Process quality $(15+5+5$ points$)$:

A production system has a x$-$bar chart of UCL $=74\mathrm{.}0135,$ LCL $=73\mathrm{.}9865$ and x$-$double bar $=74\mathrm{.}0000.$ Let the product have a normally distributed measure and the sample size is $5.$

$($a$)$ Find the warning level at a $2-$sigma distance.

b$.)$What is the probability that a single sample will fall outside the warning level?

c$.)$What is the Average Run length before a warning signal is hoisted?