Question 4: (Continuation of question 3). Consider the situation in problem 3 but with three-sigma control limits. If the mean of the quality characteristic shifts one standard deviation – that is, goes out of control by one-sigma – and remains there during the collection of the next seven samples, what is the ? risk associated with each decision rule?
Control charts for x and s have been maintained on a process and have exhibited statistical control.
The sample size is n = 7. The control chart parameters are as follows:
x Chart
UCL = 428.20
=
S Chart
UCL = 3.240
Center line 425.00 Center line = 1.765
LCL = 421.80
LCL = 0.035
(a) Estimate the mean and standard deviation of the process.
(b) Estimate the natural tolerance limits for the process.
(c) Assume that the process output is well modeled by a normal distribution. If specifications
are 421 and 428, estimate the fraction nonconforming.
(d) Suppose the process mean shifts to 421.00 while the standard deviation remains constant.
What is the probability of an out-of-control signal occurring on the first sample following
the shift?
(e) For the shift in part (d), what is the probability of detecting the shift by at least the fourth
subsequent sample?Two decision rules are given here. Assume they apply to normally distributed quality
characteristic, the control chart has 2-sigma control limits, and the sample size is n=10.
Rule 1: If one or more of the next eight samples yield values of the sample average that fall outside
the control limits, conclude that the process is out of control.
Rule 2: If all of the next eight sample averages fall on the same side of the center line, conclude
that the process is out of control.
What is the type I error probability for each of these rules?
Assume that mean shift of 8 = 0.5σ and same sample, calculate the type II error for x-bar control
chart with 2-sigma control units for Rule-2A company manufacturing oil seals wants to establish x and Ŕ control charts on the process. There
are 40 preliminary samples of size 10 on the internal diameter of the seal. The summary data (in
mm) are as follows,
1280 and Ŕ = 4.35
(a) What are the three-sigma control limits for й and Ŕ?
(b) Assume that the 40 preliminary samples plot in control on both charts. Estimate the process
standard deviation.
(c) If the process mean shifts by +5 units (mean increases by 5 units), what is the probability
of not detecting this shift on the first subsequent sample?
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