Two decision rules are given here. Assume they apply to normally distributed qua

Question

Two decision rules are given here. Assume they apply to normally distributed quality characteristic, the control chart has three-sigma control limits, and the sample size is n=5.

Rule 1: If one or more of the next eight samples yield values of the sample average that fall outside the control limits, conclude that the process is out of control.

Rule 2: If all of the next eight sample averages fall on the same side of the center line, conclude that the process is out of control.

If the mean of the quality characteristic shifts one standard deviation - that is, goes out of control by one-sigma - and remains there during the collection of the next seven samples, what is the type II error associated with each decision rule?

Explanation / Answer

Solution:

We can analyze Rule I with the help of the binomial distribution. In this context, we are to perform, seven trials and each one of them has a probability a to generate an alarm. The probability that no alarms will be generated is then (1-)^7 The probability that one or more alarms will be generated is
1 - (1-)^7 = 1 - (1-0.0027)^7= 0.0187.

We can analyze Rule 2 with the help of the binomial distribution as well. The probability that all seven points vrill be on the same side of the centerline is 0.5^6 (since we do not care on which side of the centerline the first point is, as long as the remaining six points are on the same side. The probability that an alarm will be generated is then 0.5^6 = 0.0156.

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