a$.$ Assuming that management is willing to use$$ three$-$sigma control$$ limits$,$ and using only the historical information contained in the four$$ samples, show that the call center access time is in statistical control.

Since all the sample ranges are between UCL Subscript Upper Requals

select: $36\mathrm{.}69$

$36\mathrm{.}69$ seconds and LCL Subscript Upper Requals

$$

$0$ seconds and all of the sample means are between UCL Subscript x overbarequals

select: $513\mathrm{.}08$

$513\mathrm{.}08$ seconds and LCL Subscript x overbarequals

select: $483\mathrm{.}92$

$483\mathrm{.}92$seconds$,$ the call center access time is in statistical control. $($Enter your responses rounded to two decimal places$.)$

Part $3$

b$.$ Suppose that the standard deviation of the process distribution is $7\mathrm{.}13.$ If the specifications for the access time are $500$plus or minus$18$seconds$,$ is the process$$ capable? Why or why$$ not? Assume$$ three$-$sigma quality.

A$.$

$$Yes$,$ because Upper C Subscript pk is greater than the critical value of $1\mathrm{.}0.$

B$.$

$$No$,$ because Upper C Subscript pk is less than the critical value of $1\mathrm{.}0.$ This is because the process is not centered correctly.

C$.$

$$No$,$ because Upper C Subscript pk is less than the critical value of $1\mathrm{.}0.$ This is because there are too many times outside of the allowable tolerances.

D$.$

$$No$,$ because Upper C Subscript pk is greater than the critical value of $1\mathrm{.}0.$ This is because there are too many times outside of the allowable tolerances.

E$.$

$$No$,$ because Upper C Subscript pk is greater than the critical value of $1\mathrm{.}0.$ This is because the process variability did not meet the$$ three$-$sigma target.