A printing shop that produces advertising specialties producespaper cubes of various sizes, of which the 3.5 inch cube is themost popular. The cubes are cut from a stack of paper on cuttingpresses. The two sides of the cube are determined by the distanceof stops on the press from the cutting knife and remain fairlyconstant, but the height of the cube varies depending on the numberof sheets included in a “lift†by the operator. The lift heightdoes not remain constant within an operator or between operators.The difficulty is in judging, without taking much time, whatthickness of lift will give the correct height when it is pressedby the knife and cut. The humidity in the atmosphere alsocontributes to this difficulty, because the paper swells whenhumidity is high. The operators tend to err on the safe side, bylifting a thicker stack of paper than necessary.
The company management believes the cubes are being made muchtaller than the target, thus giving away excess paper and causingloss to the company. They have received advice from a consultantthat they could install a paper-counting machine, which will givethe correct lift containing exactly the same number of sheets eachtime a lift is made. This, however, will entail a huge capitalinvestment. To see if the capital investment would be justifiable,the company management wants to assess the current loss in paperbecause of the variability of the cube heights from the target.
Data were collected by measuring the heights of 20 groups offive cubes and are provided in the table below. Estimate the lossincurred because of the cubes being taller than 3.5 inches. A cubethat is exactly 3.5 inches in height weighs 1.2 lb. The companyproduces 3 million cubes per year, and the cost of paper is $64 perhundred-weight (100lb.).
Note that the current population of cube heights has adistribution (assume this to be normal) with an average andstandard deviation, and the target population of the cubes is alsoa distribution with an average of 3.5 in. and a standard deviationto be determined. (You can not make every cube exactly 3.5 in. inheight.) The target standard deviation can be smaller than thecurrent standard deviation, especially if the current process issubject to some assignable causes.
Estimate the current loss in paper because of the cubesbeing too tall. You first may have to determine the attainablevariability before estimating the loss. If any of the informationyou need is missing, make suitable assumptions, and state themclearly.
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 |
