646 CHAPTER 16 STATISTICAL QUALITY CONTROL(a) Using all the data, find trial control limits for and R
charts. Is the process in control?
(b) Use the trial control limits from part (a) to identify out-of-
control points. If necessary, revise your control limits.
Then, estimate the process standard deviation.
(c) Suppose that the specifications are at 140 2. Using the
results from part (b), what statements can you make about
process capability? Compute estimates of the appropriate
process capability ratios.
(d) To make this process a “6-sigma process,” the variance ^2
would have to be decreased such that PCRk2.0. What
should this new variance value be?
(e) Suppose the mean shifts to 139.7. What is the probability
that this shift will be detected on the next sample? What is
the ARL after the shift?
16-52. A process is controlled by a Pchart using samples of
size 100. The center line on the chart is 0.05.
(a) What is the probability that the control chart detects a shift
to 0.08 on the first sample following the shift?
(b) What is the probability that the control chart does not de-
tect a shift to 0.07 on the first sample following the shift
but does detect it on the second sample?
(c) Suppose that instead of a shift in the mean to 0.07, the
mean shifts to 0.10. Repeat parts (a) and (b).
(d) Compare your answers for a shift to 0.07 and for a shift to
0.10. Explain why they differ. Also, explain why a shift to
0.10 is easier to detect.
16-53. Suppose the average number of defects in a unit is
known to be 8. If the mean number of defects in a unit shifts to
16, what is the probability that it will be detected by the U
chart on the first sample following the shift
(a) if the sample size is n4?
(b) if the sample size is n10?
Use a normal approximation for U.
16-54. Suppose the average number of defects in a unit isX known to be 10. If the mean number of defects in a unit shifts
to 14, what is the probability that it will be detected by the U
chart on the first sample following the shift
(a) if the sample size is n1?
(b) if the sample size is n4?
Use a normal approximation for U.
16-55. Suppose that an control chart with 2-sigma lim-
its is used to control a process. Find the probability that a
false out-of-control signal will be produced on the next sam-
ple. Compare this with the corresponding probability for the
chart with 3-sigma limits and discuss. Comment on when
you would prefer to use 2-sigma limits instead of 3-sigma
limits.
16-56. Consider the control chart with 2-sigma limits in
Exercise 16-50.
(a) Find the probability of no signal on the first sample but a
signal on the second.
(b) What is the probability that there will not be a signal in
three samples?
16-57. Suppose a process has a PCR2, but the mean is
exactly three standard deviations above the upper specifica-
tion limit. What is the probability of making a product outside
the specification limits?
16-58. Consider the hardness measurement data in Exercise
16-9. Set up a CUSUM scheme for this process using 50
and 2, so that K1 and H10. Is the process in control?
16-59. Consider the data in Exercise 16-10. Set up a
CUSUM scheme for this process assuming that 80 is the
process target. Explain how you determined your estimate of
and the CUSUM parameters Kand H.
16-60. Reconsider the data in Exercise 16-12. Construct a
CUSUM control chart for this process using 0 500 as the
process target. Explain how you determined your estimate of
and the CUSUM parameters Hand K.XX16-61. Suppose a process is in control, and 3-sigma
control limits are in use on the chart. Let the mean
shift by 1.5. What is the probability that this shift will
remain undetected for three consecutive samples? What
would its probability be if 2-sigma control limits were
used? The sample size is 4.
16-62. Consider an control chart with k-sigma con-
trol limits. Develop a general expression for the proba-
bility that a point will plot outside the control limits
when the process mean has shifted by units from the
center line.16-63. Suppose that an chart is used to control a
normally distributed process and that samples of size n
are taken every nhours and plotted on the chart, which
has k-sigma limits.
(a) Find a general expression for the expected number
of samples and time that will be taken until a false
action signal is generated.
(b) Suppose that the process mean shifts to an out-of-
control state, say. Find an expres-
sion for the expected number of samples that
will be taken until a false action is generated. 1 0 XXXMIND-EXPANDING EXERCISESc 16 .qxd 5/8/02 9:58 PM Page 646 RK UL 6 RK UL 6:Desktop Folder: