Chapter 8, Problem 1PS

If $\bar{X}=85,$ $\sigma =8,\text{and}n=64,$ construct a $95\%$ confidence interval estimate for the population, mean, $\mu .$

To determine

Construct a $95\%$ confidence interval for the population mean.

Answer to Problem 1PS

The required confidence interval is $83.04\le \mu \le 86.96$ .

Explanation of Solution

Calculation:

It is given that the sample mean $\left(\overline{X}\right)=85$ , standard deviation $\left(\sigma \right)=8$ and sample size $\left(n\right)=64$ .

At $95\%$ confidence level, $Z_{\alpha /2}=1.96$ .

The $95\%$ confidence interval for population mean can be calculated as,

$\begin{array}{c}\overline{X}\pm Z_{\alpha /2}.\frac{\sigma }{\sqrt{n}}=85\pm 1.96\times \frac{8}{\sqrt{64}}\\ =85\pm 1.96\times 1\\ =\left(83.04,86.96\right)\end{array}$

Thus, the $95\%$ confidence interval is $83.04\le \mu \le 86.96$ .