The first test a doctor would order to determine whether a person is infected with HIV (the virus that causes AIDS) is the ELISA test. It detects antibodies and antigens for HIV. A study in Statistical Science by J. Gastwirth estimated that, if the person is actually infected with HIV, this test produces a positive result 97.7% of the time. If a person is not infected with HIV, the test result is negative 92.6% of the time. According the the US Centers for Disease Control (CDC), an estimated 1.1 million Americans out of a population of 321 million were infected with HIV in 2015. Using the information above, determine the probability that a randomly selected person whose ELISA test is positive actually is infected with HIV?

What is the probability that a randomly selected American is infected with HIV?

Using the answer to part (a) and the conditional probabilities of positive and negative ELISA test results, fill out the contingency table below:

ELISA Test Result Positive Negative HIV Infected Not Infected

What is the probability that a randomly selected person whose ELISA test is positive actually is infected with HIV?

What is the probability that a randomly selected person whose ELISA test is positive actually is not infected with HIV?

Based on the answer to (c) and (d) above, is the ELISA test effective?