Score: $8 / 100 \quad 1 / 9$ answered
Question 2
A poll of 634 potential voters finds that 401 intend to vote for a particular candidate.
Construct a $90 \%$ confidence interval for the true population proportion of votes the candidate will receive.
Round your answers to four decimal places.
\[
< p<
\]
What is the margin of error for this confidence interval?
Round your answer to four decimals.
Submit

Step 1 ：First, calculate the sample proportion (p̂), which is the number of successes (votes for the candidate) divided by the number of trials (total voters). In this case, p̂ = \(\frac{401}{634} = 0.6325\).

Step 2 ：Next, calculate the standard error of the proportion. The formula for this is \(\sqrt{\frac{p̂*(1-p̂)}{n}}\), where n is the number of trials. Substituting the values, we get \(\sqrt{\frac{0.6325*(1-0.6325)}{634}} = 0.0191\).

Step 3 ：Find the z-score for a 90% confidence interval. The z-score for a 90% confidence interval is 1.645.

Step 4 ：Calculate the confidence interval using the formula p̂ ± (z-score * standard error). Substituting the values, we get \(0.6325 ± (1.645 * 0.0191)\), which gives us the confidence interval (0.6009, 0.6640).

Step 5 ：Finally, calculate the margin of error. The margin of error is the z-score times the standard error, which is \(1.645 * 0.0191 = 0.0315\).

Step 6 ：\(\boxed{(0.6009, 0.6640), 0.0315}\)