The use of tools like a one sample t test is impressive but the whole debacle would have been revealed immediately if the individuals chart had been the FIRST thing done

Hey Scott-  Another detail that often goes unmentioned is the topic of rational subgroups.  Generally speaking, Cp and CpK are calculated using an Rbar over D2 or an Xbar over D4 approximation of the sample standard distribution.  This makes sense if the data is collected in rational subgroups.  What makes up a rational subgroup?  Not the setting on the statistical software!  They are inherent in the data collection.  If an established sample size is collecteded each time samples are measured rational subgroups are established.  They cannot be "created" after random data is recorded.  

The use of rational subgroups enforces the assumption that the process is under control; variation within subgroups is considered while variation between subgroups is essentially ignored.

The alternate indicies, Pp and PpK should also be considered.  In these indicies the sample standard deviation is the actual value based on individual measurements.  All variation is considered.  One benefit of considering Pp/PpK along with Cp/CpK (which again can only be used if the data is collected in rational subgroups!!) is that individual outliers can be set aside.  When one "Plots the Dots" there may be obvious outliers from otherwise in-control data.  The special causes of the outliers can be sought (and hopefully eliminated!) while the otherwise in control process can be left alone and untampered with.