A production process, sampled $30$ times with seven items per sample, resulted in these $x-$bar$-$bar and $R$ values: $x-$bar$-$bar $=28\mathrm{.}5$ inches and R$-$bar $=1\mathrm{.}6$ inches.

a$.$ Calculate the $x-$bar and $R$ limits for this process $($round to $2$ decimals$).$

$UC{L}_{\stackrel{}{x}}=$

$LC{L}_{\stackrel{}{x}}=$

$UC{L}_R=$

$LC{L}_R=$

b$.$ Does the following set of run test sample means indicate that the process is in control or out of control: $28\mathrm{.}01,28\mathrm{.}8,28\mathrm{.}1,28\mathrm{.}9,28\mathrm{.}01,$ and $29\mathrm{.}01?$ Demonstrate and explain why or why not, do not just answer yes or no$.$

Process is $$ control for the Mean $($x$-$bar chart$).($answer options: in or out of$)$

Why? $($answer options: no rules violated; violates Rule $1,$ violates Rule $2,$ violates Rule $3,$ violates Rule $4,$ violates Rule $5,$ or violates Rule $6)-$ if it violates multiple, pick the first rule violated in time order

c$.$ After the run test above, a new run test was conducted with nine samples of the item. The run test, produced these sample means: $28\mathrm{.}55,28\mathrm{.}65,29\mathrm{.}00,28\mathrm{.}72,28\mathrm{.}9,28\mathrm{.}6,28\mathrm{.}5,28\mathrm{.}65,$ and $28\mathrm{.}9.$ Is the process in or out of control for the mean? Demonstrate and explain why or why not, do not just answer yes or no$.$

Process is $$ control for the Mean $(x-$bar chart$).($answer options: in or out of$)$

Why? $($answer options: no rules violated; violates Rule $1,$ violates Rule $2,$ violates Rule $3,$ violates Rule $4,$ violates Rule $5,$ or violates Rule $6)-$ if it violates multiple, pick the first rule violated in time order

d$.$ Is this process capable of meeting customer specifications of $28+-0\mathrm{.}5$ inches? Demonstrate and explain why or why not, do not just answer yes or no$.$

Process is $$ of meeting the customer's specifications. $($answer options: capable or not capable$)$

Why? $($answer options: both specs are inside the control limits; both specs are outside the control limits; one spec is inside and one spec is outside the control limits$)$