$(4$ Marks $=1111)$

$3.$ Consider $355$ milli$-$liter $($ ml $)$ bottles of Stroh beer. The CO $2$ content in a bottle is specified by the brew master to be at least $2\mathrm{.}50$ ml $.$ Seven samples of $3$ bottles were collected from the bottling process, and their CO $2$ content $($in ml$)$ was measured:

$\backslash$table$[[$Sample$,$Observation,Mean$],[$Range$,,,,],[$No$.,1,2,3,,],[1,2\mathrm{.}76,2\mathrm{.}72,2\mathrm{.}77,,],[2,2\mathrm{.}67,2\mathrm{.}72,2\mathrm{.}72,,],[3,2\mathrm{.}81,2\mathrm{.}75,2\mathrm{.}75,,],[4,2\mathrm{.}76,2\mathrm{.}86,2\mathrm{.}86,,],[5,2\mathrm{.}76,2\mathrm{.}76,2\mathrm{.}76,,],[6,2\mathrm{.}86,2\mathrm{.}86,2\mathrm{.}86,,],[7,2\mathrm{.}78,2\mathrm{.}75,2\mathrm{.}75,,]]$

The mean and standard deviation of the above $21$ observations are $2\mathrm{.}7757$ ml and $0\mathrm{.}0553$ ml $,$ respectively.

a$.$ A histogram of the above $21$ observations is displayed below. Is the process capable? Why? $[$note that there is only one specification limit$.]$

b$.$ Calculate the $C_{PK}.$ Is the process capable? Why?

c$.$ Using the A$2$ value $($in factors table$),$ determine the upper and lower limits for sample mean $($ X $-$bar$)$ control chart. Is the process in control?

d$.$ Using the D$3$ and D$4$ values, determine the upper and lower limits for sample range $($R$)$ control chart. Is the process in control?

$\backslash$table$[[n,A2,D3,D4$