The image is of a control chart with all samples falling within the bands of +/- 1 sigma and the LCL and UCL lines at 3 sigma

(c) Is it normal for the operation of a control chart? If it is not normal, what is the most likely reason for this situation?

To detect this situation in a control chart, a manufacturer decides to use the following rule on the x-bar chart to decide if their process is in control or not. The rule is

Rule: If at least seven among eight consecutive points lie within mu+sigma_xbar and mu-sigma_xbar, the process is considered as out of control.

Assume the quality characteristic is independent, normally distributed.The process mean is mu and the standard deviation of xbar is sigma_xbar.

Answer the following questions.

(d) What is the probability of 8 consecutive points fall between mu+sigma_xbar and mu-sigma_xbar when the process is in control?

(e) What is the Type I error of this rule?

(f) Given the sample size n=4, if the mean of the quality characteristic shifts one and a half process standard deviation, and remains there during the collection of the next 8 samples, what is the type II error associated with this rule?