MS-DS Master of Data science Statistical Inference & Regression Models 2 — Questions and Answers
Question 1: A researcher fits a multiple linear regression model and finds that adding a 5th predictor increases R² from 0.82 to 0.83 but decreases adjusted R² from 0.81 to 0.80. What does this indicate?
- The new predictor significantly improves the model
- The new predictor does not contribute enough to justify its inclusion (Correct answer)
- The model is now overfitted and should be discarded
- Adjusted R² is always lower than R² so this is expected and acceptable
Correct answer: The new predictor does not contribute enough to justify its inclusion
Adjusted R² penalizes unnecessary predictors, so a decrease indicates the added variable does not improve the model enough to offset the loss of a degree of freedom.
Question 2: In hypothesis testing, a p-value of 0.03 is obtained with α = 0.05. Which conclusion is correct?
- Fail to reject H₀ because the effect is small
- Reject H₀; there is sufficient evidence against it at the 5% level (Correct answer)
- Accept H₁ as definitively true
- The result is not statistically significant
Correct answer: Reject H₀; there is sufficient evidence against it at the 5% level
Since p = 0.03 < α = 0.05, we reject the null hypothesis at the 5% significance level.
Question 3: Which assumption of OLS regression is violated when the variance of residuals increases with fitted values?
- Linearity
- Independence of errors
- Homoscedasticity (Correct answer)
- Normality of errors
Correct answer: Homoscedasticity
Heteroscedasticity is the violation of the homoscedasticity assumption, where error variance is not constant across levels of the predictors.
Question 4: A 95% confidence interval for a population mean is (12.4, 18.6). What is the correct interpretation?
- There is a 95% probability the true mean lies in this interval
- 95% of individual observations fall between 12.4 and 18.6
- If repeated sampling is used, 95% of similarly constructed intervals will contain the true mean (Correct answer)
- The sample mean is 95% accurate
Correct answer: If repeated sampling is used, 95% of similarly constructed intervals will contain the true mean
Confidence intervals are interpreted in terms of long-run frequency: 95% of intervals constructed with this procedure will capture the true parameter.
Question 5: In logistic regression, the logit function transforms the response variable to:
- A value between -1 and 1
- The log of the odds of the outcome (Correct answer)
- The square root of the probability
- A standardized z-score
Correct answer: The log of the odds of the outcome
The logit transformation is log(p/(1-p)), the natural log of the odds, which maps probabilities from (0,1) to (-∞, +∞).
Question 6: When performing a Durbin-Watson test, a statistic close to 2 indicates:
- Strong positive autocorrelation in residuals
- Strong negative autocorrelation in residuals
- No significant autocorrelation in residuals (Correct answer)
- Multicollinearity among predictors
Correct answer: No significant autocorrelation in residuals
The Durbin-Watson statistic ranges from 0 to 4, with values near 2 indicating no autocorrelation in the residuals.
Question 7: A Variance Inflation Factor (VIF) of 12 for a predictor in a multiple regression model suggests:
- The predictor is highly significant
- Severe multicollinearity affecting that predictor (Correct answer)
- The predictor has low explanatory power
- Heteroscedasticity in the residuals
Correct answer: Severe multicollinearity affecting that predictor
VIF > 10 is a common threshold indicating severe multicollinearity, meaning the predictor is nearly linearly dependent on others in the model.
MS-DS Master of Data science Statistical Inference & Regression Models 2
A researcher fits a multiple linear regression model and finds that adding a 5th predictor increases R² from 0.82 to 0.83 but decreases adjusted R² from 0.81 to 0.80.
What does this indicate?