Question

Sports Illustrated planned to ask a random sample of Division I college athletes, “Do you believe performance-enhancing drugs are a problem in college sports?” Which of the following is the smallest number of athletes that must be interviewed to estimate the true proportion who believe performance-enhancing drugs are a problem within $\pm 2\%$ with $90\%$ confidence?

$$\begin{gathered} a.17 \\ b.21 \\ c.1680 \\ d.1702 \\ e.2401 \end{gathered}$$

Short answer

The correct optionis

a.$n=\frac{{\left[z_{\alpha /2}\right]}^{2}0.25}{M{E}^2}=\frac{1.{65}^2\times 0.25}{0.{02}^2}\approx 1702$

Step-by-step solution

Given information

We have to find the smallest number of athletes that must be interviewed to estimate the true proportion.

Explanation 

Formula sample size:

$\hat{p}\text{known:}n=\frac{{\left[z_{\alpha /2}\right]}^2\hat{p}\hat{q}}{M{E}^2}=\frac{{\left[z_{\alpha /2}\right]}^2\hat{p}(1-\hat{p})}{M{E}^2}$

$\hat{p}\text{unknown:}n=\frac{{\left[z_{\alpha /2}\right]}^{2}0.25}{M{E}^2}$

the sample size is:

$n=\frac{{\left[z_{\alpha /2}\right]}^{2}0.25}{M{E}^2}=\frac{1.{65}^2\times 0.25}{0.{02}^2}\approx 1702$