Que $2$: Quality Control in Door Panel Fitting $($See Notes $8)$

An automobile manufacturer has received customer complaints about uneven gaps in door panels, leading to improper sealing and wind noise at high speeds. The production manager suspects that a key cutting operation might be the cause of these issues.

The target specification for this cutting operation is $30\mathrm{.}000$ inches, with an allowable tolerance of $+-0\mathrm{.}020$ inches.

To investigate, the production manager decides to inspect three consecutive door panels at the start of each shift for a period of $5$ days. The dimensions of the cuts are recorded as follows:

$\backslash$table$[[$Day$,$Shift,Panel $1($inches$),$Panel $2($inches$),$Panel $3($inches$)],[1,1,30\mathrm{.}012,30\mathrm{.}008,30\mathrm{.}004],[1,2,29\mathrm{.}998,30\mathrm{.}005,30\mathrm{.}01],[2,1,30\mathrm{.}015,30\mathrm{.}003,29\mathrm{.}995],[2,2,30\mathrm{.}008,30\mathrm{.}002,30\mathrm{.}007],[3,1,30,29\mathrm{.}996,30\mathrm{.}013],[4,1,29\mathrm{.}994,30\mathrm{.}006,30\mathrm{.}009],[5,2,30\mathrm{.}01,30\mathrm{.}004,30\mathrm{.}002]]$

a$)$ Calculate the mean and range of the cutting dimensions for each sample.

b$)$ Determine the control limits for $\stackrel{}{x}-$Chart and R$-$Chart, respectively. $($Use $A_2=0\mathrm{.}255$; $D_3=0\mathrm{.}0001$ and $D_4=1\mathrm{.}85)$