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.

Short answer

The correct answer is (b)$\sqrt{2}$.

Step-by-step solution

Given Information

Random sample size, $n=750$.

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

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 $\frac{n}{2}=375$.

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 $\sqrt{2}$.