A number of states are considering legislation that calls for people to get mandatory drug testing in order to qualify for public assistance money. Some argue that mandatory testing of welfare recipients would jeopardize the livelihood of many people who do not have substance abuse problems and is therefore an unwarranted invasion of their privacy.

Consider a hypothetical situation. Suppose that the currently used test for illegal drugs is $99\%$ accurate, and suppose that in the population to be tested (in this case it is the population of people on public assistance of a given state) $2300$ out of $100,000$ people are substance abusers. The question is: what is the probability of a “false negative”? That is, the probability that the test identifies a person on public assistance as using drugs when they actually do not use drugs?

1. Make a model for this situation.

   Hint (a):

   Make a two-way table or tree diagram.
   Users/Non-users and Positive drug test/Negative drug test

   More Help (a):

   

2. How does the proportion of people who are using drugs compare with the proportion of people who will be told they are using drugs but really are not?

   Hint (b):

   The proportion of people using drugs is ²³/₁₀₀₀.
   Use your model to find the proportion of non-users that test positive (situation D).

3. If a randomly tested recipient’s test comes back positive, what is the probability that he or she is not actually using drugs?

   Partial Answer (c):

   $0.0097/(0.0097 + 0.02277)$

4. Write up your statistical conclusions.

   Hint (d):

   Use your answers to parts (b) and (c) to justify your conclusions.

5. Are the test results mathematically independent of using or not using drugs? How could you check this? Explain.

   Hint (e):

   Does the accuracy of the drug test depend on whether or not someone is a user?