# STAT 495, Fall 2003 Homework Assignment #4

```STAT 495, Fall 2003
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 &plusmn;0.001 inch.
The data are outside diameters for subgroups of four consecutive parts. The subgroups are spaced
approximately 12 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.
(a) Using all 35 subgroups, compute centerlines and control limits for X and R charts.
(b) Construct X and R charts. You may use JMP, Minitab or another computer prackage to do this.
(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. Jack Brown at the
General Motors Lansing site, collected data on the weight, in grams, of the material for each cycle.
This was done by weighing the solidified plastic that comes out of the mold. A subgroup consists of
five weight measurements. There are 45 subgroups.
(a) Using all 45, compute centerlines and control limits for X̄ and R charts.
(b) Actually construct the X̄ and R charts. You can use JMP, Minitab or another computer package
to do this.
(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 a histogram for the entire set of 225 measurements. Use cutpoints from 600 to 750 with
interval widths of 10. Comment on the shape of the histogram and what it is telling you about
the weight of material.
(f) What comes out of the mold consists of two parts and a runner. Several of the subgroups contain
weights for the two parts without the runner. These subgroups are: 1, 7, 8, 11, 12, 15, 16,
17, 18, 23, 25, 29, and 30. Assuming that this special cause can be eliminated, reconstruct the
control charts after first removing these subgroups. Use a computer package to do this. How do
the control limits on these charts compare to those in a) and b)? What can you say about the
stability of the process now? Again use the one point outside control limits alarm rule.
(g) 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 (f). How has the
estimate 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
X̄
R
47.25
19
50.25
4
50.75
16
45.50
9
55.00
11
47.50
10
52.25
9
53.25
22
51.50
8
53.25
12
50.75
20
54.25
18
44.00
9
43.75
12
51.75
18
50.75
15
45.75
10
47.25
22
45.75
22
54.00
15
55.25
6
48.50
12
45.50
13
53.25
14
47.75
15
44.50
23
43.25
14
39.00
13
44.50
31
46.50
13
41.50
18
39.00
14
42.75
13
43.00
9
44.50
12
47.807 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
6.482
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
X̄
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
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
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
9.48
```