Example $2$

A process leads to the measures presented in the table below. The standard deviation of the process $$ is ${0\mathrm{.}05}^{\text{'}\text{'}}.$ The target value set by the customer is $7,$ with a tolerance of $+-{0\mathrm{.}2}^{\text{'}\text{'}}.$

a$.$ Please calculate $UC{L}_{R^{}}$ and $LC{L}_{R^{}}($theoretical method$)$

b$.$ Please calculate $UC{L}_x$ and $LC{L}_x($practical method$)$

c$.$ Please calculate $UC{L}_R$ and $LC{L}_R($practical method$)$

d$.$ Is the process in control based on the $R$ values and control limits found in response a$)?$

e$.$ Is the process capable to fulfill customer requirements based on the $\stackrel{}{x}$ values?

f$.$ How much should the standard deviation of the process $$ be if the company wants to work according to Six Sigma quality principles?

g$.$ Based on your response in $f),$ is the current process capable to fulfill the requirements of Six Sigma quality principles?

$\backslash$table$[[$Sample #$,$Observation,$\backslash$bar $($x$),$R$],[1,2,3,4],[1,7\mathrm{.}08,6\mathrm{.}97,7\mathrm{.}11,7\mathrm{.}10,7\mathrm{.}07,0\mathrm{.}14],[2,6\mathrm{.}86,7\mathrm{.}01,7\mathrm{.}15,6\mathrm{.}92,6\mathrm{.}99,0\mathrm{.}29],[3,6\mathrm{.}98,7\mathrm{.}01,6\mathrm{.}91,6\mathrm{.}98,6\mathrm{.}97,0\mathrm{.}10],[4,6\mathrm{.}99,6\mathrm{.}99,7\mathrm{.}05,7\mathrm{.}03,7\mathrm{.}02,0\mathrm{.}06],[5,6\mathrm{.}86,6\mathrm{.}99,6\mathrm{.}91,7\mathrm{.}06,6\mathrm{.}96,0\mathrm{.}20],[,,,\backslash$bar $($x$),7\mathrm{.}00,],[,,,,\backslash$bar $($R$),0\mathrm{.}16]]$