Question

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

that these drugs are a problem.

Which of the following are the mean and standard deviation of the sampling distribution of the sample proportion $\hat{p}$?

$Mean=0.30,SD=0.017$

b. Mean $=0.30,SD=0.55$

c. Mean $=0.30,SD=0.0003$

d. Mean $=225,SD=12.5$

e. Mean $=225,SD=157.5$

Short answer

The correct option is (a) Mean $=0.30,SD=0.017$

Step-by-step solution

Given information

$$\begin{gathered} p=0.30 \\ n=750 \end{gathered}$$

Concept

${\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}$

Calculation

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

As a result, the sample percentage sampling distribution has a mean of $0.30$ and a standard deviation of $0.017$ Hence, the correct option is (a)