A quality inspector took five samples, each with four observations, of the length of a part $($in centimetres$).$ She computed the mean of each sample $($see table below$).$

a$.$ Use the following data to obtain three sigma $($i$.$e$.,z=3)$ control limits for sample mean. Is the process in control?

b$.$ If a new sample $($sample $6)$ has values $12\mathrm{.}10,12\mathrm{.}19,12\mathrm{.}12,$ and $12\mathrm{.}14,$ using the sample mean control chart in part a$,$ has the process mean changed $($i$.$e$.,$ is there an assignable variation$)?$ It is known from previous experience that the standard deviation of the process is $0\mathrm{.}02cm.$

$\backslash$table$[[,$Sample$],[$Observation$,1,1,2,3,4,5,,,,,,,],[,2,12\mathrm{.}11,12\mathrm{.}15,12\mathrm{.}09,12\mathrm{.}12,12\mathrm{.}09,,,,,,,],[,3,12\mathrm{.}11,12\mathrm{.}12,12\mathrm{.}09,12\mathrm{.}10,12\mathrm{.}14,,,,,,,],[,4,12\mathrm{.}08,12\mathrm{.}11,12\mathrm{.}11,12\mathrm{.}08,12\mathrm{.}13,,,,,,,],[,\stackrel{}{x},12\mathrm{.}10,12\mathrm{.}12,12\mathrm{.}11,12\mathrm{.}10,12\mathrm{.}12,,,,,,,]]$