The length of industrial filters is a quality characteristic of interest. Thirty samples, each of size 5, are chosen from the process. The data yield an average length of 110 mm, with the process standard deviation estimated to be 4 mm.
(a) Find the warning limits for a control chart for the average length.
(b) Find the 3σ control limits. What is the probability of a type I error?
(c) If the process mean shifts to 112 mm, what are the chances of detecting this shift by the third sample drawn after the shift?
(d) What is the chance of detecting the shift for the first time on the second sample point drawn after the shift?
(e) What is the ARL for a shift in the process mean to 112 mm? How many samples, on average, would it take to detect a change in the process mean to 116 mm?