Assignment $3$

$$6\mathrm{.}7$ Auto pistons at Yongpin Zhou'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:

$\backslash$table$[[$Day$,$Mean $($mm$),$Range $($mm$)],[1,156\mathrm{.}9,4\mathrm{.}2],[2,153\mathrm{.}2,4\mathrm{.}6],[3,153\mathrm{.}6,4\mathrm{.}1],[4,155\mathrm{.}5,5\mathrm{.}0],[5,156\mathrm{.}6,4\mathrm{.}5]]$

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

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

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

d$.$ What are the $UC{l}_R$ and $LC{L}_R$ using $3?$

e$.$ If the true diameter mean should be $155mm$ and you want this as your center $($nominal$)$ line, what are the new $UC{L}_{\frac{?}{\text{b}\text{a}\text{r}}(z)}$ and $LC{L}_{\stackrel{}{x}}?$

f$.$ Plot the chart for c$)$ and d$)$