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Answer Key: Linear Regression with the California Housing Dataset

1. Which features are most strongly correlated with the target variable?

Using the correlation matrix and heatmap, we find that:

• 'MedInc' (median income) has the strongest positive correlation with the target variable ('MedHouseVal').
• 'AveRooms' and 'HouseAge' have moderate correlations.
• 'AveOccup' and 'Population' are less correlated.

2. How does median income affect median house value?

• From the scatter plot of 'MedInc' vs 'MedHouseVal', we see a clear positive trend.
• Higher median income is associated with higher median house values.
• The relationship appears nonlinear at the high end (suggesting diminishing returns).

3. What does the residual plot tell us about model assumptions?

• The residuals show some heteroscedasticity: the spread of residuals increases with the predicted value.
• This violates the assumption of constant variance (homoscedasticity).
• Suggests the need for transforming the response or using robust regression methods.

4. Which assumptions of linear regression may be violated?

• Linearity: Mostly holds, especially for 'MedInc'. But some relationships appear nonlinear.
• Heteroscedasticity: Residual plot shows increasing variance — assumption is violated.
• Multicollinearity: Correlation heatmap suggests moderate multicollinearity between some predictors (e.g., AveRooms and AveOccup).
• Normality of residuals: The histogram shows residuals are roughly normal but with some skew.
• Outliers: A few points with large residuals suggest outliers.

5. What would improve the model?

• Feature engineering: polynomial terms or interaction terms.
• Use regularized regression (e.g., Ridge or Lasso).
• Apply log transformation to skewed predictors or response variable.
• Use tree-based models or nonlinear regression models.