This question is composed of 2 parts so pleas read carefully.

1-

a-Two decision rules are given here. Assume they apply to normally distributed quality characteristic, the control chart has 2-sigma control limits, and the sample size is n=10.

Rule 1: If one or more of the next eight 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 eight sample averages fall on the same side of the center line, conclude that the process is out of control. What is the type I error probability for each of these rules? Assume that mean shift of =0.5 and same sample, calculate the type II error for x-bar control chart with 2-sigma control units for Rule-2

b-Consider the situation in the above problem but with three-sigma control limits. If the mean of the quality characteristic shifts one standard deviation - that is, goes out of control by one-sigma - and remains there during the collection of the next seven samples, what is the risk associated with each decision rule?