Refer to Table S$6\mathrm{.}1-$ Factors for Computing Control Chart Limits $(3$ sigma$)$ for this problem.

Twelve samples, each containing five parts, were taken from a process that produces steel rods at Emmanual Kodzi's factory. The length of each rod in the samples was determined. The results were tabulated and sample means and ranges were computed. The results were:

$\backslash$table$[[$Sample$,\backslash$table$[[$Sample$],[$Mean $($in$.)]],$Range $($in$.),$Sample,Sample Mean $($in$.)$ Range $($in$.),],[1,8\mathrm{.}602,0\mathrm{.}044,7,8\mathrm{.}601,0\mathrm{.}021],[2,8\mathrm{.}600,0\mathrm{.}051,8,8\mathrm{.}605,0\mathrm{.}058],[3,8\mathrm{.}589,0\mathrm{.}042,9,8\mathrm{.}595,0\mathrm{.}039],[4,8\mathrm{.}606,0\mathrm{.}037,10,8\mathrm{.}603,0\mathrm{.}038],[5,8\mathrm{.}595,0\mathrm{.}048,11,8\mathrm{.}601,0\mathrm{.}054],[6,8\mathrm{.}599,0\mathrm{.}053,12,8\mathrm{.}604,0\mathrm{.}061]]$

For the given data, the $\frac{?}{\text{b}\text{a}\text{r}}(\stackrel{}{x})=8\mathrm{.}6$ inches $($round your response to four decimal places$).$

Based on the sampling done, the control limits for $3-$sigma $\stackrel{}{x}$ chart are:

Upper Control Limit $(UC{L}_{-})=8\mathrm{.}626$ inches $($round your response to four decimal places$).$

Lower Control Limit $(LC{L}_{\frac{?}{\text{b}\text{a}\text{r}}()})=8\mathrm{.}5740$ inches $($round your response to four decimal places$).$

Based on the $\stackrel{}{x}-$chart, is one or more samples beyond the control limits$?$

For the given data, the $\frac{?}{\text{b}\text{a}\text{r}}(R)=\mathrm{.}0455$ inches $($round your response to four decimal places$).$

The control limits for the $3-$sigma R$-$chart are:

Upper Control Limit $(UC{L}_R)=$ inches $($round your response to four decimal places$).$

Definition

$\backslash$table$[[\backslash$table$[[$Sample$],[$Size$,n$