PROBLEM 1: SHORT ANSWER (50 pts. total)
a) (20 pts.) The coach of the Red Shirts, a Multi-dimensional League Baseball team, is recruiting new players. To do so, she is conducting a home run competition. Each batter gets 100 tries to hit an identically pitched ball over a fence that is 310 yards away; a ball hit over the fence is considered sufficient for making a home run. The coach would like to recruit the top 1% of all batters.
i. A typical batter will hit 60% of the balls; of the balls the3y hit, suppose that the dis- tance traveled is normally distributed with mean of 300 yards and standard deviation 50 yards. How many home runs should the coach set as a minimum limit for being recruited to the Red Shirts?
ii. J.D. Martian-ez hits the ball 80% of the time. When he hits the ball, the distance it travels is normally distributed with mean 295 yards and standard deviation of 60 yards. Based on the minimum limit set in part i., what is the probability that J.D. will make the team?
iii. The coach plans to host tryouts every week, allowing players to participate at most 10 times. J.D. plans on trying out until he makes the team. What is the expected number of times J.D. will have to try out to make the team?
iv. If the coach allows players to try out more than once, will she actually be identifying the top 1% of all batters? Explain your answer.
b) (30 pts.) Suppose that a pharmaceutical company has asked you to work as a consultant on their COVID-19 vaccine project. They are planning to conduct a study to demonstrate that their vaccine candidate is more effective than the one Pfizer and BioNTech announced in early November 2020, which was stated to have efficacy of 90% against placebo in par- ticipants without prior evidence of SARS-CoV-2 infection. They currently plan to enroll 2,000 participants, randomizing half to receive their vaccine candidate and half to receive the vaccine candidate from Pfizer and BioNTech. They are interested in detecting a differ- ence in proportions of at least 5% on the outcome of whether a patient becomes infected with SARS-CoV-2. The results of the trial will be analyzed with a two-sided test.
Previous dose-testing trials have been conducted to determine the lowest possible dose of virus sufficient for infecting most of those exposed; this will be the dosage used in the cur- rent study. Participants in the study will be healthy volunteers ages 18-30 years who con- sent to being deliberately exposed to the SARS-CoV-2 virus in a controlled setting and have screened negative for risk factors associated with severe COVID-19. Following exposure, any participants who test positive for the virus will be immediately treated with an antiviral drug. Studies that involve deliberate exposure of participants to an infectious disease are commonly referred to as challenge trials.
You have been asked to conduct a simulation study to assess whether the study team has chosen an appropriate sample size. Based on discussions with the study team, you decide it seems feasible that they have identified a more effective vaccine and assign probability 0.75 to the alternative hypothesis.
i. Conduct a simulation incorporating your belief about the efficacy of the new vaccine candidate; be sure to clearly comment your code where necessary. In one paragraph, briefly explain the general logic and organization of your simulation approach.
ii. Based on the simulation results, comment on whether the sample size is appropriate. If you believe the sample size is not appropriate, suggest a new sample size and explain your reasoning.
iii. Suppose the study team conducts the trial with the sample size you recommend in part ii. and obtains statistically significant results at the α = 0.05 level. Compute the estimated probability that the alternative hypothesis is true, given the observed data. Explain why this estimated probability changes if you have less confidence in the effectiveness of the new vaccine candidate.
iv. Suppose the study team conducts the trial and obtains statistically significant evi- dence that the new vaccine candidate has efficacy greater than 90%, with p-value =
0.03. A news outlet reports on the research finding, claiming that based on the p- value, the new vaccine is substantially better at protecting against COVID-19 than the Pfizer/BioNTech vaccine.
In language accessible to an audience without a statistics background, explain why the news outlet’s interpretation of the p-value is flawed and provide an accurate interpre- tation. Limit your answer to no more than seven sentences.
v. The vaccine data Moderna announced on 16 Nov 2020 are from a field study, where individuals were randomized to receive either the vaccine candidate or placebo then instructed to return to their daily routine and use an online app daily to screen them- selves for COVID-19 symptoms. Both the study participants and the staff administer- ing shots were blinded to which shots contained vaccine or placebo. The New York Times press release mentioned that Moderna “slowed enrollment in September to ensure di- versity among participants, and ultimately included 37 percent from communities of color, and 42 percent from populations considered at high risk because they were over 65 or had conditions like diabetes, obesity or heart disease.”
Comment on whether the results from Moderna’s field study or the hypothetical chal- lenge trial would be more informative for understanding whether a particular vaccine candidate is effective for protecting college undergraduates from SARS-CoV-2 infection while they are living on campus. Be sure to fully explain your reasoning. Limit your answer to no more than six sentences.
PROBLEM 2: MCAS (70 pts. total)
The Massachusetts Comprehensive Assessment System (MCAS) is a standardized exam admin- istered to students in the state of Massachusetts since 1993. Under law, students educated with public funds are required to participate in statewide testing. Students take exams according to their grade level in subjects such as Mathematics and English Language Arts; passing grades on the Grade 10 MCAS are required for high school graduation. Exam results are used to check student progress as well as measure school and district performance.
However, research has shown that variation in standardized test scores is often explained by fac- tors such as race and socioeconomic class; students of color and students from economically disad- vantaged backgrounds tend to score lower on standardized tests as a result of having less access to educational resources than their peers. For example, a New York Times article from October 2020 reported the results of a study demonstrating that in the United States, students performed worse on standardized tests for every additional day that was 80 degrees Fahrenheit or higher—the as- sociation was observed for Black and Hispanic students, in addition to students with lower family income, but not for white students.
In this problem, you will use data from the 2018 Grade 10 Mathematics MCAS to investigate ev- idence of achievement gaps in standardized test score at the school level. The mcas data contains information on 355 schools in Massachusetts. Scores on the MCAS are classified into levels: warn- ing/failing, needs improvement, proficient, and advanced. Students who score in the “proficient” category are said to “demonstrate a solid understanding of challenging subject matter”, while those in the “advanced” category are said to “demonstrate a comprehensive and in-depth under- standing of rigorous subject matter”; the variable PA_perc represents the percentage of students at a school who score at the proficient or advanced level.
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