(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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