Answer to Question #187902 in Statistics and Probability for Mayondi Chikwama

Question #187902
An X-bar control chart monitors the mean of a process by checking that the average stays between control.pngand control2.png. When the process is under control,
(a) What is the probability that five consecutive sample means of n cases stay within these limits?

(b) What is the probability that all of the means for 100 days falls within the control limits?

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Expert's answer
2021-05-07T11:35:48-0400

(a)

The choice "\\alpha=0.0027" puts the control limits at "Z_{\\frac{\\alpha}{2}}" standard errors from the process target. The probability that a sample mean goes out of the limit is "0.0027" . This Gives the probability that a sample mean stay within these limits is "1-0.0027=0.9973."  


Therefore, the probability that five consecutive sample means of n cases stay within these limit is:

    "(0.9973)^5=0.986573"


(b) The choice "\\alpha=0.0027" puts the control limits at "Z_{\\frac{\\alpha}{2}}" standard errors from the process target. The probability that a sample mean goes out of the limit is "0.0027." This Gives the probability that a sample mean stay within these limits is "1-0.0027=0.9973."


Therefore, the probability that all 100 sample means of 100 days (cases) stay within these limit is:

    "(0.9973)^{100}=0.763101"


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