$7.-9.$ Donuts for July $4$th $($You have $2$ attempts for this problem$)$

$3$ points possible $($graded$)$

Dunking Donuts in New York's Times Square is preparing its shops for July $4$th$,$ Independence Day. They are introducing a special $$July $4$th Donut$$ with a red, white and blue cream inside $($is that a good marketing idea??$).$

The average and variability of the amount of cream of all produced donuts should be very consistent, and, as the $3-$color$-$cream process is new to the company X$-$bar and R$-$charts are set up to control the process. The following data was collected during June to prepare for the production for the July $4$th weekend. The target is $45\mathrm{.}0$ g$.$

$($We assume for this practice problem that these $10$ samples are sufficient for calculating the control limits$,$ although $25$ are recommended.$)$

From this data, you can calculate the following values:

X$-$bar$-$bar: $44\mathrm{.}8$ g

R$-$bar: $4\mathrm{.}9$ g

Table of Control Chart Constants

What are the upper control limit $($UCL$),$ center line $($CL$)$ and lower control limit $($LCL$)$ for X$-$bar?

UCL$($X$-$bar$)=47\mathrm{.}6$; CL$($X$-$bar$)=45$; LCL$($X$-$bar$)=42\mathrm{.}0$

UCL$($X$-$bar$)=48\mathrm{.}4$; CL$($X$-$bar$)=45$; LCL$($X$-$bar$)=41\mathrm{.}2$

UCL$($X$-$bar$)=47\mathrm{.}6$; CL$($X$-$bar$)=44\mathrm{.}8$; LCL$($X$-$bar$)=42\mathrm{.}0$

UCL$($X$-$bar$)=48\mathrm{.}4$; CL$($X$-$bar$)=44\mathrm{.}8$; LCL$($X$-$bar$)=41\mathrm{.}2$

unanswered

What are the upper control limit $($UCL$),$ center line $($CL$)$ and lower control limit $($LCL$)$ for R$?$

UCL$($R$)=11\mathrm{.}2$; CL$($R$)=4\mathrm{.}9$; LCL$($R$)=0$

UCL$($R$)=10\mathrm{.}3$; CL$($R$)=5\mathrm{.}0$; LCL$($R$)=0$

UCL$($R$)=11\mathrm{.}2$; CL$($R$)=5\mathrm{.}0$; LCL$($R$)=0$

UCL$($R$)=10\mathrm{.}4$; CL$($R$)=4\mathrm{.}9$; LCL$($R$)=0$

unanswered

What can you conclude from this data?

The production line data indicates an out$-$of$-$control sample. An assignable cause should be identified and eliminated.

The production line data is all in control, but the amount of cream should be increased by $0\mathrm{.}2$g

The production line is all in control, no further work needed