MATH4330 Assignment 1 -Solved

. Question 1: A researcher wants to study whether an existing drug causes a particular 
kind of skin reaction. She collects data from an existing medical database; she samples 
individuals who showed the skin reaction as well as individuals without a reaction. She 
then examines how many individuals had taken the drug of interest. 
(a) ] What kind of study is this? Be as specifific as possible, and explain your 
answer. 
(b) Suppose the researcher fifinds that those who took the drug had a higher 
chance of developing a skin reaction. Do you believe this result? Can you think of 
other specifific factors that might a↵ect the conclusion? Explain your answers. 
Question A researcher runs a randomized experiment where study partic
ipants are randomized to be given either a drug or a placebo. Another researcher wants 
to perform an additional study on this same study group. He asks participants whether 
they get more or less than 3 hours of exercise per week. He concludes that individuals 
with more than 3 hours of exercise per week have lower blood pressure than those who 
get less than 3 hours, on average. Are you worried about confounders in this conclusion? 
Explain. 
Question 3: We want to study the genetics of colour distribution in a population of 
unicorns. Suppose that in unicorns there is a single gene that determines colour; there 
are two variants of the gene, one called A and the other called a. Each unicorn has two 
copies of the gene (one from each parent). The combination of the gene variants for copy 
1 and copy 2 of the gene determines colour as follows: 
Copy 1 Copy 2 Colour 
A A 
Red 
A a 
Pink 
a A 
Pink 
a a 
White 
We want to determine if “random mating” is happening in this unicorn population. This 
would mean that the probability that a newly born unicorn inherits variant A or a with 
probability equal to the prevalence of each variant in the overall population. 
1(a)  Let p be the proportion of the A variant in the population, and q be the 
proportion of the variant a, so that q = 1−p
. Under random mating, each newborn 
unicorn inherits A with probability p and 
a 
with probability q and the two copies 
inherited in each unicorn are independent of one another. Calculate the expected 
proportions of unicorn colours under random mating. 
(b)  Suppose you know that p = 0.75 and q = 0.25, and that you collect the 
following data from a sample of unicorns: 
Colour Number of unicorns 
Red 45 
Pink 49 
White 12 
Use the appropriate statistical test to determine whether the random mating as
sumption holds in this population. Write out your calculations for this part; 
using R for this part will not result in credit. 
2Ravish Kamath 

213893664 
Question 1 
A researcher wants to study whether an existing drug causes a particular 
kind of skin reaction. She collects data from an existing medical database; 
she samples individuals who showed the skin reaction as well as individuals 
without a reaction. She then examines how many individuals had taken the 
drug of interest. 
(a) What kind of study is this? Be as specifific as possible, and explain your 
answer. 
(b) Suppose the researcher fifinds that those who took the drug had a higher 
chance of developing a skin reaction. Do you believe this result? Can 
you think of other specifific factors that might affffect the conclusion? 
Explain your answers. 
Solution 
(a) This study would be considered a retrospective study. The reason 
would be that the researcher is looking into an already existing database 
system, hence she is looking into the past for her subjects. Based on 
this medical database system, she is sampling individuals who already 
showed a skin reaction vs. individuals that did not have a reaction. 
(b) It is possible to have this result. However there might be certain factors 
that can affffect the conclusion by the researcher. For example, season
ality, where exposure to high levels of sunlight can cause skin reaction, 
food allergies can cause skin reactions, and fifinally prior infections and 
diseases could have caused similar skin reaction. All of these have no 
relation to the drug that was used by the patients, however they were 
still able to get skin reactions though not from the drug. This can 
cause some problems with the way the experiment was held. However 
depending on the sample size, these may be outlier situations, and the 
drug does indeed cause a development of skin reaction. 
Page 1Ravish Kamath 
4330: Assignment 1 
213893664 
Question 2 
A researcher runs a randomized experiment where study participants are 
randomized to be given either a drug or a placebo. Another researcher 
wants to perform an additional study on this same study group. He asks 
participants whether they get more or less than 3 hours of exercise per 
week. He concludes that individuals with more than 3 hours of exercise per 
week have lower blood pressure than those who get less than 3 hours, on 
average. Are you worried about confounders in this conclusion? Explain. 
Solution 
Yes there is a worry about confounders in this conclusion. The fifirst 
confounder that can be thought offff is age of the participants. Age is a 
factor because older people may have less time to exercise due to work, 
family etc. and with age, high blood pressure is quite common than younger 
people. Because of the age confounder, it may seem that less hours of exercise 
will cause higher blood pressure, however it could just be the age of those 
individuals that may be causing the high blood pressure. 
Page 2Ravish Kamath 
4330: Assignment 1 
213893664 
Question 3 
Question 3: We want to study the genetics of colour distribution in a pop
ulation of unicorns. Suppose that in unicorns there is a single gene that 
determines colour; there are two variants of the gene, one called A and the 
other called a. Each unicorn has two copies of the gene (one from each par
ent). The combination of the gene variants for copy 1 and copy 2 of the gene 
determines colour as follows: 
Copy 1 Copy 2 Colour 
A A 
Red 
A a 
Pink 
a A 
Pink 
a a 
White 
We want to determine if “random mating” is happening in this unicorn pop
ulation. This would mean that the probability that a newly born unicorn 
inherits variant A or a with probability equal to the prevalence of each variant 
in the overall population. 
(a) Let p be the proportion of the A variant in the population, and q be 
the proportion of the variant a, so that q = 1 − p . Under random 
mating, each newborn unicorn inherits A with probability 
p and a 
with probability q and the two copies inherited in each unicorn are 
independent of one another. Calculate the expected proportions of 
unicorn colours under random mating. 
(b) Suppose you know that p = 0.75 and q = 0.25, and that you collect the 
following data from a sample of unicorns: 
Colour Number of unicorns 
Red 45 
Pink 49 
White 12 
Use the appropriate statistical test to determine whether the random mating 
assumption holds in this population. Write out your calculations for 
this part; using R for this part will not result in credit. 
Page 3Ravish Kamath 
4330: Assignment 1 
213893664 
Solution 
(a) 
E(Red Unicorn Colour) = proportion of A ⇥ proportion of A 
= (p ⇥ p) 
= p2 
E(Pink Unicorn Colour) = pproportion of a ⇥ proportion of A 
= (q ⇥ p) 
= (qp) 
= p(1 − p) 
E(White Unicorn Colour) = proportion of a ⇥ proportion of a 
= (q ⇥ q) 
= q2 
= (1 − p)2 
(b) Let n = 45 + 49 + 12 = 106. 
E(Red Unicorn colour) = (0.75)2 ⇥ 106 
= 59.625 
E(Pink Unicorn Colour) = (0.75) ⇥ (0.25) ⇥ (106) 
= 19.875 
E(White Unicorn Colour) = (0.25)2 ⇥ 106 
= 59.625 
χ2 = (45 − 59.625)2 
59.625 
+ 
(49 − 19.875)2 
19.875 
+ 
(12 − 59.625)2 
59.625 
= 84.3074 
H0: Random mating does occur in the unicorn population 
Ha: Random mating does not occur 
Let ↵ = 0.05 and df = 3 − 1=2 
Page 4Ravish Kamath 
4330: Assignment 1 
213893664 
Using R to calculate th test statistic, we get that the p-value is 4.9304⇥ 
10−19 which is well below 0.05. Hence we reject H0 and say that there 
is no evidence to show that random mating does occur in the unicorn 
population. 
Page 5Question 1 
A) This study would be considered a retrospective study. The reason would be that the 
researcher is looking into an already existing database system, hence she is looking into 
the past for her subjects. Based on this medical database system, she is sampling 
individuals who already showed a skin reaction vs individuals that did not have a 
reaction. 
B) It is possible to have this result. However there might be certain factors that can 
affect the conclusion by the researcher. For example, seasonality, where exposure to high 
levels of sunlight can cause skin reaction, food allergies can cause skin reactions, and 
finally, prior infections and diseases could have caused similar skin reaction. All of these 
have no relation to the drug that was used by the patients, however they were still able 
to get skin reactions though not from the drug. This can cause some problems with the 
way the experiment was held. However depending on the sample size, these may be 
outlier situations, and the drug does indeed cause a development of skin reaction. Question 2 
Yes there is a worry about confounders in this conclusion. The first confounder that can 
be thought off is age of the participants. Age is a factor because older people may have 
less time to exercise due to work, family etc. and with age, high blood pressure is quite 
common than younger people. Because of the age confounder, it may seem that less 
hours of exercise will cause higher blood pressure, however it could just be the age of 
those individuals that may be causing the high blood pressure. Question 3