Problem $4(20$ points$)$

In a study to isolate both gauge repeatability and gauge reproducibility, two operators use the same gauge to measure $10$ parts three times each. The data are shown here.

$\backslash$table$[[,$Operator $1,$Operator $2],[$Part$,$Measurements,Measurements,$],[$Number $1,2,3,1,2,3,,,],[1,50,49,50,50,48,51,,],[2,52,52,51,51,51,51,,],[3,53,50,50,54,52,51,,],[4,49,51,50,48,50,51,,],[5,48,49,48,48,49,48,,],[6,52,50,50,52,50,50,,],[7,51,51,51,51,50,50,,],[8,52,50,49,53,48,50,,],[9,50,51,50,51,48,49,,],[10,47,46,49,46,47,48,,]]$

$($a$)$ Estimate gauge repeatability and reproducibility.

$($b$)$ Estimate the standard deviation of measurement error.

$($c$)$ If the specifications are at $50+-5,$ what can you say about gauge capability?