An $\bar { x }$ chart for a normally distributed random variable has control limits set at $\pm$3 standard errors from the centerline.One sample mean falls outside the control limits when the process is actually under control.The probability that this happened by chance is:
A) .997
B) .500
C) .050
D) .003

Question

Quizplus · Accepted Answer

The probability of a sample mean falling outside the control limits when the process is under control is given by the standard normal distribution. Since the sample mean fell outside the control limits, we can calculate the z-score as (sample mean - centerline) / (standard error). This z-score can then be looked up in a standard normal distribution table to find the probability of it occurring. Since the control limits are set at 3 standard errors from the centerline, the z-score in this case is ((3+1) standard errors from the centerline) = 4 standard errors. Looking this up in a standard normal distribution table gives a probability of approximately 0.003. Therefore, the answer is D.

An xˉ\bar { x }xˉ Chart for a Normally Distributed Random Variable Has Control Limits

An $\bar { x }$ Chart for a Normally Distributed Random Variable Has Control Limits