Suppose that when a process is in control a variable can be taken to be normally distributed with a mean of $0=10\mathrm{.}0$ and a standard deviation of $\backslash$sigma $=0\mathrm{.}2,$ and that a control chart is to be implemented by plotting the average of n $=4$ observations at each time point.

$($a$)$ What are the center line and control limits of a $3-$sigma control chart that you would construct to monitor this process?

$($b$)$ If a sample average x $=9\mathrm{.}5$ is obtained, should the process be declared to be out of control? What if x $=10\mathrm{.}25?$

$($c$)$ If the process moves out of control so that the variable mean becomes $=10\mathrm{.}15$ with $\backslash$sigma $=0\mathrm{.}2,$ what is the probability that an observation lies outside the control limits$?$ What is the average run length for detecting this change?