Q. 44 - The Practice of Statistics for AP

Question

The magazine Sports Illustrated asked a random sample of $750$ Division I college athletes, “Do you believe performance-enhancing drugs are a problem in college sports?” Suppose that $30\%$ of all Division I athletes think that these drugs are a problem. Let $\hat{p}$be the sample proportion who say that these drugs are a problem.

The standard deviation of the sampling distribution is about 

(a) $0.0006$.

(b) $0.033$.

(c) $0.017$.

(d) $0$.

(e) None of these.

Step-by-Step Solution

Answer

$\text{ (c) }{\sigma }_{\hat{p}}=0.017$

Step 1: Given Information

Given

$$\begin{gathered} p=30\% \\ =0.30 \end{gathered}$$

$n=750$

Step 2: Explanation

The standard deviation of the sampling distribution of the sample proportion $\hat{p}$ is equal to:

$$\begin{gathered} {\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}} \\ =\sqrt{\frac{0.30(1-0.30)}{750}} \\ \approx 0.0167332 \\ \approx 0.017 \end{gathered}$$