Question

1. Copper wires are coated with a thin plastic coating. and the thickness of the coating (in mils) is measured. The data from the last 30 samples are presented in table E12. The means are R 150.075, R-6.97, and s 3.082. a. Compute the 3σ limit for the R chart. Is the variance out of control at any point? If so, Samples of four wires are taken every hour delete the samples that are out of control and recomputeXand R Compute the 3ơ limits for the R control? If not, at what point is it first detected to be out of control? chart. On the basis of the 3ơ limits, is the process mean in b. c. On the basis of the Western Electric rules, is the process mean in control? If not, when is it first detected to be out of control? TABLE E12 Data for Exercise 12 Sample Sample Values 150.125 148.850 150.200 150.475 49.275 150.275 47.875 147450 148.100 147.825 150.925 151.900 149.325 151.275 151.200 150.900 151.625 152.700 151.100 147.575 49.550 151.175 150.200 147.950 150.525 51.225 151.350 47.825 153.300 150.175 46.0 147.4 151.9 155.2 47.1 147.5 151.4 149.4 148.7 148.4 149.6 154. 151.3 150.0 152.4 148.2 146.4 147.5 152.9 150.3 150.2 142.9 152.5 155.5 47.8 148.3 145.7 149. 37.1 156.6 147.2 148.9 151.1 148.1 145.6 147.6 151.3 151.3 142.5 146.2 151.3 153.5 150.2 148.7 151.9 152.2 149.3 154.2 52.8 149.1 148.5 146.9 52.9 1499 151.9 150.4 49.0 1499 153.1 152.8 53.9 150.8 153.9 145.0 50.4 151.8 151.3 153.0 157.2 152.6 148.4 152.6 152.7 156.2 146.8 148.7 50.2 148.2 149.8 142 51.0 151.7 148.5 147.0 43.8 154.5 154.8 151.6 43.0 156.4 149.2 152.2 48.8 147.7 147.1 148.2 153.8 145.4 149.5 153.4 151.6 149.3 155.0 149.0 149.4 151.4 154.6 1500 49.8 149.0 146.8 145.7 155.8 1524 150.2 154.8 153.9 145.7 150.7 150.4 1.81 2.92 5.37 1.66 8.02 19.5 4.29 2.01 1.38 2.06 4.20 1.08 3.59 4.20 3.75 2.19 15 2.6 17 21 23 25 11.0 13.4 5.64 0.72 3.93 2.77 2.32 1.90 2.51 3.38 26 5.2 8.2 30 2. Repeat Exercise 1, using the S chart in place of the R chart.

Answer 1

a) 3$\sigma$ limits for R chart are calculated via the following formula

$\bar{R}=\frac{\sum R_{i}}{n}$ , $LCL=\bar{R}-3d_{3}\sigma$ , $LCL=\bar{R}+3d_{3}\sigma$

R Chart of S1, ..., S4 20 UCL-16.25 15 刎 α 10 R-7.12 e辦 0D LCL-0 1 4 710 13 16 19 22 25 28 Sample

Yes from the above R chart variance is out of control for 8^th sample as given in the above R Chart.

After deleting of the 8^th sample the variance is in control now as given below in revised R chart

R Chart of S1, ..., S4 UCL-14.97 14 12 10 刎 R-6.56 E 6 e辦 4 LCL-0 0D 1 4710 13 16 19 2225 28 Sample

R- 6,56
and  
X-150.17

b) X bar chart is calculated with the following formula

$\bar{\bar{X}}=\frac{\sum_{i=1}^{n} \bar{x_{i}}}{n}$ , $LCL = \bar{\bar{X}}- 3\frac{\sigma }{\sqrt{n}}$ ,
LCL = X + 3

Xbar Chart of Si, ..., S4 UCL=155.26 155.0 152.5 刎 150.08 150.0 e辦 147.5 145.0 LC 144.89 1 4 7 10 13 16 19 22 25 28 Sample

The process mean is in control.

c) As per Western electric rules Rule 3 i,e (Four out of five consecutive points fall beyond the 1σ-limit (in zone B or beyond), on the same side of the center line) does fall here and the sample no is 5^th and is given below in the chart.

Zone Chart of Si,..., S4 +3 StDev-155.26 4 +2 StDev-153.53 +1 StDev-151.80 X-150.08 0D 1 StDev-148.35 22 X4 6 8 2 StDev-146.62 4 3 StDev-144.89 6 8 10 12 14 16 18 20 22 24 26 28 30 Sample

d) S.Chart is calculated via the following formula

$\bar{S}=C_{4}\sigma$,   $LCL=\bar{S}-3\sigma \sqrt{1-C_{4}^{2}}$$UCL=\bar{S}+3\sigma \sqrt{1-C_{4}^{2}}$

S Chart of S1,..., S4 UCL 7.222 S-3.187 1 4710 13 16 19 22 25 28 Sample

Here the process is out of control for sample no 8.

Now after deleting the sample no 8 the revised Chart is given below

S Chart of S1,..., S4 7 UCL-6.652 5 4 3 S 2.935 2 LCL=0 1 4710 13 16 19 22 25 28 Sample