Q.46 - The Practice of Statistics for AP

Question

The sampling distribution of $\hat{p}$ is approximately Normal because

(a) there are at least $7570$ Division I college athletes.

(b) $np=225$ and $n(1-p)=525$.

(c) a random sample was chosen.

(d) a large sample size like $n=750$ guarantees it.

(e) the sampling distribution of $\hat{p}$ always has this shape.

Step-by-Step Solution

Answer

The correct answer is (b) $np=225$ and $n(1-p)=525$.

Step 1: Given Information

Random sample size, $n=750$.

Division Proportion I believe that these drugs are an issue for athletes. $=30\%$.

Step 2: Explanation

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

Consider$p$ the percentage of Division I players who believe these drugs are an issue, and $n$ the sample size.

$p =30 \%$

   $=0.30$

$n=750$

Step 3: Explanation

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

$np=750\times 0.30$

     $=225$

Also, substitute $750$ for $\mathrm{n}$ and $0.30$ for $p$ in $n(1-p)$.

$n(1-p)=750(1-0.30)$

              $=750\times 0.70$

              $=525$

We can see that both $np$ and $n(1-p)$ are at least ten, indicating that the normal requirement is satisfied and the sampling distribution of the sample percentage is about normal.

Hence, the correct answer is (b).