Chapter 12, Problem AP4.11CPT

To determine

To find out how large the sample size is needed, if the survey firm wants the standard deviation of the sampling distribution of $\widehat{p}$ be equal $0.01$ .

Answer to Problem AP4.11CPT

Option (b) is the correct option.

Explanation of Solution

A survey firm wants to ask a random sample of adults in Ohio if they support an increase in the sales tax, with additional revenue going to education.

It is given:

  $\begin{array}{l}p=0.40\\ {\sigma }_{\widehat{p}}=0.01\end{array}$

The standard deviation of the sampling distribution is given by:

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

Now, we will evaluate the equation to find n as:

  $\begin{array}{l}{\sigma }_{\widehat{p}}=\sqrt{\frac{p(1-p)}{n}}\\ \Rightarrow {\sigma }_{\widehat{p}}\sqrt{n}=\sqrt{p(1-p)}\\ \Rightarrow \sqrt{n}=\frac{\sqrt{p(1-p)}}{{\sigma }_{\widehat{p}}}\\ \Rightarrow n={(}^{\frac{\sqrt{p(1-p)}}{{\sigma }_{\widehat{p}}}}\end{array}$

Now, putting the values in the formula above:

  $\begin{array}{l}n={(}^{\frac{\sqrt{p(1-p)}}{{\sigma }_{\widehat{p}}}}\\ ={(}^{\frac{\sqrt{0.40(1-0.40)}}{0.01}}\\ =2400\end{array}$

Thus the required sample size is $2400$ . Therefore, option (b) is the correct option.