Suppose that a process with a normally distributed output has a mean of $55$ and a variance of $4.$

a$.$ If the specifications are $55\mathrm{.}0\backslash$pm $4\mathrm{.}00,$ compute Cp and Cpk $($both Cpu and Cpl$).$

b$.$ Suppose the mean shifts to $53\mathrm{.}0$ but the variance remains unchanged. Recompute and interpret these process capability indices.

c$.$ If the variance can be reduced to $40\%$ of its original value, how do the process capability indices change $($using the original mean of $55)?$ What can be done to have the process run at six sigma level?

Variable Control Chart Constants

Sample size X$-$bar chart R$-$chart

R$-$chart

n A$2$ D$3$ D$4$

$21\mathrm{.}88003\mathrm{.}267$

$31\mathrm{.}02302\mathrm{.}574$

$40\mathrm{.}72902\mathrm{.}282$

$50\mathrm{.}57702\mathrm{.}114$

$60\mathrm{.}48302\mathrm{.}004$

$70\mathrm{.}4190\mathrm{.}0761\mathrm{.}924$

$80\mathrm{.}3730\mathrm{.}1361\mathrm{.}864$

$90\mathrm{.}3370\mathrm{.}1841\mathrm{.}816$

$100\mathrm{.}3080\mathrm{.}2231\mathrm{.}777$

$120\mathrm{.}2660\mathrm{.}2831\mathrm{.}717$

$140\mathrm{.}2350\mathrm{.}3281\mathrm{.}672$

$160\mathrm{.}2120\mathrm{.}3631\mathrm{.}637$

$2801940\mathrm{.}3911\mathrm{.}608$

$200\mathrm{.}1800\mathrm{.}4151\mathrm{.}585$

$220\mathrm{.}1670\mathrm{.}4341\mathrm{.}566$

$240\mathrm{.}1570\mathrm{.}4511\mathrm{.}548$