Lab 6-1: Bayes’ Theorem Example: Water Quality Testing

In this scenario, a water quality testing company has developed a new inexpensive technology for detecting “Chemical X” contamination in drinking water. This test can correctly identify the presence or absense of “Chemical X” above a certain threshold by giving a positive result 98% of the time (true positive). Therefore the probability of a false negative is 2%. This test can correctly identify when the water sample does not have “Chemical X” by giving a negative result 97% of the time (true negative). Therefore the probability of a false positive is 3%. For this example, assume that the observed probability of the presense of “Chemical X” in a random sample from a large number of water samples is about 1 in 1,200.

A. Draw a tree diagram for this problem as you work through this example.

B. Using Bayes’ theorem, estimate the probability that a water sample will have “Chemical X” given that the test result is positive. Are you surprised by the answer? What is the fundamental problem with this test?

C. How accurate would the test have to be to achieve a 50% probability of having “Chemical X” given that the test result is positive (true positive)? (Note: In this problem, try solving for the accuracies of both the positive and negative test results that will give you a 50% true positive rate. It may not be possible to achieve a 50% true positive rate with one or the other.)

D. Estimate the likelihood that a water sample does not have “Chemical X” given that the test result is negative (true negative). Is the test more useful in this framework as a screening step when trying to decide to use more accurate, but also more expensive, tests?