We need to establish the $3$ sigma control limits for a production or manufacturing line. We collect

$n=5$ items at periodic intervals during a production shift. The mean and variance are found to be

normally distributed. After $40$ subgroups have been analyzed, we have

${\displaystyle \underset{i=1}{\overset{i=40}{}}}\stackrel{}{x}=1600$

${\displaystyle \underset{i=1}{\overset{i=40}{}}}{s}_i=50$

$($a$)$Compute the three sigma control limit for the mean and variance.

$($b$)$ If the customer's specification limits are $40+-2\mathrm{.}0,$ what are your conclusions regarding the ability of

your production process to produce items conforming to specifications?