Chapter 7.2, Problem 45E

To determine

The correct answer from the options that decreasing the sample size from 750 to 375 would multiply the standard deviation by (a) 2, (b) $\sqrt{2}$ , (c) $\frac{1}{2}$ , (d) $\frac{1}{\sqrt{2}}$ , (e) none of these.

Answer to Problem 45E

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 standard deviation of the sampling distribution of $\widehat{p}$is, ${\sigma }_{\widehat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$ .

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 375 is half of the sample size 750.

So, if the sample size is decreased to 375 then the we can write the sample size as $\frac{n}{2}=375$ .

Know that the standard deviation of the sampling distribution of $\widehat{p}$is ${\sigma }_{\widehat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$ .

Write the standard deviation of the sampling distribution for the sample size 375 as,

  $\begin{array}{c}{\sigma }_{\widehat{p}}=\sqrt{\frac{p\left(1-p\right)}{\frac{n}{2}}}\\ =\sqrt{\frac{2p\left(1-p\right)}{n}}\\ =\sqrt{2}\sqrt{\frac{p\left(1-p\right)}{n}}\end{array}$

This means the standard deviation is multiplied by $\sqrt{2}$ .

Hence, the correct answer is (b).