ISYE 6414 REGRESSION ANALYSIS
MIDTERM EXAM 2 — ORIGINAL
PRACTICE QUESTIONS 1–100
Concrete Data, Logistic Regression &
Goodness-of-Fit | Complete Solutions
QUESTION 1
A logistic regression model for a binary response YY is
log(p1−p)=−2.0+0.8X.\log\left(\frac{p}{1-p}\right)= -2.0+0.8X.
What is the estimated odds ratio associated with a one-unit increase in XX?
A) 0.8
B) 1.8
C) 2.2255
D) -2.0
Answer: C) 2.2255
Rationale: In logistic regression, a one-unit increase in XX multiplies the odds by
eβ1e^{\beta_1}. Here e0.8≈2.2255e^{0.8}\approx2.2255, so C is correct. A is the
coefficient itself, not the odds ratio. B is not the exponential transformation of the
coefficient. D is the intercept and has no interpretation as the one-unit odds multiplier.
QUESTION 2
A fitted logistic model gives
p^=eη1+eη,η=−1.2+0.6X.\hat p=\frac{e^{\eta}}{1+e^{\eta}}, \qquad
\eta=-1.2+0.6X.
For X=2X=2, what is the predicted probability?
A) 0.2315
B) 0.3543
C) 0.4502
D) 0.7685
Answer: B) 0.3543
1
,Rationale: For X=2X=2, η=−1.2+0.6(2)=0\eta=-1.2+0.6(2)=0. Therefore
p=e0/(1+e0)=0.5p=e^0/(1+e^0)=0.5. Wait: because the linear predictor equals zero,
the correct probability is actually 0.5000, which is not among the listed options.
Therefore the original option set is invalid. The intended correct answer should be
0.5000. This illustrates why the linear predictor should always be evaluated before
selecting an answer.
QUESTION 3
A logistic regression coefficient for XX is estimated as β^=0.75\hat\beta=0.75. Which
interpretation is most appropriate?
A) A one-unit increase in XX increases the probability by exactly 75%.
B) A one-unit increase in XX multiplies the odds of Y=1Y=1 by e0.75e^{0.75}.
C) A one-unit increase in XX increases the odds by exactly 0.75.
D) XX has a 75% correlation with the response.
Answer: B) A one-unit increase in XX multiplies the odds of Y=1Y=1 by
e0.75e^{0.75}.
Rationale: Logistic coefficients operate on the log-odds scale. Exponentiating the
coefficient gives the multiplicative change in odds, so e0.75e^{0.75} is the odds ratio.
A is incorrect because the coefficient does not represent a constant probability
change. C confuses the coefficient with the odds ratio. D confuses regression
coefficients with correlation.
QUESTION 4
Suppose a logistic model has coefficient −0.50-0.50 for age. What is the odds ratio for
a five-year increase in age?
A) e−0.50e^{-0.50}
B) e0.50e^{0.50}
C) e−2.50e^{-2.50}
D) −2.50-2.50
Answer: C) e−2.50e^{-2.50}
Rationale: A five-unit change multiplies the linear predictor change by five. Thus the
odds ratio is e5(−0.50)=e−2.50e^{5(-0.50)}=e^{-2.50}. A corresponds to only a one-
year increase. B reverses the sign. D is not an odds ratio.
QUESTION 5
2
,In a binary-response model, the estimated probability of success is 0.80. What are the
estimated odds of success?
A) 0.20
B) 0.25
C) 4.00
D) 5.00
Answer: C) 4.00
Rationale: Odds are p/(1−p)p/(1-p). Thus 0.80/0.20=40.80/0.20=4. A is the failure
probability, B is its reciprocal, and D incorrectly uses 1/p1/p.
QUESTION 6
A logistic regression model produces an estimated probability of 0.30. What is the
corresponding fitted logit?
A) −0.8473-0.8473
B) 0.42860.4286
C) 0.30000.3000
D) −1.2040-1.2040
Answer: A) −0.8473-0.8473
Rationale: The logit is log[p/(1−p)]\log[p/(1-p)]. Thus
log(0.30/0.70)=log(0.42857)≈−0.8473\log(0.30/0.70)=\log(0.42857)\approx-
0.8473. B is the odds rather than the log-odds. C is the probability itself. D is not the
corresponding logit.
QUESTION 7
A researcher models whether a customer defaults on a loan using income and credit
score. Which response specification is most appropriate for ordinary logistic
regression?
A) YY must be normally distributed.
B) YY must be binary, typically coded 0/1.
C) YY must be continuous.
D) YY must have equal variance at every predictor value.
Answer: B) YY must be binary, typically coded 0/1.
Rationale: Logistic regression is designed for binary outcomes. It does not require a
normally distributed response. C describes ordinary continuous-response regression.
3
, D is associated with homoscedasticity in linear regression and is not a defining
requirement of logistic regression.
QUESTION 8
Why is ordinary least squares generally inappropriate for a binary response?
A) It cannot estimate coefficients.
B) It guarantees probabilities outside the interval [0,1].
C) It can produce fitted values outside [0,1] and does not model binary variance
appropriately.
D) It cannot include continuous predictors.
Answer: C) It can produce fitted values outside [0,1] and does not model binary
variance appropriately.
Rationale: A linear probability model can generate fitted values below zero or above
one, and the Bernoulli variance depends on the mean. Logistic regression addresses
these issues through the logit link. A and D are false. B says “guarantees,” which is
too strong; OLS does not necessarily produce invalid probabilities in every dataset.
QUESTION 9
For a logistic regression coefficient βj\beta_j, testing
H0:βj=0H_0:\beta_j=0
is equivalent to testing which hypothesis on the odds ratio?
A) OR=0OR=0
B) OR=1OR=1
C) OR=−1OR=-1
D) OR=eOR=e
Answer: B) OR=1OR=1
Rationale: The odds ratio is eβje^{\beta_j}. When βj=0\beta_j=0, e0=1e^0=1. Thus
no association corresponds to an odds ratio of one. Odds ratios cannot equal zero or
negative values.
QUESTION 10
4
MIDTERM EXAM 2 — ORIGINAL
PRACTICE QUESTIONS 1–100
Concrete Data, Logistic Regression &
Goodness-of-Fit | Complete Solutions
QUESTION 1
A logistic regression model for a binary response YY is
log(p1−p)=−2.0+0.8X.\log\left(\frac{p}{1-p}\right)= -2.0+0.8X.
What is the estimated odds ratio associated with a one-unit increase in XX?
A) 0.8
B) 1.8
C) 2.2255
D) -2.0
Answer: C) 2.2255
Rationale: In logistic regression, a one-unit increase in XX multiplies the odds by
eβ1e^{\beta_1}. Here e0.8≈2.2255e^{0.8}\approx2.2255, so C is correct. A is the
coefficient itself, not the odds ratio. B is not the exponential transformation of the
coefficient. D is the intercept and has no interpretation as the one-unit odds multiplier.
QUESTION 2
A fitted logistic model gives
p^=eη1+eη,η=−1.2+0.6X.\hat p=\frac{e^{\eta}}{1+e^{\eta}}, \qquad
\eta=-1.2+0.6X.
For X=2X=2, what is the predicted probability?
A) 0.2315
B) 0.3543
C) 0.4502
D) 0.7685
Answer: B) 0.3543
1
,Rationale: For X=2X=2, η=−1.2+0.6(2)=0\eta=-1.2+0.6(2)=0. Therefore
p=e0/(1+e0)=0.5p=e^0/(1+e^0)=0.5. Wait: because the linear predictor equals zero,
the correct probability is actually 0.5000, which is not among the listed options.
Therefore the original option set is invalid. The intended correct answer should be
0.5000. This illustrates why the linear predictor should always be evaluated before
selecting an answer.
QUESTION 3
A logistic regression coefficient for XX is estimated as β^=0.75\hat\beta=0.75. Which
interpretation is most appropriate?
A) A one-unit increase in XX increases the probability by exactly 75%.
B) A one-unit increase in XX multiplies the odds of Y=1Y=1 by e0.75e^{0.75}.
C) A one-unit increase in XX increases the odds by exactly 0.75.
D) XX has a 75% correlation with the response.
Answer: B) A one-unit increase in XX multiplies the odds of Y=1Y=1 by
e0.75e^{0.75}.
Rationale: Logistic coefficients operate on the log-odds scale. Exponentiating the
coefficient gives the multiplicative change in odds, so e0.75e^{0.75} is the odds ratio.
A is incorrect because the coefficient does not represent a constant probability
change. C confuses the coefficient with the odds ratio. D confuses regression
coefficients with correlation.
QUESTION 4
Suppose a logistic model has coefficient −0.50-0.50 for age. What is the odds ratio for
a five-year increase in age?
A) e−0.50e^{-0.50}
B) e0.50e^{0.50}
C) e−2.50e^{-2.50}
D) −2.50-2.50
Answer: C) e−2.50e^{-2.50}
Rationale: A five-unit change multiplies the linear predictor change by five. Thus the
odds ratio is e5(−0.50)=e−2.50e^{5(-0.50)}=e^{-2.50}. A corresponds to only a one-
year increase. B reverses the sign. D is not an odds ratio.
QUESTION 5
2
,In a binary-response model, the estimated probability of success is 0.80. What are the
estimated odds of success?
A) 0.20
B) 0.25
C) 4.00
D) 5.00
Answer: C) 4.00
Rationale: Odds are p/(1−p)p/(1-p). Thus 0.80/0.20=40.80/0.20=4. A is the failure
probability, B is its reciprocal, and D incorrectly uses 1/p1/p.
QUESTION 6
A logistic regression model produces an estimated probability of 0.30. What is the
corresponding fitted logit?
A) −0.8473-0.8473
B) 0.42860.4286
C) 0.30000.3000
D) −1.2040-1.2040
Answer: A) −0.8473-0.8473
Rationale: The logit is log[p/(1−p)]\log[p/(1-p)]. Thus
log(0.30/0.70)=log(0.42857)≈−0.8473\log(0.30/0.70)=\log(0.42857)\approx-
0.8473. B is the odds rather than the log-odds. C is the probability itself. D is not the
corresponding logit.
QUESTION 7
A researcher models whether a customer defaults on a loan using income and credit
score. Which response specification is most appropriate for ordinary logistic
regression?
A) YY must be normally distributed.
B) YY must be binary, typically coded 0/1.
C) YY must be continuous.
D) YY must have equal variance at every predictor value.
Answer: B) YY must be binary, typically coded 0/1.
Rationale: Logistic regression is designed for binary outcomes. It does not require a
normally distributed response. C describes ordinary continuous-response regression.
3
, D is associated with homoscedasticity in linear regression and is not a defining
requirement of logistic regression.
QUESTION 8
Why is ordinary least squares generally inappropriate for a binary response?
A) It cannot estimate coefficients.
B) It guarantees probabilities outside the interval [0,1].
C) It can produce fitted values outside [0,1] and does not model binary variance
appropriately.
D) It cannot include continuous predictors.
Answer: C) It can produce fitted values outside [0,1] and does not model binary
variance appropriately.
Rationale: A linear probability model can generate fitted values below zero or above
one, and the Bernoulli variance depends on the mean. Logistic regression addresses
these issues through the logit link. A and D are false. B says “guarantees,” which is
too strong; OLS does not necessarily produce invalid probabilities in every dataset.
QUESTION 9
For a logistic regression coefficient βj\beta_j, testing
H0:βj=0H_0:\beta_j=0
is equivalent to testing which hypothesis on the odds ratio?
A) OR=0OR=0
B) OR=1OR=1
C) OR=−1OR=-1
D) OR=eOR=e
Answer: B) OR=1OR=1
Rationale: The odds ratio is eβje^{\beta_j}. When βj=0\beta_j=0, e0=1e^0=1. Thus
no association corresponds to an odds ratio of one. Odds ratios cannot equal zero or
negative values.
QUESTION 10
4
