Statistics Regression & Correlation Analysis 2 — Questions and Answers
Question 1: What is the least squares criterion used for in regression?
- Minimizing the sum of squared residuals (Correct answer)
- Maximizing the R² value
- Minimizing the number of predictors
- Maximizing the F-statistic
Correct answer: Minimizing the sum of squared residuals
Ordinary least squares (OLS) finds the line that minimizes the total of squared vertical distances from the data points.
Question 2: What assumption about residuals is required for ordinary least squares regression to be valid?
- Residuals are approximately normally distributed with constant variance (Correct answer)
- Residuals must be zero for all observations
- Residuals must be positively skewed
- Residuals must increase as X increases
Correct answer: Residuals are approximately normally distributed with constant variance
OLS regression assumes residuals are normally distributed, have constant variance (homoscedasticity), and are independent.
Question 3: What does a leverage point in regression mean?
- An observation with an extreme X value that has potential to strongly influence the regression line (Correct answer)
- A data point with a very large residual
- An outlier in the response variable Y
- A predictor with near-zero variance
Correct answer: An observation with an extreme X value that has potential to strongly influence the regression line
High-leverage points have unusual predictor values and can disproportionately pull the regression line toward them.
Question 4: Spearman's rank correlation is used instead of Pearson's when?
- The data are ordinal or the relationship is monotonic but not necessarily linear (Correct answer)
- Both variables are normally distributed
- The sample size is very large
- The variables have no measurement error
Correct answer: The data are ordinal or the relationship is monotonic but not necessarily linear
Spearman's correlation uses ranked values and is appropriate for ordinal data or non-linear monotonic relationships.
Question 5: What does heteroscedasticity mean in the context of regression?
- The variance of residuals changes across levels of the predictor (Correct answer)
- The predictor and response variable are not correlated
- The regression coefficients are not significant
- The residuals are not independent
Correct answer: The variance of residuals changes across levels of the predictor
Heteroscedasticity means the spread of residuals is not constant, violating the equal-variance assumption of OLS.
Question 6: In simple linear regression with the equation Ŷ = 5 + 2X, what is the predicted value when X = 4?
- 13 (Correct answer)
- 10
- 9
- 18
Correct answer: 13
Substituting X = 4: Ŷ = 5 + 2(4) = 5 + 8 = 13.
Statistics Regression & Correlation Analysis 2
What is the least squares criterion used for in regression?