On November 20, 2020, Pfizer and BioNTech applied to the US Food and Drug Administration for emergency use authorization of their Covid-19 vaccine

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On November 20, 2020, Pfizer and BioNTech applied to the US Food and Drug Administration for emergency use authorization of their Covid-19 vaccine.1 The application was based on interim results from the phase 3 clinical trial. Of the 170 participants in the trial who became infected with Covid-19, 8 had received the vaccine while 162 had not (they had received a placebo instead of the vaccine). Note that study participants who were infected with Covid were exposed naturally just like everyone else in the population over the period of the clinical trial. Assume that 21,000 of the 41,135 study participants had received the vaccine while the remainder had not. 

a. Use a 2×2 contingency table with vaccine received (Yes/No) by row and subsequent Covid infection (Yes/No) by column. Complete the contingency table with joint and marginal frequencies (not probabilities) from the information above. 

b. Compute the expected frequencies, assuming independence, of receiving the vaccine (Yes/No) and subsequent Covid infection (Yes/No). 

c. Use the expected and actual frequencies to determine if Covid infection depends on whether or not one had received the vaccine. Specifically, report the appropriate test statistic, the p-value for that test statistic and your conclusion, based on that p-value. 

d. First, compute the probability that a participant who had received the vaccine was subsequently infected with Covid. Second, compute the probability that a participant who was subsequently infected with Covid had received the vaccine. Which of these probabilities, the first or the second, offers information about the effectiveness of the vaccine compared to the placebo (i.e., no vaccine)?