Executive Summary

Using the 1974 Motor Trend mtcars data (32 cars), we asked whether manual or automatic transmission is better for fuel economy (MPG) and how large the difference is. A naive comparison shows manual-transmission cars average 7.24 MPG more than automatic cars (24.4 vs. 17.1 MPG, Welch t = -3.77, p = 0.0014). However, manual cars in this sample are also systematically lighter and quicker, and both traits are strongly associated with MPG. After adjusting for vehicle weight and quarter-mile time, transmission type still matters, but the effect shrinks to +2.94 MPG for manual over automatic (95% CI: 0.05 to 5.83 MPG, p = 0.047) — an estimate of the effect holding weight and acceleration fixed, not the raw group difference. Weight is the dominant predictor: each additional 1,000 lb costs about 3.92 MPG. We conclude manual transmission is associated with better mileage, but most of the raw 7.24 MPG gap is explained by manual cars being lighter, not by the transmission itself.

Exploratory Data Analysis

The boxplot (Figure 1) shows manual cars have both a higher median and higher spread of MPG. Pairwise correlations (Appendix, Fig. A1) show wt (weight, r = -0.87) and qsec (1/4-mile time, r = 0.42) are the variables most associated with mpg besides am, and wt and am are themselves correlated (manual cars tend to be lighter) — a classic confounding pattern that motivates a multivariable model rather than the simple two-group comparison.

Model Selection Strategy

We began with the simple regression mpg ~ am, which simply reproduces the raw group means and confirms the unadjusted 7.24 MPG gap (R² = 0.36 — transmission alone explains only 36% of MPG variation). We then fit the full model with all 10 other variables, which raised R² to 0.88 but left am statistically insignificant (p = 0.21) with badly inflated standard errors — a sign of multicollinearity among the 10 correlated predictors (e.g., cyl, disp, hp, wt are all measuring “bigger/heavier car”).

To find a parsimonious model we used stepwise selection (both directions, AIC) starting from the full model, which converged on: \[\text{mpg} = \beta_0 + \beta_1\,\text{am} + \beta_2\,\text{wt} + \beta_3\,\text{qsec}\] An ANOVA comparing this model against the am-only model confirms wt and qsec jointly add significant explanatory power (F = 45.6, p < 0.001), while adding the remaining 8 variables back does not significantly improve fit over this 3-variable model (F = 0.88, p = 0.55). Variance inflation factors for wt and qsec are low (2.48 and 1.36), indicating no problematic collinearity remains. We select this as our final model.

Final model: mpg ~ am + wt + qsec
Estimate Std. Error t value Pr(>|t|)
(Intercept) 9.618 6.960 1.382 0.178
am_fManual 2.936 1.411 2.081 0.047
wt -3.917 0.711 -5.507 0.000
qsec 1.226 0.289 4.247 0.000

Model fit: R² = 0.85 (adjusted 0.83); the model explains 85% of the variability in MPG using only transmission type, weight, and quarter-mile time.

Answering the Questions of Interest

Is manual or automatic better for MPG? Manual transmission is associated with higher MPG, both unadjusted and after controlling for weight and acceleration. The adjusted effect remains statistically significant at the 5% level (p = 0.047).

Quantifying the difference: Holding weight and quarter-mile time constant, switching from automatic to manual transmission is associated with an increase of 2.94 MPG (95% CI: 0.05 to 5.83 MPG). By contrast, the unadjusted difference was 7.24 MPG — most of that gap is attributable to manual cars being lighter (each 1,000 lb of weight costs 3.92 MPG, 95% CI: -5.37 to -2.46, p < 0.001) and having faster quarter-mile times, not to the transmission mechanism itself. As a sensitivity check we also fit a model allowing transmission and weight to interact (Appendix C); it shows the manual-transmission advantage is largest for light cars and essentially disappears for the heaviest cars, reinforcing that weight — not transmission per se — drives most of the fuel-economy difference.

Residual Diagnostics

Residuals vs. fitted values show no strong pattern, supporting linearity and roughly constant variance. The Normal Q-Q plot is close to the line (a Shapiro-Wilk test on the residuals gives p = 0.08, failing to reject normality). No point has Cook’s distance near 1 (max = 0.35, for the Chrysler Imperial), so no single car unduly drives the fit, though the Imperial and Merc 230 have moderately high leverage and are worth flagging as outliers of interest in future data collection.

Conclusion

After adjusting for confounding by vehicle weight and acceleration, manual transmission cars in this sample get significantly better mileage than automatics — about 2.94 MPG more, holding weight and quarter-mile time fixed (95% CI 0.05–5.83 MPG) — but the bulk of the commonly-cited automatic-vs-manual MPG gap is really a weight effect: manual cars in this data happen to be substantially lighter.

Appendix

A. Pairwise relationships

B. Full (kitchen-sink) model, for comparison

Full model with all 10 covariates (R^2 = 0.88, adj R^2 = 0.82)
Estimate Std. Error t value Pr(>|t|)
(Intercept) 17.820 16.306 1.093 0.287
am_fManual 2.617 2.005 1.306 0.207
wt -3.806 1.847 -2.061 0.053
hp -0.046 0.027 -1.701 0.104
cyl_f6 -1.660 2.262 -0.734 0.472
cyl_f8 1.637 4.316 0.379 0.708
disp 0.014 0.017 0.799 0.433
drat 0.026 1.676 0.016 0.988
qsec 0.647 0.722 0.896 0.381
vs 1.747 2.273 0.769 0.451
gear 0.764 1.457 0.525 0.606
carb 0.509 0.942 0.540 0.595
Nested model comparison (ANOVA)
Res.Df RSS Df Sum of Sq F Pr(>F)
30 720.8966 NA NA NA NA
28 169.2859 2 551.61067 41.3738666 0.0000001
20 133.3235 8 35.96248 0.6743464 0.7083175

C. Sensitivity check: transmission x weight interaction

Interaction model: mpg ~ am * wt + qsec
Estimate Std. Error t value Pr(>|t|)
(Intercept) 9.723 5.899 1.648 0.111
am_fManual 14.079 3.435 4.099 0.000
wt -2.937 0.666 -4.409 0.000
qsec 1.017 0.252 4.035 0.000
am_fManual:wt -4.141 1.197 -3.460 0.002

The significant negative interaction term (am:wt, p = 0.002) means the manual-transmission MPG advantage declines as weight increases: at low weights manual cars have a large advantage, but for very heavy cars the predicted advantage shrinks toward zero. This is consistent with the main model’s finding that weight is the primary driver of MPG differences.

D. Group summary statistics and t-test

Group summary, and Welch two-sample t-test: t = -3.77, df = 18.3, p = 0.0014
Group Mean_MPG SD_MPG N.Var1 N.Freq
Automatic Automatic 17.15 3.83 Automatic 19
Manual Manual 24.39 6.17 Manual 13