Tablet hardness. Exercise concerns process control data on the hardness of tablets (measured in kilograms) for a pharmaceutical product. Table 27.4 gives data for 20 new samples of size 4, with the 
and s for each sample. The process has been in control with mean at the target value µ= 11.5 kg and standard deviation σ= 0.2 kg. HARDNESS2
(a) Make both and s charts for these data based on the information given about the process.
(b) At some point, the within-sample process variation increased from σ= 0.2 kg to σ= 0.4 kg. About where in the 20 samples did this happen? What is the effect on the s chart? On the x chart?
(c) At that same point, the process mean changed from µ= 11.5 kg to µ= 11.7 kg. What is the effect of this change on the s chart? On the x chart?
Exercise 27.11:
Tablet hardness. A pharmaceutical manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The hardness of a sample from each lot of tablets is measured in order to control the compression process. The process has been operating in control with mean at the target value µ= 11.5 kilograms (kg) and estimated standard deviation σ=0.2 kg. Table 27.2 gives three sets of data, each representing for 20 successive samples of n =4 tablets. One set remains in control at the target value. In a second set, the process mean shifts suddenly to a new value. In a third, the process mean drifts gradually.
(a) What are the center line and control limits for an chart for this process?
(b) Draw a separate chart for each of the three data sets. Mark any points that are beyond the control limits.
(c) Based on your work in (b) and the appearance of the control charts, which set of data comes from a process that is in control? In which case does the process mean shift suddenly, and at about which sample do you think that the mean changed? Finally, in which case does the mean drift gradually?
