The metal body for a spark plug is made by a combination of cold extrusion and machining. The occurrence of surface cracking following the extrusion process has been shown by Pareto diagrams to be responsible for producing virtually all of the defective parts. To identify opportunities for improvement, you have been using a control chart to monitor the process. The probability of a false alarm is 0.0056

a. On average, how many samples will you take before getting an out of control signal?

b. What is the probability that there will be no false alarms in the next 15 samples taken?

c. What is the probability that there will be at least one false alarm in the next 50 samples?

d. In a standard control chart the control limits are 3 standard deviations away from the mean, and the probability of a false alarm is 0.0027. Given that the probability of a false alarm for this chart is .0056, how many standard deviations away from the center line are the control limits?

e. Suppose the process mean shifts such that the probability of detecting the shift is 0.30. What is b?

f. What is the probability of failing to detect the shift by the 9th sample collected?