Q. 50 - The Practice of Statistics for AP

Question

50. Most people can roll their tongues, but many can’t. The ability to roll the tongue is genetically determined. Suppose we are interested in determining what proportion of students can roll their tongues. We test a simple random sample of $400$ students and find that $317$ can roll their tongues. The margin of error for a $95\%$ confidence interval for the true proportion of tongue rollers among students is closest to
(a) $0.008$. (c) $0.03$. (e) $0.208$.
(b) $0.02$. (d) $0.04$

Step-by-Step Solution

Answer

The correct answer is option (d)$0.04$.

Step 1: Given information

 A simple random sample of $400$ students and $317$ can roll their tongues. To find the correct option for the margin of error for a $95\%$ confidence interval for the true proportion of tongue rollers among students is closest.

Step 2: Explanation

The sample proportion is calculated by dividing the number of successes by the sample size:
$\hat{p}=\frac{x}{n}$

$=\frac{317}{400}$

$\approx 0.7925$

Using table A, for confidence level $95\%$,

$z_{\alpha /2}=z_{0.025}$

$=1.96$

Determine the margin of error as follows:

$E=z_{\alpha /2}\times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

$=1.96\times \sqrt{\frac{0.7925(1-0.7925)}{400}}$

$\approx 0.0397$

$\approx 0.04$

As a result, the correct option is (d)$0.04$.