Chapter 6, Problem 6.1.1RE

The waking times (in minutes past 5:00 A.M.) of 40 people who start work at 8:00 A.M. are shown in the table at the left. Assume the population standard deviation is 45 minutes. Find (a) the point estimate of the population mean μ and (b) the margin of error for a 90% confidence interval.

To determine

To find: The point estimate for the mean $\mu$.

Answer to Problem 6.1.1RE

The point estimate for the mean $\mu$ is 103.5 minutes.

Explanation of Solution

Given info:

The data shows that the waking times of 40 people who start work at 8:00 A.M.

Calculation:

Point estimate:

The point estimate of the population mean $\mu$ is the sample mean $\left(\overline{x}\right)$.

Software Procedure:

Step by step procedure to obtain the sample mean using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics.
• In Variables enter the columns Waking times (in minutes past 5).
• In Statistics choose mean.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the point estimate for the mean waking time is 103.5 minutes.

To determine

To find: The margin of error.

Answer to Problem 6.1.1RE

The margin of error is 11.70.

Explanation of Solution

Given info:

$c=0.90$, $\sigma =45\text{ minutes}$ and $n=40$.

Calculation:

Critical value:

Software procedure:

Step by step procedure to obtain the P-value using the MINITAB software:

• Choose Graph > Probability Distribution Plot choose View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Click the Shaded Area tab.
• Choose Probability Value and Both Tail for the region of the curve to shade.
• Enter the Probability value as 0.10.
• Click OK.

Output using the MINITAB software is given below:

From the output, the critical value $\left(z_c\right)$ is 1.645.

Margin of error:

The formula for the margin of error is,

$E=z_c\left(\frac{\sigma }{\sqrt{n}}\right)$

Substitute 1.645 for $z_c$, 45 for $\sigma$ and 40 for n in E.

$\begin{array}{c}E=1.645\left(\frac{45}{\sqrt{40}}\right)\\ =\left(1.645\right)\left(7.1151\right)\\ =11.70\end{array}$

Thus, the margin of error is 11.70.