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If you sample 450 parts per day from a daily production of 12,000 parts, what is the upper and lower control limits of an np chart if the average...

he upper control limit is 3 standard deviations above the expected number of defects, while the lower control limit is 3 standard deviations below.


When we know the proportion that will not have defects is 0.975, the proportion that will have defects is just 1 minus that: p = 0.025.

For an np chart, we calculate these figures in terms of actual numbers of items; but the process is basically the same as for a p chart where we only calculate them in terms of proportions of items (the only difference is you multiply by n).

So the expected number of defects per sample is:


`np = (450)(0.025) = 11.25`

The standard deviation is given by:


`sigma = sqrt{n p (1-p)} = sqrt{450(0.025)(0.975)} = 3.312`


Then, we can get the upper and lower control limits:
`UCL = np + 3sigma = 11.25 + (3)(3.312) = 21.19`
`LCL = np - 3sigma = 11.25 - (3)(3.312) = 1.31`` `


If the process is under control, the number of defects should be between these two values in each sample. The total amount produced turns out not to matter (as long as the sample is large and the population is much larger still).

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