$4)$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?