One common testfor HIV/AIDS is the ELISA test. A study by Gastwirth (1987)estima

Question

One common testfor HIV/AIDS is the ELISA test. A study by Gastwirth (1987)estimates

that when aperson actually carries the AIDS virus, this test produces apositive result 97.7%

of the time.When a person does not carry the AIDS virus, the test result isnegative 92.6% of

the time. Thestudy further claims that a base rate of about 0.5% of the Americanpopulation

carries the AIDSvirus. That is, the base rate is the probability a randomlyselected individual

carries thevirus. For the people who test positive, what is theprobability they actually have

thedisease?

Explanation / Answer

If we had 1000 people, we would expect 5 of them (0.5%) to be HIVpositive and 995 HIV negative. The test: 92.6% Correct Negs --> 995 * 0.926 = 921.37 7.4%   False Pos --> 995 * 0.074 = 73.63 97.7% Correct Pos --> 5 * 0.977 = 4.885 2.3%   False Neg --> 5 * 0.023 = 0.115 All the people who test positive: 4.885+73.63 = 78.515 All the people who test positive and are positive: 4.885 4.885/(4.885+73.63) = 0.062 = 6.2% are actually HIVpositive.

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