$($Problem $21-30$: $1$ point each$)$ If you need a Z $-$value, let $\backslash (\backslash$mathrm$\{$Z$\}=3\backslash )$ and carry up to four decimal places if necessary. $21.$ A quality analyst wants to construct a sample mean chart for controlling a packaging process. He knows from experience that whenever this process is under control, package weight is normally distributed with a mean of $20$ ounces and a standard deviation of one ounce. Each day last week, he randomly selected four packages and weighed each: What is the upper control limit of a process? $22.$ A design engineer wants to construct a sample mean chart for controlling the service life of a halogen headlamp his company produces. He knows from numerous previous samples that this service life is normally distributed with a mean of $500$ hours and a standard deviation of $20$ hours. On three recent production batches, he tested service life on random samples of four headlamps, with these results: If he uses upper and lower control limits of $(\backslash (\backslash$mathrm$\{$UCL$\}=520\backslash ))$ and $(\backslash (\backslash$mathrm$\{$LCL$\}=480\backslash ))$ hours, on what sample$($s$)($if any$)$ does service life appear to be out of control? The following problems refer to problems $23$ and $24.$ A quality analyst wants to construct an $\backslash ($ x $\backslash )-$bar chart to determine whether three machines, all producing the same product, are under control concerning a particular quality variable. Accordingly, he sampled four units of output from each machine, with the following results: