Chapter 19, Problem 16SE

a.

To determine

Find the estimate of the mean when the process is in control.

Answer to Problem 16SE

The estimate of the mean for the production process that is in control is 95.4.

Explanation of Solution

Calculation:

The means for 20 samples of a production process are provided.

The estimate of the mean for the production process is given below:

$\begin{array}{c}\mu =\frac{{\displaystyle \sum \overline{x}}}{n}\\ =\frac{\left(\begin{array}{l}95.72+95.44+95.40+95.50+95.56+95.72+95.60+95.24+95.46+95.44+\\ 95.80+95.22+94.82+95.78+95.18+95.32+95.08+95.22+95.04+95.46\end{array}\right)}{20}\\ =\frac{1,908}{20}\\ =95.4\end{array}$

Thus, the estimate of the mean for the production process that is in control is 95.4.

b.

To determine

Construct a $\overline{x}$ control chart for the production process.

Answer to Problem 16SE

Output obtained using MINITAB software is given below:

Explanation of Solution

Calculation:

Upper control limits for the $\overline{x}$chart:

$\begin{array}{c}\text{UCL}=\mu +3\frac{\sigma }{\sqrt{n}}\\ =95.4+3\frac{0.50}{\sqrt{5}}\\ =95.4+0.6696\\ =96.07\end{array}$

Lower control limits for the $\overline{x}$chart:

$\begin{array}{c}\text{LCL}=\mu -3\frac{\sigma }{\sqrt{n}}\\ =95.4-3\frac{0.50}{\sqrt{5}}\\ =95.4-0.6696\\ =94.73\end{array}$

X-bar chart:

Software procedure:

Step-by-step software procedure to obtain the X-bar chart using EXCEL:

• Open an EXCEL file.
• Enter the observation.
• Enter X-double bar in a new cell B1, in cell B2 enter a formula “=Average(A2:A21)”.
• Click Enter.
• Click on Insert > select Insert Line chart icon.
• Click on the chart > select Layout from the Chart Tools.
• Select Chart Title > Above Chart.
• Enter X-bar Chart in the dialog box.

Thus, the control limits for the $\overline{x}$ chart are $\text{UCL}=96.07\text{and}\text{LCL}=94.73$.

c.

To determine

Explain whether any of the 20 sample means indicate that the process is out of control or not.

Explanation of Solution

From the $\overline{x}$ chart, it is observed that all the sample means lie within UCL and LCL.

Thus, the 20 sample means do not indicate that the process was out of control.