Questions: 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=907 and x=600 who said "yes." Use a 99% confidence level. a) Find the best point estimate of the population proportion p. (Round to three decimal places as needed.) b) Identify the value of the margin of error E. E= (Round to three decimal places as needed.) c) Construct the confidence interval. <p< (Round to three decimal places as needed.) d) Write a statement that correctly interprets the confidence interval. Choose the correct answer below. A. There is a 99% chance that the true value of the population proportion will fall between the lower bound and the upper bound. B. 99% of sample proportions will fall between the lower bound and the upper bound. C. One has 99% confidence that the sample proportion is equal to the population proportion. D. One has 99% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.

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=907 and x=600 who said "yes." Use a 99% confidence level. a) Find the best point estimate of the population proportion p. (Round to three decimal places as needed.) b) Identify the value of the margin of error E. E= (Round to three decimal places as needed.) c) Construct the confidence interval. <p< (Round to three decimal places as needed.) d) Write a statement that correctly interprets the confidence interval. Choose the correct answer below. A. There is a 99% chance that the true value of the population proportion will fall between the lower bound and the upper bound. B. 99% of sample proportions will fall between the lower bound and the upper bound. C. One has 99% confidence that the sample proportion is equal to the population proportion. D. One has 99% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.
Transcript text: 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, $\mathrm{n}=907$ and $\mathrm{x}=600$ who said "yes." Use a $99 \%$ confidence level. a) Find the best point estimate of the population proportion $p$. $\square$ (Round to three decimal places as needed.) b) Identify the value of the margin of error $\mathbf{E}$. $\mathrm{E}=$ $\square$ (Round to three decimal places as needed.) c) Construct the confidence interval. $\square$ $<\mathrm{p}<\square$ $\square$ (Round to three decimal places as needed.) d) Write a statement that correctly interprets the confidence interval. Choose the correct answer below. A. There is a $99 \%$ chance that the true value of the population proportion will fall between the lower bound and the upper bound. B. $99 \%$ of sample proportions will fall between the lower bound and the upper bound. C. One has $99 \%$ confidence that the sample proportion is equal to the population proportion. D. One has $99 \%$ confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.
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Solution

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Solution Steps

Step 1: Calculate the Sample Proportion

Using the formula \(\hat{p} = \frac{X}{n}\), where \(X=600\) and \(n=907\), we find \(\hat{p} = 0.662\).

Step 2: Find the Z-score

The Z-score corresponding to the desired confidence level is given as \(Z=2.576\).

Step 3: Calculate the Standard Error of the Proportion

Using the formula \(SE = \sqrt{\hat{p}(1-\hat{p})/n}\), we find \(SE = 0.016\).

Step 4: Calculate the Margin of Error

Using the formula \(E = Z \times SE\), with \(Z=2.576\), we find \(E = 0.04\).

Step 5: Construct the Confidence Interval

The confidence interval for the population proportion \(p\) is calculated as \(\hat{p} \pm E\), which gives us (0.621, 0.702).

Final Answer:

We are confident that the true population proportion \(p\) lies within the interval (0.621, 0.702).

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