Suppose that a cookie making process has a normally distributed out put with a mean of $30$ grams and variance of $4$ grams. $(30$ points$)($a$)$ If the specifications are $30\mathrm{.}0\backslash$pm $6$ grams, compute Cp$,$ Cpu, Cpl$,$ and Cpk$.$ Is the process capable and centered?$($b$)$ Suppose the mean shifts to $27\mathrm{.}0$ but the variance remains unchanged. Recompute Cp$,$ Cpu, Cpl$,$ and Cpk$.$ and interpret these process capability indexes $2.$Suppose that a refrigeration process at Coolfoods, Ltd$.$ Has a normally distributed output with a mean of $25\mathrm{.}0$ and a variance of $1\mathrm{.}44.(60$ points$)($a$)$ If the specifications are $25\mathrm{.}0\backslash$pm $3\mathrm{.}25,$ compute Cp$,$ Cpu, Cpl$,$ and Cpk$.$ Is the process capable and centered?$($b$)$ Suppose the mean shifts to $23\mathrm{.}0$ but the variance remains unchanged. Recompute Cp$,$ Cpu, Cpl$,$ and Cpk$.$ and interpret these process capability indexes.$($c$)$ If the variance can be reduced to $0\mathrm{.}576,$ how do the process capability indices Cp$,$ Cpu, Cpl$,$ and Cpk change $($using the original mean of $25\mathrm{.}0)?$