Regressional Models Course Project
Vivek Mangipudi
October 7, 2015
Analysis of Miles per Gallon between Automatic Transmission and Manual Transmission
Executive Summary:
Here it is assumed that I work for Motor Trend, a magazine about the automobile industry.With regards to this automobile industry, Looking at a data set of these collection of cars, We will be exploring the relationship between a set of variables and miles per gallon (MPG) (outcome).The dataset for this particular project is from the 1974 Motor Trend US magazine, and contains details of the fuel consumption,ten measurements of automobile design and performance for 32 automobiles (1973–74 models).We use regression models and exploratory data analyses to mainly explore how automatic (am = 0) and manual (am = 1) transmissions features affect the MPG feature. We are particularly interested in the following two questions:
“Is an automatic or manual transmission better for MPG?”
“Quantifying the MPG difference between automatic and manual transmissions”
To answer these two questions we make use of various analyses including t test, anova and others.Then, based on the results we draw conclusions from it while determining answers to the above two questions. First step involves loading the data and looking at the summary.
library(datasets)
data(mtcars)
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. :8.000Now we will look at the various variables in this data set followed by a glance into it’s contents
names(mtcars)## [1] "mpg" "cyl" "disp" "hp" "drat" "wt" "qsec" "vs" "am" "gear"
## [11] "carb"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 1Our first step is to look at MPG for both Automatic Transmission and Manual Transmission an decide which is better. Now
boxplot(mpg~am, data = mtcars,
xlab = "Transmission",
ylab = "Miles per Gallon",
col='blue',
main = "MPG by Transmission Type",
names = c("Automatic", "Manual"))
From this box plot we can pretty much say that Manual Transmission is better than Automatic transmission. But we are going to do that in a methodical fashion. We now make use of the two sample t-test with a 95 % confidence level to confirm our above hypothesis.
t.test(mtcars$mpg~mtcars$am,conf.level=0.95)##
## Welch Two Sample t-test
##
## data: mtcars$mpg by mtcars$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 0 mean in group 1
## 17.14737 24.39231From the observation it is clear that the p value= 0.001374,so we reject the null hypothesis.
We will therefore conclude that Manual is better than automatic.
##PART 2: QUANTIFYING THE DIFFERENCE IN MPG (Exploratory analysis)
Here we have to look at the individual means. First for Automatic
summary(mtcars[mtcars$am==0,])## mpg cyl disp hp
## Min. :10.40 Min. :4.000 Min. :120.1 Min. : 62.0
## 1st Qu.:14.95 1st Qu.:6.000 1st Qu.:196.3 1st Qu.:116.5
## Median :17.30 Median :8.000 Median :275.8 Median :175.0
## Mean :17.15 Mean :6.947 Mean :290.4 Mean :160.3
## 3rd Qu.:19.20 3rd Qu.:8.000 3rd Qu.:360.0 3rd Qu.:192.5
## Max. :24.40 Max. :8.000 Max. :472.0 Max. :245.0
## drat wt qsec vs
## Min. :2.760 Min. :2.465 Min. :15.41 Min. :0.0000
## 1st Qu.:3.070 1st Qu.:3.438 1st Qu.:17.18 1st Qu.:0.0000
## Median :3.150 Median :3.520 Median :17.82 Median :0.0000
## Mean :3.286 Mean :3.769 Mean :18.18 Mean :0.3684
## 3rd Qu.:3.695 3rd Qu.:3.842 3rd Qu.:19.17 3rd Qu.:1.0000
## Max. :3.920 Max. :5.424 Max. :22.90 Max. :1.0000
## am gear carb
## Min. :0 Min. :3.000 Min. :1.000
## 1st Qu.:0 1st Qu.:3.000 1st Qu.:2.000
## Median :0 Median :3.000 Median :3.000
## Mean :0 Mean :3.211 Mean :2.737
## 3rd Qu.:0 3rd Qu.:3.000 3rd Qu.:4.000
## Max. :0 Max. :4.000 Max. :4.000The mean is : 17.15 Now for Manual
summary(mtcars[mtcars$am==1,])## mpg cyl disp hp
## Min. :15.00 Min. :4.000 Min. : 71.1 Min. : 52.0
## 1st Qu.:21.00 1st Qu.:4.000 1st Qu.: 79.0 1st Qu.: 66.0
## Median :22.80 Median :4.000 Median :120.3 Median :109.0
## Mean :24.39 Mean :5.077 Mean :143.5 Mean :126.8
## 3rd Qu.:30.40 3rd Qu.:6.000 3rd Qu.:160.0 3rd Qu.:113.0
## Max. :33.90 Max. :8.000 Max. :351.0 Max. :335.0
## drat wt qsec vs
## Min. :3.54 Min. :1.513 Min. :14.50 Min. :0.0000
## 1st Qu.:3.85 1st Qu.:1.935 1st Qu.:16.46 1st Qu.:0.0000
## Median :4.08 Median :2.320 Median :17.02 Median :1.0000
## Mean :4.05 Mean :2.411 Mean :17.36 Mean :0.5385
## 3rd Qu.:4.22 3rd Qu.:2.780 3rd Qu.:18.61 3rd Qu.:1.0000
## Max. :4.93 Max. :3.570 Max. :19.90 Max. :1.0000
## am gear carb
## Min. :1 Min. :4.000 Min. :1.000
## 1st Qu.:1 1st Qu.:4.000 1st Qu.:1.000
## Median :1 Median :4.000 Median :2.000
## Mean :1 Mean :4.385 Mean :2.923
## 3rd Qu.:1 3rd Qu.:5.000 3rd Qu.:4.000
## Max. :1 Max. :5.000 Max. :8.000The mean is : 24.39 Thus the difference between the means is approximately 7mpg.Now let us examine what factors actuallly affect the miles per gallon by using regression analysis. ##Single Variable Linear Regression model:
x<-lm(mpg~am,data=mtcars)
summary(x)##
## 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: 0.000285As it can be observed this model has a r-squared error of 35.98%. i.e this model accounts for only 35.98% of the variance. So we will try the multivariate regression. ##Multi Variate Regression:
mvr <- lm(mpg~am + wt + hp + cyl, data = mtcars)
anova(x,mvr)## Analysis of Variance Table
##
## Model 1: mpg ~ am
## Model 2: mpg ~ am + wt + hp + cyl
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 30 720.9
## 2 27 170.0 3 550.9 29.166 1.274e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Our required final model is the one shown below:
summary(mvr)##
## Call:
## lm(formula = mpg ~ am + wt + hp + cyl, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4765 -1.8471 -0.5544 1.2758 5.6608
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 36.14654 3.10478 11.642 4.94e-12 ***
## am 1.47805 1.44115 1.026 0.3142
## wt -2.60648 0.91984 -2.834 0.0086 **
## hp -0.02495 0.01365 -1.828 0.0786 .
## cyl -0.74516 0.58279 -1.279 0.2119
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.509 on 27 degrees of freedom
## Multiple R-squared: 0.849, Adjusted R-squared: 0.8267
## F-statistic: 37.96 on 4 and 27 DF, p-value: 1.025e-10Here this model explains 84.9% of the variance.
It may be concluded that on average, manual transmissions have 1.478 more mpg than automatic. ##Appendix: ###1.Scatterplot matrix of the “mtcars” dataset
pairs(mpg ~ ., data = mtcars)
###2. Various Plots (Residual plots and analysis)
plot(mvr)
par(mfrow=c(2,1))
hist(mtcars$mpg, breaks=10, xlab="MPG", main="MPG histogram")
plot(density(mtcars$mpg), main="kernel density", xlab="MPG")
End of Document.