Problem $5$

A logistics manager wants to assess how varied the weight of lorry loads is over a period of time. He takes a $5$ samples of $20$ lorry loads $(n)$ and notes the average load $($tonnes$)$ and the range within each sample.

$\backslash$table$[[$Sample$,$Weight average in tonnes $($x$),$Range $($R$)],[1,15\mathrm{.}5,4\mathrm{.}5],[2,12\mathrm{.}3,5\mathrm{.}3],[3,8\mathrm{.}8,6\mathrm{.}2],[4,10\mathrm{.}3,5\mathrm{.}2],[5,13\mathrm{.}1,3\mathrm{.}8]]$

a$)$ What are the population means for the mean and the range?

b$)$ What are the upper and lower control limits for the sample means chart?

c$)$ What are the upper and lower controls for the range chart?