Question

In order for a vaccine to be effective, it should reduce a person's chance of acquiring a disease. Consider a hypothetical vaccine for malaria-a tropical disease that kills between 1.5 and 2.7 million people every year. \(^{20}\) Suppose the vaccine is tested with 500 volunteers in a village who are malaria free at the beginning of the trial. Two hundred of the volunteers will get the experimental vaccine and the rest will not be vaccinated. Suppose that the chance of contracting malaria is \(10 \%\) for those who are not vaccinated. Construct a two-way table to show the results of the experiment if: (a) The vaccine has no effect. (b) The vaccine cuts the risk of contracting malaria in half.

Short answer

The table for the scenario where the vaccine has no effect contains the figures: Vaccinated & contracted malaria: 20, Not vaccinated & contracted malaria: 30. The table for the scenario where the vaccine cuts risk in half contains the figures: Vaccinated & contracted malaria:10, Not vaccinated & contracted malaria:30.

Step-by-step solution

Calculation of number of individuals not vaccinated

From the total of 500 volunteers subtract the 200 who received the vaccine. So, the number of individuals who are not vaccinated is \(500 - 200 = 300\).

Table for vaccine having no effect

In this scenario, the vaccine does not reduce the risk of contracting malaria. For this case, the number of individuals within each group who contract the disease can be calculated as 10% of the population of that group. \n - Vaccinated and contracted malaria: \(10\% \times 200 = 20\) - Not Vaccinated and contracted malaria: \(10\% \times 300 = 30\) All the rest in each group didn't contract the disease. So in the table there should be these values: \n | - | Vaccinated | Unvaccinated | | --- | --- | --- | | Contracted Malaria | 20 | 30 | | Didn't Contract Malaria | 180 | 270 |

Table for vaccine cutting risk in half

In this scenario, the vaccine reduces contraction risk by half. This means we divide the percentage of contracting the disease for vaccinated individuals by 2. For vaccinated individuals, risk becomes: \(10/2 = 5\% \). For not vaccinated remains 10%. The people contracting the disease in each group are: - Vaccinated and contracted malaria: \(5\% \times 200 = 10 \) - Not vaccinated and contracted malaria: \(10 \% \times 300=30 \) All the rest in each group didn't contract the disease. So in the table there should be these values: \n | - | Vaccinated | Unvaccinated | | --- | --- | --- | | Contracted Malaria | 10 | 30 | | Didn't Contract Malaria | 190 | 270 |