Drug testing in athletes. Due to inaccuracies in drug-testing procedures (e.g., false positives and false negatives), in the medical field the results of a drug test represent only one factor in a physician’s diagnosis. Yet when Olympic athletes are tested for illegal drug use (i.e., doping), the results of a single test are used to ban the athlete from competition. In Chance (Spring 2004), University of Texas biostatisticians D. A. Berry and L. Chastain demonstrated the application of Bayes’s rule to making inferences about testosterone abuse among Olympic athletes. They used the following example: In a population of 1,000 athletes, suppose 100 are illegally using testosterone. Of the users, suppose 50 would test positive for testosterone. Of the nonusers, suppose 9 would test positive.

a. Given that the athlete is a user, find the probability that a drug test for testosterone will yield a positive result. (This probability represents the sensitivity of the drug test.)

b. Given that the athlete is a nonuser, find the probability that a drug test for testosterone will yield a negative result. (This probability represents the specificity of the drug test.)

c. If an athlete tests positive for testosterone, use Bayes’s rule to find the probability that the athlete is really doping. (This probability represents the positive predictive value of the drug test.)