QUANTITATIVE REASONING WITH
DATA (GEA 1000) SEMISTER 1
MIDTERM TEST (suggested solutions)
2025-2026 UNIVERSITY OF SINGAPORE
, GEA1000 Quantitative Reasoning with Data
2025-2026 Semester 1
Midterm Test
Suggested Solutions
Q1
The table below shows male and female patients undergoing two treatment types, X or Y. The outcome of the
treatment is designated as either successful or unsuccessful. The success rates of the respective treatments across
genders are also calculated.
Male Female
No. of Patients No. of Rate of No. of Patients No. of Rate of
Treated Success Success Treated Success Success
Treatment X ? ? 50% 40 32 80%
Treatment Y ? ? ? ? ? 60%
Total 100 50 50% ? ? ?
Unfortunately, some of the data is missing. We know that all missing values are non-zero. Which of the following
statements is necessarily true?
(I) Simpson’s Paradox is observed when the subgroups of Treatment X and Treatment Y are combined, when
considering the relationship between gender and outcome.
(II) Treatment type is a confounder between the variables gender and outcome.
a) Only (I)
b) Only (II)
c) Both (I) and (II)
d) Neither (I) nor (II)
Explanation
By the basic rule of rates, among males, rate of success given Treatment Y must be 50%. By the basic rule of rates,
among females, overall rate of success must be between 60% and 80%. Simpson’s Paradox is not observed when the
subgroups are combined, hence (I) is false.
Depending on the data, treatment type may or may not be a confounder.
For the following given set of values in the table, rate(X|male) = 80/100 = 40/50 = rate(X|female), so treatment type
is not associated with gender. Treatment type is not necessarily a confounder hence (II) is false.
Male Female
Undergone Successful Rate of Undergone Successful Rate of
Treatment Success Treatment Success
Treatment X 80 40 50% 40 32 80%
Treatment Y 20 10 50% 10 6 60%
Total 100 50 50% 50 38 >60% and
<80%
DATA (GEA 1000) SEMISTER 1
MIDTERM TEST (suggested solutions)
2025-2026 UNIVERSITY OF SINGAPORE
, GEA1000 Quantitative Reasoning with Data
2025-2026 Semester 1
Midterm Test
Suggested Solutions
Q1
The table below shows male and female patients undergoing two treatment types, X or Y. The outcome of the
treatment is designated as either successful or unsuccessful. The success rates of the respective treatments across
genders are also calculated.
Male Female
No. of Patients No. of Rate of No. of Patients No. of Rate of
Treated Success Success Treated Success Success
Treatment X ? ? 50% 40 32 80%
Treatment Y ? ? ? ? ? 60%
Total 100 50 50% ? ? ?
Unfortunately, some of the data is missing. We know that all missing values are non-zero. Which of the following
statements is necessarily true?
(I) Simpson’s Paradox is observed when the subgroups of Treatment X and Treatment Y are combined, when
considering the relationship between gender and outcome.
(II) Treatment type is a confounder between the variables gender and outcome.
a) Only (I)
b) Only (II)
c) Both (I) and (II)
d) Neither (I) nor (II)
Explanation
By the basic rule of rates, among males, rate of success given Treatment Y must be 50%. By the basic rule of rates,
among females, overall rate of success must be between 60% and 80%. Simpson’s Paradox is not observed when the
subgroups are combined, hence (I) is false.
Depending on the data, treatment type may or may not be a confounder.
For the following given set of values in the table, rate(X|male) = 80/100 = 40/50 = rate(X|female), so treatment type
is not associated with gender. Treatment type is not necessarily a confounder hence (II) is false.
Male Female
Undergone Successful Rate of Undergone Successful Rate of
Treatment Success Treatment Success
Treatment X 80 40 50% 40 32 80%
Treatment Y 20 10 50% 10 6 60%
Total 100 50 50% 50 38 >60% and
<80%
