A teller at a drive$-$up window at a bank had the following service times $($in minutes$)$ for $20$ randomly selected customers.

SAMPLE

$1234$

$4\mathrm{.}54\mathrm{.}64\mathrm{.}54\mathrm{.}7$

$4\mathrm{.}24\mathrm{.}54\mathrm{.}64\mathrm{.}6$

$4\mathrm{.}24\mathrm{.}44\mathrm{.}44\mathrm{.}8$

$4\mathrm{.}34\mathrm{.}74\mathrm{.}44\mathrm{.}5$

$4\mathrm{.}34\mathrm{.}34\mathrm{.}64\mathrm{.}9$

a$)$ Determine the mean of each sample $(2$ms$).$

Mean sample $1$: $(4\mathrm{.}5+4\mathrm{.}2+4\mathrm{.}2+4\mathrm{.}3+4\mathrm{.}3)/5=4\mathrm{.}3$

Mean sample $2$: $(4\mathrm{.}6+4\mathrm{.}5+4\mathrm{.}4+4\mathrm{.}7+4\mathrm{.}3)/5=4\mathrm{.}5$

Mean sample $3$: $(4\mathrm{.}5+4\mathrm{.}6+4\mathrm{.}4+4\mathrm{.}4+4\mathrm{.}6)/5=4\mathrm{.}5$

Mean sample $4$: $(4\mathrm{.}7+4\mathrm{.}6+4\mathrm{.}8+4\mathrm{.}5+4\mathrm{.}9)/5=4\mathrm{.}7$

b$)$ Suppose the process mean is unknown, estimate it $(1$m$).$

Mean process: $(4\mathrm{.}3+4\mathrm{.}5+4\mathrm{.}5+4\mathrm{.}7)/4=4\mathrm{.}5$

c$)$ Suppose the process standard deviation is $0\mathrm{.}192,$ estimate the mean and standard deviation of the sampling distribution $(4$ms$).$

d$)$ What would be the three$-$sigma control limits for the process, based on the information in part c$)?(2$ms$)$ Using these control limits$,$ what will be the percentage that the service times are not acceptable? $(3$ms$)$

e$)$ What percentage that the service times are not acceptable would control limits of $4\mathrm{.}14$ and $4\mathrm{.}86$ provide? $(3$ms$)$

f$)$ Calculate the limits of control charts for means and ranges using the Control Chat Factor Table. Are any samples beyond the control limits$?$ If so$,$ which ones? $(4$ms$)$

g$)$ Why the control limits are different for means in parts d$)$ and f$)?(4$ms$)$

h$)$ If the process has a known mean of $4\mathrm{.}4$ and a known standard deviation of $0\mathrm{.}18,$ what would be the three$-$sigma control limits for a mean chart? Are any sample means beyond the control limits$?$ If so$,$ which one$($s$)?(4$ms$)$