Question

Forbes magazine published data on the best small firms in $2012$. These were firms that had been publicly traded for at least a year, have a stock price of at least $\backslash (5$per share, and have reported annual revenue between $\backslash )5$million and$\$1$ billion.

The Table 8.13 shows the ages of the corporate CEOs for a random sample of these firms.

Use this sample data to construct a $90\%$confidence interval for the mean age of CEO’s for these top small firms. Use the Student's t-distribution.

Short answer

We estimate that the average age of all CEOs of the greatest small businesses is between $\$54.421and\$58.713$ years, with a$95\%$ confidence level.

Step-by-step solution

Introduction

Create a $90\%$confidence interval for the average age of CEOs for these top small businesses using this sample data. The Student's-t distribution should be used.

Explanation 

Calculate the point estimate first.

$\overline{x}=56.567$

Calculate the samples' standard deviations.

$s=6.907,n=30,\text{and}df=30-1=29$

$CL=0.90$

$\alpha =1-CL$

$$\begin{gathered} =1-0.90 \\ =0.10\frac{a}{2} \\ =0.05t_{\frac{a}{2}} \\ ={\mathrm{t}}_{0.05} \\ =1.699 \end{gathered}$$

$EBM=t_{\frac{a}{2}}\left(\frac{s}{\sqrt{n}}\right)$

$EBM=1.699\left(\frac{6907}{\sqrt{30}}\right)$

$EBM=2.143$

$\overline{x}-EBM=56.567-2.146$

$\overline{x}-EBM=54.421$

$\overline{x}+EBM=56.567+2.146$

$\overline{x}+EBM=58.713$

Result

We estimate that the average age of all CEOs of the greatest small businesses is between $\$54.421and\$58.713$ years, with a $95\%$ confidence level.