The Relationship Between Miles per Gallon and Transmission Type in Automobiles
Orumwese Esosa Cyriaque
9/9/2021
Synopsis
This project involves exploring the relationship between miles per gallon (MPG) and transmission in a data set of a collection of cars. Two questions are of interest: Is an automatic or manual transmission better for MPG? and what is the quantifiable MPG difference between automatic and manual transmissions? It was found that the weight of the car (wt) is a confounding variable in the relationship between miles per gallon (mpg) and transmission (am). It was also found that manual transmission cars have, on an average, 1.55 times more mpg than automatic transmission cars.
Loading the Data
The mtcars (Motor Trend Car Road Tests) is a data set which comprises of fuel consumption and 10 aspects of automobile design and performance for 32 automobiles (1973-1974 models). It has 32 observations on 11 variables.
mpg: Miles/(US) galloncyl: Number of cylindersdisp: Displacement (cu.in.)hp: Gross horsepowerdrat: Rear axle ratiowt: Weight (1000 lbs)qsec: 1/4 mile timevs: Engine (0 = V-shaped, 1 = straight)am: Transmission (0 = automatic, 1 = manual)gear: Number of forward gearscarb: Number of carburetors
library(datasets)
library(ggplot2)## Warning: package 'ggplot2' was built under R version 4.0.5data(mtcars)
head(mtcars,11)## mpg cyl disp hp drat wt qsec vs am gear carb
## Mazda RX4 21.0 6 160.0 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160.0 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108.0 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258.0 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360.0 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225.0 105 2.76 3.460 20.22 1 0 3 1
## Duster 360 14.3 8 360.0 245 3.21 3.570 15.84 0 0 3 4
## Merc 240D 24.4 4 146.7 62 3.69 3.190 20.00 1 0 4 2
## Merc 230 22.8 4 140.8 95 3.92 3.150 22.90 1 0 4 2
## Merc 280 19.2 6 167.6 123 3.92 3.440 18.30 1 0 4 4
## Merc 280C 17.8 6 167.6 123 3.92 3.440 18.90 1 0 4 4Exploratory Data Analysis
Basic Summary Tests
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. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0
## 1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st Qu.: 96.5
## Median :19.20 Median :6.000 Median :196.3 Median :123.0
## Mean :20.09 Mean :6.188 Mean :230.7 Mean :146.7
## 3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd Qu.:180.0
## Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0
## drat wt qsec vs
## Min. :2.760 Min. :1.513 Min. :14.50 Min. :0.0000
## 1st Qu.:3.080 1st Qu.:2.581 1st Qu.:16.89 1st Qu.:0.0000
## Median :3.695 Median :3.325 Median :17.71 Median :0.0000
## Mean :3.597 Mean :3.217 Mean :17.85 Mean :0.4375
## 3rd Qu.:3.920 3rd Qu.:3.610 3rd Qu.:18.90 3rd Qu.:1.0000
## Max. :4.930 Max. :5.424 Max. :22.90 Max. :1.0000
## am gear carb
## Min. :0.0000 Min. :3.000 Min. :1.000
## 1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:2.000
## Median :0.0000 Median :4.000 Median :2.000
## Mean :0.4062 Mean :3.688 Mean :2.812
## 3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :1.0000 Max. :5.000 Max. :8.000Unique Values
unique(mtcars[,1])## [1] 21.0 22.8 21.4 18.7 18.1 14.3 24.4 19.2 17.8 16.4 17.3 15.2 10.4 14.7 32.4
## [16] 30.4 33.9 21.5 15.5 13.3 27.3 26.0 15.8 19.7 15.0unique(mtcars[,2])## [1] 6 4 8unique(mtcars[,3])## [1] 160.0 108.0 258.0 360.0 225.0 146.7 140.8 167.6 275.8 472.0 460.0 440.0
## [13] 78.7 75.7 71.1 120.1 318.0 304.0 350.0 400.0 79.0 120.3 95.1 351.0
## [25] 145.0 301.0 121.0unique(mtcars[,4])## [1] 110 93 175 105 245 62 95 123 180 205 215 230 66 52 65 97 150 91 113
## [20] 264 335 109unique(mtcars[,5])## [1] 3.90 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.07 2.93 3.00 3.23 4.08 4.93 4.22
## [16] 3.70 3.73 4.43 3.77 3.62 3.54 4.11unique(mtcars[,6])## [1] 2.620 2.875 2.320 3.215 3.440 3.460 3.570 3.190 3.150 4.070 3.730 3.780
## [13] 5.250 5.424 5.345 2.200 1.615 1.835 2.465 3.520 3.435 3.840 3.845 1.935
## [25] 2.140 1.513 3.170 2.770 2.780unique(mtcars[,7])## [1] 16.46 17.02 18.61 19.44 20.22 15.84 20.00 22.90 18.30 18.90 17.40 17.60
## [13] 18.00 17.98 17.82 17.42 19.47 18.52 19.90 20.01 16.87 17.30 15.41 17.05
## [25] 16.70 16.90 14.50 15.50 14.60 18.60unique(mtcars[,8])## [1] 0 1unique(mtcars[,9])## [1] 1 0unique(mtcars[,10])## [1] 4 3 5unique(mtcars[,11])## [1] 4 1 2 3 6 8Converting the Class of Specific Columns to Factors
#mtcars$cyl <- factor(mtcars$cyl)
mtcars$vs <- factor(mtcars$vs)
mtcars$am <- factor(mtcars$am, labels=c("automatic","manual")) ## making the labels more meaningful
#mtcars$gear <- factor(mtcars$gear)
#mtcars$carb <- factor(mtcars$carb)Boxplot
ggplot(mtcars, aes(x=am, y=mpg, fill=am)) +
geom_boxplot() +
labs(x="Transmission", y="Miles per gallon", title="Relationship Between MPG and Transmission") +
theme(plot.title=element_text(hjust=0.5, face="bold"))
From the plot above, it is expected that the manual cars generally have better mileage than automatic cars. With the aid of Statistical Inference, we will be able to have a clearer picture of the relationship.
Statistical Inference
We can use the R function t.test to find out whether our hypothesis that manual cars get better gas mileage than automatic cars is statistically significant.
t.test(data=mtcars, mpg~am)##
## Welch Two Sample t-test
##
## data: mpg by am
## t = -3.7671, df = 18.332, p-value = 0.001374
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -11.280194 -3.209684
## sample estimates:
## mean in group automatic mean in group manual
## 17.14737 24.39231From the test, since the p value is lesser than the significance level of 0.05 and the confidence interval does not contain 0, therefore our hypothesis is significant. We can conclude that the difference in mpg between the manual and automatic transmission is infact significant.
Regression Models
Linear Regression Model
We will use mpg as the dependent variable and am as the independent variable to fit a linear regression model.
am.model <- lm(mpg~am, mtcars)
summary(am.model)##
## Call:
## lm(formula = mpg ~ am, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.3923 -3.0923 -0.2974 3.2439 9.5077
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.147 1.125 15.247 1.13e-15 ***
## ammanual 7.245 1.764 4.106 0.000285 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.902 on 30 degrees of freedom
## Multiple R-squared: 0.3598, Adjusted R-squared: 0.3385
## F-statistic: 16.86 on 1 and 30 DF, p-value: 0.000285The R-squared value for this model is only 0.3598. This means that fitting mpg on am alone only explains about 36% of the variance in mpg.
Multivariable Regression Model
We will run a linear regression model for all variables against mpg. This gives us insight into variables with coefficient significance as well as an initial attempt at explaining mpg. Additionally, we will also look at the correlation of variables with mpg to help us choose an appropriate model.
Full Model
full.model <- lm(mpg~., mtcars)
summary(full.model)##
## Call:
## lm(formula = mpg ~ ., data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4506 -1.6044 -0.1196 1.2193 4.6271
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.30337 18.71788 0.657 0.5181
## cyl -0.11144 1.04502 -0.107 0.9161
## disp 0.01334 0.01786 0.747 0.4635
## hp -0.02148 0.02177 -0.987 0.3350
## drat 0.78711 1.63537 0.481 0.6353
## wt -3.71530 1.89441 -1.961 0.0633 .
## qsec 0.82104 0.73084 1.123 0.2739
## vs1 0.31776 2.10451 0.151 0.8814
## ammanual 2.52023 2.05665 1.225 0.2340
## gear 0.65541 1.49326 0.439 0.6652
## carb -0.19942 0.82875 -0.241 0.8122
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.65 on 21 degrees of freedom
## Multiple R-squared: 0.869, Adjusted R-squared: 0.8066
## F-statistic: 13.93 on 10 and 21 DF, p-value: 3.793e-07This shows that 86% or more of the variance in the data can be explained by the model but the p values of each variable is more than the 0.05 significance level. Therefore we’ll use cor() function to find the variables with strong correlations.
data(mtcars) ## Recalling the data because cor() had issues working with the former dataset due to it not being entirely numerical
cor(mtcars)[1,]## mpg cyl disp hp drat wt qsec
## 1.0000000 -0.8521620 -0.8475514 -0.7761684 0.6811719 -0.8676594 0.4186840
## vs am gear carb
## 0.6640389 0.5998324 0.4802848 -0.5509251From the above output, it can be seen that cyl, disp, hp and wt have a strong correlation with mpg. The relationship can further be visualized with a correlation plot and a pairs plot.
## Correlation Plot
library(corrplot)## corrplot 0.90 loadedcorrplot.mixed(cor(mtcars), lower="circle", upper="number")
## Pairs plot
pairs(mpg~., data=mtcars)
Final Model
The final model is given as;
final.model <- lm(mpg~am+cyl+disp+hp+wt, data=mtcars)
summary(final.model)##
## Call:
## lm(formula = mpg ~ am + cyl + disp + hp + wt, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.5952 -1.5864 -0.7157 1.2821 5.5725
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 38.20280 3.66910 10.412 9.08e-11 ***
## am 1.55649 1.44054 1.080 0.28984
## cyl -1.10638 0.67636 -1.636 0.11393
## disp 0.01226 0.01171 1.047 0.30472
## hp -0.02796 0.01392 -2.008 0.05510 .
## wt -3.30262 1.13364 -2.913 0.00726 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.505 on 26 degrees of freedom
## Multiple R-squared: 0.8551, Adjusted R-squared: 0.8273
## F-statistic: 30.7 on 5 and 26 DF, p-value: 4.029e-10The model has an R-squared value 0.8551, a high F statistic and a low p-value. Therefore, the model is a good fit.
Residual Plot and Analysis
The “Residuals vs Fitted” plot here shows us that the residuals are homoscedastic. We can also see that they are normally distributed using the quantile plot.
par(mfrow=c(2,2))
plot(final.model)
Conclusion
From our analysis, we can conclude that the weight of the car (wt) is a confounding variable in the relationship between miles per gallon (mpg) and transmission (am). It was also found that manual transmission cars have, on an average, 1.55 times more mpg than automatic transmission cars.