1. A manufacturing company monitors a critical dimension of its product by sampling 4 items per lot, and then using the sample average result (X-bar) to both evaluate whether the production process is on-target (Target = 0) or not, as well as to make a decision about the quality of the lot for further processing.

Having had training in setting up control charts, they have established control limits for their process at 3*_X-bar. They have sufficient data to characterize their process as follows:

Results are normally distributed

²_BetweenLots = 8 and ²_WithinLots = 8, so

²_{Individual Value(X)} = 8 + 8 = 16 and ²_{Sample Average(X-bar)} = 8 + 8/4 = 10

The lot size is sufficiently large that the covariance between an Individual Value (X) and the Sample Average (X-bar) obtained from the lot is _X,X-bar = ²_BetweenLots = 8.

A process engineer has just come to you with the news that the most recent lot sample average was X-bar = -8. He is concerned that while this is not outside the specs of 10 that this lot will still have too many out-of-spec parts, and asks you for an estimate of this proportion.What answer do you provide?

Comment on the control plan and the practice of comparing the lot sample averages versus specs in order to screen lots (ie, practice here is if sample average is within specs, then continue to process, if outside specs, then rework the lot).