a). Respiratory viruses like the common cold can be transmitted through respiratory droplets present in the exhaled breath of infected individuals. Hand hygiene and the use of face masks are thought to be important strategies for preventing virus transmission. A study was recently conducted to explore the potential efficacy of surgical face masks to prevent respiratory virus transmission.

Participants were recruited from an outpatient clinic of a private hospital in Taiwan. All participants were diagnosed as having the influenza virus through a rapid diagnostic test and virologic confirmation in a laboratory. Participants were randomly assigned to either wear a surgical face mask or not during the 30 minute collection phase, in which they were instructed to breathe as normal. The respiratory droplets in the exhaled breath particles were then analyzed to see whether influenza virus could be detected.

The virus was detected in 8 of the 26 samples collected from those not wearing a surgical face mask and in 2 of the of the 28 samples collected from those wearing a face mask.

i. Do these data suggest that wearing surgical face masks could be an effective public health intervention for reducing person-to-person transmission of respiratory viruses? Conduct a formal analysis and summarize the results in language accessible to someone who has not taken a course in statistics.

ii. Compute the probability that under the null hypothesis, with the stated study design and observing 10 samples overall in which the virus is detected, that 9 samples positive for influenza virus were from the individuals wearing masks.

iii. Suppose the results of this study will inform public policy on whether the use of masks and face coverings while in public should be mandatory during the COVID-19 pan- demic. Comment on whether making a Type I or Type II error in this context has more serious consequences. Limit your response to no more than five sentences.

iv. Is it possible to know whether the results from the analysis in part i. represent a Type II error? Explain your answer.

b)Researchers plan to conduct a follow-up study to the one described in part a) that specifically examines individuals infected with SARS-CoV-2, the virus that causes COVID- 19. They plan to enroll 200 participants, randomizing half to the intervention group (i.e., wearing masks) and half to the control group (i.e., not wearing masks). Under the null hypothesis, they expect that the frequency of virus detection will be equal in both groups at 30%. They are interested in detecting a difference in proportions of at least 15%; i.e., that the frequency of virus detection is 0.15 (or lower) in the intervention group.

i. Suppose that the null hypothesis of no difference is true. Simulate a possible set of results under the null hypothesis and conduct a two-sided test of the null hypothesis from the simulated data. Comment on whether this set of simulated results represents evidence against the null hypothesis.

ii. Suppose the alternative hypothesis is true. Simulate a possible set of results under the alternative hypothesis and conduct a two-sided test of the null hypothesis from the simulated data. Comment on whether this set of simulated results represents evidence against the null hypothesis.

iii. The researchers think that wearing masks is effective at limiting virus transmission and there is less chance of detecting coronavirus among samples collected from the intervention group. They decide to quantify their belief by assigning probabilities to the null and alternative hypotheses; specifically, they decide to assume the null hypothesis is true with 0.25 probability and the alternative is true with 0.75 probability.

Conduct a simulation that generates a hypothetical population of 10,000 possible sets of results according to the researchers' prior beliefs about mask effectiveness and the described study design. For each set of results, record 1) whether the results were simulated under the null hypothesis or the alternative hypothesis, and 2) whether the results represent evidence against the null hypothesis. It may be helpful to summarize the simulation results in a two-way table.

Briefly explain the general logic and organization of your simulation; i.e., provide a high level overview of the steps of your simulation.

Based on the simulation results, comment on whether the study team has chosen an appropriate sample size; if you believe the sample size is not appropriate, suggest a new sample size. Explain your answer.