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

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 $5$ samples, we have:

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

Assume that the quality characteristic is normally distributed.

Assume that the quality characteristic is normally distributed. a$.$ Compute control limits for the $\stackrel{}{x}$ and $R$ control charts. $(10$ pts$.)$

b$.$ All points on both control charts fall between the control limits on part $($a$).$ What are the natural tolerance limits of the process? $(5$ pts$.)$

c$.$ The specifications limits for this process are $41\mathrm{.}0+-5\mathrm{.}0.$ Assuming that if an item exceed the upper specification limit it can be reworked and if it is below the lower specification producing? $(10$ pts$.)$

d$.$ Make suggestions as to how process performance could be improved. $(5$ pts$.)$

Please show the solutions for b$,$c$,$ and d in detail. Thank you!

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