Chapter 8.2, Problem 56E

To determine

To explain how to adjust sample size.

Answer to Problem 56E

Multiply the sample size by 4.

Explanation of Solution

Given:

We reduce the margin of error from 3 percentage points to 1.5 percentage points.

Formula:

Margin of error:

  $E=Z_{\alpha /2}\times \sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}$

As per given information, we reduce the margin of error from 3 percentage points to 1.5 percentage points. That means, need to divide by 2 on both sides of formula.

The margin of error is,

  $\begin{array}{l}\frac{E}{2}=\frac{1}{2}\times Z_{\alpha /2}\times \sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}\\ \frac{E}{2}=Z_{\alpha /2}\times \sqrt{\frac{1}{4}\times \frac{\widehat{p}(1-\widehat{p})}{n}}\\ \frac{E}{2}=Z_{\alpha /2}\times \sqrt{\frac{\widehat{p}(1-\widehat{p})}{4n}}\end{array}$

Therefore, the sample size n should be replaced by 4n.