Question III: Consider $\stackrel{}{x}$ and $s$ charts with sample size $5.$ After $30$ samples were taken in Phase I, we have

${\displaystyle \underset{i=1}{\overset{30}{}}}{\stackrel{}{x}}_i=6000$ and ${\displaystyle \underset{i=1}{\overset{30}{}}}{s}_i=270.$ Use these numbers to make $\frac{?}{\text{b}\text{a}\text{r}}(\stackrel{}{x})+-$Lhat$({)}_{\stackrel{}{x}}$ and $\frac{?}{\text{b}\text{a}\text{r}}(s)+-$Lhat$({)}_x$ charts so that the ARL

of the $\stackrel{}{x}$ chart equals $500.$ Both charts use the same $L$ value. $(25\%)$

$($a$)$ If there are no assignable causes of variation in the system, compute control limits for the $\stackrel{}{x}$ and $s$ charts.

Hint: ${}^{-1}(0\mathrm{.}001)=-3\mathrm{.}09,{}^{-1}(0\mathrm{.}002)=-2\mathrm{.}88,$ and ${}^{-1}(0\mathrm{.}003)=-2\mathrm{.}75(10\%)$