A research institute poll asked respondents if they felt vulnerable to identity theft. In the poll, and who said “yes.” Use a confidence level.
a) Find the best point estimate of the population proportion p.
0.612
(Round to three decimal places as needed.)
b) Identify the value of the margin of error E

(Round to three decimal places as needed.)
(a scores.
(a) through (d).

Ask by Mccarthy Macdonald. in the United States

Mar 15,2025

Question

A research institute poll asked respondents if they felt vulnerable to identity theft. In the poll, and who said “yes.” Use a confidence level.
a) Find the best point estimate of the population proportion p.
0.612
(Round to three decimal places as needed.)
b) Identify the value of the margin of error E

(Round to three decimal places as needed.)
(a scores.
(a) through (d).

Ask by Mccarthy Macdonald. in the United States

Mar 15,2025

UpStudy AI · Accepted Answer

We start by finding the best point estimate for the population proportion. Since $$x=571$$ respondents answered "yes" out of $$n=933$$, the point estimate is

$$
\hat{p}=\frac{x}{n}=\frac{571}{933}\approx0.612.
$$

Next, to compute the margin of error at a $$99\%$$ confidence level, we use the formula

$$
E=z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}},
$$

where the critical value for $$99\%$$ confidence is $$z^*\approx2.576$$.

1. Compute the product $$\hat{p}(1-\hat{p})$$:

$$
\hat{p}(1-\hat{p})=0.612(1-0.612)=0.612(0.388)=0.237.
$$

2. Divide by $$n$$:

$$
\frac{\hat{p}(1-\hat{p})}{n}=\frac{0.237}{933}\approx0.000254.
$$

3. Take the square root:

$$
\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=\sqrt{0.000254}\approx0.01597.
$$

4. Multiply by $$z^*$$ to reach the margin of error:

$$
E=2.576\times 0.01597\approx0.041.
$$

Thus, the margin of error is

$$
E\approx0.041.
$$
The best point estimate of the population proportion $$ p $$ is $$ 0.612 $$. The margin of error $$ E $$ at a $$ 99\% $$ confidence level is approximately $$ 0.041 $$.

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