STAT 495, Fall 2003
Homework Assignment #7
1. A die-casting process is being used to make a cast housing that will be sold to a client who
will perform further machining operations on the cast part. The weight of the part is a good
indicator of process performance. The cast part has a target weight of 775 grams with a
tolerance of ±20 grams. The data for 40 subgroups of 5 parts are presented below. The data
file is available on the course webpage.
Subgroup
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
X1
779
775
772
779
771
781
788
780
765
772
773
773
784
788
764
786
784
776
775
779
X2
782
763
782
776
763
767
777
777
772
773
774
770
774
770
771
773
767
781
769
775
X3
771
775
770
773
779
772
770
778
775
775
772
783
778
779
765
771
771
766
783
773
X4
771
779
790
780
776
783
765
786
784
775
776
778
779
788
778
778
759
774
779
775
X5
773
774
775
779
787
771
759
780
785
765
777
785
776
778
782
771
787
773
771
774
Subgroup
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
X1
779
776
781
781
784
785
783
774
777
759
782
766
777
789
782
773
783
782
782
777
X2
786
778
788
774
765
772
772
778
773
771
773
774
773
783
778
782
769
782
770
769
X3
785
788
784
772
765
776
772
774
774
785
776
779
777
771
766
767
777
780
770
780
X4
787
775
770
778
777
772
781
782
771
774
774
781
786
775
769
773
777
770
774
768
X5
784
789
782
760
779
778
766
793
783
780
785
773
774
782
767
770
761
772
775
779
(a) Verify that the process is in a state of statistical control by constructing and interpreting
appropriate X and R control charts.
(b) Construct a histogram of the 200 individual measurements. Comment on the shape of
the histogram.
(c) What proportion of the observed measurements are above the upper specification limit?
below the lower specification limit?
(d) Give values of the estimated process potential, C p and process performance, Cpk indices
using the overall standard deviation of the 200 measurements.
(e) What are the predicted (expected) proportion of future measurements that will be above
the upper specification limit? below the lower specification limit? Use the overall standard deviation of the 200 measurements.
(f) Give values of the estimated process potential, C p and process performance, Cpk indices
using the average subgroup range, R.
(g) What are the predicted (expected) proportion of future measurements that will be above
the upper specification limit? below the lower specification limit? Use the the average
subgroup range, R.
(h) Is there much of a difference between your answers in parts d) and e) compared to those
in f) and g)?
(i) Summarize your findings in an executive summary that would be read by a person in
management. You should include any graphs that may help management understand
what is happening with this process.
2. A decision needs to be made regarding the possible purchase of some new casting equipment.
Before proceeding further data is collected to assess the performance of the current equipment.
An important part that will consume much of the run time of the equipment is to be used for
the study. The width of the cast part at a crucial spot should be 3.000 ± 0.025 centimeters.
Below are summarized data from 32 consecutive rational subgroups of size n=5. Analyze the
data and comment on the capability of this process. Again summarize your findings for a
busy executive and include any graphs you feel will help make your case.
Subgroup
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
X
2.991
3.004
2.997
3.004
2.999
3.000
2.988
2.986
2.991
2.994
3.017
2.987
3.013
2.993
3.009
2.990
Range
0.038
0.027
0.049
0.033
0.043
0.070
0.044
0.050
0.064
0.048
0.070
0.051
0.060
0.019
0.044
0.043
Subgroup
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
X
3.005
3.008
3.001
3.000
2.993
2.982
2.990
3.003
3.007
2.998
3.012
2.996
2.997
3.004
2.987
3.002
Range
0.039
0.054
0.009
0.062
0.034
0.012
0.028
0.091
0.049
0.052
0.077
0.040
0.018
0.020
0.063
0.028