Chapter 7.2, Problem 37E

To determine

To calculate: The sample size required to reduce the standard deviation.

Answer to Problem 37E

The required sample size is 4048.

Explanation of Solution

Given information:

Sample size = 1012

Proportion of condition which is met = 10%

Formula used:

The mean of the sampling distribution of $\widehat{p}$is, ${\mu }_{\widehat{p}}=p$ .

The standard deviation of the sampling distribution of $\widehat{p}$is, ${\sigma }_{\widehat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}$ .

Here, the formula of standard deviation can be written if the standard deviation needs to become half the current as,

  $\begin{array}{c}\frac{{\sigma }_{\widehat{p}}}{2}=\frac{1}{2}\times \sqrt{\frac{p\left(1-p\right)}{n}}\\ =\sqrt{\frac{p\left(1-p\right)}{4n}}\end{array}$

Here, by quadrupling we can write the sample size as 4n.

Substitute 1012 for n in the sample size 4n and simplify.

  $4\left(1012\right)=4048$

Hence, the required sample size is 4048.