1. How will staying with single suppliers benefit a company in the long run? When multiple suppliers are used, where does the emphasis tend to be placed? How can a company be sure that its single-source suppliers will not fail to meet demand?
2. A fastener company is suppling washers to a manufacturer of kitchen appliances who has specified that the outside diameter be inch. Fifty samples of washers show that the process is in good statistical control, the diameters are normally distributed, and that and for the processes are 0.505 and 0.0065 inch, respectively.
(a) What percentage of the washers made by this process are within the specifications? Sketch the appropriate graphs to represent the distribution of the process along with the specifications.
(b) What would happen if the process variability was reduced by about half (i.e., changed to 0.003 inch)?
3. Suppose you take a random sample of 25 from a lot of 1,000 items but later realize that you should have selected a sample of 40 items instead. How do you correct this error? Should you put the 25 items back and draw the required random sample of 40 from the original lot, or should you simply draw an additional random sample of 15 from the 975 remaining items?
4. A study has been initiated on a filling process. The data in the table represent the net weights in pounds (above 1 kg) for one kg bags of chemical material. The samples of represent five consecutive bags produced by filling head 2 on a four-head filling machine. The samples were collected at 30-minute intervals. Ultimately, the goal is to fill the bags as close to 1 kg possible without going under this nominal weight. Regulations allow only 0.1% of the bags to be below this value. The first step in this study was to examine the filling process from a statistical control point of view.
(a) Use the data above to establish and R charts for the process. (Use MINITAB only).
(b) Interpret the charts. If statistical signals/out-of-control conditions are present, assume that the physical cause was identified. Remove the out-of-control subgroups and recalculate the centerlines and control limits. (Use MINITAB only).
(c) Continue until the charts show good statistical control. Comment.
Sample Day Time
Range
1 1 7:30 0.92 1.01 0.95 1.04 0.90 0.964 0.14
2 1 8:00 1.15 1.02 0.98 0.94 0.99 1.016 0.21
3 1 8:30 0.94 0.91 1.00 1.05 0.95 0.970 0.14
4 1 9:00 1.11 0.94 0.89 1.11 1.00 1.010 0.22
5 1 9:30 0.95 0.97 0.97 0.98 0.86 0.946 0.12
6 1 10:00 1.02 0.89 0.97 0.95 0.97 0.960 0.13
7 1 10:30 1.18 0.84 0.95 1.39 1.03 1.078 0.55
8 1 11:30 0.94 1.15 1.07 0.99 1.03 1.036 0.21
9 1 12:00 1.03 1.20 1.00 1.10 1.09 1.084 0.20
10 1 12:30 0.98 0.82 0.98 1.02 1.13 0.986 0.31
11 1 1:00 0.98 0.95 0.97 1.04 0.89 0.966 0.15
12 1 1:30 1.10 1.12 1.01 1.12 1.04 1.078 0.11
13 1 2:00 1.10 0.94 0.88 0.92 0.91 0.950 0.22
14 1 2:30 1.01 0.99 1.11 0.96 1.05 1.024 0.15
15 1 3:00 1.17 1.30 1.21 0.69 0.82 1.038 0.61
16 2 7:30 0.97 1.03 1.09 1.04 0.94 1.014 0.15
17 2 8:00 0.92 0.88 0.83 0.94 0.87 0.888 0.11
18 2 8:30 0.99 1.00 0.95 1.00 0.90 0.968 0.1
19 2 9:00 0.88 1.09 1.05 1.05 1.01 1.016 0.21
20 2 9:30 0.87 1.08 0.99 0.97 1.04 0.990 0.21
21 2 10:00 1.08 0.99 1.18 1.02 1.07 1.068 0.19
22 2 10:30 0.60 1.28 .97 0.84 1.01 0.940 0.68
23 2 11.30 0.89 0.99 1.02 0.95 0.99 0.968 0.13
24 2 12:00 1.01 0.90 0.97 1.09 1.13 1.020 0.23
25 2 12:30 0.95 1.01 1.09 1.10 1.10 1.050 0.15
26 2 1:00 1.10 0.96 1.02 1.03 1.01 1.024 0.14
27 2 1:30 0.92 1.05 1.03 0.99 1.08 1.014 0.16
28 2 2:00 1.00 0.87 1.00 1.05 0.97 0.978 0.18
29 2 2:30 0.96 1.03 1.03 1.11 1.05 1.036 0.15
30 2 3:00 1.15 0.84 1.02 1.18 1.05 1.048 0.34
31 3 7:30 0.91 0.85 0.89 0.82 0.95 0.884 0.13
32 3 8:30 0.95 0.92 0.95 0.84 0.92 0.916 0.11
33 3 8:30 0.98 0.98 1.01 1.12 1.19 1.056 0.21
34 3 9:00 0.89 0.90 1.05 1.05 0.87 0.952 0.18
35 3 9:30 1.16 0.96 0.96 1.06 1.00 1.028 0.20
36 3 10:00 1.10 0.87 0.95 1.05 1.14 1.022 0.27
37 3 10:30 0.83 0.75 1.04 1.25 0.77 0.928 0.50
38 3 11:30 0.98 1.02 1.06 0.87 1.00 0.986 0.19
39 3 12:00 0.95 0.88 0.97 1.01 0.85 0.932 0.16
40 3 12:30 1.04 0.95 1.00 1.14 1.06 1.038 0.19
41 3 1.00 0.98 0.96 1.04 1.09 1.05 1.024 0.13
42 3 1.30 0.80 0.99 0.98 1.03 0.89 0.938 0.23
43 3 2:00 1.04 1.00 0.87 1.02 0.91 0.968 0.17
44 3 2:30 1.04 1.02 0.92 1.00 1.01 0.978 0.10
45 3 4:00 0.71 1.36 1.10 1.24 0.81 1.044 0.65
After these steps were taken a group was formed to study the process further with hopes of reducing the variation in the process. One of the first steps taken was to create a cause-and-effect diagram, as shown.
Using this as a guide, it was determined that the most likely cause of variation centered around the fact that adjustments were made assuming a much smaller particle size than actually was present in the chemical material. Once this common-cause/chronic problem was addressed, the charting was continued with the following data being collected.
Sample Day Time
Range
1 4 7:30 0.97 0.99 1.05 0.98 0.96 0.990 0.09
2 4 8:00 1.02 0.99 1.00 1.00 1.00 1.002 0.03
3 4 8:30 0.96 1.05 0.96 1.02 0.97 0.992 0.09
4 4 9:00 1.05 1.00 1.00 0.98 0.99 1.004 0.07
5 4 9:30 0.98 1.00 0.95 0.97 0.97 0.974 0.05
6 4 10:00 0.98 0.97 0.98 1.04 0.99 0.992 0.07
7 4 10:30 0.99 1.05 1.03 0.99 0.99 1.010 0.06
8 4 11:30 0.95 0.94 0.99 1.03 1.00 0.982 0.09
9 4 12:00 0.97 1.01 1.01 1.01 1.02 1.004 0.05
10 4 12:30 1.02 0.99 0.97 0.99 1.02 0.998 0.05
11 4 1:00 0.99 1.00 1.01 1.05 1.02 1.014 0.06
12 4 1:30 0.99 0.99 1.00 0.98 1.01 0.994 0.03
13 4 2:00 1.02 1.00 1.01 0.99 0.97 0.998 0.05
14 4 2:30 0.99 0.94 0.98 0.99 0.95 0.970 0.05
15 4 3:00 1.04 1.00 1.01 0.98 0.98 1.002 0.06
(d) Plot these new data in the chart that you set up in part ( c ). Is this new chart in control?
5. In casting a certain type of flywheel, it was observed that many parts were defective and being scrapped because of hot tears. Hot tears are voids in the cast metal that are likely to occur at weak points in the shape as shown below.
The data shown in the table summarize the number of parts produced and the number of defectives observed for 25 consecutive days of operation for the process.
(a) Construct a P chart for these data using MINITAB.
(b) Comment on the appearance of the P-chart and note any statistical signal that
appears. If necessary revise the trial limits (use the appropriate subcommand.)
Sample Number, i Sample Size
Number of Nonconforming
Units,
1 100 12
2 80 8
3 80 6
4 100 9
5 110 10
6 110 12
7 100 11
8 100 16
9 90 10
10 90 6
11 110 20
12 120 15
13 120 9
14 120 8
15 110 6
16 80 8
17 80 10
18 80 7
19 90 5
20 100 8
21 100 5
22 100 8
23 100 10
24 90 6
25 90 9
6. Use Cusum and EWMA charts to control the data given in question 1. For Cusum use h = 5 and k = 0.5 and for EWMA use = .05 and 0.9.
(a) Do the charts indicate that the process is in control?
(b) Interpret the charts and comment on the patterns and / or trend you see.
( c ) Compare the performance of Cusum , X-bar and EWMA with different .
7. In a railroad-track cross section a configuration was tested to determine the
resistant to breakage under used conditions. 10 miles of track was laid in 25 locations.
X-bar and R charts are being used to monitor the resistant to breakage. On each location 4 pieces of one mile track is selected at random (sample size is 4) and the number of cracks and other fracture related conditions (x) was measured over a two year usage period. However a new operator inadvertently has begun to use a sample of size 5.
(a) What impact does this have on the ongoing charting process?
(b) What could be the consequences if the operator had inadvertently changed to a sample
size of n=3?