motor_trend_mpg_data_analysis.Rmd

---
title: "Motor Trend MPG Data Analysis"
author: "Matthew Bender"
date: "September 19, 2015"
output: html_document
---
*Created with knitr*

#### Executive Summary
This report analyzed the relationship between transmission type (manual or 
automatic) and miles per gallon (MPG). The report set out to determine which 
transmission type produces a higher MPG. The `mtcars` dataset was used for this
analysis. A t-test between automatic and manual transmission vehicles shows that
manual transmission vehicles have a 7.245 greater MPG than automatic 
transmission vehicles. After fitting multiple linear regressions, analysis 
showed that the manual transmission contributed less significantly to MPG, only
an improvement of 1.81 MPG.  Other variables, weight, horsepower, and number of 
cylinders contributed more significantly to the overall MPG of vehicles.

#### Load Data
Load the dataset and convert categorical variables to factors.
```{r results='hide', message=FALSE}
library(ggplot2)
data(mtcars)
head(mtcars, n=3)
dim(mtcars)
mtcars$cyl <- as.factor(mtcars$cyl)
mtcars$vs <- as.factor(mtcars$vs)
mtcars$am <- factor(mtcars$am)
mtcars$gear <- factor(mtcars$gear)
mtcars$carb <- factor(mtcars$carb)
attach(mtcars)
```

#### Exploratory Analysis
**See Appendix Figure I** Exploratory Box graph that compares Automatic and Manual 
transmission MPG. The graph leads us to believe that there is a significant 
increase in MPG when for vehicles with a manual transmission vs automatic.

##### Statistical Inference
T-Test transmission type and MPG
```{r}
testResults <- t.test(mpg ~ am)
testResults$p.value
```
The T-Test rejects the null hypothesis that the difference between transmission
types is 0.  
```{r}
testResults$estimate
```
The difference estimate between the 2 transmissions is 7.24494 MPG in favor of 
manual.

##### Regression Analysis
Fit the full model of the data
```{r results='hide'}
fullModelFit <- lm(mpg ~ ., data = mtcars)
summary(fullModelFit)  # results hidden
summary(fullModelFit)$coeff  # results hidden
```
Since none of the coefficients have a p-value less than 0.05 we cannot conclude
which variables are more statistically significant. 

Backward selection to determine which variables are most statistically 
significant
```{r results='hide'}
stepFit <- step(fullModelFit)
summary(stepFit) # results hidden
summary(stepFit)$coeff # results hidden
```

The new model has 4 variables (cylinders, horsepower, weight, transmission). The
R-squared value of 0.8659 confirms that this model explains about 87% of the 
variance in MPG. The p-values also are statistically significantly because they
have a p-value less than 0.05. The coefficients conclude that increasing the 
number of cylinders from 4 to 6 with decrease the MPG by 3.03.  Further 
increasing the cylinders to 8 with decrease the MPG by 2.16.  Increasing the 
horsepower is decreases MPG 3.21 for every 100 horsepower.  Weight decreases the
MPG by 2.5 for each 1000 lbs increase. A Manual transmission improves the MPG by
1.81.

#### Residuals & Diagnostics
Residual Plot
**See Appendix Figure II**

The plots conclude:

1. The randomness of the Residuals vs. Fitted plot supports the assumption of
independence
2. The points of the Normal Q-Q plot following closely to the line conclude that
the distribution of residuals is normal
3. The Scale-Location plot random distribution confirms the constant variance 
assumption
4. Since all points are within the 0.05 lines, the Residuals vs. Leverage 
concludes that there are no outliers
```{r}
sum((abs(dfbetas(stepFit)))>1)
```

#### Conclusion
There is a difference in MPG based on transmission type. A manual transmission
will have a slight MPG boost. However, it seems that weight, horsepower, & 
number of cylinders are more statistically significant when determining MPG.

### Appendix Figures

#### I
```{r echo=FALSE}
  boxplot(mpg ~ am, 
          xlab="Transmission Type (0 = Automatic, 1 = Manual)", 
          ylab="MPG",
          main="MPG by Transmission Type")
```

#### II
```{r echo=FALSE}
par(mfrow = c(2, 2))
plot(stepFit)
```