Question $2.(50$ points$)$

Telstar Appliance Company uses a process to paint refrigerators with a coat of enamel. During each shift, a sample of $5$ refrigerators is selected $(1\mathrm{.}4$ hours apart$)$ and the thickness of the paint $($in mm$)$ is determined. If the enamel is too thin, it will not provide enough protection. If it's too thick, it will result in an uneven appearance with running and wasted paint. The following table below lists the measurements from $20$ consecutive shifts. Use the Excel Software or Minitab software to find the X bar$-$R chart and X bar$-$S chart. Upload your excel or Minitab files on brightspace and attach their hard copies to your final exam. Put answers in this document as well.

a$)$ Construct the X bar$-$R charts. Are there any points outside the control limits$?$ Please discuss your result. $(25$ points$)$

b$)$ Construct the X bar$-$S charts. Are there any points outside the control limits$?$ Please discuss your result. $(25$ points$)\backslash$begin$\{$tabular$\}\{|$c$|$c$|\}$

$\backslash$hline Subgroup $($Shift & Thickness $($in mm$)\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$.3.\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline & $0$ & & & & $\backslash \backslash$

$\backslash$hline $\backslash$multirow$[$t$]\{2\}\{*\}\{15\}$ & $\backslash$multirow$[$t$]\{2\}\{*\}\{3.\}$ & $\backslash$multirow$[$b$]\{2\}\{*\}\{3\mathrm{.}4\}$ & $\backslash$multirow$[$b$]\{2\}\{*\}\{3\mathrm{.}5\}$ & $\backslash$multirow$[$b$]\{2\}\{*\}\{3\mathrm{.}5\}$ & $\backslash$multirow$[$b$]\{2\}\{*\}\{3\}\backslash \backslash$

$\backslash$hline & & & & & $\backslash \backslash$

$\backslash$hline $\backslash$multirow$[$t$]\{2\}\{*\}\{16\}$ & $3.$ & & & & $\backslash \backslash$

$\backslash$hline & $4$ & $3\mathrm{.}6$ & $3\mathrm{.}6$ & $3\mathrm{.}8$ & $3\mathrm{.}1\backslash \backslash$

$\backslash$hline $\backslash$multirow$[$t$]\{2\}\{*\}\{17\}$ & & & & & $\backslash \backslash$

$\backslash$hline & & $2\mathrm{.}4$ & $1\mathrm{.}5$ & $1\mathrm{.}3$ & $1\mathrm{.}8\backslash \backslash$

$\backslash$hline $\backslash$multirow$[$t$]\{2\}\{*\}\{18\}$ & $3.$ & & & & $\backslash \backslash$

$\backslash$hline & & $1\mathrm{.}6$ & $1\mathrm{.}6$ & $3\mathrm{.}3$ & $3\mathrm{.}3\backslash \backslash$

$\backslash$hline $\backslash$multirow$[$t$]\{2\}\{*\}\{19\}$ & $2.$ & & & & $\backslash \backslash$

$\backslash$hline & & $2\mathrm{.}6$ & $2\mathrm{.}7$ & $2\mathrm{.}8$ & $3\mathrm{.}2\backslash \backslash$

$\backslash$hline $\backslash$multirow$[$t$]\{2\}\{*\}\{20\}$ & $2.$ & & & & $\backslash \backslash$

$\backslash$hline & & $2\mathrm{.}8$ & $2\mathrm{.}3$ & $2$ & $2\mathrm{.}9\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$