A process is normally distributed and in control, with known mean and variance,

Question

A process is normally distributed and in control, with known mean and variance, and the usual three-sigma limits are used on the x control chart, so that the probability of a single point plotting outside the control limits when the process is in control is 0.0027. Suppose that this chart is being used in phase I and the averages from a set of m samples or subgroups from this process are plotted on this chart. What is the probability that at least one of the averages will plot outside the control limits when m=5? Repeat these calculations for the cases where m=10, m=20, m=30, and m=50. Discuss the results that you have obtained.

Explanation / Answer

For m=5;

T: # of units outside the control limits.

T~Binomial(n=5;p=0.0027)

P(T>0)=1-P(T=0) = 1-0.99735=0.0134

For m=10;

P(T>0)=1-P(T=0) = 1-0.997310=0.0267

For m=20;

P(T>0)=1-P(T=0) = 1-0.997320=0.0526

For m=30;

P(T>0)=1-P(T=0) = 1-0.997330=0.0779

For m=50;

P(T>0)=1-P(T=0) = 1-0.997350=0.1264

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