ISYE 6414 Midterm Exam 1 Part 2 - Complete R Solutions, Code
& Interpretations | Questions and Answers | 2026 Update | 100%
Correct - GT. - 114 Questions and Answers Already Graded A+
Premium Exam Tested And Verified
Subject Area Regression Analysis and R Programming
Description This exam assesses mastery of multiple linear regression, model diagnostics,
variable selection, and interpretation of R output. It covers transformations,
interactions, multicollinearity, residual analysis, and model comparison
techniques using R.
Expected Grade A+
Total Questions 114
Duration 3 hours
Learning Outcomes 1. Interpret regression coefficients in various functional forms
2. Diagnose model assumptions using residual plots and statistics
3. Select appropriate models using information criteria and adjusted R²
4. Detect and address multicollinearity, outliers, and influential points
Accreditation This exam adheres to the rigorous standards of top US R1 research universities,
emphasizing both theoretical understanding and practical application in R.
Page 1
,Question 1 of 114
In a multiple linear regression model fit in R, the summary output shows a large p-value for the
overall F-test but one individual predictor has a very small p-value. Which of the following is the
most likely explanation?
A. The model suffers from severe multicollinearity that inflates the standard errors of all predictors.
B. The significant predictor is highly correlated with the response but the others contribute little, yet the F-test
remains non-significant due to low sample size.
C. The predictors are orthogonal and the F-test is incorrectly computed by R.
D. The residual degrees of freedom are zero because the model is saturated.
The correct answer is:
Correct Action: The significant predictor is highly correlated with the response but the others contribute
little, yet the F-test remains non-significant due to low sample size.
Rationales
• The significant predictor is highly correlated with the response but the others contribute little, yet the F-test
remains non-significant due to low sample size. (Correct):
This is the correct action. A non-significant overall F-test despite a significant individual predictor can occur when the
sample size is small and the other predictors add little explanatory power. The F-test tests the joint null that all coefficients
are zero; if only one predictor is strong, the F-test may not reject because the others dilute the overall effect and the test
• The model suffers from severe multicollinearity that inflates the standard errors of all predictors. (Incorrect):
This option is not appropriate. The F-test tests the joint null that all coefficients are zero; if only one predictor is strong, the
F-test may not reject because the others dilute the overall effect and the test lacks power. Multicollinearity (A) would
inflate standard errors of correlated predictors, but the significant predictor would likely have a large SE too
• The predictors are orthogonal and the F-test is incorrectly computed by R. (Incorrect):
This option is not appropriate. The F-test tests the joint null that all coefficients are zero; if only one predictor is strong, the
F-test may not reject because the others dilute the overall effect and the test lacks power. Multicollinearity (A) would
inflate standard errors of correlated predictors, but the significant predictor would likely have a large SE too
• The residual degrees of freedom are zero because the model is saturated. (Incorrect):
This option is not appropriate. The F-test tests the joint null that all coefficients are zero; if only one predictor is strong, the
F-test may not reject because the others dilute the overall effect and the test lacks power. Multicollinearity (A) would
inflate standard errors of correlated predictors, but the significant predictor would likely have a large SE too
Page 2
,Question 2 of 114
A linear regression of log(y) on log(x1) and x2 (with x2 untransformed) yields coefficients: log(y) =
0.5*log(x1) + 0.3*x2. Which statement correctly interprets the effect of x1 on y?
A. A 1% increase in x1 is associated with a 0.5% increase in y, holding x2 constant.
B. A 1-unit increase in x1 is associated with a 0.5% increase in y, holding x2 constant.
C. A 1% increase in x1 is associated with a 0.5-unit increase in log(y) holding x2 constant.
D. A 1% increase in x1 is associated with a 50% increase in y, holding x2 constant.
The correct answer is:
Correct Action: A 1% increase in x1 is associated with a 0.5% increase in y, holding x2 constant.
Rationales
• A 1% increase in x1 is associated with a 0.5% increase in y, holding x2 constant. (Correct):
This is the correct action. In a log-log model, the coefficient for a log-transformed predictor represents elasticity: a 1%
change in the predictor leads to a (coefficient) % change in the response. Here, 0.5 means a 1% increase in x1 is associated
with a 0.5% increase in y, holding x2 constant. Option B confuses unit and percent changes. Option C misstates the
• A 1-unit increase in x1 is associated with a 0.5% increase in y, holding x2 constant. (Incorrect):
This option is not appropriate. Here, 0.5 means a 1% increase in x1 is associated with a 0
• A 1% increase in x1 is associated with a 0.5-unit increase in log(y) holding x2 constant. (Incorrect):
This option is not appropriate. Here, 0.5 means a 1% increase in x1 is associated with a 0
• A 1% increase in x1 is associated with a 50% increase in y, holding x2 constant. (Incorrect):
This option is not appropriate. Here, 0.5 means a 1% increase in x1 is associated with a 0
Page 3
, Question 3 of 114
You fit a linear model and examine residual vs. fitted plot and the scale-location plot. The residual vs.
fitted plot shows a clear funnel shape (spread increasing with fitted values), and the scale-location
plot shows an upward trend. Which assumption is most likely violated?
A. Linearity of the relationship
B. Normality of errors
C. Constant variance (homoscedasticity)
D. Independence of errors
The correct answer is:
Correct Action: Constant variance (homoscedasticity)
Rationales
• Constant variance (homoscedasticity) (Correct):
This is the correct action. A funnel shape in the residual vs. fitted plot (increasing spread) indicates heteroscedasticity: the
variance of errors changes with the fitted values. The scale-location plot (square root of standardized residuals vs. fitted)
showing an upward trend confirms increasing variance. Linearity would show a curved pattern in the residual plot, not
• Linearity of the relationship (Incorrect):
This option is not appropriate. fitted plot (increasing spread) indicates heteroscedasticity: the variance of errors changes
with the fitted values. The scale-location plot (square root of standardized residuals vs
• Normality of errors (Incorrect):
This option is not appropriate. fitted plot (increasing spread) indicates heteroscedasticity: the variance of errors changes
with the fitted values. The scale-location plot (square root of standardized residuals vs
• Independence of errors (Incorrect):
This option is not appropriate. fitted plot (increasing spread) indicates heteroscedasticity: the variance of errors changes
with the fitted values. The scale-location plot (square root of standardized residuals vs
Page 4
& Interpretations | Questions and Answers | 2026 Update | 100%
Correct - GT. - 114 Questions and Answers Already Graded A+
Premium Exam Tested And Verified
Subject Area Regression Analysis and R Programming
Description This exam assesses mastery of multiple linear regression, model diagnostics,
variable selection, and interpretation of R output. It covers transformations,
interactions, multicollinearity, residual analysis, and model comparison
techniques using R.
Expected Grade A+
Total Questions 114
Duration 3 hours
Learning Outcomes 1. Interpret regression coefficients in various functional forms
2. Diagnose model assumptions using residual plots and statistics
3. Select appropriate models using information criteria and adjusted R²
4. Detect and address multicollinearity, outliers, and influential points
Accreditation This exam adheres to the rigorous standards of top US R1 research universities,
emphasizing both theoretical understanding and practical application in R.
Page 1
,Question 1 of 114
In a multiple linear regression model fit in R, the summary output shows a large p-value for the
overall F-test but one individual predictor has a very small p-value. Which of the following is the
most likely explanation?
A. The model suffers from severe multicollinearity that inflates the standard errors of all predictors.
B. The significant predictor is highly correlated with the response but the others contribute little, yet the F-test
remains non-significant due to low sample size.
C. The predictors are orthogonal and the F-test is incorrectly computed by R.
D. The residual degrees of freedom are zero because the model is saturated.
The correct answer is:
Correct Action: The significant predictor is highly correlated with the response but the others contribute
little, yet the F-test remains non-significant due to low sample size.
Rationales
• The significant predictor is highly correlated with the response but the others contribute little, yet the F-test
remains non-significant due to low sample size. (Correct):
This is the correct action. A non-significant overall F-test despite a significant individual predictor can occur when the
sample size is small and the other predictors add little explanatory power. The F-test tests the joint null that all coefficients
are zero; if only one predictor is strong, the F-test may not reject because the others dilute the overall effect and the test
• The model suffers from severe multicollinearity that inflates the standard errors of all predictors. (Incorrect):
This option is not appropriate. The F-test tests the joint null that all coefficients are zero; if only one predictor is strong, the
F-test may not reject because the others dilute the overall effect and the test lacks power. Multicollinearity (A) would
inflate standard errors of correlated predictors, but the significant predictor would likely have a large SE too
• The predictors are orthogonal and the F-test is incorrectly computed by R. (Incorrect):
This option is not appropriate. The F-test tests the joint null that all coefficients are zero; if only one predictor is strong, the
F-test may not reject because the others dilute the overall effect and the test lacks power. Multicollinearity (A) would
inflate standard errors of correlated predictors, but the significant predictor would likely have a large SE too
• The residual degrees of freedom are zero because the model is saturated. (Incorrect):
This option is not appropriate. The F-test tests the joint null that all coefficients are zero; if only one predictor is strong, the
F-test may not reject because the others dilute the overall effect and the test lacks power. Multicollinearity (A) would
inflate standard errors of correlated predictors, but the significant predictor would likely have a large SE too
Page 2
,Question 2 of 114
A linear regression of log(y) on log(x1) and x2 (with x2 untransformed) yields coefficients: log(y) =
0.5*log(x1) + 0.3*x2. Which statement correctly interprets the effect of x1 on y?
A. A 1% increase in x1 is associated with a 0.5% increase in y, holding x2 constant.
B. A 1-unit increase in x1 is associated with a 0.5% increase in y, holding x2 constant.
C. A 1% increase in x1 is associated with a 0.5-unit increase in log(y) holding x2 constant.
D. A 1% increase in x1 is associated with a 50% increase in y, holding x2 constant.
The correct answer is:
Correct Action: A 1% increase in x1 is associated with a 0.5% increase in y, holding x2 constant.
Rationales
• A 1% increase in x1 is associated with a 0.5% increase in y, holding x2 constant. (Correct):
This is the correct action. In a log-log model, the coefficient for a log-transformed predictor represents elasticity: a 1%
change in the predictor leads to a (coefficient) % change in the response. Here, 0.5 means a 1% increase in x1 is associated
with a 0.5% increase in y, holding x2 constant. Option B confuses unit and percent changes. Option C misstates the
• A 1-unit increase in x1 is associated with a 0.5% increase in y, holding x2 constant. (Incorrect):
This option is not appropriate. Here, 0.5 means a 1% increase in x1 is associated with a 0
• A 1% increase in x1 is associated with a 0.5-unit increase in log(y) holding x2 constant. (Incorrect):
This option is not appropriate. Here, 0.5 means a 1% increase in x1 is associated with a 0
• A 1% increase in x1 is associated with a 50% increase in y, holding x2 constant. (Incorrect):
This option is not appropriate. Here, 0.5 means a 1% increase in x1 is associated with a 0
Page 3
, Question 3 of 114
You fit a linear model and examine residual vs. fitted plot and the scale-location plot. The residual vs.
fitted plot shows a clear funnel shape (spread increasing with fitted values), and the scale-location
plot shows an upward trend. Which assumption is most likely violated?
A. Linearity of the relationship
B. Normality of errors
C. Constant variance (homoscedasticity)
D. Independence of errors
The correct answer is:
Correct Action: Constant variance (homoscedasticity)
Rationales
• Constant variance (homoscedasticity) (Correct):
This is the correct action. A funnel shape in the residual vs. fitted plot (increasing spread) indicates heteroscedasticity: the
variance of errors changes with the fitted values. The scale-location plot (square root of standardized residuals vs. fitted)
showing an upward trend confirms increasing variance. Linearity would show a curved pattern in the residual plot, not
• Linearity of the relationship (Incorrect):
This option is not appropriate. fitted plot (increasing spread) indicates heteroscedasticity: the variance of errors changes
with the fitted values. The scale-location plot (square root of standardized residuals vs
• Normality of errors (Incorrect):
This option is not appropriate. fitted plot (increasing spread) indicates heteroscedasticity: the variance of errors changes
with the fitted values. The scale-location plot (square root of standardized residuals vs
• Independence of errors (Incorrect):
This option is not appropriate. fitted plot (increasing spread) indicates heteroscedasticity: the variance of errors changes
with the fitted values. The scale-location plot (square root of standardized residuals vs
Page 4
