Step $3$: Monitoring the process

$5.$ Subsequently, samples of $4$ bottles were taken from the filling line. The fillings were measured, and the following results were obtained. Use this data to check for process control. Is the process in control? Why?

$\backslash$table$[[$Sample$,$Units,Sample stats,Hypothesis conclusions$],[1,2,3,4,$X$-$bar,R$,\backslash$table$[[$Decision$],[$about the$],[$X$-$bar chart$]],\backslash$table$[[$Decision$],[$about the$],[R$ chart$]],\backslash$table$[[$Decision about$],[$the process$],[($in control or$],[$out of control$)]]],[1,12\mathrm{.}00,11\mathrm{.}97,12\mathrm{.}1,12\mathrm{.}08,,,,,],[2,11\mathrm{.}91,11\mathrm{.}94,12\mathrm{.}1,11\mathrm{.}96,,,,,],[3,12\mathrm{.}10,11\mathrm{.}7,12\mathrm{.}0,12\mathrm{.}10,,,,,]]$

Concepts:

$6.$ What is the possible error type committed in drawing conclusions drawn for the $3$ samples?

Sample I:

Sample $2$:

Sample $3$:

$7.$ What causes of variation $($common or assignable$)$ are present in the process now, after sample $3?($Based on your answer to #$3$ above.$)$

$8.$ What should the manager's next step$($s$)$ be$?$

Page $2$

The production manager at Sunny Soda, Inc. is interested in tracking the quality of the company's $12-oz$ bottle filling line. The manager collected $15$ samples of $4$ bottles each and calculated the overall mean filling was $12\mathrm{.}0095oz$ and the overall range was $0\mathrm{.}1267oz.($This is based on raw sample data. I have performed the calculations for the $x-$bar and $R$ to save time. See problem $24$ at the end of the chapter.$)$