Q$3)$: In an X$-$bar chart for manufacturing steel rivets, the UCL is $5cm$ and the LCL is $3cm.$ The manufacturing process has a standard deviation of $0\mathrm{.}2cm.$

$($a$)$ What is the maximum possible value of the process capability index for this manufacturing process?

$($b$)$ Is this manufacturing process six$-$sigma capable $?$

$($c$)$ Several samples $($each of the same size$)$ were taken. The average rivet size across the samples was $4\mathrm{.}4cm.$ What is the process capability index $?$