Question

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 = 1033 ) and ( x = 593 ) who said \yes.\ use a ( 95% ) confidence level
click the icon to view a table of z scores.
a) find the best point estimate of the population proportion ( p ).
0.574
(round to three decimal places as needed.)
b) identify the value of the margin of error ( e ).
( e=square )
(round to three decimal places as needed.)

Explanation:

Step1: Calculate the best point estimate

The best - point estimate of the population proportion \(p\) is the sample proportion \(\hat{p}\). The formula for \(\hat{p}\) is \(\hat{p}=\frac{x}{n}\), where \(x\) is the number of successes in the sample and \(n\) is the sample size.
Given \(x = 593\) and \(n=1033\), then \(\hat{p}=\frac{593}{1033}\approx0.574\)

Step2: Calculate the margin of error

For a \(95\%\) confidence level, the critical value \(z_{\alpha/2}\) is \(1.96\) (from the standard normal distribution table). The formula for the margin of error \(E\) for a proportion is \(E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)
First, calculate \(\hat{p}(1-\hat{p})=0.574\times(1 - 0.574)=0.574\times0.426 = 0.244524\)
Then \(\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=\sqrt{\frac{0.244524}{1033}}\approx\sqrt{0.0002367}\approx0.0154\)
Finally, \(E=1.96\times0.0154\approx0.030\)

Answer:

a) The best - point estimate of the population proportion \(p\) is \(0.574\)
b) The margin of error \(E\) is \(0.030\)