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

Auto pistons at Wenming Chung's plant in Shanghai are produced in a forging process, and the diameter is a critical factor that must be controlled. From

sample sizes of $10$ pistons produced each day, the mean and the range of this diameter have been as follows:

a$)$ What is the value of $\frac{?}{\text{b}\text{a}\text{r}}(\stackrel{}{x})?$

$\frac{?}{\text{b}\text{a}\text{r}}(\stackrel{}{x})=$

mm $($round your response to two decimal places$).$

b$)$ What is the value of $\frac{?}{\text{b}\text{a}\text{r}}(R)?$

$\frac{?}{\text{b}\text{a}\text{r}}(R)=$

mm $($round your response to two decimal places$).$

c$)$ What are the $UC{L}_{\stackrel{}{x}}$ and $LC{L}_{\stackrel{}{x}}$ using $3-$sigma?

Upper Control Limit $(UC{L}_{\stackrel{}{x}})=156\mathrm{.}53mm($round your response to two decimal places$).$

Lower Control Limit $(LC{L}_{\frac{?}{\text{b}\text{a}\text{r}}(-)})=153\mathrm{.}79mm($round your response to two decimal places$).$

d$)$ What are the $UC{L}_R$ and $LC{L}_R$ using $3-$sigma?

Upper Control Limit $(UC{L}_R)=$

mm $($round your response to two decimal places$).$