solution

For a particular process the mean value of the output is known to vary throughout the day. As this is a bottleneck machine that will effectively shutdown the plant when not operating and as resetting the machine would require several hours, management has decided to allow this machine variation to continue. Fortunately, the variation in the output is predictable as it begins at one value and increase, approximately uniformly, throughout the day. Also, the standard deviation for this machine is constant over the day so management is able to predict the output. At the present time, the average mean is 1.610, the standard deviation, is 0.020 and the variation over the 8 hour work day is 0.040. The lower specification for this product is 1.535 and the upper specification is 1.655. The daily production is 1200 units.

To estimate the output, the day is divided into 8 hours and the output from that hour is calculated assuming all of the output is produced on the half hour. For example, the output from 8 AM until 9 AM is calculating using the estimated value of the mean at 8:30. Some fraction of this output will be below the lower specification limit and some fraction will be above the upper specification limit. As the day goes on, the amount below the lower limit will decrease and the amount above will increase. Estimate the number of items produced that are outside the specification for one day. Assume the calculations are performed at each half hour during the day.