Consider $355$ milli$-$liter $(ml)$ bottles of Stroh beer. The $CO2$ content in a bottle is specified by the brew master to be at least $2\mathrm{.}50ml.$ Seven samples of $3$ bottles were collected from the bottling process, and their $CO2$ content $($in ml$)$ was measured:

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

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