A company has just negotiated a contract to produce a part for another company. In the process of manufacturing the part, the inside diameter of successive parts becomes smaller and smaller as the cutting tool wears. However, the specification is so wide relative to machine capability that it is possible to set tire diameter initially at a large value and let the process run for a while before replacing the cutting tool.
Tire inside diameter decreases at an average rate of .001 cm per part, and the process has a standard deviation of .01 cm and the variability is approximately Normally distributed. After how many parts must the tool be replaced if the design specification is between 3 cm and 3.5cm, and the initial setting is 3 standard deviations below the upper design specification?