\( 1 \leftarrow \) 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, \( \mathrm{n}=933 \) and \( \mathrm{x}=571 \) who said "yes. " Use a \( 99 \% \) confidence level. a) Flick the icon to view a table of \( z \) scores. 0.612 (Round to three decimal places as needed.) b) Identify the value of the margin of error E . E= 0.041 (Round to three decimal places as needed.) c) Construct the confidence interval. \( \square

Ask by Hanson Medina. in the United States

Mar 15,2025

To solve this problem, let's break down each part step by step. **a)** Given that the sample proportion \( p̂ = \frac{x}{n} = \frac{571}{933} \approx 0.612 \). **b)** The margin of error \( E \) is calculated with the formula: \[ E = z \times \sqrt{\frac{p̂(1 - p̂)}{n}} \] Using a \( z \) score for a \( 99\% \) confidence level, which is approximately \( 2.576 \), we substitute our values: \[ E \approx 2.576 \times \sqrt{\frac{0.612(1 - 0.612)}{933}} \approx 0.041 \] **c)** Now we can construct the confidence interval: First, we calculate the confidence interval limits: - Lower limit = \( p̂ - E \) - Upper limit = \( p̂ + E \) Substituting our values: - Lower limit = \( 0.612 - 0.041 \approx 0.571 \) - Upper limit = \( 0.612 + 0.041 \approx 0.653 \) Thus, the confidence interval is: \[ 0.571 < p < 0.653 \] So, the final answer is \( 0.571 < p < 0.653 \) (rounded to three decimal places).