Question

The data represents the copper concentration in a plating process. Copper concentration in a plating pool is controlled by an automated colorimeter that takes readings and adjusts the concentration as required. To verify the correct functioning of a new colorimeter system copper concentration is measured manually six times per shift. (a) Plot an x-bar and R control chart for this process. Perform runs tests to Westem Electric rules 1 through 5 in Table 5.1, p. 205 of the textbook. Is the process in control? Revise the control limits as necessary. (b) Estimate the mean and standard deviation of the revised process. (c) Is the layer thickness of the revised process normally distributed? (d) If the specifications are 9.0 t1.50 ppm, estimate the process capability 8 Subgroup Subgroup Subgroup Subgroup Subgroup Cu Cu Cu Cu Cu ppm ppm ppm ppm 9.38 ppm 8.50 1 7.96 9.36 13 10.66 17 1 8.52 5 7.38 9.08 13 9.10 17 8.02 7.22 9.24 5 7.86 13 10.04 17 8.04 8.28 9.24 1 7.96 5 13 9.64 17 8.14 10.04 8.48 9.86 1 5 8.46 13 17 7.50 1 8.68 5 8.26 9 8.68 13 8.62 17 8.60 8.32 2 7.46 6 9.50 10 9.38 14 18 8.16 2 8.84 9.38 10 8.12 14 8.14 18 8.34 9.00 7.98 2 8.90 6 10 14 7.80 18 8.20 9.28 2 6 9.38 10 8.72 14 8.54 18 8.14 8.96 2 7.34 6 10 8.66 14 8.38 18 8.42 2 9.24 6 8.74 10 9.88 14 8.38 18 7.88 9.04 3 7.82 7 11 9.68 15 8.12 19 9.14 3 9.46 7 9.28 11 9.78 15 8.94 19 9.06 3 7.30 7 8.08 11 9.88 15 7.54 19 9.40 8.12 3 10.38 7 9.42 11 9.96 15 19 9.66 9.28 3 7 8.72 11 10.34 15 8.88 19 9.48 8.72 9.58 3 10.14 11 15 8.86 19 8.82 8.74 4 8 9.24 12 7.86 16 8.04 20 7.34 9.32 4 8.66 12 8.80 16 8.92 20 10.08 4 9.26 9.14 12 9.42 16 8.82 20 7.94 4 9.26 8.92 7.68 8.5 12 7.90 16 20 9.00 4 8.82 12 9.00 16 8.80 20 8.12 4 9.08 10.18 12 8.56 16 8.46 20 9.40 00 00

Answer 1

ANSWER:

Given that,

a)

First we need to find the R chart and see whether the process is in control with respect to the R chart.

Lower Control Limit for R chart

D3R

Upper Control Limit for R chart

D&R

where
R R = m -1
and
R
is the range of the
th
subgroup.

Here,

D3(n6 0

D4(n6 2.004

R=1.4855

Lower Control Limit for $\bar{x}$ chart

A2 R

Upper Control Limit for $\bar{x}$ chart $=\bar{\bar{x}}+A_2\bar{R}$

$\bar{\bar{x}}=\frac{1}{m}\sum_{i=1}^{m}\bar{x_i}$

where $\bar{x_i}$ is the sample mean of subgroup

th

A2 (n 6) 0.483

The data is given as:

The $\bar{x}$ chart is given as:

Xbar chart 10 9.8 9.6 9.4 9.2 9 8.8 8.6 8.4 8.2 8 10 25 15 20

The R chart is given as:

R chart 3.5 3 2.5 2 1.5 1 0.5 0 10 15 25 20

(Table 5.1 required to complete the problem)

The process is out of control because for both the $\bar{x}$ chart and R chart some of the points fall out of the control limits.

Now, remove the subgroups which do not comply to the Western Electric Rules with respect to the R chart. Again compute the value for $\bar{R}$ and hence the upper and lower limit with less number of subgroups. Check if the process is in control according to the R chart then move forward to find the xbar chart, if it is not then remove subgroups again. Repeat procedure till R chart is in control.

When R chart is in control, plot the $\bar{x}$ chart by finding the values of $\bar{\bar{x}}$ and $\bar{R}$ then forming the control limits for the modified dataset. If the $\bar{x}$ chart is in control complying to the Western Electric Rules, then it can be said that the process is in control for the modified dataset, else remove subgroups not complying to the rules.

b)

When both R-chart and $\bar{x}$ chart is in control, move forward for the following calculations:

Estimate of mean = $\bar{\bar{x}}$

Estimate of standard deviation = $\frac{\bar{R}}{d_2}$

where d2 (n =6) =2.534

c)

Take the entire values of the revised process and plot a histogram along with a normal curve. If the normal curve fits the histogram well, then it can be said that the revised process is normally distribution.

d)

Upper Specification Limit (USL) = 9+1.5 = 10.5

Lower Specification Limit LSL) = 9 - 1.5 = 7.5

Process Capability =

60 CP 6a

Take the estimate of process standard deviation of as the value of $\sigma$.