Chapter 10.2, Problem 10.58E

To determine

To Explain: that the director says about the mean score of the population of all 5000 freshman.

Answer to Problem 10.58E

(107.5601, 116.4399)

There is 95% confident that the mean IQ score of the 5000 freshmen is between $107.5601$ and $116.4399$ .

Explanation of Solution

Given:

  $\begin{array}{c}n=50\\ \overline{x}=112\\ s=15\end{array}$

Formula used:

  $E=z_{\alpha /2}\times \frac{\sigma }{\sqrt{n}}$

For confidence interval

  $\begin{array}{c}\overline{x}-E\\ \overline{x}+E\end{array}$

Calculation:

Suppose

  $c=95\%=0.95$

Since the sample size of 50 is large so using the normal distribution.

For confidence level $1-\alpha =0.95,$

  $z_{\alpha /2}=z_{0.025}$ using the normal probability

  $z_{\alpha /2}=1.96$

The margin of error is

  $E=z_{\alpha /2}\times \frac{\sigma }{\sqrt{n}}=2.093\times \frac{15}{\sqrt{50}}=4.4399$

Confidence intervals are

  $\begin{array}{c}\overline{x}-E=112-4.4399=107.5601\\ \overline{x}+E=112+4.4399=116.4399\end{array}$

(107.5601, 116.4399)

There is 95% confident that the mean IQ score of the 5000 freshmen is between $107.5601$ and $116.4399$ .