Management Summary

Motor Trend, an automobile trend magazine is interested in exploring the relationship between a set of variables and miles per gallon (MPG) outcome. In this project, we will analyze the mtcars dataset from the 1974 Motor Trend US magazine to answer the following questions:

Using simple linear regression analysis, we determine that there is a signficant difference between the mean MPG for automatic and manual transmission cars. Manual transmissions achieve a higher value of MPG compared to automatic transmission. This increase is approximately 2.1 MPG when switching from an automatic transmission to a manual one, with the weight, horsepower and discplacement held constant.

Exploratory analysis and visualizations are located in the Appendix to this document.

The data “mtcars”

First have a look on the first three lines of the data “mtcars”, that is described here and its structure:

##                mpg cyl disp  hp drat    wt  qsec vs am gear carb
## Mazda RX4     21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
## Mazda RX4 Wag 21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
## Datsun 710    22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
## '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 ...

Exploratory Data Analysis

Let’s have a look on the range of the variable “mpg” and its quantiles:

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   10.40   15.42   19.20   20.09   22.80   33.90

Let’s separate between automatic and manual transmission and calculate their relative mean of the variable “mpg”:

Automatic Transmission

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   10.40   14.95   17.30   17.15   19.20   24.40

Manual Transmission

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   15.00   21.00   22.80   24.39   30.40   33.90

Let’s do a t Test, see Appendix A.1 and we see, that there is an obvious and significant difference in the mean of “mpg” for automatic transmission with a mean of 17.1473684 and manual transmission with a mean of 24.3923077. See Appendix A.2 for the boxplot. Let’s have a closer look to the corelations of ‘mpg’ to the other variables of “mtcars”:

Correlation Table
mpg cyl disp hp drat wt qsec vs am gear carb
1 -0.852 -0.848 -0.776 0.681 -0.868 0.419 0.664 0.6 0.48 -0.551

According to the correlation table, there are at least four variables with a high correlation to our outcome variable “mpg”. The highest value comes from the weight variable “wt”. Let’s have a look to this variable separately for automatic (0) and manual (1) transmission in Appendix A.3.

Linear Models

The linear dependencies suggest toanalyse linear models as follows:

fit1 <- lm(mpg ~ am , data = mtcars) 
fit2 <- lm(mpg ~ am + wt, data = mtcars) 
fit3 <- lm(mpg ~ am + wt + hp , data = mtcars) 
fit4 <- lm(mpg ~ am + wt + hp+ disp, data = mtcars) 
fit5 <- lm(mpg ~ ., data = mtcars)

We start with the variable “mpg” as a function of the vairable “am” add one variable after another and do the ANOVA routine (see Appendix A.4) to find the simplest modell that explains significantly the change in “mpg”. I didn’t take the variable “cyl” for its high corellation with the variable “disp”. We see that adding the variables “wt” and “hp” significantly improve the model, so it’s the modell “fit3” which we use further. In Appendix A.5 you find the correlations of the four variables “used”. In Appendix A.6 you find the summay of the modell “fit3” that explans about 84% of the variability of the variable “mpg”.

Let’s turn to the resuduals of modell “fit3”. In Appendix A.7 you find the plot of the residuals. It seems that some “outliers” should be analized more carefully but over all the fit of modell “fit3” and its residuals seem to satisfy basic requirement for a linear modell to explain the variation of the variable “mpg”.

Conclusion

Is an automatic or manual transmission better for MPG?

It appears that manual transmission cars are better for MPG compared to automatic cars. However when modeled with confounding variables like displacement, HP and weight, the difference is not as significant as it seems in the beginning: a big part of the difference is explained by other variables.

Quantify the MPG difference between automatic and manual transmissions

Analysis shows that when only transmission was used in the model manual cars have an mpg increase of 7.245. However, when variables wt and hp are included, the manual car advantage drops to 2.084 with other variables contributing, sometimes more (e.g. weight) to the effect.

Appendix

A.1 t-Test for the variable “mpg” for Automatic and Manual Transmission

## 
##  Welch Two Sample t-test
## 
## data:  auto$mpg and manual$mpg
## 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 of x mean of y 
##  17.14737  24.39231

A.2 Boxplot: Mean of the variable “mpg” for Automatic and Manual Transmission

A.3 Coplot: Dependencies of weight for Automatic and Manual Transmission

coplot(mpg ~ wt | as.factor(am), data = mtcars,
       panel = panel.smooth, rows = 1)

There seems to be a quite linear dependency that differs in function of the variable “am”.

A.4 ANOVA

anova(fit1, fit2, fit3, fit4, fit5)
## Analysis of Variance Table
## 
## Model 1: mpg ~ am
## Model 2: mpg ~ am + wt
## Model 3: mpg ~ am + wt + hp
## Model 4: mpg ~ am + wt + hp + disp
## Model 5: mpg ~ cyl + disp + hp + drat + wt + qsec + vs + am + gear + carb
##   Res.Df    RSS Df Sum of Sq       F    Pr(>F)    
## 1     30 720.90                                   
## 2     29 278.32  1    442.58 63.0133 9.325e-08 ***
## 3     28 180.29  1     98.03 13.9571  0.001219 ** 
## 4     27 179.91  1      0.38  0.0546  0.817510    
## 5     21 147.49  6     32.41  0.7692  0.602559    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

A.5 Correlation of the variables of the Modell “fit3”

# Corelations
mtcars_vars <- mtcars[, c(1, 9, 6, 4)]
mar.orig <- par()$mar  # save the original values 
par(mar = c(1, 1, 1, 1))  # set your new values 
pairs(mtcars_vars, panel = panel.smooth, col = 9 + mtcars$wt)

A.6 Summary of the Modell “fit3”

## 
## Call:
## lm(formula = mpg ~ am + wt + hp, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.4221 -1.7924 -0.3788  1.2249  5.5317 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 34.002875   2.642659  12.867 2.82e-13 ***
## am           2.083710   1.376420   1.514 0.141268    
## wt          -2.878575   0.904971  -3.181 0.003574 ** 
## hp          -0.037479   0.009605  -3.902 0.000546 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.538 on 28 degrees of freedom
## Multiple R-squared:  0.8399, Adjusted R-squared:  0.8227 
## F-statistic: 48.96 on 3 and 28 DF,  p-value: 2.908e-11

A.7 Plot of the Residuals of the Modell “fit3”

par(mfrow = c(2,2))
plot(fit3)