Turning. Allan, Robbins, and Wycoff worked with a machine shop that employs a CNC (computer numerically controlled) lathe in the machining of a part for a heavy equipment manufacturer. Some summary statistics for a particular part diameter (x) obtained from 25 subgroups of n = 4 parts turned on the lathe are given in Table 3.15. The units are inches.
(a) Find retrospective control limits for the values (both means and ranges). What do the x and R values indicate about the stability of the turning process?
(b) Suppose that one wishes to apply the four Western Electric alarm rules to the x values. Specify the zones to be used for the mean diameters.
Are any of the rules violated in the first 10 samples? (If you find any violations, say which rule is violated for the first time where.)
(c) Give an estimate of the process short-term standard deviation derived from the ranges (use all 25 subgroups and the assumption that σ is constant over the study period).
(d) Engineering specifications on the diameter in question were in fact 1.1809 ± .005 in. Suppose that over short production runs, diameters can be described as normally distributed and that your estimate of σ from (c), is an appropriate description of the variation seen in short runs. Give an estimate of the best possible fraction of diameters meeting specifications available using this particular lathe.
(e) Make further use of your estimate of σ from (c), and set up control limits that could be used in the future monitoring of the process standard deviation via Shewhart charting of s based on samples of size n = 5.
