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PROBLEM 1 a) (12 pts.) Pre-term birth is a common cause of neonatal mortality, especially in low-income and middle-income countries . A large randomized study was conducted across six countries with high rates of pre-term birth to investigate the effectiveness of taking daily low-dose aspirin far reducing the risk of giving birth befare 37 weeks. The 5,780 women randomized to the treatment group received a daily 81 milligram aspirin tablet at the start of the third trimester, while the 5,764 women randomized to the control group took a placebo. Pre-term birth befare 37 weeks occurred in 668 of the women who took aspirin and 754 of those who took placebo. 1. Compute and interpret an appropriate summary statistic comparing the difference in outcomes between the two groups. Be sure to clearly show your work. n. Using language accessible to a general audience without a statistics background, explain whether this study design has the potential to demonstrate that taking daily lowdose aspirin leads to a reduced risk of pre-term birth. Limit your answer to at most seven sentences. b) (30 pts.) lnvasive melanoma of the skin is a type of malignant skin cancer; melanoma accounts far 1 % of all skin cancers diagnosed in the United Sta tes, but causes the most deaths out of all types of skin cancer. In the United States, the overall incidence rate of melanoma is 21.8 cases per 100,000 people per year. The highest incidence rate occurs among nonHispanic white males: in this subgroup, melanoma occurs with rate 34.9 per 100,000 people annually. 1. In 2018, the population of Washington, D.C. was 702,000 residents, with 36.8% nonHispanic white residents. Compute the expected number of melanoma cases among non-Hispanic white males in a given year. You may assume that the number of men and women in Washington, D.C. is equal. n. Compute the probability of observing at least 60 cases of melanoma among nonHispanic white males in a given year. m. What is an implicit assumption required to make the calculations in parts i. and ii.? Explain your answer and comment on whether the assumption seems reasonable. 1v. Suppose that in a given year, a total of 60 cases of melanoma were observed in Washington, D.C., and all cases occurred among non-Hispanic white males. Does the probability of this event equal the probability computed in part ii .? Explain your answer. v. The five-year-survival rate indicates the proportion of people who remain alive at least five years after initial diagnosis; far melanoma, the five-year-survival rate is about 92%. What is the probability that, five years after initial diagnosis, more than 1 O deaths occur among 60 cases of melanoma? c) (28 pt s.) Man y diagno stic tests for detectin g th e pres enc e of d ise ase are ba sed on bi oma rk ers; broadly sp eaking , a biom ark er is any biolo gical ind icato r th at can be measur ed , such as blood pr essur e or fa sting blood gluco se level. Consid er a diagno stic t est for a hypoth etical di se ase base d on meas ur ing th e amount of a cert ain biomark er pr ese nt in blood . High leve ls of th e bi omark er are oft en found in individu als with th e disease, but a number of non-di sease condition s can also cau se high levels of th e biomark er. Indi vidu als wi th out th e di sease ha ve bi omark er leve ls tha t are norm ally distribut ed wi th mean 1.6 ng /mL (n anogr ams per mill ilit er of blood) , an d st and ar d deviati on 0.50 ng / mL. Individu als with th e d ise ase h av e biomark er levels that are n orm ally distribut ed with m ean 5 ng / mL and stand ard deviation 1.2 ng/ mL. Valu es of 2.5 n g/mL and higher con stitut e a po siti ve test result. 1. Comput e the accurac y of th e test for tho se who have th e di sease and th e accu racy of th e t est for th ose who do n ot have th e di sease . n. In a popul ation wh ere 6% of indi vidua ls are th ou ght to have th e hyp oth etic al disease , calcula te th e probabilit y th at an indi vidual who tests p ositiv e has th e di sease . m. In ard er to account for lab error , indi vidu als who test positive are aske d to re turn for an oth er test at a lat er tim e. Wh at is th e p robabilit y th at an ind ividu al wh o tests p osi tive on two tests actuall y do es not have th e disease ? If you ma ke an y assumpt ions in yo u r calculation, b e sur e to stat e th em and comm ent on wh eth er th ey are reaso n able . 1v. Clini cian s would like to incr ease th e prob ab ilit y from part ii. by ch ang ing th e cut off valu e th at con stitut es a po sitiv e t est res ult . Sugg est a reaso n able n ew cut off val ue x, such th at va lu es of x n g/ mL and h igh er qualif y as a p osit ive test res ult. Exp la in th e reason ing behind your an swe r. Limit you r an swe r to at most 7 sent enc es. PROBLEM 2: APPALOOSA HORSES (70 pts. total) The Appaloosa is an American horse breed known for its distinctive spotted coat pattern. Predecessors of the Appaloosa can be traced back to horses brought to America in the early 1600s by Spanish explorers. The Nez Perce tribe in the Northwest became widely recognized for their work in developing the Appaloosa breed. Appaloosa coat color is derived from a base color with an overlaid spotting pattern. For example, a leopard coat pattern is considered to be a mainly white body with dark spots that flow out over the entire body. All spotting patterns on an Appaloosa are caused by having at least one allele of a gene called the leopard-complex gene. However, Appaloosas show a wide range of spotting patterns: sorne have spots that extend all over the body, sorne have spots limited to the hip area, while others appear solid colored (and have only mottled skin). This is thought to be due to the effects of a separate gene that controls patterning. Consider a patterning gene with alleles A and a, where the particular genotype (AA, Aa, or aa) determines the extent to which an Appaloosa displays spotting. Suppose that 64% of Appaloosas are homozygous dominant (with genotype AA) and that 32% of Appaloosas are heterozygous (with genotype Aa) . Among Appaloosas with genotype AA, there is a 70% of full body spotting and 20% chance of partial body spotting. Appaloosas with genotype Aa have a 50% chance of full body spotting and 25% chance of partial body spotting. Appaloosas with genotype aa have a 35% chance of no spotting and 45% of partial body spotting. Let X represent the event that an Appaloosa displays spotting, where X2 represents full body spotting, X1 represents partial body spotting, and X0 indicates lack of spotting. Let A2 = {genotype AA), A1 = {genotype Aa), and A0 = {genotype aa). ~) (16 pts. ) Based on the above description, write a simulation that models a hypothetical population of 10,000 Appaloosas. In one paragraph, briefly explain the general logic and organization of your simulation; i.e., provide a high-level overview of the steps of your simulation. b) (6 pts.) From the simulated results, approximate the proportion of Appaloosas that exhibit spotting (i.e., do not appear salid colored). c) (8 pts .) From the simulated results, approximate the probability that an Appaloosa is heterozygous with genotype Aa far the patterning gene given that it exhibits spotting. For the following questions, you may use either a non-simulation or simulation approach. Be sure to clearly show your work and define any notation that you introduce in your solution. d) ( 40 pts. ) Night blindness refers toan inherited condition characterized by impaired vision in dark conditions; in dim light or darkness, affected horses experience loss of vision. Night blindness is thought to be associated with genes involved in producing the spotted coat pattern in Appaloosas. Suppose that among Appaloosas that have genotype AA and display spotting, night blindness occurs with probability 0.45; among Appaloosas that have genotype AA and do not display spotting, night blindness occurs with probability 0.08. About 9% of all Appaloosas are genotype Aa, display spotting, and have night blindness. In Appaloosas that are genotype aa and display spotting, night blindness occurs with probability 0.12; 1 % of Appaloosas that are genotype aa and do not display spotting have night blindness. Let B represent the event that an Appaloosa has night blindness . 1. Calculate the probability that a randomly selected Appaloosa is genotype AA and has night blindness. n. Given that an Appaloosa is genotype AA and has night blindness, what is the probability that it exhibits spotting? m. If an Appaloosa is spotted and known to be genotype Aa, what is the probability that it <loes not have night blindness? 1v. What is the probability that an Appaloosa has night blindness, if it is known to be genotype aa? PROBLEM 3: JUDICIAL EMPATHY (60 pts. total) Do judges exhibit empathetic feelings that might be based on personal relationships? Specifically, might a judge 's personal relationships affect judicial decision making? A study team decided to investigate this question in the specific context of one type of personal relationship: having daughters. The team collected data on 224 judges on the United States Courts of Appeals and their rulings on 990 gender-related cases. These cases involved discrimination against women or women's rights, such as cases about reproductive rights or employment discrimination on the basis of pregnancy. A judge was considered to have made a feminist decision in a case if they voted in favor of the plaintiff representing women 's interests. The study team also collected data on the number of biological children for each judge, and the sex of each child. Oher attributes were also collected, such as religion and year of birth. Note that the party of the president who appointed a judge can be used as a proxy for the judge 's political party . Data from the study are in the file judges. Rdata. The following table provides a list of the variables in the dataset and their descriptions . Variable name circuit children daughters sons sex year.born religion republican vote . prop Descri ption name feder al circuit court the judge serves in, of the 13 circuit s tot al number of biologi cal childr en total numb er of biological daught ers total numb er of biological sons sex of th e judge , eith er Female or Mal e year of birth religiou s affiliation recorded as 1 if appointed by a Republican pre sid ent, and 0 if by a Democrat proportion of femini st-lea ning votes in gend er-relat ed cases Use th ese d ata to an swer the following question s. a) (12 pt s.) Explore the variables sex and repub lican. 1. Describe the di stribution of se x, ref erencing appropriate numerical summaries . Do the same for the distribution of republican. 11. How much mor e likely is a female jud ge to have been appointed by a Democrat pr esident than a Republican pre sident ? b) (8 pt s.) Explore the variable vote . prop. 1. Describe the distribution of proportion of femini st-leanin g votes m gender -relat ed cases, refer encing appropriate numerical and grap hic al summaries . 11. Are judg es observed to ha ve vote proportion s close to 1 n ecessari ly mor e likely to make femini st decision s in future gender-related cases than those observed to have vote pro portion s closer to O.SO? Explain your answer. c) (10 pt s.) Crea te a single informati ve plot th at illu str ates the differ ence in votin g beha vior in gender-related cases between Republican male jud ges, Republic an female judges, Democ rat male judges, and Democrat female jud ges . Summarize your findings. Be sure to address whether the data sugge st political party, sex, or both are related to progressive voting be hav ior. d) (18 pts.) In ves tiga te the association betwee n pro gress ive votin g behav ior and having chil dr en. 1. Do judges with childr en tend to vote diff erentl y in gend er-related cas es than th ose with out childr en? Explain your answer. 11. How many jud ges only have sons and how many judges only have daughters? Do these two gro up s tend to vote differently in gende r-re lated cases? Expl ain your answer. m. Of jud ges with childr en, do those with at least one daughter tend to vot e differently in gend er-related cases than th ose who only have sons? Exp lain your answer. e) (8 pts .) A news outlet is in terested in reporti ng on the result s of your ana lysis . Write a brief summ ary of th e main findin gs in language accessible to a non -sta ti stici an , addressing the main question, "Do the se data suggest th at a judge 's personal relation ships affec t judicial decision makin g?" Limit your an swer to no more than seven sentences . f) ( 4 pts. ) Based on the way the st ud y data were collected, why might it be reasonable to int erp ret the findin gs in a cau sal manner; i.e ., why might it be reasonable to believe that the data can demonstra te a causal r elationship between having daughters and more freq uently voting in favor of women 's interests? Exp lain your an swer .

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