STAT 495, Fall 2010
Homework Assignment #5
1. On homework 4, you looked at data on the outside diameter of a motor shaft machined on a
CNC turning center. The data consisted of outside diameters for subgroups of four
consecutive parts. The measurements were made in inches to the nearest ten-thousandth of an
inch. Only the last two digits were displayed, i.e. 2.1247 appears as 47 in the data set. Below
is the X chart for the 35 subgroups of size 4 with the 1- 2- and 3-“sigma” zones. The data
are given on the last page of the homework assignment.
XBar of Outside Diameter
Note: The sigma was calculated using the range.
a) Compute the limits for each of the Zones A, B and C.
b) Apply all eight of Nelson’s tests by hand and indicate if, and where, subgroups violate
each test.
c) Use JMP to answer b). Include the output from JMP. The data are available on WebCT
and the course webpage under Homework 4.
d) Comment on any differences between what you did by hand in b) and what the computer
program does.
2. For the following problem assume we have a process that is described theoretically by a
normal distribution with process mean μ and process standard deviation σ . For such a
process, the theoretical control limits on the X chart are μ ± 3
σ
n
. The “in control” ARL is
385.
a) Use the rule that sounds an alarm whenever a subgroup mean plots outside the control
limits. If the process level shifts by 0.75 σ , on average, how long will it take the chart to
sound an alarm using subgroups of size 4? size 16? size 25?
1
b) We wish to have a chart that sounds an alarm, on average, within 5 subgroups if the
process shifts 0.75 σ . What subgroup size should be used?
c) For a new rule that sounds an alarm whenever a subgroup mean plots
outside μ ± 2.5
σ
n
. What is the “in control” ARL for a chart using this rule?
d) Use the new rule that sounds an alarm whenever a subgroup mean plots
outside μ ± 2.5
σ
n
. If the process level shifts by 0.75 σ , on average, how long will it
take the chart to sound an alarm using subgroups of size 4? size 16? size 25?
e) Besides a shift in level, the process can also become more variable. Suppose we are using
a chart that sounds an alarm whenever a subgroup mean plots outside μ ± 3
σ
n
. If the
process standard deviation doubles, on average, how long will it take for an alarm to
sound with subgroups of size 4?
f) What effect will changing the subgroup size to 25 have on your answer in e)?
3. A polymer product is made in a batch reaction process. The primary quality characteristic is
the viscosity. Product specifications are for viscosity between 0.95 and 1.25. During the
initial monitoring of the process 64 consecutive batches are made and the viscosity for each
batch is determined. The data are listed below and are available on WebCT and the course
webpage.
Batch
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Viscosity
1.04
1.03
1.00
1.00
0.96
0.92
1.00
1.05
1.03
1.07
1.00
0.91
0.94
1.04
1.20
1.11
Batch
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
Viscosity
1.17
1.17
1.28
1.17
1.35
1.06
1.03
1.25
1.14
1.09
1.13
0.86
1.07
1.09
1.22
1.11
Batch
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
Viscosity
1.11
1.13
1.25
1.21
1.07
1.18
1.05
1.03
0.96
1.16
1.04
0.95
0.84
1.24
1.08
1.01
Batch
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
Viscosity
1.01
1.14
1.07
1.14
1.14
1.10
1.01
1.01
0.94
1.04
1.03
0.88
0.92
1.09
1.24
1.10
a) Construct Individual and Moving Range charts for the viscosity. You can do this using
JMP.
b) Use the average moving range to come up with an estimate of the process standard
deviation.
c) Are there any points that plot outside the control limits for the moving range chart? If so,
which batches are contributing to that moving range? What does this indicate about the
stability of the process?
2
d) Are there any points that plot outside the control limits for the individuals chart? If so,
which batches are sounding the alarms? What does this indicate about the stability of the
process?
e) What other tests for special causes are failed on the individuals chart? Be sure to indicate
the test and which batches are sounding alarms.
f) Given your answers in c), d) and e) do you think that the estimate of the process standard
deviation is appropriate in b)? Explain briefly. If you feel it is not appropriate, do you
suspect it is an over- or under-estimate of the process standard deviation?
Outside Diameter Motor Shafts
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
26
27
28
29
30
31
32
33
34
35
X1
48
48
50
50
51
44
51
46
54
54
57
51
42
43
41
60
48
46
52
62
58
50
47
56
39
40
48
36
51
43
43
29
36
48
45
X2
59
52
49
41
49
47
57
63
53
46
56
49
47
50
54
45
50
39
32
50
52
56
53
55
54
43
48
45
48
50
43
42
43
42
51
X3
42
51
60
48
60
54
53
63
53
55
53
67
48
44
59
52
45
43
54
47
57
44
42
44
54
59
34
43
24
40
49
43
43
43
43
X4
40
50
44
43
60
45
48
41
46
58
37
50
39
38
53
46
40
61
45
57
54
44
40
58
44
36
43
32
55
53
31
42
49
39
39
X
47.25
50.25
50.75
45.50
55.00
47.50
52.25
53.25
51.50
53.25
50.75
54.25
44.00
43.75
51.75
50.75
45.75
47.25
45.75
54.00
55.25
48.50
45.50
53.25
47.75
44.50
43.25
39.00
44.50
46.50
41.50
39.00
42.75
43.00
44.50
X = 47.807
R
19
4
16
9
11
10
9
22
8
12
20
18
9
12
18
15
10
22
22
15
6
12
13
14
15
23
14
13
31
13
18
14
13
9
12
R = 14.314
s
8.54
1.71
6.70
4.20
5.83
4.51
3.77
11.44
3.70
5.12
9.32
8.54
4.24
4.92
7.63
6.90
4.35
9.60
9.95
6.78
2.75
5.74
5.80
6.29
7.50
10.08
6.60
6.06
13.96
6.03
7.55
6.68
5.32
3.74
5.00
s = 6.482
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