Question $2(15\%)$:

The operations manager of Green Apple Juice Co$.$ is concerned about the quality of the company$$s $8-$oz

bottle filling line and will track it$.$ The bottles need to be filled according to the specified tolerances for this

product because the nutritional details on the label indicate that the serving size is $8$ ounces. The product's

design standard specifies a fill level of $8\mathrm{.}00$ ounces, with a tolerance of $\backslash$pm $0\mathrm{.}067$ ounces. The manager

gathered nine observations and obtained the next samples data during the production process:

Sample

Fluid oz per bottle

Average Range

$18\mathrm{.}030\mathrm{.}09$

$27\mathrm{.}990\mathrm{.}13$

$37\mathrm{.}980\mathrm{.}09$

$48\mathrm{.}030\mathrm{.}10$

$58\mathrm{.}030\mathrm{.}13$

$68\mathrm{.}010\mathrm{.}09$

$78\mathrm{.}020\mathrm{.}06$

$88\mathrm{.}000\mathrm{.}06$

$97\mathrm{.}990\mathrm{.}05$

$107\mathrm{.}990\mathrm{.}11$

a$.$ Calculate all the $-$ chart and R$-$chart limits for this process. Is the process in control? Explain.

b$.$ The results for new three samples of $8-$oz bottle juice filling are:

Page $11$ of $12$

Sample

Observation

$123456789$

$17\mathrm{.}987\mathrm{.}937\mathrm{.}938\mathrm{.}027\mathrm{.}957\mathrm{.}967\mathrm{.}957\mathrm{.}988\mathrm{.}00$

$27\mathrm{.}977\mathrm{.}968\mathrm{.}048\mathrm{.}028\mathrm{.}017\mathrm{.}978\mathrm{.}058\mathrm{.}047\mathrm{.}99$

$37\mathrm{.}957\mathrm{.}967\mathrm{.}907\mathrm{.}958\mathrm{.}008\mathrm{.}027\mathrm{.}967\mathrm{.}978\mathrm{.}01$

Is the process in control? Explain.

c$.$ Calculate the process capability ratio and the process capability index. Is the process capable of meeting

the design standard? Explain.

d$.$ What percentage of $8-$oz bottle filling will lie within the control limits obtained in part a$,$ given the mean

product$$s design standard $($process distribution$)?$