Two decision rules are given here. Assume they apply to a normally distributed quality

characteristic and the sample size is n$=5.$

Rule $1$: The control chart uses the three$-$sigma control limits$.$ If one or more of the next five

samples yield values of the sample average that fall outside the control limits$,$ conclude that the

process is out of control.

Rule $2$: If all of the next five sample averages fall on the same side of the center line, conclude

that the process is out of control.

What is the expected in$-$control average run length $($i$.$e$.$ ARL$0)$ for the control chart under each of

these rules?