15-12 IMPLEMENTING SPC 647(c) Evaluate the in-control ARL for k3. How, does
this change if k2? What do you think about the
use of 2-sigma limits in practice?
(d) Evaluate the out-of-control ARL for a shift of 1
sigma, given that n5.
16-64. Suppose a Pchart with center line at with k-
sigma control limits is used to control a process. There is
a critical fraction defective pcthat must be detected with
probability 0.50 on the first sample following the shift to
this state. Derive a general formula for the sample size
that should be used on this chart.
16-65. Suppose that a Pchart with center line at and
k-sigma control limits is used to control a process. What
is the smallest sample size that can be used on this control
chart to ensure that the lower control limit is positive?
16-66. A process is controlled by a Pchart using sam-
ples of size 100. The center line on the chart is 0.05.
What is the probability that the control chart detects a
shift to 0.08 on the first sample following the shift?
What is the probability that the shift is detected by at
least the third sample following the shift?
16-67. Consider a process where specifications on a
quality characteristic are 100 15. We know that the stan-
dard deviation of this normally distributed quality charac-
teristic is 5. Where should we center the process to mini-
mize the fraction defective produced? Now suppose the
mean shifts to 105 and we are using a sample size of 4 on
an chart. What is the probability that such a shift will be
detected on the first sample following the shift? What is the
average number of samples until an out-of-control point
occurs? Compare this result to the average number of ob-
servations until a defective occurs (assuming normality).
16-68. The NPControl Chart.An alternative to the
control chart for fraction defective is a control chart based
on the number of defectives, or the NPcontrol chart. The
chart has centerline at n, and the control limits areand the number of defectives for each sample is plotted
on the chart.
(a) Verify that the control limits given above are correct.(b) Apply this control chart to the data in Example 16- 4.
(c) Will this chart always provide results that are equiv-
alent to the usual Pchart?
16-69. The EWMA Control Chart.The exponen-
tially weighted moving average (or EWMA) is defined
as follows:where 0
1, and the starting value of the EWMA at
time t0 is (the process target). An EWMA
control chart is constructed by plotting the Ztvalues on a
chart with center line at 0 and appropriate control limits.
(a) Verify that
(b) Let be , and show that(c) Use the results of part (b) to determine the control
limits for the EWMA chart.
(d) As , the EWMA control chart should perform
like a standard Shewhart chart. Do you agree with
this statement? Why?
(e) As ,the EWMA control chart should perform
like a CUSUM. Provide an argument as to why this
is so.
(f) Apply this procedure to the data in Example 16-2.
16-70. Standardized Control Chart.Consider the P
chart with the usual 3-sigma control limits. Suppose that
we define a new variable:as the quantity to plot on a control chart. It is proposed that
this new chart will have a center line at 0 with the upper
and lower control limits at 3. Verify that this standard-
ized control chart will be equivalent to the original pchart.
16-71. Unequal Sample Sizes.One application of the
standardized control chart introduced in Exercise 16-70
is to allow unequal sample sizes on the control chart.
Provide details concerning how this procedure would be
implemented and illustrate using the following data:ZiPˆiPCP 11 P 2
nS 0XS 1^2 zt
^2
n^ a
2 b 31 11 22 t 4^2 zt V 1 Zt 2E 1 Zt 2 0Z 0 0Zt Xt 11 2 Zt 1LCLnp 32 np 11 p 2UCLnp 32 np 11 p 2pXppMIND-EXPANDING EXERCISESSample, i 12345678910
ni 20 25 20 25 50 30 25 25 25 20
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