Chapter 8.2, Problem 51E

To determine

To find: The sample size to have the margin of error of at most 0.04 at 90% confidence level.

Answer to Problem 51E

The sample size is 423.

Explanation of Solution

Given:

  $\begin{array}{l}\text{Confidence level }=90\%\\ \text{Margin of error}\left(E\right)=0.04\end{array}$

Formula used:

The formula to compute the sample size is:

  $n=\widehat{p}\left(1-\widehat{p}\right){\left(\frac{z_{\frac{\alpha }{2}}}{E}\right)}^2$

Calculation:

Here, no information about $\widehat{p}$ has been provided. Thus, assume is to be 0.50.

The value of $z_{\frac{\alpha }{2}}$ corresponding to 90% confidence level is 1.645, which is obtained from the standard normal table.

The sample size can be calculated as:

  $\begin{array}{c}n=\widehat{p}\left(1-\widehat{p}\right){\left(\frac{z_{\frac{\alpha }{2}}}{E}\right)}^2\\ =0.50\left(1-0.50\right){\left(\frac{1.645}{0.04}\right)}^2\\ =422.816\\ \approx 423\end{array}$

Hence, the sample size is 423.