16-12 IMPLEMENTING SPC 645(a) Suppose that a shift in the process mean of magnitude
occurs. Verify that the ARL for detecting the shift is
ARL 43.9.
(b) Find the ARL for detecting a shift of magnitude 2in the
process mean.
(c) Find the ARL for detecting a shift of magnitude 3in the
process mean.
(d) Compare your answers to parts (a), (b), and (c) and ex-
plain why the ARL for detection is decreasing as the mag-
nitude of the shift increases.
16-50. Consider a control chart for individuals, applied to a
continuous 24-hour chemical process with observations taken
every hour.
(a) If the chart has 3-sigma limits, verify that the in-control
ARL is ARL 370. How many false alarms would occur
each 30-day month, on the average, with this chart?
(b) Suppose that the chart has 2-sigma limits. Does this re-
duce the ARL for detecting a shift in the mean of magni-
tude ? (Recall that the ARL for detecting this shift with
3-sigma limits is 43.9.)
(c) Find the in-control ARL if 2-sigma limits are used on the
chart. How many false alarms would occur each month
with this chart? Is this in-control ARL performance satis-
factory? Explain your answer.
16-51. The depth of a keyway is an important part quality
characteristic. Samples of size n5 are taken every four hours
from the process and 20 samples are summarized as follows:16-46. Cover cases for a personal computer are manufac-
tured by injection molding. Samples of five cases are taken
from the process periodically, and the number of defects is
noted. Twenty-five samples follow:Sample r
1 139.7 1.1
2 139.8 1.4
3 140.0 1.3
4 140.1 1.6
5 139.8 0.9
6 139.9 1.0
7 139.7 1.4
8 140.2 1.2
9 139.3 1.1
10 140.7 1.0
11 138.4 0.8
12 138.5 0.9
13 137.9 1.2
14 138.5 1.1
15 140.8 1.0
16 140.5 1.3
17 139.4 1.4
18 139.9 1.0
19 137.5 1.5
20 139.2 1.3X(a) Using all the data, find trial control limits for this Uchart
for the process.
(b) Use the trial control limits from part (a) to identify out-of-
control points. If necessary, revise your control limits.
(c) Suppose that instead of samples of 5 cases, the sample
size was 10. Repeat parts (a) and (b). Explain how this
change alters your answers to parts (a) and (b).
16-47. Consider the data in Exercise 16-46.
(a) Using all the data, find trial control limits for a Cchart for
this process.
(b) Use the trial control limits of part (a) to identify out-of-
control points. If necessary, revise your control limits.
(c) Suppose that instead of samples of 5 cases, the sample
was 10 cases. Repeat parts (a) and (b). Explain how this
alters your answers to parts (a) and (b).
16-48. Suppose that a process is in control and an chart is
used with a sample size of 4 to monitor the process. Suddenly
there is a mean shift of 1.5.
(a) If 3-sigma control limits are in use on the chart, what is
the probability that this shift will remain undetected for
three consecutive samples?
(b) If 2-sigma control limits are in use on the chart, what is
the probability that this shift will remain undetected for
three consecutive samples?
(c) Compare your answers to parts (a) and (b) and explain
why they differ. Also, which limits you would recommend
using and why?
16-49. Consider the control chart for individuals with
3-sigma limits.XXXSample No. of Defects Sample No. of Defects
1 3 14 8
2 2 15 0
3 0 16 2
4 1 17 4
5 4 18 3
6 3 19 5
7 2 20 0
8 4 21 2
9 1 22 1
10 0 23 9
11 2 24 3
12 3 25 2
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