HIV testing is essential for slowing the spread of HIV infection. Many people are unaware that they're infected with HIV, so they may be less likely to take precautions to help prevent spreading the virus to others. However, when screening for diseases the prevalence rate of the disease (i.e. how big of a proportion of the population that has the disease) plays an important role for how the test results should be interpreted. Generally, HIV tests have a very high specificity. We will assume a specificity of 99.8%. The sensitivity also tends to be high, as long as the test is taken after the few weeks following an infection called the window period. We will assume a sensitivity of 99.9%.

(a) Assume we are screening a population where the prevalence of HIV is 2%. How big of a proportion of those that test positive will have HIV?

(b) Assume we are screening a population where the prevalence of HIV is 0.1 %. How big of a proportion of those that test positive will have HIV?