$9-$

A process is in statistical control with $\stackrel{}{x}=39\mathrm{.}7$ and $\frac{?}{\text{b}\text{a}\text{r}}(R)=2\mathrm{.}5.$ The control chart uses a sample size of $n=2.$

Specifications are at $40+-5.$ The quality characteristic is normally distributed.

$n=2$;$,$hat$()\frac{?}{b}=ar(\stackrel{}{x})=39\mathrm{.}7$;$\frac{,}{b}ar(R)=2\mathrm{.}5$;$,$hat$({)}_x\frac{?}{b}=ar\frac{R}{d_2}=\frac{2\mathrm{.}5}{1\mathrm{.}128}=2\mathrm{.}216$

USL$=40+5=45$;$LSL=40-5=35$

$($a$)$ Estimate the potential capability of the process.

$($b$)$ Estimate the actual process capability.

$($c$)$ Calculate and compare the PCRs Cpm and Cpkm$.$

$($d$)$ How much improvement could be made in process performance if the mean could be centered at the

nominal value?