Chapter 4, Problem 4.71P

To determine

The process is in control or not.

Answer to Problem 4.71P

The range is not within the control limits; therefore the process is not in control.

Explanation of Solution

Given:

The sample size is, $n=7$ .

The average of averages of sample size is, $\overline{\overline{x}}=125$

The average range is, $\overline{R}=17.82$

The standard deviation is, $\sigma =5.85$

Formula used:

The formula for the upper control limit for the averages is given as,

  $UC{L}_{\overline{x}}=\overline{\overline{x}}+A_2\overline{R}$

Here, $A_2$ is the control chart constant.

The formula for the lower control limit for the averages is given as,

  $LC{L}_{\overline{x}}=\overline{\overline{x}}-A_2\overline{R}$

The formula for the upper control limit for the ranges is given as,

  $UC{L}_{\overline{R}}=D_4\overline{R}$

Here, $D_4$ is the control chart constant.

The formula for the lower control limit for the ranges is given as,

  $LC{L}_{\overline{R}}=D_3\overline{R}$

Here, $D_3$ is the control chart constant.

Calculation:

Refer to the Table 4.2 to obtain the value of “Constants for control charts” for the sample size of 7.

The values of constants are given as,

  $\begin{array}{l}{A}_2=0.419\\ D_4=1.924\\ D_3=0.078\end{array}$

The value of the upper control limit for the averages can be calculated as,

  $\begin{array}{l}UC{L}_{\overline{x}}=\overline{\overline{x}}+A_2\overline{R}\\ UC{L}_{\overline{x}}=125+0.419\times 17.82\\ UC{L}_{\overline{x}}=132.46\end{array}$

The value of the lower control limit for the averages can be calculated as,

  $\begin{array}{l}LC{L}_{\overline{x}}=\overline{\overline{x}}-A_2\overline{R}\\ LC{L}_{\overline{x}}=125-0.419\times 17.82\\ LC{L}_{\overline{x}}=117.53\end{array}$

The value for the upper control limit for the ranges can be calculated as,

  $\begin{array}{l}UC{L}_{\overline{R}}=D_4\overline{R}\\ UC{L}_{\overline{R}}=1.924\times 17.82\\ UC{L}_{\overline{R}}=34.28\end{array}$

The value for the lower control limit for the ranges can be calculated as,

  $\begin{array}{l}LC{L}_{\overline{R}}=D_3\overline{R}\\ LC{L}_{\overline{R}}=0.078\times 17.82\\ LC{L}_{\overline{R}}=1.39\end{array}$

The value of average for the given sample can be calculated as,

  $\begin{array}{c}\overline{x}=\frac{120+143+124+130+105+132+121+127}{8}\\ =125.25\end{array}$

The value of range for the given sample can be calculated as,

  $\begin{array}{l}R=\text{maximum}\text{value}-\text{minimum}\text{value}\\ R=143-105\\ R=38\end{array}$

The average is within the limit as,

  $UC{L}_{\overline{x}}<\overline{x}<LC{L}_{\overline{R}}$

But the range is not within the upper and lower limit as,

  $UC{L}_{\overline{x}}<R<LC{L}_{\overline{R}}$

Conclusion:

Thus, the range is not in control limits, therefore the process is not in control.