PLEASE ANSWER A$,$B$,$ and C$.$ A manufacturing process produces bottles that are supposed to have a volume of $500$ milliliters. A sample of $4$ bottles is taken every day, and the average and range of the samples are recorded. The following data was collected over a period of $5$ days: The company wants to design an X$-$bar and R chart that will have a Type II error rate of $0\mathrm{.}1$ when the mean volume is shifted by $3$ milliliters.

$($a$)(10$ pts$)$ Determine the center line and control limits for the X$-$bar and R charts based on the in$-$control parameters for six$-$sigma limits$,$ i$.$e$.,$ L$=\backslash$pm $3\backslash$sigma $.$

$($b$)(5$ pts$)$ Determine the value of delta $($the shift in the mean volume$)$ that will result in a Type II error rate of $0\mathrm{.}1.$

$($c$)(10$ pts$)$ Redesign the X$-$bar and R charts to achieve a Type II error rate of $0\mathrm{.}1$ for the shift of $3$ millimeters in the mean volume. What is the new UCL for the X$-$bar char? What is the new LCL for the X$-$bar chart? What is the new UCL for the R chart? What is the new LCL for the R chart?