Automatic or Manual transmission? Which one should you get?
Executive summary
This study attempts to determine whether automatic transmission cars are less fuel efficient compared to manual transmission cars.
In this study, we apply a parametric test (t-test) and regression analysis to analyze this question.
Results of the statistical tests show that without controlling for other car features manual cars are more fuel-efficient than automatic cars.
On average, they run about 7 miles more than automatic cars per gallon. However Regression Analysis indicates that taking into account other car features such as displacement, rear axle ratio and car weight, manual transmission cars are no longer signficiantly better for MPG compared to automatic cars.
Section I. Data processing and summary statistics
The mtcar dataset for this study is from the 1974 “Motor Trend US”“ magazine, consisting of fuel consumption measurement (mpg) for 32 automobiles for models ranging from 1973 to 1974.
#Information about the cars dataset.
str(mtcars)## 'data.frame': 32 obs. of 11 variables:
## $ mpg : num 21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
## $ cyl : num 6 6 4 6 8 6 8 4 4 6 ...
## $ disp: num 160 160 108 258 360 ...
## $ hp : num 110 110 93 110 175 105 245 62 95 123 ...
## $ drat: num 3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
## $ wt : num 2.62 2.88 2.32 3.21 3.44 ...
## $ qsec: num 16.5 17 18.6 19.4 17 ...
## $ vs : num 0 0 1 1 0 1 0 1 1 1 ...
## $ am : num 1 1 1 0 0 0 0 0 0 0 ...
## $ gear: num 4 4 4 3 3 3 3 4 4 4 ...
## $ carb: num 4 4 1 1 2 1 4 2 2 4 ...summary(mtcars)## mpg cyl disp hp
## Min. :10.4 Min. :4.00 Min. : 71.1 Min. : 52.0
## 1st Qu.:15.4 1st Qu.:4.00 1st Qu.:120.8 1st Qu.: 96.5
## Median :19.2 Median :6.00 Median :196.3 Median :123.0
## Mean :20.1 Mean :6.19 Mean :230.7 Mean :146.7
## 3rd Qu.:22.8 3rd Qu.:8.00 3rd Qu.:326.0 3rd Qu.:180.0
## Max. :33.9 Max. :8.00 Max. :472.0 Max. :335.0
## drat wt qsec vs
## Min. :2.76 Min. :1.51 Min. :14.5 Min. :0.000
## 1st Qu.:3.08 1st Qu.:2.58 1st Qu.:16.9 1st Qu.:0.000
## Median :3.69 Median :3.33 Median :17.7 Median :0.000
## Mean :3.60 Mean :3.22 Mean :17.8 Mean :0.438
## 3rd Qu.:3.92 3rd Qu.:3.61 3rd Qu.:18.9 3rd Qu.:1.000
## Max. :4.93 Max. :5.42 Max. :22.9 Max. :1.000
## am gear carb
## Min. :0.000 Min. :3.00 Min. :1.00
## 1st Qu.:0.000 1st Qu.:3.00 1st Qu.:2.00
## Median :0.000 Median :4.00 Median :2.00
## Mean :0.406 Mean :3.69 Mean :2.81
## 3rd Qu.:1.000 3rd Qu.:4.00 3rd Qu.:4.00
## Max. :1.000 Max. :5.00 Max. :8.00Section II: Statistical Analysis
#pre-process the data for statistical analysis
aggmpg <- tapply(mtcars$mpg, mtcars$am, mean, na.rm = TRUE)
sdmpg <- tapply(mtcars$mpg, mtcars$am, sd, na.rm = TRUE)
aggam <- unique(factor(c("automatic", "manual")))
#validate pre-processing
barmpg <- barplot(aggmpg, names = aggam, ylim = c(0, 35), main = paste("Average Miles per Gallon by Transmission Type"),
space = 0.4, axes = TRUE, axis.lty = 10, col = "white", xlab = "Transmission Type",
ylab = "MPG")
box()
segments(barmpg, aggmpg - sdmpg, barmpg, aggmpg + sdmpg, lwd = 3)
segments(barmpg - 0.05, aggmpg - sdmpg, barmpg + 0.05, aggmpg - sdmpg, lwd = 2)
segments(barmpg - 0.05, aggmpg + sdmpg, barmpg + 0.05, aggmpg + sdmpg, lwd = 2)
Statistical t-test to compare mpg between automatic vs manual transmission. The results shows that manual transmission are more gas efficient than automatic cars.
t.test(mpg ~ factor(am), data = mtcars)##
## Welch Two Sample t-test
##
## data: mpg by factor(am)
## t = -3.767, df = 18.33, p-value = 0.001374
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -11.28 -3.21
## sample estimates:
## mean in group 0 mean in group 1
## 17.15 24.39Section III. Regressional Analysis
The results from the statistical tests focus on mpg and am only, without controlling for influence from other variables. If we apply a multivariate regression, the marginal impact of automatic vs manual transmission cars does not turn out to be significant. The confounding variables include displacement (disp), rear axle ratio (drat) and car weight(wt). Using car weight as an example:
fit0 <- lm(mpg ~ factor(am), data = mtcars)
summary(fit0)##
## Call:
## lm(formula = mpg ~ factor(am), data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.392 -3.092 -0.297 3.244 9.508
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.15 1.12 15.25 1.1e-15 ***
## factor(am)1 7.24 1.76 4.11 0.00029 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.9 on 30 degrees of freedom
## Multiple R-squared: 0.36, Adjusted R-squared: 0.338
## F-statistic: 16.9 on 1 and 30 DF, p-value: 0.000285fit1 <- lm(mpg ~ factor(am) + wt, data = mtcars)
summary(fit1)##
## Call:
## lm(formula = mpg ~ factor(am) + wt, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.530 -2.362 -0.132 1.403 6.878
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 37.3216 3.0546 12.22 5.8e-13 ***
## factor(am)1 -0.0236 1.5456 -0.02 0.99
## wt -5.3528 0.7882 -6.79 1.9e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.1 on 29 degrees of freedom
## Multiple R-squared: 0.753, Adjusted R-squared: 0.736
## F-statistic: 44.2 on 2 and 29 DF, p-value: 1.58e-09#The regression suggests that holding other variables constant that manual transmission cars consume on avarage more gallons of gas per mile and the results are not statistically significant.
anova(fit0, fit1)## Analysis of Variance Table
##
## Model 1: mpg ~ factor(am)
## Model 2: mpg ~ factor(am) + wt
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 721
## 2 29 278 1 443 46.1 1.9e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1##QQ plots of the residuals suggest that the errors scatter around zero.
plot(residuals(fit1), main = "QQ plot of the residuals of model 1")
qqnorm(residuals(fit1))
qqline(residuals(fit1))
#Diagnostics (dfbeta and hatvalues) indicates that there are no outlier in the dataset - individual coefficients and predicted response do not change much no matter which point is removed.
round(dfbetas(fit1)[1:32, 2], 4)## Mazda RX4 Mazda RX4 Wag Datsun 710
## -0.1512 -0.0749 -0.0996
## Hornet 4 Drive Hornet Sportabout Valiant
## -0.0897 0.0115 0.0378
## Duster 360 Merc 240D Merc 230
## 0.1902 -0.3057 -0.1745
## Merc 280 Merc 280C Merc 450SE
## -0.0161 0.0612 -0.0153
## Merc 450SL Merc 450SLC Cadillac Fleetwood
## 0.0021 0.0659 0.0725
## Lincoln Continental Chrysler Imperial Fiat 128
## 0.1637 0.4549 0.3072
## Honda Civic Toyota Corolla Toyota Corona
## 0.0088 0.1303 0.3386
## Dodge Challenger AMC Javelin Camaro Z28
## 0.1522 0.2128 0.1105
## Pontiac Firebird Fiat X1-9 Porsche 914-2
## -0.0768 0.0088 0.0058
## Lotus Europa Ford Pantera L Ferrari Dino
## -0.0018 -0.4877 -0.2121
## Maserati Bora Volvo 142E
## -0.4433 -0.0775round(hatvalues(fit1)[1:32], 4)## Mazda RX4 Mazda RX4 Wag Datsun 710
## 0.0798 0.0909 0.0775
## Hornet 4 Drive Hornet Sportabout Valiant
## 0.0725 0.0596 0.0588
## Duster 360 Merc 240D Merc 230
## 0.0552 0.0743 0.0774
## Merc 280 Merc 280C Merc 450SE
## 0.0596 0.0596 0.0585
## Merc 450SL Merc 450SLC Cadillac Fleetwood
## 0.0527 0.0526 0.1947
## Lincoln Continental Chrysler Imperial Fiat 128
## 0.2300 0.2135 0.0798
## Honda Civic Toyota Corolla Toyota Corona
## 0.1179 0.0984 0.1627
## Dodge Challenger AMC Javelin Camaro Z28
## 0.0566 0.0598 0.0530
## Pontiac Firebird Fiat X1-9 Porsche 914-2
## 0.0530 0.0916 0.0817
## Lotus Europa Ford Pantera L Ferrari Dino
## 0.1291 0.1142 0.0853
## Maserati Bora Volvo 142E
## 0.1639 0.0857