Chapter 31, Problem 31.60E

To determine

The control limits for $\overline{x}$ chart and describe the control of short term process variation by the construction of $\overline{x}$ chart after the removal of initial 10 samples.

Answer to Problem 31.60E

The upper control limit and lower control limit for s are 847.136 and 821.864 respectively.

Explanation of Solution

Given info:

The 22 samples of three films each are measured at 10 minutes intervals. In samples 1 to 10, mistakes in operating the interferometer resulted in one high outlier thickness reading that was clearly incorrect.

Calculation:

Since the points for samples 1 and 10 lie above the upper control limit, thus the two values of s are to be removed that were out of control.

Thus the values for rest 20 samples are,

Since the total past samples are 20, thus the formula to calculate the mean is,

$\overline{\overline{x}}=\frac{{\displaystyle \sum \text{number of possible outcomes}}}{\text{total number of outcomes}}$

Thus, the overall mean of the sample means is,

$\overline{\overline{x}}=834.5$

The overall mean of the standard deviation is,

$\overline{s}=6.465$

The formula to calculate the process standard deviation is,

$\stackrel{\wedge }{\sigma }=\frac{\overline{s}}{c_4}$ (1)

Substitute 6.465 for $\overline{s}$ and 0.8862 for $c_4$ in equation (1).

$\begin{array}{c}\stackrel{\wedge }{\sigma }=\frac{6.465}{0.8862}\\ =7.2952\end{array}$

The formula to calculate the center line for s chart is,

${\mu }_s=c_4\stackrel{\wedge }{\sigma }$ (2)

Substitute 0.8862 for $c_4$ and 7.2952 for $\stackrel{\wedge }{\sigma }$ in equation (2).

$\begin{array}{c}CL=\left(0.8862\right)\left(7.2952\right)\\ =6.465\end{array}$

The formula to calculate the upper control limit for an s chart is,

$\text{UCL=}{B}_6\stackrel{\wedge }{\sigma }$ (3)

Substitute 2.276 for $B_6$ and 7.2952 for $\stackrel{\wedge }{\sigma }$ in equation (3).

$\begin{array}{c}\text{UCL=}\left(2.276\right)\left(7.2952\right)\\ =16.6038\end{array}$

Thus the UCL is 16.6038.

The formula to calculate the lower control limit for s chart is,

$\text{LCL=}{B}_5\stackrel{\wedge }{\sigma }$ (4)

Substitute 0 for $B_5$ and 7.2952 for $\stackrel{\wedge }{\sigma }$ in equation (4).

$\begin{array}{c}\text{LCL=}\left(0\right)\left(7.2952\right)\\ =0\end{array}$

Thus the LCL is 0.

Thus the center line and control limits for an s chart are 6.465, 16.6038 and 0 respectively.

The x axis of the s chart shows the sample and y axis shows standard deviation.’

Upper control limit is 16.6038.

Lower control limit is 0.

Centerline of the s chart is 6.465.

This shows that all the 20 samples are within the control limits, that is the data is in control.

The formula to calculate the center line for $\overline{x}$ chart is,

$CL=\overline{\overline{x}}$ (5)

Substitute 834.5 for $\overline{\overline{x}}$ in equation (5).

$CL=834.5$

The formula to calculate the upper control limit for $\overline{x}$ chart is,

$\text{UCL=}\overline{\overline{x}}+3\left(\frac{\stackrel{\wedge }{\sigma }}{\sqrt{n}}\right)$ (6)

Substitute 834.5 for $\overline{\overline{x}}$, 3 for n and 7.2952 for $\stackrel{\wedge }{\sigma }$ in equation (6).

$\begin{array}{c}\text{UCL=}834.5+3\left(\frac{7.2952}{\sqrt{3}}\right)\\ =834.5+12.6356\\ =847.136\end{array}$

Thus the UCL is 847.136.

The formula to calculate the lower control limit for $\overline{x}$ chart is,

$\text{LCL=}\overline{\overline{x}}+3\left(\frac{\stackrel{\wedge }{\sigma }}{\sqrt{n}}\right)$ (7)

Substitute 834.5 for $\overline{\overline{x}}$, 3 for n and 7.2952 for $\stackrel{\wedge }{\sigma }$ in equation (7).

$\begin{array}{c}\text{LCL=}834.5-3\left(\frac{7.2952}{\sqrt{3}}\right)\\ =834.5-12.6356\\ =821.864\end{array}$

Thus the LCL is 821.864.

Thus the center line and control limits for an s chart are 834.5, 847.136 and 821.864 respectively.

The x axis of the $\overline{x}$ chart shows the sample and y axis shows mean.

Upper control limit is 847.136.

Lower control limit is 821.864.

Centerline of the s chart is 834.5.

This shows that all the 20 samples are within the control limits, which are the data is in control.

Thus, s and $\overline{x}$ charts are in control.