Summary

In this project we are comparing automatic and manual car transmission through miles per gallon (mpg) driven and how it correspond to several other variables.In general tqo questions are answered namely “Is an automatic or manual transmission better for MPG” and “Quantify the MPG difference between automatic and manual transmissions”.Results suggests that manual cars are efficent in fuel average per miles taking into account cofounding variables of weight and number of cylinders.

library(ggplot2)
library(datasets)
data("mtcars")
head(mtcars)
##                    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
## Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
## Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
## Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1
# Transform certain variables into factors
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"))

Regression Analysis

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

So it is evident that MPG gor manual cars is (24.3) which greater than automatic cars.To verify whether there is a signifciant difference we employed t- test

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

Pvalue is 0.001374 demonstrating a significant difference. #Linear regression We will use mpg as the dependent variable and am as the independent variable to fit a linear regression, where Beta1 is the group mean for automatic and Beta0 is the intercept.

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

This shows us that the average MPG for automatic is 17.1 MPG, while manual is 7.2 MPG. We can see that the adjusted R squared value is only 0.3385 which means that only 33.8% of the regression variance can be explained by our model. Therefore there is need to include several other predictor variables in the model.

#Multivariable Regression Model

mlr <- lm(mpg~am + cyl + disp + hp + wt, data = mtcars)
summary(mlr)
## 
## Call:
## lm(formula = mpg ~ am + cyl + disp + hp + wt, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.9374 -1.3347 -0.3903  1.1910  5.0757 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 33.864276   2.695416  12.564 2.67e-12 ***
## amManual     1.806099   1.421079   1.271   0.2155    
## cyl6        -3.136067   1.469090  -2.135   0.0428 *  
## cyl8        -2.717781   2.898149  -0.938   0.3573    
## disp         0.004088   0.012767   0.320   0.7515    
## hp          -0.032480   0.013983  -2.323   0.0286 *  
## wt          -2.738695   1.175978  -2.329   0.0282 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.453 on 25 degrees of freedom
## Multiple R-squared:  0.8664, Adjusted R-squared:  0.8344 
## F-statistic: 27.03 on 6 and 25 DF,  p-value: 8.861e-10
anova(lrm, mlr)
## Analysis of Variance Table
## 
## Model 1: mpg ~ am
## Model 2: mpg ~ am + cyl + disp + hp + wt
##   Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
## 1     30 720.90                                  
## 2     25 150.41  5    570.49 18.965 8.637e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

This results in a p-value of 8.861e-08, and we can claim that mlr is significantly better than our lrm simple model.The model explains 86.64% of the variance and as a result, cyl, disp, hp, wt did affect the correlation between mpg and am. Thus, we can say the difference between automatic and manual transmissions is 1.80 MPG.

#Graphs Plot 1 - Boxplot of MPG by transmission type

boxplot(mpg ~ am, data = mtcars, col = (c("red","blue")), ylab = "Miles Per Gallon", xlab = "Transmission Type")

#Plot 2 - Pairs (Correlation plot) plot for the data set

We explore the other variable via a pairs plot to see how all the variables correlate with mpg. From this we see that cyl, disp, hp, wt have the strongest correlation with mpg depicted below as,

pairs(mpg ~ ., data = mtcars)

#Plot 3 Residuals We check the residuals for non-normality and deduce that they are all normally distributed and homoskedastic.

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