The New York State Department of Health estimates a 10% rate of HIV for the at-risk population and a 0.3% rate for the general population. Tests for HIV are 95% accurate in detecting both true negatives and true positives. Random selection and testing of 5,000 at-risk people and 20,000 people from the general population results in the following table.

a. Verify that incidence rates for the general and at-risk populations are 0.3% and 10%, respectively. Also verify that detection rates for the general and at-risk populations are 95%.

b. Consider the at-risk population. Of those with HIV, what percentage test positive? Of those who test positive, what percentage has HIV? Explain why these two percentages are different.

c. Suppose a patient in the at-risk category tests positive for the disease. As a doctor using this table, how would you describe the patient's chance of actually having the disease? Compare this figure with the overall incidence rate of the disease.

d. Consider the general population. Of those with HIV, what percentage test positive? Of those who test positive, what percentage have HIV? Explain why these two percentages are different.

e. Suppose a patient in the general population tests positive for the disease. As a doctor using this table, how would you describe the patient's chance of actually having the disease? Compare this figure to the overall incidence rate of the disease.

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At-risk population Test positive 475 225 Test negative Infected 25 Not infected 4,275 General population Test positive Test negative Infected 57 3 Not infected 997 18,943