Executive Summary

This report investigates two questions using the mtcars dataset:

  1. Is automatic or manual transmission associated with better fuel economy?
  2. What is the MPG difference between automatic and manual cars?

The analysis uses exploratory data analysis, three linear regression models, confidence intervals, and residual diagnostics. Manual cars have higher average MPG in this sample. However, vehicle weight, horsepower, and displacement may also explain some of the observed difference.

The results describe associations in a small observational dataset and should not be interpreted as proof that transmission type causes higher fuel economy.

1. Exploratory Data Analysis

The dataset contains 32 cars, including 19 automatic cars and 13 manual cars.

Summary statistics

Summary statistics by transmission type
transmission Cars Mean_MPG Median_MPG SD_MPG Mean_Weight Mean_Horsepower
Automatic 19 17.15 17.3 3.83 3.77 160.26
Manual 13 24.39 22.8 6.17 2.41 126.85

MPG distribution

The plot compares the MPG distributions. Manual cars generally show higher fuel economy in this sample. This initial comparison does not control for differences in vehicle characteristics.

2. Fit Multiple Regression Models

We fit three models with progressively more explanatory variables.

Comparison of candidate regression models
Model Adjusted_R2 AIC Residual_SE
Model 1: Transmission only 0.338 196.484 4.902
Model 2: + Weight + Horsepower 0.823 156.135 2.538
Model 3: + Displacement 0.817 158.067 2.581

Model selection strategy

Model 1 estimates the unadjusted transmission difference. Model 2 controls for vehicle weight and horsepower, two important predictors of fuel economy. Model 3 adds displacement to examine whether the estimated transmission association changes.

We compare adjusted \(R^2\), AIC, residual standard error, and coefficient stability. A lower AIC and higher adjusted \(R^2\) can indicate better fit, but additional predictors also increase model complexity.

Because the dataset contains only 32 observations and weight, horsepower, and displacement may be correlated, we prioritize interpretability and parsimony. Model 2 is our primary adjusted model, while Model 3 is a sensitivity analysis.

3. Interpret the Regression Coefficients

In all three models, automatic transmission is the reference category. Therefore, the coefficient for transmissionManual represents the estimated MPG difference between manual and automatic cars, holding the other included variables constant.

Estimated regression coefficients
Model Term Estimate Std_Error t_value p_value
Model 1 (Intercept) 17.1474 1.1246 15.2475 0.0000
Model 1 transmissionManual 7.2449 1.7644 4.1061 0.0003
Model 2 (Intercept) 34.0029 2.6427 12.8669 0.0000
Model 2 transmissionManual 2.0837 1.3764 1.5139 0.1413
Model 2 wt -2.8786 0.9050 -3.1808 0.0036
Model 2 hp -0.0375 0.0096 -3.9018 0.0005
Model 3 (Intercept) 34.2094 2.8228 12.1189 0.0000
Model 3 transmissionManual 2.1593 1.4352 1.5045 0.1441
Model 3 wt -3.0467 1.1571 -2.6330 0.0138
Model 3 hp -0.0393 0.0124 -3.1627 0.0038
Model 3 disp 0.0025 0.0104 0.2399 0.8122

How to interpret the coefficients

  • TransmissionManual: The estimated MPG difference between manual and automatic cars, after controlling for the other predictors in the model.
  • wt: The expected MPG change for a one-unit increase in vehicle weight, holding other variables constant. Weight is measured in 1,000 pounds in mtcars.
  • hp: The expected MPG change for a one-unit increase in horsepower, holding other variables constant.
  • disp: The expected MPG change for a one-unit increase in displacement, when included in Model 3.

A coefficient’s sign indicates the direction of the association. Its magnitude gives the estimated change in MPG per unit of the predictor. A p-value alone does not measure practical importance or establish causation.

4. Quantify the MPG Difference

First, calculate the difference in group means.

## Mean MPG, automatic: 17.15
## Mean MPG, manual: 24.39
## Manual minus automatic: 7.24 MPG

Next, calculate a 95% confidence interval using Welch’s two-sample t-test. This method does not assume equal population variances.

Unadjusted MPG difference and 95% confidence interval
Difference_Manual_Minus_Automatic CI_Lower CI_Upper P_Value
7.2449 -11.2802 -3.2097 0.0014

The confidence interval quantifies uncertainty in the unadjusted difference in group means. If it excludes zero, the data provide evidence of a difference in mean MPG between the groups under the test’s assumptions.

This interval does not adjust for vehicle weight, horsepower, or displacement. The regression coefficients address those variables separately.

Adjusted transmission effect

Adjusted transmission effect from Model 2
Estimate CI_Lower CI_Upper P_Value
transmissionManual 2.0837 -0.7358 4.9032 0.1413

The adjusted estimate is the expected MPG difference associated with manual rather than automatic transmission among cars with the same modeled weight and horsepower. Its confidence interval describes statistical uncertainty under the regression assumptions.

5. Residual Diagnostics

We inspect the diagnostic plots for Model 2, our primary adjusted model.

The diagnostic plots help assess:

The small sample size limits how confidently these plots can establish that assumptions hold. Any visible patterns or influential observations should be considered when interpreting the results.

6. Conclusion

Question 1: Is automatic or manual transmission better for MPG?

In the mtcars sample, manual cars have higher average MPG than automatic cars. This is an observed association, not evidence that changing transmission alone would cause the difference.

Question 2: What is the MPG difference?

The unadjusted difference, calculated as manual minus automatic, is 7.24 MPG.

The 95% confidence interval is -11.28 to -3.21 MPG.

After adjusting for weight and horsepower, the estimated transmission difference is 2.08 MPG, with a 95% confidence interval from -0.74 to 4.9 MPG.

Final takeaway

Manual transmission is associated with higher MPG in this dataset. The unadjusted comparison quantifies the raw difference, while the adjusted regression estimates the association after accounting for selected vehicle characteristics.

The conclusions should be treated cautiously because the dataset is small and observational, and the predictors may be correlated. The analysis cannot establish a causal effect of transmission type.