Many manufacturing problems involve the matching of machine parts, such as shafts that fit into a valve hole. A particular design requires a shaft with a diameter of 22.000 mm, but shafts with diameters between 21.990 mm and 22.010 mm are acceptable. Suppose that the manufacturing process yields shafts with diameters normally distributed, with a mean of 22.002 mm and a standard deviation of 0.005 mm. For this process, what is

a. The proportion of shafts with a diameter between 21.99 mm and 22.00 mm?

b. The probability that a shaft is acceptable?

c. The diameter that will be exceeded by only two percent of the shafts?

d. What would be your answers in (a) through (c) if the standard deviation of the shaft diameters were 0.004 mm?