STAT 495, Fall 2010
Homework Assignment #4
1. Engineers at a manufacturing plant wish to monitor the outside diameter of a motor shaft that
is machined on a CNC turning center. The target value is 2.125 inches with a tolerance of ±
0.001 inch. The data consists of outside diameters for subgroups of four consecutive parts.
The subgroups are spaced approximately ½ hour apart. The measurements are made in inches
to the nearest ten-thousandth of an inch. Only the last two digits are displayed, i.e. 2.1247
appears as 47 in the data table.
a) Using the summaries provided for all 35 subgroups, compute centerlines and control
limits for X and R charts by hand.
b) Construct X and R charts using JMP or another computer package.
c) Identify any subgroups that plot outside the control limits on either of the charts.
d) What can be said about the stability of the process? Our operational definition of stability
is no points outside control limits.
e) Construct X and s charts. How do these charts differ from those in c) in terms of
evaluating the current process? How different are the control limits on the X charts using
s instead of R?
f) Come up with two estimates of the process standard deviation, one based on subgroup
ranges and the other based on subgroup standard deviations. How do these two estimates
compare?
2. A study was conducted on the injection molding process of a vehicle hush panel. Briefly, the
process starts with material being fed into the back end of the barrel and screw mechanism.
The screw rotates and heat is added, melting the material and moving it to the front. The
molten material is injected into a mold. The plastic is cured until it is solidified. This
constitutes a cycle. Data on the weight, in grams, of the material for each mold cycle was
collected at the General Motors East Lansing plant. This was done by weighing the solidified
plastic that comes out of the mold. A subgroup consists of five consecutive cycles. There are
45 subgroups.
a) Using the summaries provided for all 45 subgroups, compute centerlines and control
limits for X and R charts by hand.
b) Construct X and R charts using JMP or another computer package.
c) Looking at the R chart, are there any subgroups that plot outside the control limits? If so,
what subgroups plot outside? Are there unusual individual values within these
subgroups? Explain briefly.
d) Based on the R chart what can be said about the stability of the process? Our operational
definition of stability is no points outside control limits.
e) Looking at the X chart, are there any subgroups that plot outside the control limits? If so,
what subgroups plot outside? Are there unusual individual values within these
subgroups? Explain briefly. It may be easier to separate subgroups that plot above the
upper control limit from those that plot below the lower control limit.
f) Based on the X chart what can be said about the stability of the process? Our operational
definition of stability is no points outside control limits.
g) Use JMP to construct a histogram for the entire set of 225 measurements. Orient the
histogram so that the weight intervals are on the horizontal axis. Be sure to include an
axis label and scale for the vertical axis. Comment on the shape of the histogram and
what it is telling you about the weight of material.
h) What comes out of the mold consists of two parts and a runner. Several of the subgroups
contain some mold cycles where only the two parts without the runner were weighed.
1
i)
j)
These subgroups are: 1, 7, 8, 11, 12, 15, 16, 17, 18, 23, 25, 29, and 30. What is the
relationship between these subgroups and the X and R charts you constructed in b)?
Assuming that having no runner with the two parts is a special cause that can be
eliminated, use JMP to reconstruct the control charts after excluding these subgroups.
How do the control limits on these charts compare to those in b)? What can you say about
the stability of the process now? Again use the one point outside control limits alarm rule.
Compute an estimate of the process standard deviation using all 45 subgroups. Compute
an estimate of the process standard deviation, excluding the subgroups given in h). How
has the estimate of the process standard deviation changed?
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
2
Injection Mold Weights
Subgroup
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
36
37
38
39
40
41
42
43
44
45
X1
690
695
690
690
718
715
713
717
713
705
650
708
712
713
704
707
718
709
709
710
714
714
700
705
641
706
707
707
703
703
705
707
701
700
694
694
700
699
701
689
700
702
703
694
704
X2
690
696
690
690
719
716
712
717
714
704
711
707
710
713
645
710
719
650
708
714
713
713
699
706
701
706
705
707
702
693
703
706
700
700
695
694
699
699
698
690
699
702
703
696
703
X3
690
696
690
690
719
716
713
718
715
703
650
646
712
712
703
648
655
710
708
712
714
714
700
704
701
706
706
706
705
626
705
706
699
700
696
695
698
699
698
691
700
703
702
694
700
X4
690
694
692
689
722
717
654
658
715
704
709
704
710
712
703
709
719
647
709
708
714
712
700
703
703
706
704
705
705
619
706
705
699
699
696
695
699
698
698
694
700
705
700
697
700
X5
629
695
692
687
721
715
716
720
713
704
710
707
710
712
702
709
716
709
700
704
713
710
640
702
643
706
706
706
643
707
706
705
700
699
695
695
699
699
697
697
702
705
700
696
698
X
677.8
695.2
690.8
689.2
719.8
715.8
701.6
706.0
714.0
704.0
686.0
694.4
710.8
712.4
691.4
696.6
705.4
685.0
706.8
709.6
713.6
712.6
687.8
704.0
677.8
706.0
705.6
706.2
691.6
669.6
705.0
705.8
699.8
699.6
695.2
694.6
699.0
698.8
698.4
692.2
700.2
703.4
701.6
695.4
701.0
X = 699.5
R
61
2
2
3
4
2
62
62
2
2
61
62
2
1
59
62
64
63
9
10
1
4
60
4
62
0
3
2
62
88
3
2
2
1
2
1
2
1
4
8
3
3
3
3
6
R = 20.56
s
27.28
0.84
1.10
1.30
1.64
0.84
26.65
26.86
1.00
0.71
32.87
27.10
1.10
0.55
25.95
27.19
28.20
33.34
3.83
3.85
0.55
1.67
26.72
1.58
32.70
0.00
1.14
0.84
27.20
43.37
1.22
0.84
0.84
0.55
0.84
0.55
0.71
0.45
1.52
3.27
1.10
1.52
1.52
1.34
2.45
s = 9.48
3