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^bar 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. Reconsider the situation in Question 17. Suppose that the process mean and variance were unknown and had to be estimated from the data available from the m subgroups. What complications would this introduce in the calculations that you performed in Question 17?

Explanation / Answer

required answer

P(X >=1) = 1 - P(X =0)

= 1 - (1-p)^m

Use Binomial Distribution function formula and use m=5,10,..

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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