$[20$ points$]$ A production manager at Uitra Clean Dishwashing Company is monitoring the quality of the company's production process. There has been concern relative to the quality of the operation in accurately filling the $16$ ounces of dishwashing liquid. The product is designed for a fill level of $16\mathrm{.}00+-0\mathrm{.}30.$ The company collected the following sample data on the production process:

MGMT $3106$: Management Science and Operations Management

$\backslash$table$[[$Sample #$,$Observation $1,$Observation $2,$Observation $3,$Observation $4],[1,16\mathrm{.}40,16\mathrm{.}11,15\mathrm{.}90,15\mathrm{.}78],[2,15\mathrm{.}97,16\mathrm{.}10,16\mathrm{.}20,15\mathrm{.}81],[3,15\mathrm{.}91,16\mathrm{.}00,16\mathrm{.}04,15\mathrm{.}92],[4,16\mathrm{.}20,16\mathrm{.}21,15\mathrm{.}93,15\mathrm{.}95],[5,15\mathrm{.}87,16\mathrm{.}21,16\mathrm{.}34,16\mathrm{.}43],[5,15\mathrm{.}43,15\mathrm{.}49,15\mathrm{.}55,15\mathrm{.}92],[7,16\mathrm{.}43,16\mathrm{.}21,15\mathrm{.}99,16\mathrm{.}00],[8,15\mathrm{.}50,15\mathrm{.}92,16\mathrm{.}12,16\mathrm{.}02],[,16\mathrm{.}13,16\mathrm{.}21,16\mathrm{.}05,16\mathrm{.}01],[10,15\mathrm{.}68,16\mathrm{.}43,16\mathrm{.}20,15\mathrm{.}97]]$

a$)$ Calculate Mean, Average Range, Upper and lower control limits for $x-$Bar and R charts. Show all calculations. $($You may use Excel tools to verify your answers$).$

b$)$ Plot the xBar and R charts. $($You may use Excel tools to plot the charts.$)$

c$)$ Based on the XBar and $R$ charts, discuss whether the process mean and range in statistical control.

d$)$ Do you think this process is capable of meeting the design standard?