Consider the following example. Medical investigators in a developing country are considering using a live vaccine for HIV infection. The risks of this procedure are considerable, but a high prevalence of the disease in their population would justify the intervention. For instance, if only 10% of the population were infected, the number of complications from the vaccine would exceed the number of infections prevented. However, if the prevalence were as high as 30%, the distribution of the vaccine would be justified.
The investigators know something about the prevalence of HIV in this country, and they think that there is only a one-in-five chance that 10% of the population is infected.
To test their "null" hypothesis of a low, 10% prevalence versus their "alternative" hypothesis of a 30% prevalence, they conduct a random survey of 20 individuals, and measure their HIV status. They then compute the proportion of the infected persons in the sample.
To test their hypothesis, they compute something called a "p-value" based on the known distribution of the proportion under the "null" hypothesis. When this number is small, the "null" hypothesis is unlikely to be true. Typically, investigators require that the p-value be less than .05 to choose the alternative hypothesis over the null hypothesis.
By pressing the "Generate Data" button, you can see how the p-value is calculated from the tail area of the null distribution of the proportion. You can choose to generate data under the null or alternative hypothesis, to see how well the p-value discriminates between the two scenarios.
In the above scenario, the investigators suspected that 10% prevalence is less likely before conducting their survey. They could formally incorporate this belief into their decision making process. The data from the survey is used to revise their "prior" probabilities for the two scenarios. The revised ("posterior") probabilities are used to decide whether 10% or 30% of the population is infected. The program illustrates how this procedure of probability revision is done.