$..56\mathrm{.}8$ Bill Kime's bowling ball factory makes bowling adult size and weight only. The standard deviation in the weight of a bowling ball produced at the factory is known to be $0\mathrm{.}12$ pounds. Each day for $24$ days, the average weight, in pounds, of nine of the bowling balls produced that day has been assessed as follows:

$\backslash$table$[[$Day$,$Average $($lb$),$Day,Average $($lb$)],[1,16\mathrm{.}3,13,16\mathrm{.}3],[2,15\mathrm{.}9,14,15\mathrm{.}9],[3,15\mathrm{.}8,15,16\mathrm{.}3],[4,15\mathrm{.}5,16,16\mathrm{.}2],[5,16\mathrm{.}3,17,16\mathrm{.}1],[6,16\mathrm{.}2,18,15\mathrm{.}9],[7,16\mathrm{.}0,19,16\mathrm{.}2],[8,16\mathrm{.}1,20,15\mathrm{.}9],[9,15\mathrm{.}9,21,15\mathrm{.}9],[10,16\mathrm{.}2,22,16\mathrm{.}0],[11,15\mathrm{.}9,23,15\mathrm{.}5],[12,15\mathrm{.}9,24,15\mathrm{.}8]]$

a$)$ Establish a control chart for monitoring the average weights of the bowling balls in which the upper and lower control limits are each two standard deviations from the mean. What are the values of the control limits$?$

b$)$ If three standard deviations are used in the chart, how do these values change? Why?

S$6\mathrm{.}9$ Whole Grains LLC uses statistical process control to ensure that its health$-$conscious, low$-$fat, multigrain sandwich loaves have the proper weight. Based on a previously stable and in$-$control process, the control limits of the $\stackrel{}{x}-$ and $R-$charts are: $UC{L}_{\stackrel{}{x}}=6\mathrm{.}56.LC{L}_{\stackrel{}{x}}=5\mathrm{.}84,UC{L}_R=1\mathrm{.}141,LC{L}_R=0.$ Over the past few days, they have taken five random samples of four loaves each and have found the following: