It is required to establish Upper and Lower Quality Control limits on the weight of a semolina product produced in a Flour Mills Factory. $12$ samples of size $5$ are taken and the following Average Weight values $($grams$)$ as well as Range values $($MaximumMinimum$)$ are determined for each sample.

$\backslash$table$[[,74\mathrm{.}0,75\mathrm{.}5,72\mathrm{.}5,74\mathrm{.}2,72\mathrm{.}5,73\mathrm{.}2,72,73\mathrm{.}0,74\mathrm{.}2,74\mathrm{.}7],[$i: $3.,4\mathrm{.}0,3\mathrm{.}0,3\mathrm{.}5,2\mathrm{.}5,3\mathrm{.}0,4\mathrm{.}0,3\mathrm{.}0,2\mathrm{.}5,2\mathrm{.}0,2\mathrm{.}5]]$

a$)$ Establish $3{}_x$ control limits for $x$ and $R$ charts to be used for future process control. $($Note: Delete the points that are out of control limits and reestablish the limits$)$

b$)$ If the specifications are $75+-3$ grams on the product weight, what would be process capability? What do you think of these specifications as related to the process itself?

c$)$ If the sample size of $5$ is reduced to $3,$ what would be the new control limits for the $\stackrel{}{x}$ chart?

d$)$ What would be the control limits for $n=5$ and type I error of $=0\mathrm{.}005$ for the same data above?