6-15 The length of industrial filters is a quality characteristic of interest. T
ID: 3043866 • Letter: 6
Question
6-15 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 4mm. (a) Find the warning limits for a control chart for the average length. (b) Find the 3 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?
Explanation / Answer
Here X: “ Length of industrial filter” is a quality characteristic. Given that thirty samples each of size five gave the process average = 110 mm with process standard = 4 mm.
UCL of warning limit =Process Average + 2 Process Standard Deviation
= 110+2*4= 118
LCL of warning limit = Process Average - 2 Process Standard Deviation
= 110-2*4= 102
UCL of 3 limit =Process Average + 3 Process Standard Deviation
= 110+3*4= 122
LCL of 3 limit = Process Average - 3 Process Standard Deviation
= 110-3*4= 98
Probability of Type I error = P[ X > UCL] + P[X < LCL] = 0.0027
P[Z > (112-110)/4] = P[Z > 0.5] =0.691462
Probability that shift in process average is detected on third sample =
Probability that shift is not detected on first, second but detected on third sample= (1-0.691462)2(0.691462) = 0.065824
Probability that shift is not detected on first, but detected on second sample
= (0.691462)(0.691462) = 0.213342
Therefore ARL = 1/0.691462= 1.4462. It means shift in process will be detected on first or second sample only.
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