Let X  = ( X 1 ,...,Xn ) be the blood pressure (measured in mmHg ) and let Y = ( Y 1 ,...,Yn ) be the cortisol level (measured in mcg/dL ) recorded for n = 79 patients recruited for a study in a hospital ( Xi and Yi are measurements for the same patient). What test is most appropriate to gather evidence towards the alternative hypothesis that blood pressure is associated with cortisol level? Please provide the reasoning in detail for your answer. A)  The two-sample paired t -test with the null hypothesis that the means of X and Y differ. B)  The test with the null hypothesis that the Pearson correlation coefficient between X and Y is zero. C)  The test with the null hypothesis that the regression coefficient is zero in a linear regression with response variable X (blood pressure) and explanatory variable Y (cortisol level). (5 points) ii)  Suppose that a treatment is proposed to reduce the duration from the time of infection date, to the time at which a first negative test is recorded in people with mild COVID-19 (call this time period the duration ). Suppose that 27 people with mild COVID-19 (the study population) are administered the treatment and 73 people with mild COVID-19 are not administered the treatment (the control population). Both populations are sampled from patients tested at the same clinic over the same period. Let the durations for the study sample be X = ( X 1 , X 2 ,... ), and the durations for the control sample be Y = ( Y 1 , Y 2 ,... ). What test is most appropriate to gather evidence towards the alternative hypothesis that the treatment reduces the duration ? Please provide the reasoning for your answer. Please provide the reasoning in detail for your answer. A)  The one-sided two-sample unpaired t -test with H 0: The mean of X is greater than or equal to the mean of Y. B)  The one-sided two-sample unpaired t -test with the null hypothesis that the mean of X is less than or equal to the mean of Y. C)  The test against the null hypothesis that the Spearman’s ranked correlation coefficient between X and Y is zero. D)  The one-sided two-sample paired t -test against H 0: The mean difference between Xi and Yi is less than or equal to zero . E)  The two-sided two-sample paired t -test with the null hypothesis that the mean difference between Xi and Yi is zero . (5 points) iii)  Road vehicle accidents involving ambulances have more detrimental outcomes than accidents involving other similarly sized vehicles (Ray and Kupas, 2005). Measures to avoid such accidents are continually being refined by organizations involved in emergency medical services. Suppose that a city council is interested in knowing if adoption of such measures has lead to an improvement over the last decade. Suppose that the ratio between the number of accidents  iv)  involving ambulances (the numerator) and the number of kilometers driven by ambulances (the denominator) has be.

1. Let
X
= (
X
1
,...,Xn
) be the blood pressure (measured in
mmHg
) and let
Y
= (
Y
1
,...,Yn
) be the cortisol level (measured in
mcg/dL
) recorded for
n
= 79 patients recruited for a study in a hospital (
Xi
and
Yi
are measurements for the same patient). What test is most
appropriate to gather evidence towards the alternative
hypothesis that blood pressure is associated with cortisol level?
Please provide the reasoning in detail for your answer.
A) The two-sample paired
t
-test with the null hypothesis that the means of
X
and
Y
differ.

2. B) The test with the null hypothesis that the Pearson
correlation coefficient between
X
and
Y
is zero.
C) The test with the null hypothesis that the regression
coefficient is zero in a linear regression with response variable
X
(blood pressure) and explanatory variable
Y
(cortisol level).
(5 points)
ii) Suppose that a treatment is proposed to reduce the duration
from the time of infection date, to the time at which a first
negative test is recorded in people with mild COVID-19 (call
this time period
the duration
). Suppose that 27 people with mild COVID-19 (the study
population) are administered the treatment and 73 people with
mild COVID-19 are not administered the
treatment (the control population). Both populations are
sampled from patients tested at the same clinic over the same
period. Let
the durations
for the study sample be
X
= (
X
1
, X
2

3. ,...
), and
the durations
for the control sample be
Y
= (
Y
1
, Y
2
,...
). What test is most appropriate to gather evidence towards the
alternative hypothesis that the treatment reduces
the duration
? Please provide the reasoning for your answer. Please provide
the reasoning in detail for your answer.
A) The
one-sided
two-sample
unpaired t
-test with
H
0: The mean of
X
is
greater than or equal to the mean of Y.
B) The
one-sided
two-sample
unpaired t
-test with the null hypothesis that the mean of
X
is
less than or equal to the mean of Y.

4. C) The test against the null hypothesis that the Spearman’s
ranked correlation coefficient between
X
and
Y
is zero.
D) The
one-sided
two-sample
paired t
-test against
H
0: The mean difference between
Xi
and
Yi
is
less than or equal to zero
.
E) The
two-sided
two-sample
paired t
-test with the null hypothesis that the mean difference between
Xi
and
Yi
is
zero
.
(5 points)

5. iii) Road vehicle accidents involving ambulances have more
detrimental outcomes than accidents involving other similarly
sized vehicles (Ray and Kupas, 2005). Measures to avoid such
accidents are continually being refined by organizations
involved in emergency medical services. Suppose that a city
council is interested in knowing if adoption of such measures
has lead to an improvement over the last decade. Suppose that
the ratio between the number of accidents
iv) involving ambulances (the numerator) and the number of
kilometers driven by ambulances (the denominator) has been
recorded (
rt
with units’ number of accidents per kilometer year) for each
year
t
over the past decade. Which single one of the following
statistical quantities is most relevant
for investigating whether or not measures are leading to
improvements? Please provide the reasoning in detail for your
answer.
A) The sample standard deviation of
rt
.
B) The sample mean of
rt
.
C) The Pearson correlation coefficient
ρ
between
rt
and

6. t
.
D) The regression coefficient for
t
in a linear regression with
rt
as the response variable and
t
as the explanatory variable.
E) The regression coefficient for
rt
in a linear regression with
rt
as the explanatory variable and
t
as the response variable.
Problem 2: Bayes’ rule
A study was conducted to assess the sensitivity and specificity
of four different human immunodeficiency virus (HIV) serology
tests (Koblavi-D`eme et al. 2001). The
Determine
test was among the four, it was developed by Abbott
Laboratories (an American provider of health care, medical
devices and pharmaceuticals) and was found to have a true
negative rate (the true negative rate is also called specificity) of
99.4% and a true positive rate (the true positive rate is also
called sensitivity) of 100%. The true negative rate of a test for a
disease is the probability that someone without the disease tests
negative. The true positive rate of a test for a disease is the
probability that someone with the disease tests positive. HIV
may be transmitted from an expecting parent to their child by
transmission during childbirth or by transmission to the fetus

7. during pregnancy (throughout, assume that there’s no other way
for a newborn to be infected). Treatment by the drugs
zidovudine
or
nevirapine
has been shown to reduce the rate of these sorts of transmission
of HIV by 38% to 50% in the absence of other intervention
(Koblavi-D`eme et al. 2001).
a) Suppose that an expecting parent is infected with HIV and
they are treated with
zidovudine
or
nevirapine
during pregnancy. Suppose that after they give birth, a
Determine
serology test reports a positive test for HIV. What is the
probability that the child does not have HIV? Round your
answer to the nearest 10-th of a percent.
(6 points)
b) UNAIDS (an organization established by the United Nations
Economic and Social Council) estimates the prevalence of HIV
in Cˆote d’Ivoire among people aged 15-49 to be 2.6%. If a
Determine
serology test reported a positive test for HIV in someone
selected uniformly at random among all people in Cˆote d’Ivoire
aged 15-49, what is the probability that the person does not
have HIV? Round your answer to the nearest 10-th of a percent.
(4 points)
c) In the USA, according to the Centers for Disease Control (a
public health institute within the United States Department of
Health and Human Services), if someone has a positive serology

8. test for HIV they are not diagnosed as HIV-positive until a
second follow-up test also yields a positive test result. What is
the probability that someone is incorrectly diagnosed as HIV-
positive (
i.e.
, if someone is
not
infected with HIV, what is the probability that their first test
and also their second follow-up test are both positive)? Suppose
that both tests are
Determine
serology tests, and also assume that the test results are
statistically independent. Express your answer in expected
number of events in a million (
i.e.
something like ‘a 36 in a million chance’ or ‘a one in a million
chance’). Also: In one sentence, what is a
possible
argument as to why the assumption of independence of the two
test results might be wrong? (Your argument does not have to
be sound, but it must be valid without being tautological).
(3 points)
d) What is the probability that an HIV infected expecting parent
transmits HIV to their child either during childbirth or through
transmitting HIV to the fetus during pregnancy, given that the
parent has
not
received treatment with the drugs
zidovudine
or
nevirapine
, and in the absence of other intervention, according to the
preamble of this problem (in concordance with Koblavi-D`eme
et al. 2001)?

9. (2 points) Let
X
= (
X
1
,...,Xn
) be the blood pressure (measured in
mmHg
) and let
Y
= (
Y
1
,...,Yn
) be the cortisol level (measured in
mcg/dL
) recorded for
n
= 79 patients recruited for a study in a hospital (
Xi
and
Yi
are measurements for the same patient). What test is most
appropriate to gather evidence towards the alternative
hypothesis that blood pressure is associated with cortisol level?
Please provide the reasoning in detail for your answer.
A) The two-sample paired
t
-test with the null hypothesis that the means of
X
and
Y
differ.

10. B) The test with the null hypothesis that the Pearson
correlation coefficient between
X
and
Y
is zero.
C) The test with the null hypothesis that the regression
coefficient is zero in a linear regression with response variable
X
(blood pressure) and explanatory variable
Y
(cortisol level).
(5 points)
ii) Suppose that a treatment is proposed to reduce the duration
from the time of infection date, to the time at which a first
negative test is recorded in people with mild COVID-19 (call
this time period
the duration
). Suppose that 27 people with mild COVID-19 (the study
population) are administered the treatment and 73 people with
mild COVID-19 are not administered the
treatment (the control population). Both populations are
sampled from patients tested at the same clinic over the same
period. Let
the durations
for the study sample be
X
= (
X
1
, X
2

11. ,...
), and
the durations
for the control sample be
Y
= (
Y
1
, Y
2
,...
). What test is most appropriate to gather evidence towards the
alternative hypothesis that the treatment reduces
the duration
? Please provide the reasoning for your answer. Please provide
the reasoning in detail for your answer.
A) The
one-sided
two-sample
unpaired t
-test with
H
0: The mean of
X
is
greater than or equal to the mean of Y.
B) The
one-sided
two-sample
unpaired t
-test with the null hypothesis that the mean of
X
is
less than or equal to the mean of Y.

12. C) The test against the null hypothesis that the Spearman’s
ranked correlation coefficient between
X
and
Y
is zero.
D) The
one-sided
two-sample
paired t
-test against
H
0: The mean difference between
Xi
and
Yi
is
less than or equal to zero
.
E) The
two-sided
two-sample
paired t
-test with the null hypothesis that the mean difference between
Xi
and
Yi
is
zero
.
(5 points)

13. iii) Road vehicle accidents involving ambulances have more
detrimental outcomes than accidents involving other similarly
sized vehicles (Ray and Kupas, 2005). Measures to avoid such
accidents are continually being refined by organizations
involved in emergency medical services. Suppose that a city
council is interested in knowing if adoption of such measures
has lead to an improvement over the last decade. Suppose that
the ratio between the number of accidents
iv) involving ambulances (the numerator) and the number of
kilometers driven by ambulances (the denominator) has been
recorded (
rt
with units’ number of accidents per kilometer year) for each
year
t
over the past decade. Which single one of the following
statistical quantities is most relevant
for investigating whether or not measures are leading to
improvements? Please provide the reasoning in detail for your
answer.
A) The sample standard deviation of
rt
.
B) The sample mean of
rt
.
C) The Pearson correlation coefficient
ρ
between
rt
and

14. t
.
D) The regression coefficient for
t
in a linear regression with
rt
as the response variable and
t
as the explanatory variable.
E) The regression coefficient for
rt
in a linear regression with
rt
as the explanatory variable and
t
as the response variable.
Problem 2: Bayes’ rule
A study was conducted to assess the sensitivity and specificity
of four different human immunodeficiency virus (HIV) serology
tests (Koblavi-D`eme et al. 2001). The
Determine
test was among the four, it was developed by Abbott
Laboratories (an American provider of health care, medical
devices and pharmaceuticals) and was found to have a true
negative rate (the true negative rate is also called specificity) of
99.4% and a true positive rate (the true positive rate is also
called sensitivity) of 100%. The true negative rate of a test for a
disease is the probability that someone without the disease tests
negative. The true positive rate of a test for a disease is the
probability that someone with the disease tests positive. HIV
may be transmitted from an expecting parent to their child by
transmission during childbirth or by transmission to the fetus

15. during pregnancy (throughout, assume that there’s no other way
for a newborn to be infected). Treatment by the drugs
zidovudine
or
nevirapine
has been shown to reduce the rate of these sorts of transmission
of HIV by 38% to 50% in the absence of other intervention
(Koblavi-D`eme et al. 2001).
a) Suppose that an expecting parent is infected with HIV and
they are treated with
zidovudine
or
nevirapine
during pregnancy. Suppose that after they give birth, a
Determine
serology test reports a positive test for HIV. What is the
probability that the child does not have HIV? Round your
answer to the nearest 10-th of a percent.
(6 points)
b) UNAIDS (an organization established by the United Nations
Economic and Social Council) estimates the prevalence of HIV
in Cˆote d’Ivoire among people aged 15-49 to be 2.6%. If a
Determine
serology test reported a positive test for HIV in someone
selected uniformly at random among all people in Cˆote d’Ivoire
aged 15-49, what is the probability that the person does not
have HIV? Round your answer to the nearest 10-th of a percent.
(4 points)
c) In the USA, according to the Centers for Disease Control (a
public health institute within the United States Department of
Health and Human Services), if someone has a positive serology

16. test for HIV they are not diagnosed as HIV-positive until a
second follow-up test also yields a positive test result. What is
the probability that someone is incorrectly diagnosed as HIV-
positive (
i.e.
, if someone is
not
infected with HIV, what is the probability that their first test
and also their second follow-up test are both positive)? Suppose
that both tests are
Determine
serology tests, and also assume that the test results are
statistically independent. Express your answer in expected
number of events in a million (
i.e.
something like ‘a 36 in a million chance’ or ‘a one in a million
chance’). Also: In one sentence, what is a
possible
argument as to why the assumption of independence of the two
test results might be wrong? (Your argument does not have to
be sound, but it must be valid without being tautological).
(3 points)
d) What is the probability that an HIV infected expecting parent
transmits HIV to their child either during childbirth or through
transmitting HIV to the fetus during pregnancy, given that the
parent has
not
received treatment with the drugs
zidovudine
or
nevirapine
, and in the absence of other intervention, according to the
preamble of this problem (in concordance with Koblavi-D`eme
et al. 2001)?

17. (2 points)