$1.$ The quality inspector took observations, each with four samples, of the time for the glue to dry. The analyst computed the mean of each sample and then computed the grand mean. All values are in minutes. Use this information to obtain the process's upper control limit and lower control lime$.$

Sample$/$Observation S$1$ S$2$ S$3$ S$4$ S$5$ X$-$bar R

$113\mathrm{.}1112\mathrm{.}1513\mathrm{.}0911\mathrm{.}1215\mathrm{.}09=$

$211\mathrm{.}1011\mathrm{.}1212\mathrm{.}0913\mathrm{.}1014\mathrm{.}14$

$314\mathrm{.}1110\mathrm{.}1014\mathrm{.}1514\mathrm{.}0813\mathrm{.}13$

$411\mathrm{.}0814\mathrm{.}1113\mathrm{.}1115\mathrm{.}1012\mathrm{.}12$

a$)$ Determining control limits for means

Double X bar$=?$

R$-$bar

UCL$=$ double X bar $+($ R$-$bar$*$A$2)$

LCL $=$double X bar $-($R$-$bar$*$A$2)$

A$2=$ Find a factor from the table

b$)$ Determining control limits for ranges

R$-$bar

UCL$=$R$-$BAR$*$D$4$

LCL$=$ R$-$bar$*$D$3$

D$3,$ and D$4=$ Find factors from table