A manufacturing process involved in the production of plastic molds for a specific automotive application is under study. To control the mean thickness of this (normally distributed) process with a standard deviation, s = 3.0, chart is used with n = 5, LCL = 14.15, and UCL = 15.85. If the process means shifts to 15.2, find the probability that the shift will be detected on the:
(a) first subsequent sample of 5 observations
(b) second subsequent sample of 5 observations
 Given the following information with regards to ( , S)-chart with n = 4:
-chart: UCL = 510, Center line = 500, and LCL = 490
S-chart: UCL = 12.9709, Center line = 6.5842, and LCL = 0.19753
(a) what fraction of items are nonconforming if specifications call for 501 ± 5?
(b) find the ARL if the process mean shifts to 493.
Please see the attached file.
Here it is given that the process standard deviation s = 3, subgroup size n = 5, LCL = 14.15 and UCL = 15.85
Now the Central Line (mean) is CL =(UCL+LCL)/2 =15
Let denote the process mean. Suppose the process mean shift from 15 to 15.2.
Let and denote the sample mean of first and second subsequent sample of 5 observations.
Now the probability that the shift will be ...
The solution describes the method to find the average run length and probability of detecting a shift in a manufacturing process.