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$.),,],[1,8\mathrm{.}604,0\mathrm{.}033,7,8\mathrm{.}601,0\mathrm{.}041],[2,8\mathrm{.}604,0\mathrm{.}041,8,8\mathrm{.}603,0\mathrm{.}034],[3,8\mathrm{.}589,0\mathrm{.}034,9,8\mathrm{.}593,0\mathrm{.}027],[4,8\mathrm{.}604,0\mathrm{.}051,10,8\mathrm{.}601,0\mathrm{.}029],[5,8\mathrm{.}595,0\mathrm{.}031,11,8\mathrm{.}601,0\mathrm{.}039],[6,8\mathrm{.}599,0\mathrm{.}036,12,8\mathrm{.}608,0\mathrm{.}047]]$

A Screen Shot $2024-02-03$ at $5\mathrm{.}04\mathrm{.}48$ PM

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Use the following equations to set the upper and lower control limits for mean and range charts, based on the average range values:

$UC{L}_{\stackrel{}{x}}\frac{?}{b}=ar(\stackrel{}{x})+\frac{A_2}{b}ar(R),LC{L}_{\stackrel{}{x}}=$max$[0\frac{,}{b}ar(\stackrel{}{x})-A_2(\frac{?}{\text{b}\text{a}\text{r}}(R))]$

$UC{L}_R=\frac{D_4}{b}ar(R),LC{L}_R=$max$[0,D_3(\frac{?}{\text{b}\text{a}\text{r}}(R))],$

where $\frac{?}{\text{b}\text{a}\text{r}}(R)$ is the average range of the samples,

$\stackrel{}{x}$ is the mean of the sample means,

$A_2,D_3,$ and $D_4$ are the values found in Table $6$S$\mathrm{.}1.$

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

For the given sample size, the Mean Factor $(A_2)=0\mathrm{.}577($round your response to three decimal places$).$

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

Upper Control Limit $(UC{L}_{\stackrel{}{x}})=10\mathrm{.}2271$ inches $($round your response to four decimal places$).$

Lower Control Limit $(LC{L}_{\frac{?}{\text{b}\text{a}\text{r}}()})=10\mathrm{.}1745$ inches $($round your response to four decimal places$).$

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