Questions: Question 7, 7.1.13 Jaela Bowman Part 2 of 4 HW Score: 40%, 6 of 15 points Points: 0 of 1 Save 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=1070 and x=595 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. 0.556 (Round to three decimal places as needed.) b) Identify the value of the margin of error E. E=0.019 (Round to three decimal places as needed.)

Question 7, 7.1.13 Jaela Bowman Part 2 of 4 HW Score: 40%, 6 of 15 points Points: 0 of 1 Save 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=1070 and x=595 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. 0.556 (Round to three decimal places as needed.) b) Identify the value of the margin of error E. E=0.019 (Round to three decimal places as needed.)
Transcript text: Question 7, 7.1.13 Jaela Bowman Part 2 of 4 HW Score: $40 \%, 6$ of 15 points Points: 0 of 1 Save 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, $n=1070$ and $x=595$ 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 $p$. \[ 0.556 \] (Round to three decimal places as needed.) b) Identify the value of the margin of error $E$. \[ E=0.019 \] (Round to three decimal places as needed.) example Get more help - Clear all Check answer
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Solution

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Solution Steps

Step 1: Point Estimate of the Population Proportion

The best point estimate of the population proportion \( p \) is calculated as follows:

\[ \hat{p} = \frac{x}{n} = \frac{595}{1070} \approx 0.556 \]

Step 2: Standard Deviation of the Sample Proportion

The standard deviation of the sample proportion is given by:

\[ \sigma = \sqrt{\frac{\hat{p} \cdot (1 - \hat{p})}{n}} = \sqrt{\frac{0.556 \cdot (1 - 0.556)}{1070}} \approx 0.0152 \]

Step 3: Margin of Error Calculation

To find the margin of error \( E \), we use the formula:

\[ E = Z \cdot \frac{\sigma}{\sqrt{n}} \]

Where \( Z \) is the z-score corresponding to the confidence level of \( 99\% \), which is \( Z \approx 2.576 \). Thus, we have:

\[ E = 2.576 \cdot \frac{0.0152}{\sqrt{1070}} \approx 0.001 \]

Final Answer

The results are as follows:

  • Best point estimate of the population proportion \( p \): \( \hat{p} \approx 0.556 \)
  • Margin of error \( E \): \( E \approx 0.001 \)

\[ \boxed{\hat{p} = 0.556, \quad E = 0.001} \]

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