The linear regression model explains 85.5% of variance in median home values (R² = 0.8549, Adjusted R² = 0.8501). The three strongest predictors by random forest importance are: lower_status_pct, distance_employment, crime_rate. On the held-out test set (101 properties) the model achieves RMSE = $3.3k. The Charles River location effect is small or negligible in this dataset.

Random forest permutation importance (%IncMSE) ranks all 13 predictors by how much test error increases when each feature is randomly shuffled. 'lower_status_pct' is the strongest driver with an importance score of 96.3 — scrambling its values hurts prediction accuracy most. Features with low scores contribute little to predictive power beyond what other variables already capture. Note that importance is a measure of predictive signal, not causal direction.

The heatmap displays Pearson correlations among all 14 variables (features + target). 'lower_status_pct' has the strongest correlation with home value (r = -0.873). Dark red cells signal strong positive co-movement; dark blue signals strong negative association. Pairs of predictors with |r| > 0.7 are collinear and may inflate each other's regression standard errors.

Each point represents one census tract. The scatter reveals a strong negative relationship (r = -0.873) between the percentage of lower-status residents and median home values. The relationship curves downward most steeply at low lower_status_pct values and flattens near the extremes, suggesting a non-linear component that the random forest captures better than linear regression alone. The clustering near the $50k ceiling reflects a top-coding artefact in the original Boston Housing dataset.

Standardised OLS coefficients allow fair comparison across predictors with different units. 'distance_employment' has the strongest positive effect on home values. 'lower_status_pct' has the largest negative effect. 4 of 13 predictors are statistically significant (p < 0.05). Bars pointing right indicate value-raising features; bars pointing left indicate value-reducing ones.

Predicted vs. actual home values on the held-out test set (101 properties). Points on the 45° diagonal are perfect predictions; points above overestimate, points below underestimate. Test-set RMSE is $3.3k and R² = 0.8181, confirming the model generalises well beyond training data. Systematic under-prediction near the $50k upper bound reflects the price ceiling in the dataset — the model cannot predict values above the ceiling it was trained on.

Median home value across the 506 tracts is $24.4k (mean: $22.9k). The distribution is right-skewed with a distinct cluster of high-value tracts. No strong price ceiling artefact is visible in this dataset. Distinct low-value and mid-value clusters likely correspond to inner-city and suburban neighborhoods.

Tracts bordering the Charles River (n = 38) have a mean home value of $22.71k vs. $22.88k for non-riverside tracts (n = 468). This $0.17k difference is not statistically significant at the 5% level by Welch's two-sample t-test (p = 0.8972). The premium likely reflects waterfront amenity value, though the small sample of riverside tracts means estimates are less precise than for non-riverside properties.

Descriptive statistics for all 14 variables (506 tracts). Comparing mean to median reveals distributional skew: 'residential_zone_pct' shows the largest mean-to-median gap relative to its spread, indicating heavy skew that may influence regression assumptions. The wide ranges across features (crime_rate spans orders of magnitude; rooms_avg is tightly bounded) explain why standardisation is essential before comparing coefficient magnitudes.