Question $2-$ ARL

$2$ points possible $($graded$)$

We have been using an $\stackrel{}{x}$ chart with $3-$ limits with the parameters:

UCL $=108$

Center Line $=104$

$LCL=100$

$n=5$

After running for some time, the process has changed such that the output is still normally distributed but with

a true mean of $=102$ and a standard deviation of $=8.$

a$)$ What is average run length $($ARL$)$ for this chart with this change? $($NB: your answer should be an integer.$)$

b$)$ The quality loss function $($QLF$)=(x-x^{*}{)}^2,$ i$.$e$.k=1.$

What is the expected quality loss for our process given that the target value is the centerline of the control

chart?

Hint: If you are having trouble with this problem, you may find it helpful to revisit the note on expectation

operators from the readings in week $2.$