Please provide R code for each Question.

a) Under a ketogenic diet, consumption of carbohydrates is severely limited while fat

intake occurs in high amounts, with as much as 90% of daily calories coming from fats. This

diet triggers the body to begin breaking down stored fat in the liver to produce energy rather

than metabolizing circulating blood sugar.

Some studies have suggested that ketogenic diets might affect bone health. A study was conducted

to examine bone health in athletes adhering to a ketogenic diet versus a traditional

high-carbohydrate diet while undergoing intense training for upcoming competitions. Prior

to the study, all participants were on a high-carbohydrate diet. Participants were allocated

into diet groups based on preference; about half the participants chose to try a ketogenic

diet. For four weeks, the study participants followed study diets developed by the research

team; the two treatment diets provided equal amounts of calories and differed only in the

ratio of carbohydrate versus fat intake.

The researchers assessed bone health by measuring the concentration of biomarkers related

to bone formation and bone breakdown. Researchers found that while the biomarkers of participants

in the high-carbohydrate group showed little change over the course of the study,

those of participants on the ketogenic diet showed signs indicative of impaired bone health:

higher concentration of biomarkers related to bone breakdown and lower concentration of

biomarkers related to bone growth.

i. In the paper reporting the results of the analysis, the authors claim "We have shown

that a 4-week ketogenic diet in athletes has negative effects on bone health during periods

of high intensity exercise". Provide a short response to the publication editor explaining

clearly why the claim is potentially misleading. Be sure to use language accessible

to a general audience without a statistics background. Limit your answer to at most

five sentences.

ii. Suppose that a friend of yours has been on a ketogenic diet for three months while

training for a marathon. While he found it challenging initially to adapt to the high-fat

diet, he is now accustomed to it and thinks the diet regimen has helped boost his energy

levels. Your friend reads about the study and wonders whether the results imply that

it might be advisable to switch to a high-carbohydrate diet for the remaining month

before the marathon takes place.

You are not convinced the study results suggest that your friend should consider stopping

the ketogenic diet. Explain two reasons supporting your viewpoint. Limit your

answer to at most eight sentences.

b) (8 pts.) Fasting blood glucose (FBG) levels can be used to diagnose whether someone has

Type 2 diabetes. Among non-diabetic individuals, FBG levels are approximately normally

distributed with mean 100 mg/dL and standard deviation 12 mg/dL.

i. What FBG level is needed to be in the upper 1% of the distribution for non-diabetic

individuals?

ii. An individual is classified as pre-diabetic if two consecutive FBG measurements are

between 100 and 125 mg/dL. Calculate the probability of a non-diabetic being classified

as pre-diabetic, under the assumption that these two consecutive measurements

are independent.

c) Consider the scenario of repeatedly rolling a 6-sided die. Starting with roll i = 1, let

Ri denote the result of roll i. If Ri > i, the die is rolled again; if Ri i, the die is not rolled

again. Let N denote the number of times the die is rolled.

Calculate P (N > 3).

d) (18 pts.) Suppose that at the beginning of March, a new strain of coronavirus, Coronavirus

Disease 2020 (COVID-20), emerges and begins to spread through Harvard College undergraduates.

The symptoms of COVID-20 are similar to the flu and other respiratory illnesses.

It is thought that the infection rate of COVID-20 will be similar to that of COVID-19 in

Wuhan, but that the disease will be much less deadly.

COVID-20 is expected to infect about 20 out of every 1,000 undergraduate students per

month for the rest of the semester, while the flu is expected to infect about 30 out of every

1,000 undergraduates per month for the rest of the semester. Assume that this rate remains

the same for each of the remaining months in the semester (March, April, and May). There

are currently 6,800 Harvard College undergraduates on campus.

You may assume that infection with COVID-20 is independent of infection with the flu.

i. You and your two roommates are concerned about becoming ill, either with flu or with

COVID-20. Calculate the probability that all of you will successfully avoid contracting

either flu or COVID-20 over the next month (i.e., by the end of March).

ii. Comment on two assumptions required to make the calculation in part i. and explain

whether they are reasonable. Limit your answer to at most five sentences.

iii. Calculate the total expected number of COVID-20 cases among Harvard College undergraduates

over the next three months.

iv. If more than 150 cases of COVID-20 are observed among the undergraduates during

the next month, the University will go into shut-down mode and cancel class meetings.

What is the probability the University goes into shut-down mode by the end of March?