Q.45 - The Practice of Statistics for AP

Question

Decreasing the sample size from $750$ to $375$ would multiply the standard deviation by

(a) $2$.

(b) $\sqrt{2}$.

(c) $1/2$.

(d) $1/\sqrt{2}$.

(e) none of these.

Step-by-Step Solution

Answer

The correct answer is (b) $\sqrt{2}$.

Step 1: Given Information

Random sample size, $n=750$.

The percentage of Division I athletes that believe these drugs are a concern. $=30\%$.

Step 2: Explanation

Consider $p$ the proportion of Division I athletes who believe these drugs are a problem and $n$ the sample size.

$p=30\%$

$=0.30$

$n=750$

$375$ is half of the sample size of $750$.

If the sample size is reduced to $375$, the sample size is written as $\frac{n}{2}=375$.

Step 3: Explanation

Know what the sampling distribution's standard deviation of $\hat{p}$ is ${\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}$.

For a sample size of $375$, write the standard deviation of the sampling distribution as follows:

${\sigma }_{\hat{p}}=\sqrt{\frac{p(1-p)}{\frac{n}{2}}}$

$=\sqrt{\frac{2p(1-p)}{n}}$

$=\sqrt{2}\sqrt{\frac{p(1-p)}{n}}$

The standard deviation is then multiplied by $\sqrt{2}$.