Question $2(15\%)$:

The operations manager of Green Apple Juice Co$.$ is concerned about the quality of the company's $8-0$z 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 $+-0\mathrm{.}067$ ounces. The manager gathered nine observations and obtained the next samples data during the production process:

$\backslash$table$[[$Sample$,$Fluid oz per bottle$],[$Average$,$Range$],[1,8\mathrm{.}03,0\mathrm{.}09],[2,7\mathrm{.}99,0\mathrm{.}13],[3,7\mathrm{.}98,0\mathrm{.}09],[4,8\mathrm{.}03,0\mathrm{.}10],[5,8\mathrm{.}03,0\mathrm{.}13],[6,8\mathrm{.}01,0\mathrm{.}09],[7,8\mathrm{.}02,0\mathrm{.}06],[8,8\mathrm{.}00,0\mathrm{.}06],[9,7\mathrm{.}99,0\mathrm{.}05],[10,7\mathrm{.}99,0\mathrm{.}11]]$

a$.$ Calculate all the $\stackrel{}{x}-$chart and $R-$chart limits for this process. Is the process in control? Explain.

b$.$ The results for new three samples of $8-0z$ bottle juice filling are:

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$\backslash$table$[[$Sample$,$Observation$],[1,2,3,4,5,6,7,8,9,],[1,7\mathrm{.}98,7\mathrm{.}93,7\mathrm{.}93,8\mathrm{.}02,7\mathrm{.}95,7\mathrm{.}96,7\mathrm{.}95,7\mathrm{.}98,8\mathrm{.}00,],[2,7\mathrm{.}97,7\mathrm{.}96,8\mathrm{.}04,8\mathrm{.}02,8\mathrm{.}01,7\mathrm{.}97,8\mathrm{.}05,8\mathrm{.}04,7\mathrm{.}99,],[3,7\mathrm{.}95,7\mathrm{.}96,7\mathrm{.}90,7\mathrm{.}95,8\mathrm{.}00,8\mathrm{.}02,7\mathrm{.}96,7\mathrm{.}97,8\mathrm{.}01,]]$

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 $8-0z$ bottle filling will lie within the control limits obtained in part a$,$ given the mean product's design standard $($process distribution$)?$