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 = 978 and x = 525 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. 0.537 (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. 0.496 < p < 0.578 (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 99% chance that the true value of the population proportion will fall between the lower bound and the upper bound. b. one has 99% confidence that the sample proportion is equal to the population proportion. c. one has 99% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion. d. 99% of sample proportions will fall between the lower bound and the upper bound.

Explanation:

Step1: Calculate point - estimate

The best point - estimate of the population proportion $p$ is the sample proportion $\hat{p}=\frac{x}{n}$. Given $n = 978$ and $x = 525$, then $\hat{p}=\frac{525}{978}\approx0.537$.

Step2: Find z - score for 99% confidence level

For a 99% confidence level, the significance level $\alpha=1 - 0.99 = 0.01$, and $\alpha/2=0.005$. The $z$ - score $z_{\alpha/2}=z_{0.005} = 2.576$.

Step3: Calculate margin of error

The formula for the margin of error $E$ for a proportion is $E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$. Substitute $\hat{p}=0.537$, $n = 978$, and $z_{\alpha/2}=2.576$ into the formula:
\[

$$\begin{align*} E&=2.576\sqrt{\frac{0.537\times(1 - 0.537)}{978}}\\ &=2.576\sqrt{\frac{0.537\times0.463}{978}}\\ &=2.576\sqrt{\frac{0.248631}{978}}\\ &=2.576\sqrt{0.000254224}\\ &=2.576\times0.01594\\ &\approx0.041 \end{align*}$$

\]

Step4: Construct confidence interval

The confidence interval for the population proportion $p$ is $\hat{p}-E

Step5: Interpret confidence interval

The correct interpretation of a confidence interval is that one has 99% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.

Answer:

a) 0.537
b) 0.041
c) $0.496 < p < 0.578$
d) C. One has 99% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.