Chapter 7.2, Problem 1.4CYU

To determine

To discuss: The effect of change in sample size from 9000 to 1000 on the sampling distribution of $\widehat{p}$

Explanation of Solution

Given:

  $\begin{array}{l}\text{Population proportion}\left(p\right)=75\%=0.75\\ \text{Sample size}\left(n\right)=9000\end{array}$

Calculation:

The mean and standard deviation for the sample size 9000 can be calculated as:

  $\begin{array}{c}{\mu }_{\widehat{p}}=p\\ =0.75\end{array}$

  $\begin{array}{c}{\sigma }_{\widehat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}\\ =\sqrt{\frac{0.75\left(1-0.75\right)}{9000}}\\ =0.0046\end{array}$

Conclusion:

From above calculations, it is clear that for the sample size mean would be same and standard deviation would become smaller.