Chapter 7.2, Problem 46E

To determine

The correct answer from the options that the sampling distribution of $\widehat{p}$is approximately Normal because (a) there are at least 7570 Division I college athletes, (b) $np=225$and $n\left(1-p\right)=525$ , (c) a random sample was chosen, (d) a large sample size like n = 750 guarantees it, (e) the sampling distribution of $\widehat{p}$always has this shape.

Answer to Problem 46E

The correct answer is (b).

Explanation of Solution

Given information:

Size of the random sample, $n=750$

Proportion of Division I athletes who think that these drugs are a problem $=30\%$

Formula used:

The sampling distribution of a sample proportion is approximately normal if $np$and $n\left(1-p\right)$are at least 10.

Consider that proportion of Division I athletes who think that these drugs are a problem be p and the sample size be n.

Rewrite it as:

  $\begin{array}{c}p=30\%\\ =0.30\end{array}$

And, $n=750$

Know that the sampling distribution of a sample proportion is approximately normal if $np$and $n\left(1-p\right)$are at least 10.

Substitute 750 for n and 0.30 for p in $np$ .

  $\begin{array}{c}np=750\times 0.30\\ =225\end{array}$

Also, substitute 750 for n and 0.30 for p in $n\left(1-p\right)$ .

  $\begin{array}{c}n\left(1-p\right)=750\left(1-0.30\right)\\ =750\times 0.70\\ =525\end{array}$

Here, we see that $np$and $n\left(1-p\right)$are both at least 10; this means the normal condition is met and the sampling distribution of the sample proportion is approximately normal.

Hence, the correct answer is (b).