$1.$ 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

$2.$ A p$-$chart is a control chart for attributes used to monitor the proportion of defective items in a process. An inspector counted the number of effective monthly billing statements of the telephone company the in each of the $6$ samples. Using the following information, construct a control chart. Each sample contained $20$ statements $($observation$).$ Z value is $0\mathrm{.}02$

Sample Number of defectives

$17$

$210$

$312$

$44$

$59$

$611$

total

Computing control limits for the fraction defective

P$-$bar$=$

SDp$=$

UCL

LCL

$3.$ When the process is not centered, a Process has a mean of $9\mathrm{.}20$ grams and a standard deviation of $0\mathrm{.}30.$ The lower specification limit is $7\mathrm{.}50$ grams, and the upper specification limit is $10\mathrm{.}45$ grams. Comput Cpk

Cpk$=\{$UCL$-$x$-$bar$/3$sd$,$ X$-$bar$-$LCL$/3$sd$\}$

Sd: standard deviation

$4.$ If the process is centered, suppose the specifications are $10$ and $11,$ and the standard deviation of the process is equal to $0\mathrm{.}10.$ Comput Cp

Cp$=$UCL$-$LCL$/6$sd