Executive Summary

Manual transmission better for MPG (miles per gallon) than automatic transmission

Using hypothesis testing and simple linear regression, we determine that there is a signficant difference between the mean MPG for automatic and manual transmission cars. And that manual transmission has 7.245 more MPGs on average than automatic. We did a multivariate regression to improve estimate of transmission types on MPG. Results after anova from the multivariate regression shows that, on average, manual transmission cars get 2.084 miles per gallon more than automatic transmission cars.

Processing data

data(mtcars)
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'))
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 : Factor w/ 3 levels "4","6","8": 2 2 1 2 3 2 3 1 1 2 ...
##  $ 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  : Factor w/ 2 levels "0","1": 1 1 2 2 1 2 1 2 2 2 ...
##  $ am  : Factor w/ 2 levels "Automatic","Manual": 2 2 2 1 1 1 1 1 1 1 ...
##  $ gear: Factor w/ 3 levels "3","4","5": 2 2 2 1 1 1 1 2 2 2 ...
##  $ carb: Factor w/ 6 levels "1","2","3","4",..: 4 4 1 1 2 1 4 2 2 4 ...

Exploratory data analysis

Do a boxplot to examine car transmission types on mpg. We can say there is increase in mpg for manual transmission vs automatic transmission. Also, plot histogram to check for normal curve.

par(mfrow = c(1, 2))
# Histogram with Normal Curve
x <- mtcars$mpg
h<-hist(x, breaks=10, col="salmon", xlab="Miles Per Gallon",
   main="Histogram of Miles per Gallon")
xfit<-seq(min(x),max(x),length=40)
yfit<-dnorm(xfit,mean=mean(x),sd=sd(x))
yfit <- yfit*diff(h$mids[1:2])*length(x)
lines(xfit, yfit, col="blue", lwd=2) 

boxplot(mpg~am, data = mtcars,
        col = c("light blue", "light grey"),
        xlab = "Transmission",
        ylab = "Miles per Gallon",
        main = "MPG by Transmission Type")

Hypothesis testing

aggregate(mpg ~ am, data = mtcars, mean) 
##          am      mpg
## 1 Automatic 17.14737
## 2    Manual 24.39231

Seems that mean MPG of manual transmission cars is 7.24 MPGs higher than that of automatic transmission cars. We need to check whether thIs is a significant difference. Set alpha-value at 0.5 and run a t-test to find out.

autoData <- mtcars[mtcars$am == "Automatic",]
manualData <- mtcars[mtcars$am == "Manual",]
t.test(autoData$mpg, manualData$mpg) 
## 
##  Welch Two Sample t-test
## 
## data:  autoData$mpg and manualData$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

Since the p-value is 0.00137, we reject the null hypothesis and claim that there is a signficiant difference in the mean MPG between manual transmission cars and that of automatic transmission cars. Now we must quantify that difference.

Regression analysis

mfit <- lm(mpg~., data=mtcars)
pairs (mtcars) 

data(mtcars)
sort(cor(mtcars)[1,])
##         wt        cyl       disp         hp       carb       qsec 
## -0.8676594 -0.8521620 -0.8475514 -0.7761684 -0.5509251  0.4186840 
##       gear         am         vs       drat        mpg 
##  0.4802848  0.5998324  0.6640389  0.6811719  1.0000000

Based on pairwise correlation of variables with mpg, we see that there is little linear correlation between mpg and the variables qsec, gear, and carb.

In addition, we see that wt, cyl, disp, and hp are highly correlated with our dependent variable mpg. As such, they may be good candidates to include in our model. However, if we look at the correlation matrix, we also see that cyl and disp are highly correlated with each other. Since predictors should not exhibit collinearity, we should not have cyl and disp in in our model.

If, including wt and hp in our regression equation makes sense intuitively - heavier cars and cars that have more horsepower should have lower MPGs.

Model building and selection

fit <- lm(mpg~am, data = mtcars)
summary(lm(mpg~am, data = mtcars)) 
## 
## 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 ***
## am             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

On average, automatic cars have 17.147 MPG and manual transmission cars have 7.245 MPGs more. In addition, we see that the R2 value is 0.3598, implying this model explains only 35.98% of the variance.

Multivariate linear regression

Multivariate linear regression for mpg on am, wt, and hp. Since we have two models of the same data, we run an ANOVA to compare the two models and see if they are significantly different.

bestfit <- lm(mpg~am + wt + hp, data = mtcars)
anova(fit, bestfit) 
## Analysis of Variance Table
## 
## Model 1: mpg ~ am
## Model 2: mpg ~ am + wt + hp
##   Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
## 1     30 720.90                                  
## 2     28 180.29  2    540.61 41.979 3.745e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

With a p-value of 3.745e-09, we reject the null hypothesis and claim that our multivariate model is significantly different from our simple model.

Now, check residuals to see whether they are normally distributed and examine residual vs fitted values plot to spot for heteroskedasticity.

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

Residual are normally distributed and homoskedastic. So, we can report the estimates from our report.

summary(bestfit) 
## 
## 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

Conclusion

In Summary, Manual transmission cars have a better mpg than automatic transmission cars based on our linear model. Based on the coefficients given in our new model, manual transmission cars have a higher mpg value than automatic transmission cars by 2.09.