Chapter 31, Problem 31.61E

To determine

To calculate: The control limits for s when removing sample 2 and 17. Also add the new control limits to the s chart.

To find: The control limits for $\overline{x}$ when removing the sample 2 and 17. Make the $\overline{x}$ chart and comment on the control chart.

Answer to Problem 31.61E

For the s chart, the center line is 7.73, the lower control limit is 0 and the upper control limit is 19.85.

Output using the MINITAB software is given below:

From the s charts, it can be observed that the process is within control.

For the $\overline{x}$ chart, the center line is 672.95, the lower control limit is 657.8465 and the upper control limit is 688.0535.

Output using the MINITAB software is given below:

From the $\overline{x}$ charts, it can be observed that the process is not within control.

Explanation of Solution

Given info:

The data shows the mean and standard deviation for 22 samples of three films.

Calculation:

s chart:

Center line for an s chart:

$\begin{array}{c}CL=\overline{s}\\ =\frac{\left(\begin{array}{l}6.8+10.6+12+5.3+4.5+11+6+11.3+8.3+4.1+\\ 3.9+6.1+7.9+8.4+6.2+8.3+11.6+5.7+5.8+10.7\end{array}\right)}{20}\\ =\frac{154.6}{20}\\ =7.73\end{array}$

The value of $\widehat{\sigma }$ is

$\begin{array}{c}\widehat{\sigma }=\frac{\overline{s}}{c_4}\\ =\frac{7.73}{\text{0}\text{.8862}}\left(\text{From table 31}{\text{.3, c}}_4=\text{0}\text{.8862}\right)\\ =8.72\end{array}$

Upper control limits:

$\begin{array}{c}UCL=B_6\widehat{\sigma }\\ =2.276\left(8.72\right)\left(\text{From table 31}\text{.3,}{B}_6\left(=3\right)\text{=2}\text{.276}\right)\\ =\text{19}\text{.85}\end{array}$

Lower control limits:

$\begin{array}{c}LCL=B_5\widehat{\sigma }\\ =0\left(8.72\right)\left(\text{From table 31}\text{.3,}{B}_5\left(=3\right)\text{=0}\right)\\ =0\end{array}$

Thus, the center line is 7.73, the lower control limit is 0, and the upper control limit is 19.85.

Software procedure:

Step-by-step procedure to construct s charts using the MINITAB software:

• Choose Graph > Time Series Plot or Stat > Time Series > Time Series Plot.
• Choose With Groups and click OK.
• In Series, select the column of standard deviation.
• In Show reference lines at Y values, enter 0, 7.73 and 19.85.
• Click OK.

Observation:

Since all observations lie within the lower and upper control limit, the s chart is within control.

$\overline{x}$ chart:

Center line for a $\overline{x}$ chart:

$\begin{array}{c}CL=\overline{\overline{x}}\\ \overline{=}\frac{13,459}{20}\\ =672.95\end{array}$

The value of $\widehat{\sigma }$ is

$\begin{array}{c}\widehat{\sigma }=\frac{\overline{s}}{c_4}\\ =\frac{7.73}{\text{0}\text{.8862}}\left(\text{From table 31}{\text{.3, c}}_4=\text{0}\text{.8862}\right)\\ =8.72\end{array}$

Upper control limits:

$\begin{array}{c}UCL=\overline{\overline{x}}+3\frac{\widehat{\sigma }}{\sqrt{n}}\\ =672.95+3\frac{8.72}{\sqrt{3}}\\ =672.95+15.1035\\ =\text{688}\text{.05}\end{array}$

Lower control limits:

$\begin{array}{c}LCL=\overline{\overline{x}}-3\frac{\widehat{\sigma }}{\sqrt{n}}\\ =672.95-3\frac{8.72}{\sqrt{3}}\\ =672.95-15.1035\\ =\text{657}\text{.85}\end{array}$

Thus, the center line is 672.95, the lower control limit is 657.85 and the upper control limit is 688.05.

Software procedure:

Step-by-step procedure to construct $\overline{x}$ charts using the MINITAB software:

• Choose Graph > Time Series Plot or Stat > Time Series > Time Series Plot.
• Choose With Groups and click OK.
• In Series, select the column of mean.
• In Show reference lines at Y values, enter 672.95, 657.85 and 688.05.
• Click OK.

Observation:

Since all the observations do not lie within the lower and upper control limit, the $\overline{x}$ chart is not within control.