8.2 - Prob. 47UYKCh. 8.2 - Prob. 48UYKCh. 8.2 - Prob. 49UYKCh. 8.2 - Prob. 50UYKCh. 8.2 - Prob. 51UYKCh. 8.2 - Prob. 52ECh. 8.2 - Prob. 53ECh. 8.2 - Prob. 54ECh. 8.2 - Prob. 55ECh. 8.2 - Prob. 56ECh. 8.2 - Prob. 57ECh. 8.2 - Prob. 58ECh. 8.2 - Prob. 59ECh. 8.2 - Prob. 60ECh. 8.2 - Prob. 61ECh. 8.2 - Prob. 62ECh. 8.2 - Prob. 63ECh. 8.2 - Prob. 64ECh. 8.2 - Prob. 65ECh. 8.2 - Prob. 66ECh. 8.2 - Prob. 67ECh. 8.2 - Prob. 69ECh. 8.2 - Prob. 68ECh. 8.2 - Prob. 70ECh. 8.2 - Prob. 71ECh. 8 - Prob. 72ECh. 8 - Prob. 73ECh. 8 - Prob. 74ECh. 8 - Prob. 75ECh. 8 - Prob. 76ECh. 8 - Prob. 77ECh. 8 - Prob. 94ECh. 8 - Prob. 79ECh. 8 - Prob. 80ECh. 8 - Prob. 81ECh. 8 - Prob. 82ECh. 8 - Prob. 83ECh. 8 - Prob. 84ECh. 8 - Prob. 85ECh. 8 - Prob. 86ECh. 8 - Prob. 87ECh. 8 - Prob. 88ECh. 8 - Prob. 89ECh. 8 - Prob. 90ECh. 8 - Prob. 95ECh. 8 - Prob. 96ECh. 8 - Prob. 97ECh. 8 - Prob. 98ECh. 8 - Prob. 99ECh. 8 - Prob. 92ECh. 8 - Prob. 93ECh. 8 - Prob. 78ECh. 8 - Prob. 100ECh. 8 - Prob. 101E
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