Question
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,
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?
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