$1.(10$ points$)$ Consider a semiconductor manufacturing process that produces microchips with a

thickness specification of $0\mathrm{.}5\backslash$pm $0\mathrm{.}06$ mm$.$ The process mean and process standard deviation

are $0\mathrm{.}49$ mm and $0\mathrm{.}025$ mm$,$ respectively. However, due to the critical nature of these

microchips, any deviation beyond the specification limits is considered unacceptable.

Calculate the process capability indices C p $,$ C pu $,$ C pl $,$ Cpk $,$ and C pm for this process, and assess

whether the process meets the stringent requirement of producing microchips.

$2.(10$ points$)$ The specification limits for a process are given as $50\backslash$pm $5.$ A random sample of $25$

parts yields x$=48$ and s $=0\mathrm{.}9.$ Determine

a$)$ an estimate of Cpk

b$)$ a $90\%$ confidence interval for Cpk