Part $2$ Calculation questions $(80$ marks$)$

Question $1$: Suppose that a critical dimension of the service quality of Special Product Company is the delivery time of an iWatch to its customer. Periodically, random samples of three delivery times are measured $($in minutes$).$ The results of $4$ samples are in the following table.

$\backslash$table$[[$Samples$,$Time $($minutes$)],[1,503,480,481],[2,532,486,494],[3,484,492,488],[4,497,505,510]]$

a$)$ Please choose the control chart$($s$)$ and use $3$ sigma control limits to develop the upper and lower control limits for the control chart$($s$)$ and judge whether the process is in control or not. $(12$ marks$).$

b$)$ Based on the data in part a$),$ the specification for the delivery time is $465$ minutes to $550$ minutes, is the specification being met, calculate both $C_p$ and $C_{pk}?$ Using the formulation $=\frac{\sqrt[\phantom{2}]{n}}{3}\frac{A_2}{b}ar(R)(n$ is the sample size$)$ to estimate the standard deviation from average range. $(8$ marks$).$