Samples of $n=6$ items each are taken from a process at regular intervals. A quality

characteristic is measured, and $\stackrel{}{x}$ and $R$ values are calculated for each sample. After $50$

samples, we have

${\displaystyle \underset{i=1}{\overset{50}{}}}{\stackrel{}{x}}_i=2000,{\displaystyle \underset{i=1}{\overset{50}{}}}{R}_i=200$

a$)$ Find control limits for $\stackrel{}{x}$ and $R$ Charts

b$)$ What are the natural tolerance limits of the process?

c$)$ If the specification limits are $41+-5\mathrm{.}0,$ what are your conclusions regarding the ability

of the process to produce items within these specifications?

d$)$ Assuming that if an item exceeds the upper specification limit it can be reworked, and

if it is below the lower specification limit it must be scrapped, what percentage scrap

and rework is the process producing?