ISEN 614 Homework #1
(Due: Feb 10)
Question 1 (6 points).
The thickness of a printed circuit board is an important quality parameter. Data on board thickness (in
inches) are given in the following table for 25 samples of three boards each.
Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
x1
0.0629
0.0630
0.0628
0.0634
0.0619
0.0613
0.0630
0.0628
0.0623
0.0631
0.0635
0.0623
0.0635
0.0645
0.0619
0.0631
0.0616
0.0630
0.0636
0.0640
0.0628
0.0615
0.0630
0.0635
0.0623
x2
0.0636
0.0631
0.0631
0.063
0.0628
0.0629
0.0639
0.0627
0.0626
0.0631
0.063
0.063
0.0631
0.064
0.0644
0.0627
0.0623
0.063
0.0631
0.0635
0.0625
0.0625
0.0632
0.0629
0.0629
x3
0.064
0.0622
0.0633
0.0631
0.063
0.0634
0.0625
0.0622
0.0633
0.0633
0.0638
0.063
0.063
0.0631
0.0632
0.063
0.0631
0.0626
0.0629
0.0629
0.0616
0.0619
0.063
0.0635
0.063
(a) Set up x-bar and R control charts for =0.0027. Is the process in statistical control?
(b) Set up an S chart, and compare with the R chart in part (a).
Question 2 (6 points).
Given the in-control mean 0 and the process standard deviation  (i.e., the standard deviation of
individual observations), determine the UCL and LCL for an x-bar chart with the type I error of 0.001
and sample size of 4. Assume that a mean shift is 1.5 , what is the type II error under this mean shift?
Question 3 (7 points) [From Montgomery 2001, pp 204, Ex 4-26].
Two decision rules are given here. Assume they apply to a normally distributed quality characteristic and
the sample size is n=5.
Rule 1: The control chart uses the three-sigma control limits. If one or more of the next seven samples
yield values of the sample average that fall outside the control limits, conclude that the process is out of
control.
Rule 2: If all of the next seven sample averages fall on the same side of the center line, conclude that the
process is out of control.
What is the -error probability for the control chart under each of these rules?
Question 4. [6 pts]
The following data represent individual observations on bath concentration from a chemical process. The
target value is 10 and the process standard deviation is given as  = 0.8.
(a) Use k = 0.5 and h =5, built a tabular CUSUM to quickly detect a shift of about 1 in the process
mean. Based on the tabular CUSUM, please conclude whether the process has a mean shift using
h=5. If the process has a mean shift, please indicate at which sample this mean shift started.
(b) Estimate the magnitude of the mean shift and calculate the  = 1   0 /  .
(c) Calculate out-of-control ARL1 (using the approximated formula) for the two-sided CUSUM.
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
xi
9.45
7.99
9.29
11.66
12.16
10.18
8.04
11.46
9.2
10.34
9.03
11.47
10.51
9.4
10.08
i
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
xi
9.37
10.62
10.31
8.52
10.84
10.9
9.33
12.29
11.5
10.6
11.08
10.38
11.62
11.31
10.52