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 prob- ability of a single point plotting outside the control Suppose that this chart is being used in phase I and the averages from a set of m samples or subgroups from this process is plotted on this chart. What is the probability that at least one of the averages will plot outside the control limits when ,n 5? Repeat these calculations for the cases where m - 10, m - 20, m-30, and n 50. Discuss the results that you have obtained.

Explanation / Answer

Solution:

Given that p = 0.0027

When m = 5,

P (x 1) = 1 – P (x = 0) = 1 – 5C0 (0.0027)^0 (1 – 0.0027)^5 = 1 – 0.9866 = 0.0134

When m = 10

, P (x 1) = 1 – P (x = 0) = 1 – 10C0 (0.0027)^0 (1 – 0.0027)^10 = 1 – 0.9733 = 0.0267

When m = 20

, P (x 1) = 1 – P (x = 0) = 1 – 20C0 (0.0027)^0 (1 – 0.0027)^20 = 1 – 0.9474 = 0.0526

when m = 30

p (x 1) = 1 - P (x = 0) = 1 - 30C0 (0.0027)^0 (1 – 0.0027)^30 = 1 - 0.9221 = 0.0779

WHEN M= 50

p (x 1) = 1 - P (x = 0) = 1 - 50C0 (0.0027)^0 (1 – 0.0027)^50 = 1 - 0.8736 = 0.1264

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