Statistics: Concepts and Controversies
Statistics: Concepts and Controversies
9th Edition
ISBN: 9781464192937
Author: David S. Moore, William I. Notz
Publisher: W. H. Freeman
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Chapter 21, Problem 29E

Section 1

To determine

To find: The 70% confidence interval for the proportion of American adults who reported looking for information on food labels about whether the food they are purchasing is organic.

Expert Solution & Answer
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Answer to Problem 29E

Solution: The confidence interval is (0.474,0.506).

Explanation of Solution

Calculation:

Compute the confidence interval using formula as follows:

Confidence interval=[p^±z(p^(1p^)n)]

where

p^ is the sample proportionz is the tabulated value for desired confidence intervaln is the sample size

Now, compute the 70% confidence interval as follows:

Confidence interval=[p^±z(p^(1p^)n)]=[0.49±1.04(0.49(10.49)1004)]=(0.49±0.016)=(0.474,0.506)

Interpretation: It can be concluded with 70% confidence that proportion of American adults who reported looking for information on food labels about whether the food they are purchasing is organic lies between 47.4% and 50.6%.

Section 2

To find: The 80% confidence interval for the proportion of American adults who reported looking for information on food labels about whether the food they are purchasing is organic.

Solution: The confidence interval is (0.470,0.510).

Explanation:

Calculation:

Compute the confidence interval using formula as follows:

Confidence interval=[p^±z(p^(1p^)n)]

where

p^ is the sample proportionz is the tabulated value for desired confidence intervaln is the sample size

Now, compute the 80% confidence interval as follows:

Confidence interval=[p^±z(p^(1p^)n)]=[0.49±1.24(0.49(10.49)1004)]=(0.49±0.02)=(0.47,0.51)

Interpretation: It can be concluded with 80% confidence that proportion of American adults who reported looking for information on food labels about whether the food they are purchasing is organic lies between 47% and 51%.

Section 3

To find: The 90% confidence interval for the proportion of American adults who reported looking for information on food labels about whether the food they are purchasing is organic.

Solution: The confidence interval is (0.464,516).

Explanation:

Calculation:

Compute the confidence interval using formula as follows:

Confidence interval=[p^±z(p^(1p^)n)]

where

p^ is the sample proportionz is the tabulated value for desired confidence intervaln is the sample size

Now, compute the 90% confidence interval as follows:

Confidence interval=[p^±z(p^(1p^)n)]=[0.49±1.64(0.49(10.49)1004)]=(0.49±0.026)=(0.464,0.516)

Interpretation: It can be concluded with 90% confidence that proportion of American adults who reported looking for information on food labels about whether the food they are purchasing is organic lies between 46% and 52%.

Section 4

To find: The 99% confidence interval for the proportion of American adults who reported looking for information on food labels about whether the food they are purchasing is organic.

Solution: The confidence interval is (0.449,0.531).

Explanation:

Calculation:

Compute the confidence interval using formula as follows:

Confidence interval=[p^±z(p^(1p^)n)]

where

p^ is the sample proportionz is the tabulated value for desired confidence intervaln is the sample size

Now, compute the 99% confidence interval as follows:

Confidence interval=[p^±z(p^(1p^)n)]=[0.49±2.58(0.49(10.49)1004)]=(0.49±0.041)=(0.449,0.531)

Interpretation: It can be concluded with 99% confidence that proportion of American adults who reported looking for information on food labels about whether the food they are purchasing is organic lies between 45% and 53%.

Section 5

To find: The comparison of confidence intervals obtained in sections 1, 2, 3, and 4.

Solution: The width of confidence interval widens with increase in confidence level.

Explanation:

Calculation:

Now, compute the width of the 70%, 80%, 90%, and 99% confidence intervals.

The width of 70% confidence interval is calculated as shown below:

70% width=Upper limitLower limit=0.5060.474=0.032

The width of 80% confidence interval is calculated as shown below:

80% width=Upper limitLower limit=0.5100.470=0.04

The width of 90% confidence interval is calculated as shown below:

90% width=Upper limitLower limit=0.5160.464=0.052

The width of 99% confidence interval is calculated as shown below:

99% width=Upper limitLower limit=0.5310.449=0.082

From the above calculations, it is clear that if there is an increase in the confidence level, then there is an increase in the width of the confidence interval.

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