Chapter 6.4, Problem 26E

Solution

To find: It needs to be determined the expression in terms of n for the sample size needed to reduce the margin of error to $\pm \frac{1}{2}E$ and by how many times the sample size must be increased to cut the margin of error in half.

The sample size must be 4n to get the margin of error as $\frac{1}{2}E$ and this means that the sample size must be increased 4 times.

Given information: A random sample of size n is given which is required to produce a margin of error of $\pm E$ .

Formula used: The formula that is used for evaluating the margin of error is $\pm \frac{1}{\sqrt{n}}$ .

Calculation: It can be assumed that the margin of error is $\pm E$ and when each side is multiplied by $\frac{1}{2}$

then the equality won’t change so one gets as:

  $\begin{array}{c}\pm E=\pm \frac{1}{\sqrt{n}}\\ \pm \frac{1}{2}E=\pm \frac{1}{2}\cdot \frac{1}{\sqrt{n}}\\ \pm \frac{1}{2}E=\frac{1}{\sqrt{4}}\cdot \frac{1}{\sqrt{n}}\end{array}$

Further evaluation results in as:

  $\begin{array}{c}\pm \frac{1}{2}E=\frac{1\cdot 1}{\sqrt{4}\cdot \sqrt{n}}\\ \pm \frac{1}{2}E=\frac{1}{\sqrt{4n}}\end{array}$

Hence the sample size must be 4n to get the margin of error as $\frac{1}{2}E$ and this means that the sample size must be increased 4 times.