Avoid Fines by Warming - Up the Machines ∗

A company produces 1000 metal cylinders a day for 10 days. The government Safety Authority will take a random sample of cylinders, take a random sample of locations within each of these cylinders, and test for percentage Nickel (Ni) and percentage Chromium (Cr). For each sample in which the percentages are outside some intervals, they fine the company e1,000,000. If the proportion of percentages outside of some tighter intervals is too high, they fine the company e1,000,000. For quality control, the company first takes their own sample of the percentage Ni and Cr at some locations on some cylinders. Based on this data, they wish to know the expected value of the fine they must pay. Based on the data collected by the company, we determine the functional dependence of the percentages Ni and Cr on the day, cylinder number within that day, and location of the sample on that cylinder. We then used two different approaches to find the expected fine. By determining the frequencies of cells violating the specification intervals and making simplifying assumptions on the sampling method, we used properties of Binomial distributions to determine the expected value of the fine to be e430,000. By simulating the sampling and fine method of the Safety Authority, we determine a Monte Carlo estimate of the expected fine to be e429732; by performing a bootstrap analysis, we obtain a 95% confidence interval of [427879, 431588]. (Note: This paper is an entry in a game.)