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 = 1096 ) and ( x = 507 ) 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 ).
(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.)
c) construct the confidence interval.
( square<p<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 ( 95% ) chance that the true value of the population proportion will fall between the lower bound and the upper
bound.
b. one has ( 95% ) confidence that the sample proportion is equal to the population proportion.
c. ( 95% ) of sample proportions will fall between the lower bound and the upper bound.
d. one has ( 95% ) confidence that the interval from the lower bound to the upper bound actually does contain the true value
of the population proportion.

Explanation:

Step1: Calculate point estimate

The best point estimate of population proportion \( p \) is sample proportion \( \hat{p}=\frac{x}{n} \).
Given \( n = 1096 \) and \( x = 507 \), then \( \hat{p}=\frac{507}{1096}\approx0.463 \).

Step2: Find \( z - \)score

For \( 95\% \) confidence level, \( \alpha=1 - 0.95=0.05 \), \( \alpha/2=0.025 \).
From \( z - \)table, \( z_{\alpha/2}=z_{0.025}=1.96 \).

Step3: Calculate margin of error \( E \)

\( E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \).
Substitute \( \hat{p}=0.463 \), \( n = 1096 \), \( z_{\alpha/2}=1.96 \)
\( E=1.96\sqrt{\frac{0.463\times(1 - 0.463)}{1096}}\)
\(=1.96\sqrt{\frac{0.463\times0.537}{1096}}\)
\(=1.96\sqrt{\frac{0.248631}{1096}}\)
\(=1.96\times\sqrt{0.000227}\)
\(=1.96\times0.0151\)
\(E\approx0.030\)

Step4: Construct confidence interval

Confidence interval is \( \hat{p}-ESubstitute \( \hat{p}=0.463 \), \( E = 0.030 \)
\(0.463-0.030 < p<0.463 + 0.030\)
\(0.433

Step5: Interpret confidence interval

A confidence interval gives a range of values within which we are confident the true population parameter lies.
For a \( 95\% \) confidence interval, one has \( 95\% \) confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.

Answer:

a) \( 0.463 \)
b) \( 0.030 \)
c) \( 0.433 < p<0.493 \)
d) D. One has \( 95\% \) confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.