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:

MEAN $($MM$)$ RANGE $($MM$).$

$1$ I

I $156\mathrm{.}9$ I $4\mathrm{.}2$ ~$------------------$ ~ $---\_\_.--,-----.-----.....------$~$---------------------------------"\text{'}---\_\_,--$

$2153\mathrm{.}2$ : $4\mathrm{.}6$ i $.------------------------------------$~

i $---.-,----------.-----....-$

$3153\mathrm{.}64\mathrm{.}1------$~$$~$--.-...---$~$\_.,,.,\_.....---...-.$~$--------------\_\_\_\_.,...,.--$~$-----1$

$155\mathrm{.}5$ : $5\mathrm{.}0$ I $...---------$l$---------------1$

$\backslash .\text{'}----------$~ $....!4$

$5156\mathrm{.}614\mathrm{.}5$ i

$-------$r$.$ill!ll$!$S$.\_,$BJI$\_$W$.$:I. llll$!$JI $----$DMWIIS:~ililllllll!la$-$Ui$\text{'}$$i~il$!$l'iCIIIGlli!lllllllfflli'!i $-$~$$ S$.$dl $,$t

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

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

c$)$ What are the UCLx and LCLx$,$ using $3$cr$?$ Plot the data.

d$)$ What are the UCL R and LCL R$,$ using $3$ er$?$ Plot the data.

e$)$ If the true diameter mean should be $155$ mm and you want

this as your center $($nominal$)$ line, what are the new UCL.x and

LCLx$?$ P