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:

$$

Day

Mean x

$($mm$)$

Range R

$($mm$)$

$1$

$156\mathrm{.}9$

$4\mathrm{.}0$

$2$

$151\mathrm{.}2$

$4\mathrm{.}6$

$3$

$153\mathrm{.}6$

$4\mathrm{.}1$

$4$

$157\mathrm{.}5$

$5\mathrm{.}2$

$5$

$156\mathrm{.}6$

$4\mathrm{.}5$

Part $2$

$$a$)$ What is the value of x$?$

x$=155\mathrm{.}16155\mathrm{.}16$ mm $($round your response to two decimal$$ places$).$

Part $3$

$$b$)$ What is the value of R$?$

R$=4\mathrm{.}484\mathrm{.}48$ mm $($round your response to two decimal$$ places$).$

Part $4$

$$c$)$ What are the UCLx and LCLx using $3-$sigma$?$

Upper Control Limit $($UCLx$)=156\mathrm{.}54156\mathrm{.}54$ mm $($round your response to two decimal$$ places$).$

Part $5$

Lower Control Limit $($LCLx$)=153\mathrm{.}78153\mathrm{.}78$ mm $($round your response to two decimal$$ places$).$

Part $6$

$$d$)$ What are the UCLR and LCLR using $3-$sigma$?$

Upper Control Limit $($UCLR$)=7\mathrm{.}967\mathrm{.}96$ mm $($round your response to two decimal$$ places$).$

Part $7$

Lower Control Limit $($LCLR$)=1\mathrm{.}001\mathrm{.}00$ mm $($round your response to two decimal$$ places$).$

Part $8$

$$e$)$ If the true diameter mean should be $155$ mm and you want this as your center$($nominal$)$ line, what are the new UCLx and LCLx$?$

Upper Control Limit $($UCLx$)=156\mathrm{.}38156\mathrm{.}38$ mm $($round your response to two decimal$$ places$).$

Part $9$

Lower Control Limit $($LCLx$)=$ enter your response here mm $($round your response to two decimal$$ places$).$