Test Questions for University Statistics: Addressing Regression Assumptions

📚 Quick Study Guide

🔍 Linearity: The relationship between the independent and dependent variables must be linear. Check scatterplots for non-linear patterns. 📏 Independence of Errors: The errors (residuals) should be independent of each other. Look for patterns in residual plots; Durbin-Watson test can also be used. 📦 Homoscedasticity: The variance of the errors should be constant across all levels of the independent variables. Assess using scatterplots of residuals against predicted values; Breusch-Pagan test is useful. 🍎 Normality of Errors: The errors should be normally distributed. Check histograms or Q-Q plots of the residuals; Shapiro-Wilk test can be employed. 💡 Multicollinearity: High correlation among independent variables can distort regression results. Check Variance Inflation Factor (VIF); VIF > 5 or 10 often indicates multicollinearity. The formula for VIF is $VIF = \frac{1}{1-R^2}$, where $R^2$ is the R-squared value from regressing one independent variable against the others. 📝 Outliers: Data points with large residuals can disproportionately influence the regression model. Identify outliers using Cook's distance or leverage values.

🧪 Practice Quiz

1. Which of the following is NOT a key assumption of linear regression?
   1. Linearity
   2. Independence of errors
   3. Heteroscedasticity
   4. Normality of errors
2. What does heteroscedasticity violate?
   1. The errors have constant variance
   2. The relationship is linear
   3. The errors are normally distributed
   4. The errors are independent
3. Which test is commonly used to detect heteroscedasticity?
   1. T-test
   2. F-test
   3. Breusch-Pagan test
   4. Chi-squared test
4. High VIF values indicate the presence of which issue in regression?
   1. Autocorrelation
   2. Multicollinearity
   3. Heteroscedasticity
   4. Non-linearity
5. Which plot is most suitable for visually assessing the linearity assumption?
   1. Histogram of residuals
   2. Scatterplot of residuals vs. fitted values
   3. Q-Q plot of residuals
   4. Scatterplot of independent variable vs. dependent variable
6. What is the consequence of violating the assumption of independence of errors (autocorrelation)?
   1. Biased coefficient estimates
   2. Inefficient coefficient estimates
   3. Inflated standard errors
   4. All of the above
7. Which of the following is a method to address multicollinearity?
   1. Adding more variables
   2. Removing one of the correlated variables
   3. Transforming the dependent variable
   4. Using robust standard errors