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

$\backslash$table$[[$Sample$,$Observation,Mean,Range$],[$No$.,1,2,3],[1,2\mathrm{.}76,272,2\mathrm{.}77,,],[2,2\mathrm{.}67,2\mathrm{.}72,2\mathrm{.}72,,],[3,281,2\mathrm{.}75,2\mathrm{.}75,,],[4,276,2\mathrm{.}86,2\mathrm{.}86,,],[5,2\mathrm{.}76,2\mathrm{.}76,2\mathrm{.}76,,],[6,2\mathrm{.}86,2\mathrm{.}86,2\mathrm{.}86,,],[7,2\mathrm{.}78,2\mathrm{.}75,2\mathrm{.}75,,]]$

The mean and standard deviation of the above $21$ observations are $2\mathrm{.}7757ml$ 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 $C_{px}.$ Is the process capable? Why?

c$.$ Using the $A_2$ value $($in factors table$),$ determine the upper and lower limits for sample matan $($X$-$bax$)$ contol char Is the process in coutrol?