Chapter 10, Problem 22PT3

To determine

To explain: The effect of increase in sample size from 50 to 200.

Answer to Problem 22PT3

It would be divided by 2.

Explanation of Solution

Given:

Sample size is increased from 50 to 200.

Confidence level is 95%.

Formula used:

The formula to compute the margin of error is:

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

Here,

n is the sample size

  $\widehat{p}$ is the sample proportion.

E is the margin of error.

The above-mentioned formula indicates that there is an inverse relationship between the margin of error and the sample size.

If sample size increases from 50 to 200 then, it means that there is 4 times increase in the sample size. So, the formula can be written as:

  $\begin{array}{c}E=z_{0.05/2}\times \sqrt{\frac{\widehat{p}\left(1-\widehat{p}\right)}{4n}}\\ =\frac{1}{2}\times z_{0.05/2}\times \sqrt{\frac{\widehat{p}\left(1-\widehat{p}\right)}{n}}\end{array}$

So, it could be concluded that the new margin of error is half of the old margin of error and it has been divided by 2.

Hence, the correct option is (d).