Montgomery exercise $7\mathrm{.}12$ in the $7$th edition. A process that produces

titanium forgings for automobile turbocharger wheels is to be controlled through the use of a fraction

nonconforming control chart. Initially, one sample of size $150$ is taken each day for $20$ days, and the

results are shown in the table below

$($a$)$ Establish a control chart for the fraction nonconforming to monitor future production. $($This

means compute the center line and the control limits$.)$

$($b$)$ What is the smallest sample size that could be used for this process and still give a positive lower

control limit on the chart?