a$)$ Determine the upper and lower control limits of the $\stackrel{}{x}-$chart, where $=3.$

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

$UC{L}_{\stackrel{}{x}}\frac{?}{b}=ar(z){?}_x+\frac{A_2}{b}ar(R)LC{L}_{\stackrel{}{x}}=$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 $6\mathrm{.}5\mathrm{.}1.$

For the given sample size, Mean Factor $(A_2)={0\mathrm{.}419}^{\text{'}}($round your response to three decimal places$).$

For the given sample size, Upper Range $(D_4)={1\mathrm{.}924}^{TT}($round your response to three decimal places$)$