A process has mean 12 and standard deviation 3. The process is monitored by taking samples of size 5 at regular intervals. The process is declared to be out of control if a point plots outside the 3σ control limits on an X̅ chart.

a. If the process mean shifts to 14, what is the average number of samples that will be drawn before the shift is detected on an X̅ chart?

b. An upward shift to what value will be detected with an ARL of 4?

c. If the sample size remains at 5, to what value must the standard deviation be reduced to produce an ARL of 4 when the process mean shifts to 14?

d. If the standard deviation remains at 3, what sample size must be used to produce an ARL no greater than 4 when the process mean shifts to 14?