Simulation is useful to see how long it might take before an out-of-control condition is recognized by an X or R control chart. Proceed as follows:

a. Generate 30 subsamples of size n = 5 each, where each observation comes from a normal distribution with a given mean μ and standard deviation ­. (You can choose μ and σ­.) Then "freeze" these values (with the Copy and PasteSpecial/Values commands), and form X̅ and R control charts based on all 30 subsamples.

b. Below the subsamples in part a, generate 30 more subsamples of size n = 5 each, where each observation comes from a normal distribution with mean μ + k1σ­ and standard deviation k2σ. That is, the mean has shifted by an amount k1σ ­, and the standard deviation has been multiplied by a factor k2σ. The values k1 and k2 should be entered as input parameters that you can change easily. Initially, set k1 = 1.5 and k2 = 1, although you can try other values in a follow-up sensitivity analysis. Do not freeze the observations in these 30 new subsamples. Create new X̅ and R control charts that plot all 60 subsamples but have the same control limits from part a. By pressing the F9 key and/or changing the values in the k1 and k2 input cells, you change the behavior of the control charts.

c. Write up your results. In particular, indicate how long it takes for the control charts to realize that the process is out of control with respect to the original control limits, and how this depends on k1 and k2