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Prediction Process

Once the model is trained, we use it to make predictions on the test set that was held out during training.

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Making Predictions

# Make predictions on the test data
y_pred = lr_model.predict(X_test)
The predict() method:
  • Takes the test features (X_test) as input
  • Applies the learned linear equation
  • Returns predicted values for Yearly Amount Spent
  • Produces 150 predictions (one for each test sample)

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Example Predictions

Here’s a sample of the first 5 predictions: 
y_pred[:5]
# Output: array([498.82, 519.53, 562.95, 478.91, 423.82])
These values represent the model’s predicted yearly spending in dollars for the first five customers in the test set.

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Evaluation Metrics

We use two primary metrics to assess model performance: Mean Squared Error (MSE) and R-squared (R²).

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Computing Metrics

from sklearn.metrics import mean_squared_error, r2_score

# Evaluate the model
mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)

print("Mean Squared Error:", mse)
print("R-squared:", r2)

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Results

Based on the model evaluation: 
# Output:
Mean Squared Error: 80.90
R-squared: 0.9885

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Understanding Mean Squared Error (MSE)

MSE: 80.90 Mean Squared Error measures the average squared difference between actual and predicted values.

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Formula

MSE = (1/n) × Σ(actual - predicted)²

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Interpretation

  • Lower is better: MSE of 80.90 is relatively low
  • Units: MSE is in squared dollars (dollars²)
  • Root MSE: √80.90 ≈ 9.00,meaningpredictionsareoffbyabout9.00, meaning predictions are off by about 9 on average
  • Context: Given that yearly spending ranges from 256to256 to 765, an RMSE of $9 represents excellent accuracy
An MSE of 80.90 indicates that the model’s predictions are very close to actual values, with an average error of approximately $9 per prediction.

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Understanding R-squared (R²)

R²: 0.9885 R-squared measures the proportion of variance in the target variable explained by the model.

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Formula

R² = 1 - (SS_res / SS_tot)
Where:
  • SS_res: Sum of squared residuals
  • SS_tot: Total sum of squares

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Interpretation

  • Range: 0 to 1 (higher is better)
  • Result: 0.9885 means the model explains 98.85% of the variance
  • Excellent Fit: Values above 0.95 indicate a very strong model
  • Implication: Only 1.15% of variance remains unexplained
An R² of 0.9885 is exceptionally high, indicating that the four features (Session Length, Time on App, Time on Website, and Membership Length) are excellent predictors of yearly customer spending.

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Model Coefficients

The coefficients reveal how each feature impacts the prediction: 
# Extract model coefficients and intercept
print("Coefficients:", lr_model.coef_)
print("Intercept:", lr_model.intercept_)

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Results

# Output:
Coefficients: [25.83, 38.81, 0.28, 61.30]
Intercept: -1048.82

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Coefficient Interpretation

FeatureCoefficientImpact
Avg. Session Length25.83+$25.83 per additional minute
Time on App38.81+$38.81 per additional minute
Time on Website0.28+$0.28 per additional minute
Length of Membership61.30+$61.30 per additional year

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Key Insights

  1. Length of Membership has the strongest impact ($61.30/year)
  2. Time on App is the second most important feature ($38.81/minute)
  3. Time on Website has minimal impact ($0.28/minute)
  4. Avg. Session Length has moderate impact ($25.83/minute)
The large difference between Time on App (38.81)andTimeonWebsite(38.81) and Time on Website (0.28) suggests that the mobile app is significantly more effective at driving customer spending than the website.

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Interpreting the Intercept

Intercept: -1048.82 The intercept represents the predicted yearly spending when all features equal zero.
  • Theoretical Value: Not practically meaningful (customers can’t have zero engagement)
  • Mathematical Role: Adjusts the baseline of the prediction equation
  • Don’t Overthink: Negative intercepts are common and don’t indicate a problem

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How to Interpret Metrics

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Is This a Good Model?

Yes! The model shows excellent performance: ✅ High R² (0.9885): Explains nearly 99% of variance ✅ Low MSE (80.90): Predictions are accurate within ~$9 ✅ Meaningful Coefficients: Features align with business intuition ✅ Interpretable: Clear insights for decision-making

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Business Implications

1

Mobile App Priority

Time on App has 138x more impact than Time on Website (38.81vs38.81 vs 0.28), suggesting the company should prioritize mobile app improvements.
2

Customer Retention

Length of Membership has the highest coefficient ($61.30), indicating that customer loyalty programs and retention strategies are crucial.
3

Website Enhancement

The low website coefficient suggests potential for improvement - making the website more engaging could increase its contribution to spending.

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Complete Evaluation Code

Here’s the full code for model evaluation: 
from sklearn.metrics import mean_squared_error, r2_score

# Make predictions
y_pred = lr_model.predict(X_test)

# Calculate metrics
mse = mean_squared_error(y_test, y_pred)
r2 = r2_score(y_test, y_pred)

# Display results
print("Mean Squared Error:", mse)
print("R-squared:", r2)
print("Coefficients:", lr_model.coef_)
print("Intercept:", lr_model.intercept_)

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Summary

The linear regression model demonstrates excellent predictive performance:
  • Accuracy: RMSE of ~$9 on yearly spending predictions
  • Explanatory Power: Explains 98.85% of variance in customer spending
  • Actionable Insights: Clear guidance on resource allocation (prioritize app over website)
  • Customer Focus: Strong emphasis on membership length highlights importance of retention
This model provides a solid foundation for business decision-making regarding mobile app vs. website investment strategies.

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