a) 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 for reducing the risk of giving birth before 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 before 37 weeks occurred in 668 of the women who took aspirin and 754 of those who took placebo.

i. Compute and interpret an appropriate summary statistic comparing the difference in outcomes between the two groups. Be sure to clearly show your work.

ii. 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) Invasive melanoma of the skin is a type of malignant skin cancer; melanoma accounts for 1% of all skin cancers diagnosed in the United States, 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 non- Hispanic white males: in this subgroup, melanoma occurs with rate 34.9 per 100,000 people annually.

i. In 2018, the population of Washington, D.C. was 702,000 residents, with 36.8% non- Hispanic 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.

ii. Compute the probability of observing at least 60 cases of melanoma among non- Hispanic white males in a given year.

iii. 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.

iv. 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; for melanoma, the five-year-survival rate is about 92%. What is the probability that, five years after initial diagnosis, more than 10 deaths occur among 60 cases of melanoma?

c) Many diagnostic tests for detecting the presence of disease are based on biomarkers; broadly speaking, a biomarker is any biological indicator that can be measured, such as blood pressure or fasting blood glucose level. Consider a diagnostic test for a hypothetical disease based on measuring the amount of a certain biomarker present in blood. High levels of the biomarker are often found in individuals with the disease, but a number of non-disease conditions can also cause high levels of the biomarker. Individuals without the disease have biomarker levels that are normally distributed with mean 1.6 ng/mL (nanograms per milliliter of blood), and standard deviation 0.50 ng/mL. Individuals with the disease have biomarker levels that are normally distributed with mean 5 ng/mL and standard deviation 1.2 ng/mL. Values of 2.5 ng/mL and higher constitute a positive test result.

i. Compute the accuracy of the test for those who have the disease and the accuracy of the test for those who do not have the disease.

ii. In a population where 6% of individuals are thought to have the hypothetical disease, calculate the probability that an individual who tests positive has the disease.

iii. In order to account for lab error, individuals who test positive are asked to return for another test at a later time. What is the probability that an individual who tests positive on two tests actually does not have the disease? If you make any assumptions in your calculation, be sure to state them and comment on whether they are reasonable.

iv. Clinicians would like to increase the probability from part ii. by changing the cutoff value that constitutes a positive test result. Suggest a reasonable new cutoff value x, such that values of x ng/mL and higher qualify as a positive test result. Explain the reasoning behind your answer. Limit your answer to at most 7 sentences.