Question

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?

The following content

Answer 1

(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:

    $(0.9973)^{100}=0.763101$