$4)$ A process for producing electron circuits has achieved very high yield levels. An average of only $8$ defective $$parts per million is currently produced.

a$)$ what are the upper and lower control limits for a sample size of $100?$

b$)$ Recompute the upper and lower control limits for a sample size of $10,000.$

c$)$ Which of these two sample sizes would you recommend? Explain why.

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$9)$ The Robin Hood Bank has noticed a decline in daily deposits. The average daily deposit has been running at $$109$ Million with an average range of $$15$ Million over the past year. The deposits over the past $6$ days have been $110,102,96,87,115$ and $106.$

a$)$ What are the CL$,$ UCL, and LCL for the x$$ and R charts based on a sample size of $6$ days?

b$)$ Compute an average and range for the sample that consists of the past $6$ days. Do the figures for the past six days suggest a change in the average or range from the past year?

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$12)$ A certain process has an upper specification limit of $220$ and a lower specification limit of $160.$ The process standard deviation is $6,$ and the mean is $170.$

a$)$ Calculate the Cp and the Cpk for this process.

b$)$ What could be done to improve the process capability Cpk to $1\mathrm{.}0?$

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$14)$ A customer has specified that they require a process capability of Cp$=1\mathrm{.}5$ for a certain product. Assume that the USL$=1,100,$ assume that the LSL$=700,$ and the process is centered within the specification range.

a$)$ What standard deviation should the process have?

b$)$ What is the process mean value?

c$)$ What can the company do if it is not capable of meeting these requirements?