$3.$ 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:

Sample Observation Mean Range

No$.123$

$12\mathrm{.}762\mathrm{.}722\mathrm{.}77$

$22\mathrm{.}672\mathrm{.}722\mathrm{.}72$

$32\mathrm{.}812\mathrm{.}752\mathrm{.}75$

$42\mathrm{.}762\mathrm{.}862\mathrm{.}86$

$52\mathrm{.}762\mathrm{.}762\mathrm{.}76$

$62\mathrm{.}862\mathrm{.}862\mathrm{.}86$

$72\mathrm{.}782\mathrm{.}752\mathrm{.}75$

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

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?