For a combined ELISA–Western blot blood test for HIV positive status, the sensitivity is about 0.999 and the specificity is about 0.9999 (Gigerenzer 2002, pp. 124 , 126 ).
a. Consider a high-risk group in which 10% are truly HIV positive. Construct a tree diagram to summarize this diagnostic test.
b. Display the intersection probabilities from part a in a contingency table.
c. A person from this high-risk group has a positive test result. Using the tree diagram or the contingency table, find the probability that this person is truly HIV positive.
d. Explain why a positive test result is more likely to be in error when the prevalence is lower. Use tree diagrams or contingency tables with frequencies for 10,000 people with 10% and 1% prevalence rates to illustrate your arguments.