Linear Regression Analysis in Python

Let’s start by loading the data and inspecting it first. After that, we can fit the simple linear regression using RM as the feature and MEDV as the target.

The data is in place and the two columns needed for the simple regression are confirmed: RM for average rooms and MEDV for median home value. The next missing piece is the actual fitted linear relationship.

Using the Boston housing data with RM as the single predictor and MEDV as the target, the fitted simple linear regression is:

\hat{\text{MEDV}} = -34.67 + 9.10 × \text{RM}

From the observed model output:

• Intercept: -34.67
• Slope for RM: 9.10
• R^2: 0.484

What this means:

• For each additional room, the model predicts MEDV increases by about 9.10.
• The negative intercept is just where the fitted line crosses when RM = 0; it is not practically meaningful here because homes with 0 rooms are outside the data range.
• The R^2 of about 0.484 means RM alone explains roughly 48.4% of the variation in housing prices, so rooms are an important predictor, but not the whole story.

So, there is a clear positive linear relationship: houses with more rooms tend to have higher median value.

We can fit an OLS model with all 13 predictors and MEDV as the target. There are no obvious ID-like columns here, so I’m keeping all non-target features.

The multiple linear regression with all features has already been fit with statsmodels and the summary output confirms a strong overall model.

From the observed summary:

• Dependent variable: MEDV
• Observations: 506
• Predictors: 13
• R^2 = 0.741
• Adjusted R^2 = 0.734
• F-statistic: 108.1
• Prob(F-statistic): 6.72e-135

What this means:

• Using all available predictors improves fit substantially versus the simple regression with only RM (R^2 ≈ 0.484 vs. $0.741$).
• The model explains about 74.1% of the variation in MEDV.
• The adjusted R^2 staying close to R^2 suggests the added features are contributing useful signal overall rather than only inflating fit.
• The extremely small F-test p-value shows the full regression is statistically significant as a whole.

This is the statsmodels summary output you requested, and it indicates that the full multiple linear regression provides a much better explanation of housing prices than the one-feature model.

Let’s start with the most useful diagnostic: a residuals-vs-fitted plot. It helps check linearity and gives a first look at constant variance.

The residuals-vs-fitted diagnostic has been created, but the notebook output here only shows the figure object, so it isn’t enough by itself to judge the pattern. The Q-Q plot and a formal homoscedasticity check are still needed.

Based on the outputs you now have:

• Homoscedasticity: the Breusch–Pagan test is clearly significant, with p-values around \6.27 × 10^{-9} and \1.50 × 10^{-9}. That means you should reject constant variance. In practical terms, the residual variance changes with the fitted values, so the model shows heteroscedasticity.

• Q-Q plot: the plot itself is present, but not numerically summarized. Its purpose is to check whether residuals follow a normal distribution. From the available notebook evidence, this assumption was checked visually, but the strongest concrete conclusion in the current outputs is about variance, not normality.

• Residual plot: you produced the residuals-vs-fitted plot, which is the right diagnostic for linearity and variance patterns. Since the notebook only shows the figure object rather than the rendered pattern here, the key confirmed conclusion comes from the formal test: the residual spread is not constant.

So, against your original goal:

• Residual plot: created and appropriate for assumption checking
• Q-Q plot: created and appropriate for checking residual normality
• Homoscedasticity: violated, supported by the significant Breusch–Pagan result

Overall, the main regression assumption issue visible from your outputs is non-constant error variance. This means standard OLS coefficient estimates are still unbiased under usual conditions, but the usual standard errors, confidence intervals, and p-values may be unreliable unless you use a correction such as heteroscedasticity-robust standard errors.

The simple regression using only RM has:

• R^2 = 0.4835

The multiple regression using all predictors has:

• R^2 = 0.7410

So the multiple regression fits the data better:

• Absolute increase in R^2: about $0.2575$
• That means the explained variance rises from about 48.4% to 74.1%

Interpretation:

• The simple model explains roughly half of the variation in MEDV
• The multiple model explains about three-quarters of the variation
• So adding the other predictors provides a substantial improvement over using RM alone

One caution: a higher R^2 in the multiple model is expected when adding more variables, so for a fairer comparison it can also help to look at adjusted R^2. In your OLS summary, the multiple model’s adjusted R^2 is 0.734, which is still strong.