Reconsider Prob. 22.6-4. Suppose now that more careful statistical analysis has provided new estimates of the probability distributions of the radii of the shafts and bushings. In actuality, the probability distribution of the radius of a shaft (in inches) has the probability density function
Obtaining uniform random numbers as instructed at the beginning of the Problems section, perform a simulated experiment for estimating the probability of interference. Notice that almost all cases of interference will occur when the radius of the bushing is much closer to 1.0000 inch than to 1.0100 inches. Therefore, it appears that an efficient experiment would generate most of the simulated bushings from this critical portion of the distribution. Take this observation into account in part (b). For each of the following cases, use the same 10 pairs of uniform random numbers to generate random observations, and calculate the resulting estimate of the probability of interference.
(a) Use the crude Monte Carlo technique.
(b) Develop and apply a stratified sampling approach to this problem.
(c) Use the method of complementary random numbers.
Prob. 22.6-4
For one new product to be produced by the Aplus Company, bushings will need to be drilled into a metal block and cylindrical shafts inserted into the bushings. The shafts are required to have a radius of at least 1.0000 inch, but the radius should be as little larger than this as possible. With the proposed production process for producing the shafts, the probability distribution of the radius of a shaft has a triangular distribution with a minimum of 1.0000 inch, a most likely value of 1.0010 inches, and a maximum value of 1.0020 inches. With the proposed method of drilling the bushings, the probability distribution of the radius of a bushing has a normal distribution with a mean of 1.0020 inches and a standard deviation of 0.0010 inch. The clearance between a bushing and a shaft is the difference in their radii. Because they are selected at random, there occasionally is interference (i.e., negative clearance) between a bushing and a shaft to be mated. Management is concerned about the disruption in the production of the new product that would be caused by this occasional interference. Perhaps the production processes for the shafts and bushings should be improved (at considerable cost) to lessen the chance of interference. To evaluate the need for such improvements, management has asked you to determine how frequently interference would occur with the currently proposed production processes. Estimate the probability of interference by performing 500 iterations of a simulation on a spreadsheet. Also generate other available results regarding the frequency distribution of the clearance (positive or negative).