IV$.$ Cpk $($Process Capability Index$)$ of capable processes are ones with a ratio$/$index of $1\mathrm{.}33(4$ sigma$)$ or

greater. $(25$ points$)$

You are the process improvement manager and have developed a new machine to cut insoles for the

company's top$-$of$-$the$-$line running shoes. The company's goal is no more than $3\mathrm{.}4$ defects per million.

The insoles cannot be more than $\frac{+}{.}001$ of an inch from the thickness specification of $\mathrm{.}250".$ The new

process has a mean $=\mathrm{.}250$inch and a standard deviation $=\mathrm{.}0005$ inch.

a$)$ Should you replace the existing machine which has a Cpk of $1\mathrm{.}0?$

b$)$ If the insoles can be $\frac{+}{-}{\mathrm{.}002}^{\text{'}}($instead of ${\mathrm{.}001}^{\text{'}\text{'}})$ from the required ${\mathrm{.}250}^{\text{'}\text{'}},$ what is new the Cpk and

would you replace the existing machine?

c$)$ What else can be considered to improve the Cpk$?$