A company that manufactures soap bars finds that the bars tend to be underweight if too much air is blown into the soap solution so that its density is too low. It is decided to set up a control chart to monitor the density of the soap solution, and at regular time intervals the soap solution density is measured and plotted on a control chart. Experience indicates that the process is in control when the solution density has a mean value of μ0 = 1250 and a standard deviation of σ = 12. Suppose that it is a reasonable approximation to take the soap solution density as being normally distributed.
(a) What are the center line and control limits of a 3-sigma control chart that you would construct to monitor the soap solution density?
(b) If a randomly sampled solution had a density of 1300 would you take this as evidence that the production process had moved out of control? What if the density is 1210?
(c) If the soap production process moves out of control so that the soap solution density has a mean μ = 1240 with σ = 12, what is the probability that a randomly sampled solution has a density that lies outside the control limits? What is the average run length for detecting this change?