Question

The manager of the Scissors Department walks into your office. His needs are simple. Due to staffing levels, he wants his day shift to inspect 30 scissors, his evening shift 25 scissors and his midnight shift 20 scissors. They are asked to count the number of scratches on each pair of scissors. He tells you that typically there are 14 scratches on each pair of scissors and wants the control limits applicable for each shift’s inspection totals. That afternoon he returns and asks you to also calculate the limits for a control chart where he takes the number of scratches for the entire day, that is, 75 scissors are inspected each day. He reminds you that there are 14 scratches on each pair of scissors.

Answer 1

We model the number of scratches on a pair of scissors as a Poisson random variable with mean 14. Such an assumption is based on the faith that the number of scratches on a scissors increases at a high rate if one goes too away to the right of the mean.

Now we know that the mean and variance of a poisson random variable are both same, here 14.

Let be the i-th scissor that is tested for scratches, these are all independently and identically distributed Poisson (14) random variables.

Hence expected number of scratches on scissors for day shift is

Similarly, the expected number of scratches are for evening shift and for night shift, and

The variances are also the same as the expectations, and hence is for day shift, for evening shift and   for night shift, and finally for the entire day.

Hence the control limits, calculated as are given by

	LCL	CL	UCL
Day shift	358.52	420	481.48
Evening shift	293.88	350	406.124
Night shift	229.80	280	330.20
Entire Day	952.79	1050	1147.21