$($a$)$ Name and explain major cost categories identified by COQ.

$($b$)$ Why is quality at the source better than quality through inspection?

A company produces a part that goes into precision testing equipment. The specifications for the part are $8\mathrm{.}0+-3\mathrm{.}0$ millimeters $(mm).$ Management is concerned about the capability of the process to produce that part. The following data were randomly collected during test runs of the process:

$\backslash$table$[[$Sample$,$Observation $($Millimeters$)],[1,2,3,4,5,6,7,8],[1,9\mathrm{.}1,8\mathrm{.}9,8\mathrm{.}8,9\mathrm{.}2,8\mathrm{.}1,6\mathrm{.}9,9\mathrm{.}3,9\mathrm{.}1],[2,7\mathrm{.}6,8\mathrm{.}0,9\mathrm{.}0,10\mathrm{.}1,7\mathrm{.}9,9\mathrm{.}0,8\mathrm{.}0,8\mathrm{.}8],[3,8\mathrm{.}2,9\mathrm{.}1,8\mathrm{.}2,8\mathrm{.}7,9\mathrm{.}0,7\mathrm{.}0,8\mathrm{.}8,10\mathrm{.}8],[4,8\mathrm{.}2,8\mathrm{.}3,7\mathrm{.}9,7\mathrm{.}5,8\mathrm{.}9,7\mathrm{.}8,10\mathrm{.}1,7\mathrm{.}7],[5,10\mathrm{.}0,8\mathrm{.}1,8\mathrm{.}9,9\mathrm{.}0,9\mathrm{.}3,9\mathrm{.}0,8\mathrm{.}7,10\mathrm{.}0]]$

$($a$)$ Construct an $R-$ and an $\stackrel{}{x}-$chart for this process. Is the process in statistical control?

$($b$)$ Suppose that the standard deviation of the process distribution is $0\mathrm{.}851.$ Is the process capable of producing the part at the three$-$sigma level? Why or why not? If not capable, is the problem variability or the centering of the process?