Question

1. The thickness of a printed circuit board is an important quality parameter. Data on board thickness (in inches) are given in Table 6 for 25 samples of three boards each.
   (a) Set up x-bar and R control charts. Is the process in statistical control?
   (b) Estimate the process standard deviation.
   (c) What are the limits that you would expect to contain nearly all the process measurements?
   (d) If the specifications are at 0.0630 in. ± 0.0015 in.,
   what is the value of the PCR Cp?     

                                Table 6

   Sample No.       X1                  X2              X3

   1                         0.0629       0.0636       0.0640

   2                         0.0630       0.0631      0.0622

   3                        0.0628        0.0631    0.0633

   4                        0.0634        0.0630      0.0631

   5                        0.0619       0.0628       0.0630

   6                       0.0613       0.0629      0.0634

   7                        0.0630       0.0639      0.0625

   8                        0.0628       0.0627      0.0623                  

   9                         0.0623      0.0627      0.0633

   10                       0.0631      0.0631     0.0633

   11                      0.0635      0.0630     0.0638

   12                       0.0623      0.0630     0.0630

   13                      0.0635      0.0631     0.0630

   14                       0.0645      0.0640     0.0631

   15                       0.0619      0.0644     0.0632

   16                      0.0631     0.0627      0.0630

   17                      0.0616       0.0623     0.0631

   18                      0.0630       0.0630     0.0626

   19                      0.0636       0.0631     0.0629

   20                      0.0640       0.0635    0.0629

   21                      0.0628       0.0625     0.0616

   22                      0.0615       0.0625     0.0619

   23                      0.0630       0.0632     0.0630

   24                      0.0635       0.0629    0.0635

   25                       0.0623      0.0629    0.0630

Answer 1

(a) Set up x-bar and R control charts. Is the process in statistical control?

The X-bar and R-chart are obtained in the following steps,

Step 1: Calculate the sample average, and range, for each sample.

Sample No.	X1	X2	X3
1	0.0629	0.0636	0.04	0.0555	0.0236
2	0.063	0.0631	0.02	0.0487	0.0431
3	0.0628	0.0631	0.063	0.0630	0.0003
4	0.0634	0.063	0.061	0.0625	0.0024
5	0.0619	0.0628	0.06	0.0616	0.0028
6	0.0613	0.0629	0.0613	0.0618	0.0016
7	0.063	0.0639	0.062	0.0630	0.0019
8	0.0628	0.0627	0.062	0.0625	0.0008
9	0.0623	0.0627	0.03	0.0517	0.0327
10	0.0631	0.0631	0.063	0.0631	0.0001
11	0.0635	0.063	0.0633	0.0633	0.0005
12	0.0623	0.063	0.063	0.0628	0.0007
13	0.0635	0.0631	0.0633	0.0633	0.0004
14	0.0645	0.064	0.063	0.0638	0.0015
15	0.0619	0.0644	0.063	0.0631	0.0025
16	0.0631	0.0627	0.0633	0.0630	0.0006
17	0.0616	0.0623	0.0613	0.0617	0.0010
18	0.063	0.063	0.0632	0.0631	0.0002
19	0.0636	0.0631	0.0632	0.0633	0.0005
20	0.064	0.0635	0.0642	0.0639	0.0007
21	0.0628	0.0625	0.0621	0.0625	0.0007
22	0.0615	0.0625	0.0611	0.0617	0.0014
23	0.063	0.0632	0.0633	0.0632	0.0003
24	0.0635	0.0629	0.0633	0.0632	0.0006
25	0.0623	0.0629	0.063	0.0627	0.0007
			Sum	1.5378	0.1216

Step 2: Calculate the grand average, and range,

Step 3: The control limits for and, charts are obtained as follow,

Chart

Where the value of is obtained from the control chart factor table,

For sample size = 3 ,

The X bar chart is obtained in excel. The screenshot is shown below,

Chart

Where, the value of and are obtained from the control chart factor table,

For sample size = 3 ,

We can see that, there are few data points outside the control limit hence the process is not in control.

(b)

The process standard deviation for the x-bar values is obtained in excel using the function =STDEV().

c)

We expect all the data would lie with three standard deviations from the mean data values.

The control limits are defined as,

Where,

d)

The process capability measure by process capability ratio,

Where,