Use the sample data and confidence level given below to complete parts (a) through (d). A research institute poll asked respondents if they felt vulnerable to identity theft. In the poll, n= 1048 and x = 568 who said "yes." Use a 95% confidence level. E Click the icon to view a table of z scores. a) Find the best point estimate of the population proportion p. 0.542 (Round to three decimal places as needed.) b) Identify the value of the margin of error E. E= 0.030 (Round to three decimal places as needed.) c) Construct the confidence interval. 0.512
Use the sample data and confidence level given below to complete parts (a) through (d). A research institute poll asked respondents if they felt vulnerable to identity theft. In the poll, n= 1048 and x = 568 who said "yes." Use a 95% confidence level. E Click the icon to view a table of z scores. a) Find the best point estimate of the population proportion p. 0.542 (Round to three decimal places as needed.) b) Identify the value of the margin of error E. E= 0.030 (Round to three decimal places as needed.) c) Construct the confidence interval. 0.512
Big Ideas Math A Bridge To Success Algebra 1: Student Edition 2015
1st Edition
ISBN:9781680331141
Author:HOUGHTON MIFFLIN HARCOURT
Publisher:HOUGHTON MIFFLIN HARCOURT
Chapter11: Data Analysis And Displays
Section11.3: Shapes Of Distributions
Problem 14E
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Question
Please answer only part "D"
![### Confidence Interval Analysis: Population Proportion
**Objective:** Use sample data and a specified confidence level to determine and interpret the confidence interval for a population proportion.
**Scenario:**
A research institute conducted a poll asking respondents if they felt vulnerable to identity theft. In the poll,
- \( n = 1048 \) respondents
- \( x = 568 \) respondents said "yes."
- Use a 95% confidence level.
**Steps:**
1. **Find the best point estimate of the population proportion \( p \).**
\[
\hat{p} = \frac{x}{n} = \frac{568}{1048} = 0.542
\]
*(Round to three decimal places as needed.)*
2. **Identify the value of the margin of error \( E \).**
- Given: \( E = 0.030 \)
\[
E = 0.030
\]
*(Round to three decimal places as needed.)*
3. **Construct the confidence interval.**
\[
\hat{p} - E < p < \hat{p} + E
\]
\[
0.542 - 0.030 < p < 0.542 + 0.030
\]
\[
0.512 < p < 0.572
\]
*(Round to three decimal places as needed.)*
4. **Write a statement that correctly interprets the confidence interval.**
\[
\text{Choose the correct answer from the options below:}
\]
- **A:** One has 95% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.
- **B:** There is a 95% chance that the true value of the population proportion will fall between the lower bound and the upper bound.
- **C:** One has 95% confidence that the sample proportion is equal to the population proportion.
- **D:** 95% of sample proportions will fall between the lower bound and the upper bound.
The correct answer is **A**, which accurately reflects the concept of confidence intervals in statistics.
This analysis helps quantify the uncertainty around the point estimate of the population proportion, providing a range within which the true population proportion is likely to fall with a specified](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6eeb107-3976-477c-976e-1778ebbdfd8b%2F08446920-00d3-4b43-ad70-731d0856916b%2Fucgvxji_processed.png&w=3840&q=75)
Transcribed Image Text:### Confidence Interval Analysis: Population Proportion
**Objective:** Use sample data and a specified confidence level to determine and interpret the confidence interval for a population proportion.
**Scenario:**
A research institute conducted a poll asking respondents if they felt vulnerable to identity theft. In the poll,
- \( n = 1048 \) respondents
- \( x = 568 \) respondents said "yes."
- Use a 95% confidence level.
**Steps:**
1. **Find the best point estimate of the population proportion \( p \).**
\[
\hat{p} = \frac{x}{n} = \frac{568}{1048} = 0.542
\]
*(Round to three decimal places as needed.)*
2. **Identify the value of the margin of error \( E \).**
- Given: \( E = 0.030 \)
\[
E = 0.030
\]
*(Round to three decimal places as needed.)*
3. **Construct the confidence interval.**
\[
\hat{p} - E < p < \hat{p} + E
\]
\[
0.542 - 0.030 < p < 0.542 + 0.030
\]
\[
0.512 < p < 0.572
\]
*(Round to three decimal places as needed.)*
4. **Write a statement that correctly interprets the confidence interval.**
\[
\text{Choose the correct answer from the options below:}
\]
- **A:** One has 95% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.
- **B:** There is a 95% chance that the true value of the population proportion will fall between the lower bound and the upper bound.
- **C:** One has 95% confidence that the sample proportion is equal to the population proportion.
- **D:** 95% of sample proportions will fall between the lower bound and the upper bound.
The correct answer is **A**, which accurately reflects the concept of confidence intervals in statistics.
This analysis helps quantify the uncertainty around the point estimate of the population proportion, providing a range within which the true population proportion is likely to fall with a specified
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