what could explain the discrepancy between age groups and the overall population

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

1. Compute P(Hospitalized | Fully Vax) and P(Hospitalized | Not Vax) for the overall population, and display your result as a percentage with at least four decimal places.

2. Calculate conditional probabilities from part 1 for the two age groups displayed in the table. Again, show your result as a percentage with at least four decimal places.

3. Let’s define the vaccine “efficacy” as Efficacy = 1 − P(Hospitalized | Fully Vax) P(Hospitalized | Not Vax) . This measure captures the percentage reduction in hospitalizations in the vaccinated group relative to the unvaccinated group. Compute the efficacy among the overall population.

4. Also compute the efficacy for the two age groups. How does the efficacy among the overall population compare to the breakdown by age? What could explain the discrepancy between age groups and the overall population?

5. Suppose in a counterfactual world the conditional probabilities computed in part 1 remain the same, but now we swap the vaccination rate, i.e., only 1, 302, 912 individuals are fully vaccinated and 5, 634, 634 are not vaccinated. Compute the expected number of hospitalizations for the vaccinated and unvaccinated population implied by the conditional probabilities. How many more hospitalizations would there have been?

6. Suppose someone observes the data and argues: “Most of the hospitalized are fully vaccinated individuals. Therefore, vaccines are not very effective.” Do you find this argument convincing? In 1-2 sentences, explain why or why not. What other factors besides vaccine efficacy could explain that the majority of the hospitalized are vaccinated? 

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