$8\mathrm{.}1$ A factory has $10$ departments, all of which have quality problems leading to delays in shipping products to customers. A breakdown of the number of quality problems for each department $($listed alphabetically$)$ is as follows: $(1)$ assembly, $16$ problems; $(2)$ final packaging, $9$ problems; $(3)$ finishing, $37$ problems; $(4)$ forging, $73$problems; $(5)$ foundry, $362$ problems; $(6)$ machine shop, $294$problems ; $(7)$ plastic molding, $120$ problems; $(8)$ receiving inspection, $124$ problems ; $(9)$ sheet metalworking, $86$ problems; and $(10)$ tool$-$making, $42$ problems. $($a$)$ Construct a Pareto chart for this data. $($b$)$ Assuming that all quality problems are of equal value, in which department would you start to take corrective action to reduce the quality problems? $($c$)$ Determine the percentage of total quality problems that are attributable$/$

$8\mathrm{.}2$ Using your answer to part $($c$)$ of the preceding problem, $($a$)$ determine the parameter A in Eq$.(8\mathrm{.}1$ see notes$)$ representing the Pareto cumulative distribution. Use $20\%$ of the departments as the x value in your computations. $($b$)$ Construct the idealized Pareto chart based on your answer to part $($a$)$ and discuss the comparison between this idealized chart and the actual data in the previous problem. Use a spreadsheet program to calculate the data for part $($b$).$

$8\mathrm{.}3$ Assume that $75\%$ of the sales in a retail company are accounted for by $25\%$ of the customers. $($a$)$ Determine the parameter A in the Pareto cumulative distribution equation. $($b$)$ Given that the relationship is valid for the remaining sales, how much of the sales value is accounted for by $50\%$ of the customers?