Summary

In this project we are going to work for Motor Trend, a magazine about the automobile industry. They are interested in exploring the relationship between a set of variables and miles per gallon. They are particularly interested in the following two questions:

  1. Is an automatic or manual transmission better for MPG?
  2. Quantify the MPG difference between automatic and manual transmissions

Exploratory Data Analysis

The data was extracted from the 1974 Motor Trend US magazine, and comprises fuel consumption and 10 aspects of automobile design and performance for 32 automobiles (1973-74 models).

library(datasets) 
data(mtcars)

Some exploratory data analysis:

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 ...

For the analysis we have to transform the class of some variables variable into a factor:

mtcars$cyl <- factor(mtcars$cyl)
mtcars$vs <- factor(mtcars$vs)
mtcars$gear <- factor(mtcars$gear)
mtcars$carb <- factor(mtcars$carb)
mtcars$am <- factor(mtcars$am,labels=c('Automatic','Manual'))

1.Is an automatic or manual transmission better for MPG?

In Appendix 1 we can see a boxplot of MPG by transmission types (Appendix 2). According to Appendix 2 boxplot, it seems that manual transmission is better in miles/(US) gallon than automatic transmission. We can do a t-test to reject the null hypothesis (there is no difference in MPG between manual and automatic transmission).

t.test(mpg ~ am,data=mtcars)
## 
##  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                24.39231

Based on t-test, p-value = 0.001374 < 0.05, so we can reject the null hypothesis and hence manual transmission is better than automatic transmission for MPG.

2.Quantify the MPG difference between automatic and manual transmissions

To quantify the MPG difference between automatic and manual transmissions we are going to fit a lineal model between mpg and am variables.

fit1=lm(mpg ~ am,data=mtcars)
summary(fit1)
## 
## 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: 0.000285

From this model, we can say that automatic run at 17.15 mpg, while manual have 7.24 more mpg and about 35.98% of the variance is explained by this model.

Multivariable linear regression model:

fit2=lm(mpg ~ am + wt, data=mtcars)
fit3=lm(mpg ~ am + wt + cyl, data=mtcars)
fit4=lm(mpg ~ am + wt + cyl + hp, data=mtcars)
anova(fit1,fit2,fit3,fit4)
## Analysis of Variance Table
## 
## Model 1: mpg ~ am
## Model 2: mpg ~ am + wt
## Model 3: mpg ~ am + wt + cyl
## Model 4: mpg ~ am + wt + cyl + hp
##   Res.Df    RSS Df Sum of Sq       F   Pr(>F)    
## 1     30 720.90                                  
## 2     29 278.32  1    442.58 76.1924 3.32e-09 ***
## 3     27 182.97  2     95.35  8.2077 0.001725 ** 
## 4     26 151.03  1     31.94  5.4991 0.026935 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The final model:

summary(fit4)
## 
## Call:
## lm(formula = mpg ~ am + wt + cyl + hp, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.9387 -1.2560 -0.4013  1.1253  5.0513 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 33.70832    2.60489  12.940 7.73e-13 ***
## amManual     1.80921    1.39630   1.296  0.20646    
## wt          -2.49683    0.88559  -2.819  0.00908 ** 
## cyl6        -3.03134    1.40728  -2.154  0.04068 *  
## cyl8        -2.16368    2.28425  -0.947  0.35225    
## hp          -0.03211    0.01369  -2.345  0.02693 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.41 on 26 degrees of freedom
## Multiple R-squared:  0.8659, Adjusted R-squared:  0.8401 
## F-statistic: 33.57 on 5 and 26 DF,  p-value: 1.506e-10

This model explains 86.59% of the variance. It may be concluded that on average, manual transmissions have 1.81 more mpg than automatic.

Appendix

Appendix 1:

library(ggplot2)
g=ggplot(mtcars, aes(x=am,y=mpg,fill=am)) +
        geom_boxplot(varwidth=T,width=0.5)+
        geom_jitter(shape=16,position=position_jitter(0.1),alpha=0.5)+
        labs(x="Transmission",y="Miles per galon")+
        scale_x_discrete(labels=c("Manual","Automatic"))
g

Appendix 2:

Residual plots:

plot(fit4)