Question

For a fixed confidence level, show that (approximately) quadrupling the sample size is necessary to halve the margin of error. (Hint: Use Formula 8.2.)

Short answer

Quadrupling the sample size is required to cut the margin of error in half for a fixed confidence level.

Step-by-step solution

Concept

The formula used :$n={\left(\frac{z_{\frac{a}{2}}\sigma }{E}\right)}^2$

Calculation

Show that halving the margin of error requires quadrupling the sample size. The sample formula is as follows:

$n={\left(\frac{z_{\frac{a}{2}}\sigma }{E}\right)}^2$

Substitute $E$by $\frac{E}{2}$in the formula for sample size.

$$\begin{gathered} n=\left(\frac{z_a\sigma }{\left(\frac{E}{2}\right)}\right) \\ =4{\left(\frac{z_a}{\frac{z^2}{E}}\right)}^2 \end{gathered}$$

As a result, for a fixed confidence level, quadrupling the sample size is required to cut the margin of error in half.