Call:
lm(formula = `Energy Efficiency (MPG)` ~ Horsepower, data = car_data_clean)
Residuals:
Min 1Q Median 3Q Max
-9.968 -1.966 0.683 1.933 9.241
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 41.270812 1.864643 22.133 < 2e-16 ***
Horsepower -0.053695 0.006956 -7.719 2.68e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3.689 on 38 degrees of freedom
Multiple R-squared: 0.6106, Adjusted R-squared: 0.6003
F-statistic: 59.58 on 1 and 38 DF, p-value: 2.677e-09Lecture 5: Interpreting the Model Results
Overview:
In this lecture, we will interpret the results of a simple linear regression model. The model predicts Energy Efficiency (MPG) using Horsepower as the predictor variable. We will explain how to interpret the coefficients, R-squared value, p-values, and discuss what these metrics reveal about the relationship between Horsepower and MPG.
Key Learning Outcomes:
By the end of this lecture, students will be able to: - Interpret the coefficients from a linear regression model. - Understand the significance of the R-squared value and residuals. - Evaluate p-values to assess statistical significance. - Connect the model output to real-world contexts, focusing on the trade-off between engine power and fuel efficiency.
Interpreting the Model Output:
The regression output provides a summary of the model fitted between Energy Efficiency (MPG) and Horsepower:
Call:
lm(formula = `Energy Efficiency (MPG)` ~ Horsepower, data = car_data_clean)Residuals:
- Residuals are the differences between the actual and predicted values of MPG.
- The smallest residual is -9.968, and the largest is 9.241.
- The median residual is close to zero (0.683), suggesting that most predictions are fairly accurate.
- The first quartile (1Q) and third quartile (3Q) residuals are -1.966 and 1.933, showing that most residuals are within this range.
Coefficients:
The table below shows the relationship between Horsepower and MPG:
| Coefficient | Estimate | Std. Error | t value | Pr(> |
|---|---|---|---|---|
| Intercept | 41.270812 | 1.864643 | 22.133 | < 2e-16 *** |
| Horsepower | -0.053695 | 0.006956 | -7.719 | 2.68e-09 *** |
- Intercept:
- The intercept is 41.27, which represents the estimated MPG when Horsepower is zero. While this isn’t realistic, it serves as a starting point for the model.
- Slope (Horsepower):
- The slope is -0.0537. This means that for every additional unit of Horsepower, MPG decreases by about 0.054.
- A negative slope is expected, as higher horsepower generally results in lower fuel efficiency.
Statistical Significance:
- P-value for Horsepower:
- The p-value for Horsepower is 2.68e-09, which is much smaller than 0.05. This means the relationship between Horsepower and MPG is statistically significant.
- The three asterisks (’*’**) indicate high statistical significance (p < 0.001).
Goodness of Fit:
The Goodness of Fit measures tell us how well the model explains the data:
Residual standard error: 3.689 on 38 degrees of freedom
Multiple R-squared: 0.6106, Adjusted R-squared: 0.6003
F-statistic: 59.58 on 1 and 38 DF, p-value: 2.677e-09- Residual Standard Error (RSE):
- The residual standard error is 3.689, meaning the typical prediction error is about 3.69 MPG.
- A smaller RSE would mean the model is more accurate.
- R-squared:
- The R-squared value is 0.6106, meaning the model explains 61.06% of the variation in MPG.
- This shows that Horsepower is a fairly strong predictor of MPG, but 38.94% of the variation remains unexplained.
- Adjusted R-squared:
- The Adjusted R-squared is 0.6003, which is close to R-squared since we have only one predictor. This confirms the model isn’t overfitting.
- F-statistic:
- The F-statistic is 59.58, with a p-value of 2.677e-09. This shows the model as a whole is statistically significant, meaning Horsepower is a good predictor of MPG.
Connecting the Output to the Real World:
- Application of Results:
- The model confirms the trade-off between Horsepower and MPG. More powerful vehicles tend to be less fuel-efficient.
- This insight is useful for both manufacturers and consumers. Manufacturers can design cars with this trade-off in mind, while consumers can choose based on their needs—either fuel efficiency or performance.
- Trade-offs:
- The negative slope shows the trade-off between engine power and fuel efficiency. Manufacturers must balance Horsepower and MPG when designing vehicles.
- Consumers must decide whether they want more horsepower at the cost of lower MPG or vice versa.
Potential Limitations of the Model:
- Linearity Assumption:
- The model assumes a linear relationship between Horsepower and MPG. However, this relationship might not be linear, especially for very high or very low horsepower values.
- More complex models might better capture this relationship.
- Unexplained Variance:
- While the model explains 61% of the variation in MPG, the remaining 39% is unexplained. Other factors, like vehicle weight or aerodynamics, might also affect MPG.
Summary of Key Concepts:
- Coefficients: The slope (-0.0537) shows that MPG decreases by about 0.054 for each additional unit of Horsepower.
- R-squared: The model explains 61.06% of the variance in MPG, meaning Horsepower is a strong predictor, but other factors may still be at play.
- P-values: The p-value for Horsepower (2.68e-09) indicates that Horsepower significantly impacts MPG.
- Real-World Application: The model provides insights into the trade-offs between engine performance and fuel efficiency, important for both manufacturers and consumers.
Next Steps:
In the next lecture, we will explore strategies for dealing with outliers and refining the model for better accuracy. We will discuss how to identify and address outliers and how doing so can improve model performance.
Assignment for Students:
- Use the model output from your own regression to interpret the coefficients and R-squared values.
- Discuss the real-world implications of the relationship between Horsepower and MPG.
- Identify any potential limitations of your model and suggest possible ways to refine it.