Summary

This document is intended to be a concise report to explain hypothesis testings performed on a dataset containing information about cars (available here). The analysis was created as part of the Data Science Certificate in the class Methods for Data Analysis at University of Washington.

The idea is to show some findings regarding the significance of the price of the cars, more especifically when relating to fuel type, aspiration and drive wheels. Some functions created for this purpose are included in the appendix.

This report considers data already cleaned, as described by “Auto Exploration Report”, and starts with the analysis of the distribution of the prices of the cars, followed by the hypothesis testing of price and the variables of interest: fuel type, aspiration and drive wheels.

Price Analysis

Considering that our objective is to later perform hypothesis testing of some variables in the auto data in relation to prices, we should be concerned about the distribution of price, to then be able to use the most suitable type of test.

First we load the functions created (loading omitted from the report, loaded behind the scenes but included in the appendix) for this report. Then the data is loaded using the function read.auto and we run a quick summary of price to have a first look at the data.

# read.auto function loads and cleans the data
Auto.Price = read.auto(path = '.') # function read.auto is included in the appendix
summary(Auto.Price$price)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##    5118    7775   10300   13210   16500   45400       4

We can see that NA’s exist and we will have to deal with them. Now the first thing to do when analysing distributions is to check its normality. To do this we will use the Q-Q Plot (for reference, go to: Q-Q Plot).

qqPlotOfPrice() # function qqPlotOfPrice is included in the appendix

The qqPlotOfPrice function (included in the appendix) scales the data for better visualization, and also deals with the NA’s in the price data, by removing them. Looking at the plot produced above, we can see that the sample and theoretical quantiles are not very close to a linear relationship, which is represented by the red dashed line. With this in mind we check the normality of the distribution of the Log of the price.

qqPlotOfPrice(log) # function qqPlotOfPrice is included in the appendix

From this plot we can see that the log of the price is very close to the red dashed line, leading to closer normality. Also, when compared to the Q-Q Plot of price alone, it surely shows a better result. Therefore, we will use the log of the price in our hypothesis tests.

Hypothesis Testing

Following we have three sections with the testings for hypothesis of relations between price and fuel type, aspiration and drive wheels.

Fuel Type

The fuel of a vehicle is the liquid that supplies the engine. In this dataset we have two types of fuel: diesel and gas.

table(Auto.Price$fuel.type)
## 
## diesel    gas 
##     20    185
sort(tapply(Auto.Price$price, Auto.Price$fuel.type, mean, na.rm=TRUE), decreasing=TRUE)
##   diesel      gas 
## 15838.15 12916.41

At first, we notice that the type gas has fewer observations (20 out of 205, ~10%) when compared to diesel type (185 out of 205, ~90%).

By analyzing the price for each type, we see that they are very close and possibly our test may be inconclusive. To assert this hypothesis, we do a Welch Two Sample t-test, defining the following null and alternative hypothesis:

  • Null Hypothesis - H0: Mean price of diesel is equal to the mean price of gas
  • Alternative Hypothesis - HA: Mean price of diesel is different than the mean price of gas
t.test(Auto.Price$price[Auto.Price$fuel.type == 'diesel'], 
       Auto.Price$price[Auto.Price$fuel.type == 'gas'], "two.sided", 0, FALSE, FALSE, 0.95)
## 
##  Welch Two Sample t-test
## 
## data:  Auto.Price$price[Auto.Price$fuel.type == "diesel"] and Auto.Price$price[Auto.Price$fuel.type == "gas"]
## t = 1.5943, df = 23.611, p-value = 0.1242
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -863.9723 6707.4546
## sample estimates:
## mean of x mean of y 
##  15838.15  12916.41

With a not so small p-value, we can not reject the null hypothesis. But this does not mean that we can accept the null hypothesis, since there are two possible reasons as to why we failed: the alternative hypothesis was false to begin with; or we did not collect enough evidence for the alternative hypothesis.

Aspiration

Aspiration basically relates to how the engine’s internal combustion works, being, for example, power enhanced with chargers. In our dataset there are only two types of aspiration: std (standard) or turbo (with turbocharger).

table(Auto.Price$aspiration)
## 
##   std turbo 
##   168    37
sort(tapply(Auto.Price$price, Auto.Price$aspiration, mean, na.rm=TRUE), decreasing=TRUE)
##    turbo      std 
## 16254.81 12542.18

We can notice that turbo has fewer observations (37 out of 205, ~18%) than std (168 out of 205, ~82%), but these do not seem to be too extreme to affect our results.

Also, by analyzing the price for each type of aspiration, we see that turbo has a higher mean price than std. To assert this hypothesis, we do a Welch Two Sample t-test, defining the following null and alternative hypothesis:

  • Null Hypothesis - H0: Mean price of turbo is lesser or equal than the mean price of std
  • Alternative Hypothesis - HA: Mean price of turbo is greater than the mean price of std
t.test(Auto.Price$price[Auto.Price$aspiration == 'turbo'], 
       Auto.Price$price[Auto.Price$aspiration == 'std'], "greater", 0, FALSE, FALSE, 0.95)
## 
##  Welch Two Sample t-test
## 
## data:  Auto.Price$price[Auto.Price$aspiration == "turbo"] and Auto.Price$price[Auto.Price$aspiration == "std"]
## t = 3.0693, df = 64.602, p-value = 0.001568
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
##  1694.08     Inf
## sample estimates:
## mean of x mean of y 
##  16254.81  12542.18

With a very small p-value, we can reject the null hypothesis, confirming our finding that the mean price of turbo aspiration is greater than the mean price of the std aspiration.

Drive Wheels

Drive wheels essentially dictates the traction of the cars, into 4 wheels (4wd), two forward wheels (fwd) or two rear wheels (rwd). Observing the variable drive.wheels, we see it has exactly three levels: 4wd, fwd and rwd.

table(Auto.Price$drive.wheels)
## 
## 4wd fwd rwd 
##   9 120  76
sort(tapply(Auto.Price$price, Auto.Price$drive.wheels, mean, na.rm=TRUE), decreasing=TRUE)
##      rwd      4wd      fwd 
## 19757.61 10241.00  9244.78

We can also notice that the level 4wd has very few observations (9 out of 205, ~4%) and for this reason may be hard to account while making sure to not overfit. Therefore, we will only compare rwd and fwd.

By analyzing the price for each level, we see that rwd has a higher mean price than fwd. The following plot also seems to confirm this hypothesis.

library(ggplot2)
ggplot(Auto.Price[Auto.Price$drive.wheels %in% c('fwd','rwd'),], aes(price)) + 
    geom_histogram(binwidth=1000, na.rm=TRUE) + facet_grid(. ~ drive.wheels) + 
    labs(title = "Histogram of Price by Drive Wheels fwd and rwd") + labs(x = "Price (US$)", y = "Frequency")

To assert this hypothesis, we do a Welch Two Sample t-test, defining the following null and alternative hypothesis:

  • Null Hypothesis - H0: Mean price of rwd is lesser or equal than the mean price of fwd
  • Alternative Hypothesis - HA: Mean price of rwd is greater than the mean price of fwd
t.test(Auto.Price$price[Auto.Price$drive.wheels == 'rwd'], 
       Auto.Price$price[Auto.Price$drive.wheels == 'fwd'], "greater", 0, FALSE, FALSE, 0.95)
## 
##  Welch Two Sample t-test
## 
## data:  Auto.Price$price[Auto.Price$drive.wheels == "rwd"] and Auto.Price$price[Auto.Price$drive.wheels == "fwd"]
## t = 9.6178, df = 86.907, p-value = 1.232e-15
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
##  8695.536      Inf
## sample estimates:
## mean of x mean of y 
##  19757.61   9244.78

With a very small p-value, we can reject the null hypothesis, confirming our finding that the mean price of rwd drive wheel is greater than the mean price of the fwd drive wheel.

Appendix

Functions loaded during the initialization of this report are showed below.

##  Read the csv file into a data frame
read.auto <- function(path = 'SET-YOUR-PATH-HERE'){
    ## Function to read the csv file
    filePath <- file.path(path, 'Automobile price data _Raw_.csv')
    auto.price <- read.csv(filePath, header = TRUE, 
                           stringsAsFactors = TRUE, na.strings = "?")
    
    ## Coerce some character columns to numeric
    numcols <- c('price', 'bore', 'stroke', 'horsepower', 'peak.rpm',
                 'highway.mpg', 'city.mpg', 'compression.ratio',
                 'engine.size', 'curb.weight', 'height', 'width',
                 'length', 'wheel.base', 'normalized.losses',
                 'symboling')
    auto.price[, numcols] <- lapply(auto.price[, numcols], as.numeric)
    
    ## Clean and tidy num.of.doors
    auto.price$num.of.doors <- as.character(auto.price$num.of.doors)
    auto.price$num.of.doors[auto.price$num.of.doors == 'four'] <- 4
    auto.price$num.of.doors[auto.price$num.of.doors == 'two'] <- 2
    auto.price$num.of.doors <- as.integer(auto.price$num.of.doors)
    
    ## Clean and tidy num.of.cylinders
    auto.price$num.of.cylinders <- as.character(auto.price$num.of.cylinders)
    auto.price$num.of.cylinders[auto.price$num.of.cylinders == 'eight'] <- 8
    auto.price$num.of.cylinders[auto.price$num.of.cylinders == 'five'] <- 5
    auto.price$num.of.cylinders[auto.price$num.of.cylinders == 'four'] <- 4
    auto.price$num.of.cylinders[auto.price$num.of.cylinders == 'six'] <- 6
    auto.price$num.of.cylinders[auto.price$num.of.cylinders == 'three'] <- 3
    auto.price$num.of.cylinders[auto.price$num.of.cylinders == 'twelve'] <- 12
    auto.price$num.of.cylinders[auto.price$num.of.cylinders == 'two'] <- 2
    auto.price$num.of.cylinders <- as.integer(auto.price$num.of.cylinders)
    
    auto.price
}

# Outputs the Q-Q Plot of Price with a function applied to it
qqPlotOfPrice <- function(price.function=I){
  library(ggplot2)
  scaled_price <- as.data.frame(scale(price.function(Auto.Price$price))) # Scales price and applies function
  names(scaled_price) <- 'price'
  if(!identical(price.function, I)){ # Defines one of the words of the main title of the plot
    if(identical(price.function, log)){
      title_price <- 'Log of Price'
    }
    else if(identical(price.function, sqrt)){
      title_price <- 'Sqrt of Price'
    }
    else {
      title_price <- 'Transformation of Price'
    }
  }
  else {
    title_price <- 'Price'
  }
  # Plots the Q-Q Plot
  ggplot(scaled_price, aes(sample=price)) + stat_qq(na.rm = TRUE) + 
    geom_abline(aes(slope=1,intercept=0), color="red", lty="dashed") + 
    labs(title=paste0("Q-Q Plot of the distribution of ", title_price, " in Auto Data"), y="Sample Quantiles", x="Theoretical Quantiles")
}