Reoall that the Pzer-Biontech vaccine trial modeled the number of study participants di- agnosed with CVlg as binomial with probabilities in: and are in the vaccine and placebo arms, respectively. Recall, also, that they reparameterized the problem in terms of H = ELI/{17V + are), the conditional probability that a participant diagnosed with (JV-19 was in the vaccinated arm. The study would continue until 164 conrmed {JV-19 cases were observed. The vaccine would be declared effective if, at the end of the study [i.e, when there are 154 CV-IQ cases), the posterior probability that vaccine efcacy (VE) exceeds 30% is at least 98.5%. Recall that VE : (1 23),\":1 :9). An interim analysis of VE was planned when 32 (JV-19' diagnoses were observed. The stopping rules for this interim analysis were: 0 Stop for Eicacy: if the probability that VE > 30% given the current data exceeds 99.5%, i.e., PI(VE > 0.30 I current data) 22- 0.995. 0 Stop for Futility: if either of the following holds: (a) The probability that VE > 0.30 given the current data is less than 5%. 0R (b) The projected probability that the vaccine would be declared effective at the end of the trial is less than 5%. The prior distribution is H m Beta(o:, 15'). Suppose we adopt this prior with o: = 3 and ,6 = 1; here we are being pessimistic about the performance of the vaccine, saying, on average, that a person with (JV-19 is 3 times more likely to be in the vaccinated group. When 32 CVlQ cases were diagnosed, '6 were in the vaccinated group and 32 6 = 25 were in the placebo group. An interim analysis was then performed. {a} Is the criterion to stop for ecicacy met? Explain your work and conclusion. (b) Is criterion (a) to stop for futility met? Explain your work and conclusion. (c) When 164 (JV-19 cases have been observed [end of study), how many must have occured in the vaccinated arm for the vaccine to be declared a success? (d) Given that 6 of the current 32 (JV-10 cases are in the vaccinated arm, is criterion (b) to stop for futility met? Again, explain your work and conclusions