$\backslash$table$[[4\mathrm{.}5,4\mathrm{.}6,4\mathrm{.}5,4\mathrm{.}7],[4\mathrm{.}2,4\mathrm{.}5,4\mathrm{.}6,4\mathrm{.}6],[4\mathrm{.}2,4\mathrm{.}4,4\mathrm{.}4,4\mathrm{.}8],[4\mathrm{.}3,4\mathrm{.}7,4\mathrm{.}4,4\mathrm{.}5],[4\mathrm{.}3,4\mathrm{.}3,4\mathrm{.}6,4\mathrm{.}9]]$

a$.$ Determine the mean of each sample.

b$.$ If the process parameters are unknown, estimate its mean and standard deviation.

c$.$ Estimate the mean and standard deviation of the sampling distribution.

d$.$ What would three$-$sigma control limits for the process be$?$ What alpha risk would they provide?

e$.$ What alpha risk would control limits of $4\mathrm{.}14$ and $4\mathrm{.}86$ provide?

f$.$ Using limits of $4\mathrm{.}14$ and $4\mathrm{.}86,$ are any sample means beyond the control limits$?$ If so$,$ which one$($s$)?$

g$.$ Construct control charts for means and ranges using $$ Table $10\mathrm{.}3.$ Are any samples beyond the control limits$?$ If so$,$ which one$($s$)?$

h$.$ Explain why the control limits are different for means in parts $d$ and $g.$

i$.$ If the process has a known mean of $4\mathrm{.}4$ and a known standard deviation of $.18,$ what would three$-$sigma control limits be for a mean chart? Are any sample means beyond the control limits$?$ If so$,$ which one$($s$)?$