A technician in a quick oil change shop had the following service times $($in minutes$)$ for $20$ randomly selected cars:

Sample

$1$

$4\mathrm{.}5$

$4\mathrm{.}2$

$4\mathrm{.}2$

$4\mathrm{.}3$

$4\mathrm{.}3$

$2$

$4\mathrm{.}6$

$4\mathrm{.}5$

$3$

$4\mathrm{.}4$

$4\mathrm{.}7$

$4\mathrm{.}3$

$4\mathrm{.}5$

$4\mathrm{.}6$

$4\mathrm{.}4$

$4\mathrm{.}4$

$4\mathrm{.}6$

$4$

$4\mathrm{.}7$

$4\mathrm{.}6$

$4\mathrm{.}8$

$4\mathrm{.}5$

$4\mathrm{.}9$

$*$ Determine the mean of each sample.

$*$ Estimate the mean and standard deviation of the process.

$*$ Estimate the mean and standard deviation of the sampling distribution of the sample mean.

$*$ What would three$-$sigma control limits for the sample mean be$?$ What alpha risk would they provide?

$*$ What alpha risk would control limits of $4\mathrm{.}14$ and $4\mathrm{.}86$ for the sample mean provide?

$*$ Using the control limits of $4\mathrm{.}14$ and $4\mathrm{.}86,$ are any sample means beyond these control limits$?$ If so$,$ which one$($s$)?$

$*$ Construct the control charts for sample mean and sample range using Table $10-3.$ Are any sample means and ranges beyond the control limits$?$ If so$,$ which one$($s$)?$

$*$ Explain why the control limits are different for sample mean in parts d and g$.$

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