Question

A printing shop that produces advertising specialties produces paper cubes of various sizes, of which the 3.5 inch cube is the most popular. The cubes are cut from a stack of paper on cutting presses. The two sides of the cube are determined by the distance of stops on the press from the cutting knife and remain fairly constant, but the height of the cube varies depending on the number of sheets included in a “lift” by the operator. The lift height does not remain constant within an operator or between operators. The difficulty is in judging, without taking much time, what thickness of lift will give the correct height when it is pressed by the knife and cut. The humidity in the atmosphere also contributes to this difficulty, because the paper swells when humidity is high. The operators tend to err on the safe side, by lifting a thicker stack of paper than necessary.

The company management believes the cubes are being made much taller than the target, thus giving away excess paper and causing loss to the company. They have received advice from a consultant that they could install a paper-counting machine, which will give the correct lift containing exactly the same number of sheets each time a lift is made. This, however, will entail a huge capital investment. To see if the capital investment would be justifiable, the company management wants to assess the current loss in paper because of the variability of the cube heights from the target.

Data were collected by measuring the heights of 20 groups of five cubes and are provided in the table below. Estimate the loss incurred because of the cubes being taller than 3.5 inches. A cube that is exactly 3.5 inches in height weighs 1.2 lb. The company produces 3 million cubes per year, and the cost of paper is $64 per hundred-weight (100lb.).

Note that the current population of cube heights has a distribution (assume this to be normal) with an average and standard deviation, and the target population of the cubes is also a distribution with an average of 3.5 in. and a standard deviation to be determined. (You can not make every cube exactly 3.5 in. in height.) The target standard deviation can be smaller than the current standard deviation, especially if the current process is subject to some assignable causes.

Estimate the current loss in paper because of the cubes being too tall. You first may have to determine the attainable variability before estimating the loss. If any of the information you need is missing, make suitable assumptions, and state them clearly.

3.61	3.59	3.53	3.63	3.63	3.57	3.61	3.52	3.47	3.53	3.61	3.65	3.30	3.52	3.60
3.59	3.59	3.58	3.58	3.57	3.60	3.64	3.44	3.59	3.61	3.49	3.65	3.52	3.50	3.53
3.61	3.58	3.58	3.64	3.57	3.59	3.54	3.52	3.60	3.58	3.44	3.54	3.60	3.51	3.63
3.62	3.63	3.52	3.63	3.53	3.55	3.64	3.60	3.59	3.63	3.59	3.54	3.57	3.59	3.63
3.60	3.62	3.60	3.62	3.57	3.54	3.58	3.49	3.60	3.55	3.53	3.62	3.51	3.49	3.63

3.63	3.65	3.60	3.61	3.63
3.66	3.64	3.63	3.62	3.61
3.68	3.65	3.63	3.64	3.63
3.63	3.64	3.61	3.61	3.63
3.64	3.61	3.63	3.61	3.62

Answer 1