The next two questions involve the following data:

The phase III trial for the Pfizer vaccine against SARS-CoV-2 had a total of 21,720 individuals receiving the vaccine and 21,720 individuals receiving a placebo. During the monitoring period that started 7 days after the second dose, the vaccine group had a total of 8 cases of COVID-19 while the placebo group had 162 cases. Cases of severe COVID-19 were also reduced with 1 severe case of the 8 cases in the vaccine group being severe and 4 severe cases of the 162 cases in the placebo group.

1. Design and execute statistical tests of the null hypothesis that the vaccine had no effect on disease prevention as well as the null hypothesis that it had no effect on severe disease (that is, conduct two tests). Show your work and explain your rationale for any choices you make along the way (e.g., one-sided or two-sided test).

2. On an absolute scale, the case counts (and particularly the severe cases) during the Pfizer trial were low. Generate 95% confidence intervals for the absolute risk of all four categories (cases w/ vaccine, severe cases w/ vaccine, cases w/ placebo, severe cases w/ placebo). Absolute risk for each category is estimated by dividing the number of cases by the total individuals in that category. Show your work and note any assumptions you're making.