MotorTrend Analysis
Ashutosh
16 September 2017
Executive Summary
Motor Trend is interested in exploring the relationship between a set of variables and miles per gallon (MPG). They are particularly interested in the following two questions:
- “Is an automatic or manual transmission better for MPG”
- “Quantify the MPG difference between automatic and manual transmissions”
This analysis uses mtcars dataset to explore these relationships by using statitical inference first and then building a model using Linear Regression and interpreting the results.
The analysis shows that when we create a linear model between mpg and am, on an average manual transmission scores over automatic transmission by about 7.24494 mpg. However this model does not cover decent variability. The best model (found using stepwise regression) shows that mpg depends on wt, qsec and am. In presence of qsec and am, the expected difference in mpg drops to 2.9358.
Reading mtcars dataset
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 1PreProcessing
Let’s convert am into a categorical variable. 0 = AT(Automatic Transmission), 1 = MT(Manual Transmission)
input<-mtcars
input$am <- as.factor(input$am)
levels(input$am) <-c("AT", "MT")Exploratory Analysis
- The distribution of outcome variable (mpg) is plotted using a histogram which suggests resemblance with normal distribution. See Appendix 1.
- A boxplot of outcome variable (mpg) is plotted with am. It suggests manual transmission is better for mpg. See Appendix 2.
- A Pair plot is plotted for all the variables of the dataset. The Pair plot shows strong relationship between different variables and miles per gallon. See Appendix 3
Statistical Inference to identify which transmission is better
Boxplot suggests that manual transmission is better than automatic transmission for miles per Gallon. Let’s try to prove it using t-test.
set.seed(12345)
t.test(input$mpg~input$am)##
## Welch Two Sample t-test
##
## data: input$mpg by input$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 AT mean in group MT
## 17.14737 24.39231Mean in group AT is 17.14737 while mean in group MT is 24.39231. The p-value is 0.001374 and thus the result is significant.
The null hypothesis can be rejected and we can say that manual transmission is better than automatic transmission for miles per Gallon.
Model Building
Let’s build a regression model between mpg as outcome and am as predictor
fit<-lm(mpg~am,data=input)
summary(fit)##
## Call:
## lm(formula = mpg ~ am, data = input)
##
## 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 ***
## amMT 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- The regression model shows that Mean of Automatic tranmission is 17.147 and mean of Manual Tranmission is 17.147 + 7.245 = 24.392.
- The p-values for coefficients are significant but the R2 value is 0.36 and thus this model caputures very less variability.
Let’s try to build a model using mpg as outcome and all remaining variables as predictors.
fit_all<-lm(mpg~.,data=input)- The fit all model shows high R squared values, but the p-value of coefficients are clearly not significant.
Let’s find the best model using stepwise Regression which choses the best model by AIC. We will use the fit_all model and then go backwards to the fitest model.
fit_best<-step(fit_all,direction = "backward",trace = 0)
summary(fit_best)##
## Call:
## lm(formula = mpg ~ wt + qsec + am, data = input)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4811 -1.5555 -0.7257 1.4110 4.6610
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.6178 6.9596 1.382 0.177915
## wt -3.9165 0.7112 -5.507 6.95e-06 ***
## qsec 1.2259 0.2887 4.247 0.000216 ***
## amMT 2.9358 1.4109 2.081 0.046716 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.459 on 28 degrees of freedom
## Multiple R-squared: 0.8497, Adjusted R-squared: 0.8336
## F-statistic: 52.75 on 3 and 28 DF, p-value: 1.21e-11- The fittest model has just 3 predictors :
mpg ~ wt + qsec + am
- The p-value of the coefficients are significant and Residual standard error: 2.459
- R Squared Value = 0.8497 and thus captures good variablility of the data.
- The coefficient of amMT = 2.9358 and it represents the expected change in mpg when comparing AT vs MT in presence of two other predictors wt and qsec.
- Residual plot is shown in Appendix 4. The plot shows existence of some outliers. But overall the model captures good variablility.
Conclusion
- Manual transmission is better than automatic transmission for miles per Gallon.
- Miles per gallon mostly depends on three variables - wt, qsec, am
- Expected increase in mpg when transmission is changed from automatic to manual is 2.9358, keeping wt and qsec as adjusting variables.
Appendix
Appendix 1 Histogram of Mpg
hist(mtcars$mpg,breaks = 10,xlab="MPG")
Appendix 2 Boxplot of mpg and am
library(ggplot2)
library(caret)
ggplot(input, aes(x=am, y=mpg)) + geom_boxplot()
Appendix 3 Plotting pairs for mtcars data
pairs(mpg ~ ., data = mtcars)
Appendix 4 Best model residuals
par(mfrow = c(2,2))
plot(fit_best)