$(4$ Marks$=1+1+1+1)$

$3.$

The mean and standard deviation of the above $21$ observations are $2\mathrm{.}7757$ ml and $0\mathrm{.}0553$ ml$,$ respectively.

a A histogram of the above $21$ observations is displayed below. Is the process capable? Why? $[$note that there is only one specification limit$.]$

The process is capable as the Mean of the total is higher than $2\mathrm{.}5$ml

b$.$ Calculate the $.$ Is the process capable? Why?

$$LSL $=2\mathrm{.}5,$ there is no USL

mean $=2\mathrm{.}7757$

Standard devation $=0\mathrm{.}0553$

Cpk $=(2\mathrm{.}7757-2\mathrm{.}5)/(3$X$0\mathrm{.}0553)=1\mathrm{.}66$

A process having Cpk greater than $1\mathrm{.}33$ is capable. So this process is capable

c$.$ Using the A$2$ value $($in factors table$),$ determine the upper and lower limits for sample mean $($X$-$bar$)$ control chart. Is the process in control?

d$.$ Using the D$3$ and D$4$ values, determine the upper and lower limits for sample range $($R$)$ control chart.

Is the process in control?

N A$2$ D$3$ D$4$

$21\mathrm{.}8803\mathrm{.}27$

$31\mathrm{.}0202\mathrm{.}57$

$40\mathrm{.}7302\mathrm{.}28$

$50\mathrm{.}5802\mathrm{.}28$

$60\mathrm{.}4802\mathrm{.}00$

$70\mathrm{.}420\mathrm{.}081\mathrm{.}92$

$80\mathrm{.}370\mathrm{.}141\mathrm{.}86$

$90\mathrm{.}340\mathrm{.}181\mathrm{.}82$

$100\mathrm{.}310\mathrm{.}221\mathrm{.}78$

$110\mathrm{.}290\mathrm{.}261\mathrm{.}74$