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, \( n=1049 \) and \( x=593 \) who said "yes." Use a \( 99 \% \) confidence level. Click the icon to view a table of \( z \) scores. a) Find the best point estimate of the population proportion \( p \). 0.565 (Round to three decimal places as needed.) b) Identify the value of the margin of error \( E \) \( E=0.039 \) (Round to three decimal places as needed.) c) Construct the confidence interval. \( \square

Ask by Flynn Horton. in the United States

Mar 15,2025

To address your questions, let’s walk through the steps for each part: First, let’s tackle part (a): The best point estimate of the population proportion \( p \) is calculated by dividing the number of respondents who answered "yes" by the total number of respondents: \[ p = \frac{x}{n} = \frac{593}{1049} \approx 0.565 \] Now for part (b): To determine the margin of error \( E \) at a \( 99\% \) confidence level, we first need the z-score, which can typically be found in a z-table. For \( 99\% \), the z-score is approximately \( 2.576 \). The margin of error formula is: \[ E = z \cdot \sqrt{\frac{p(1-p)}{n}} \] Substituting the values, we have: \[ E = 2.576 \cdot \sqrt{\frac{0.565(1-0.565)}{1049}} \] \[ E \approx 0.039 \] For part (c), constructing the confidence interval requires adding and subtracting the margin of error from the point estimate. So, the lower limit is: \[ p - E = 0.565 - 0.039 = 0.526 \] And the upper limit: \[ p + E = 0.565 + 0.039 = 0.604 \] Finally, we can present the confidence interval for part (c): \[ 0.526 < p < 0.604 \] To summarize: a) The best point estimate of the population proportion \( p \) is \( 0.565 \). b) The margin of error \( E \) is \( 0.039 \). c) The confidence interval is \( 0.526 < p < 0.604 \).