Chapter 14.5, Problem 9CFU

a.

To determine

Find the standard error of the mean.

Answer to Problem 9CFU

  $0.28$

Explanation of Solution

Given information:

An aptitude test was taken by an employment agency. $150$ clients were randomly selected and the time to complete the test of each client was calculated. The average time was $27.5\mathrm{mins}$ with the standard deviation $3.5\mathrm{mins}$ .

Calculation:

Here, we have to find the standard error of the mean:

  $\begin{array}{l}{\sigma }_{\overline{X}}=\frac{Standarddeviation}{\sqrt{N}}\\ {\sigma }_{\overline{X}}=\frac{3.5}{\sqrt{150}}=0.28\end{array}$

Hence, the standard error is $0.28$ .

b.

To determine

Find the interval about the sample mean with $50\%$ chance of having true mean within the interval.

Answer to Problem 9CFU

  $\left[27.30,27.70\right]$

Explanation of Solution

Given information:

An aptitude test was taken by an employment agency. $150$ clients were randomly selected and the time to complete the test of each client was calculated. The average time was $27.5\mathrm{mins}$ with the standard deviation $3.5\mathrm{mins}$ .

Calculation:

Here, we will find the interval:

  $\begin{array}{l}P=50\%correspondstot=0.7\\ Interval:\\ \left[\overline{X}-t{\sigma }_{\overline{X}},\overline{X}+t{\sigma }_{\overline{X}}\right]\\ \left[27.30,27.70\right]\end{array}$

Hence, the required answer is $\left[27.30,27.70\right]$ .

c.

To determine

Find the interval about the sample mean having $1\%$ confidence level.

Answer to Problem 9CFU

  $\left[27.75,28.24\right]$

Explanation of Solution

Given information:

An aptitude test was taken by an employment agency. $150$ clients were randomly selected and the time to complete the test of each client was calculated. The average time was $27.5\mathrm{mins}$ with the standard deviation $3.5\mathrm{mins}$ .

Calculation:

Here, $1\%$ confidence level means when $P=99\%$ .

  $\begin{array}{l}P=99\%correspondstot=2.58\\ \mathrm{interval}:\\ \left[\overline{X}-t{\sigma }_X,\overline{X}+t{\sigma }_X\right]\\ \left[27.75,28.24\right]\end{array}$

Hence, the required answer is $\left[27.75,28.24\right]$ .