Import the Data

library(tidyverse)
library(readr)

co2<-read.csv("https://raw.githubusercontent.com/kitadasmalley/MATH239/main/data/CO2_Emissions_Midterm.csv",
              header = TRUE)

co2<-co2%>%
  filter(Fuel.Type!="")

Part 1 (20 points): Properties of Linear Models

Question 1: (5 points)

Please write the form of a linear model and annotate the following components: explanatory variables (\(x_i\)), response variable (\(y\)), parameters (\(\beta_0, \beta_1, ...\)), error term (\(\epsilon_i\)).

Y(response variable) = B0(Y-intercept)+B1(Slope)*X1(Explanatory Variable)+e(error term) #### Question 2: (5 points)

List the assumptions for fitting a linear model.

Hint: There are five that we talked about in class - the relationship between x and y is linear - the variance for the residual is the same for every value of x(normal dist I’m pretty sure) - independent variables are not highly coorelated with each other

Question 3: (5 points)

Create a pairs plot for the numeric variables in these dataset. What relationships do you observe between the potential explanatory variables and the response (carbon emissions)?

Most of the explanatory variables seem to have a linear relationship to carbon emissions. The explanatory variables that are categorical seem to have more of a staircase look to them, but that just might be the different levels it goes up by for the different values.

Hint: In order to accomplish this, you might want to use the select function first for the numeric variables. Then call the pairs() function.

## SPACE FOR CODE

numeric = unlist(lapply(co2, is.numeric)) 
pairs(co2[,numeric])

Question 4: (5 points)

In this dataset we have both Fuel Consumption Comb (L/100km) and Fuel Consumption Comb (mpg). Look at the pairs plot for the relationships between these variables and the response. Which one of these would you want to include as an explanatory variable in your model? Why?

Hint: Think about the model assumptions from Question 8

Extra Credit: (5 points)

In the data description, it says that Fuel Consumption Comb (L/100km) is a combined rating (55% city, 45% hwy). Why would it not be appropriate to include Fuel Consumption City (L/100km), Fuel Consumption Hwy (L/100km), and Fuel Consumption Comb (L/100km) in the same model? How is this related to VIF?

Including the Fuel Consumption Comb (L/100km) would give the model the same data twice with the others included in fuel consumption, and there is a an uneven ratio which would already make the model weigh the two differences unevenly. It seems that this would increase the variance in the model giving a worse VIF.

Part 2 (25 points): Working with a Categorical Explanatory Variable

How does fuel type affect CO2 emissions?

Question 5: (5 points)

Fuel type is a categorical variable. What are the levels and how many are there? Which group will R choose to be the reference group? Does this make sense in the context of these data?

There are 5 levels and they are “Z” “D” “X” “E” “N”

## SPACE FOR CODE
co2$Fuel.Type
##    [1] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
##   [19] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "D" "Z" "D" "Z" "Z" "D" "Z" "Z"
##   [37] "Z" "D" "Z" "Z" "Z" "D" "Z" "Z" "D" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
##   [55] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
##   [73] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "D" "D" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
##   [91] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "D" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
##  [109] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
##  [127] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X" "X" "X" "X"
##  [145] "X" "E" "X" "E" "X" "Z" "Z" "Z" "X" "Z" "Z" "X" "Z" "Z" "X" "X" "Z" "X"
##  [163] "X" "X" "Z" "X" "Z" "X" "Z" "X" "X" "X" "X" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
##  [181] "E" "Z" "E" "X" "X" "X" "X" "Z" "X" "X" "X" "Z" "Z" "Z" "Z" "Z" "Z" "X"
##  [199] "X" "X" "X" "D" "X" "X" "X" "X" "E" "X" "X" "E" "X" "X" "E" "X" "E" "X"
##  [217] "E" "X" "E" "X" "E" "X" "E" "X" "X" "E" "X" "X" "E" "X" "X" "E" "X" "Z"
##  [235] "X" "X" "X" "X" "E" "X" "E" "Z" "X" "E" "X" "E" "Z" "X" "X" "X" "X" "X"
##  [253] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "E" "X" "E" "X" "E" "X" "E" "X"
##  [271] "X" "X" "X" "X" "X" "X" "X" "X" "E" "X" "X" "X" "E" "X" "X" "X" "X" "E"
##  [289] "X" "E" "Z" "X" "E" "X" "X" "X" "E" "X" "X" "X" "Z" "Z" "X" "X" "X" "X"
##  [307] "X" "X" "E" "X" "E" "X" "E" "Z" "X" "X" "X" "X" "E" "X" "X" "X" "X" "X"
##  [325] "X" "E" "X" "E" "X" "X" "X" "E" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
##  [343] "X" "X" "E" "X" "E" "X" "E" "X" "X" "E" "X" "X" "X" "X" "X" "X" "X" "X"
##  [361] "X" "X" "E" "X" "X" "X" "E" "X" "E" "X" "X" "X" "X" "X" "X" "X" "E" "X"
##  [379] "E" "X" "E" "X" "E" "X" "E" "X" "E" "X" "E" "X" "E" "X" "X" "X" "X" "X"
##  [397] "X" "X" "X" "X" "X" "X" "E" "X" "E" "X" "E" "X" "E" "X" "X" "X" "X" "X"
##  [415] "X" "X" "X" "X" "X" "X" "Z" "X" "X" "X" "X" "E" "X" "E" "X" "X" "X" "X"
##  [433] "X" "X" "X" "X" "E" "X" "E" "X" "E" "X" "E" "X" "E" "X" "E" "X" "X" "E"
##  [451] "X" "X" "E" "X" "E" "X" "E" "Z" "X" "E" "X" "E" "Z" "X" "X" "E" "X" "X"
##  [469] "E" "X" "E" "X" "E" "Z" "E" "Z" "E" "X" "E" "X" "E" "X" "X" "X" "X" "X"
##  [487] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
##  [505] "X" "X" "Z" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
##  [523] "X" "X" "X" "X" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
##  [541] "Z" "Z" "Z" "X" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "E" "Z" "Z" "Z" "E"
##  [559] "Z" "Z" "E" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X" "X" "X" "X" "X" "X" "X"
##  [577] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "D" "X" "E" "Z" "X" "X" "X" "X"
##  [595] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
##  [613] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
##  [631] "X" "X" "X" "X" "X" "X" "X" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "E" "Z"
##  [649] "Z" "E" "Z" "Z" "E" "Z" "Z" "E" "Z" "Z" "E" "X" "X" "X" "Z" "Z" "Z" "Z"
##  [667] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X" "X" "Z" "X" "X" "X"
##  [685] "X" "X" "X" "X" "X" "X" "X" "E" "Z" "Z" "Z" "Z" "Z" "Z" "X" "X" "X" "X"
##  [703] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
##  [721] "X" "X" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "E" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
##  [739] "Z" "Z" "Z" "Z" "Z" "D" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
##  [757] "Z" "Z" "Z" "Z" "D" "Z" "Z" "Z" "D" "Z" "Z" "E" "D" "Z" "Z" "Z" "Z" "Z"
##  [775] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
##  [793] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
##  [811] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X" "X" "X" "Z" "Z" "Z" "X" "X"
##  [829] "X" "X" "X" "X" "Z" "X" "X" "X" "Z" "Z" "Z" "Z" "X" "X" "X" "X" "X" "X"
##  [847] "X" "X" "Z" "Z" "Z" "Z" "Z" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
##  [865] "X" "X" "X" "X" "X" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
##  [883] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "D" "Z" "Z" "Z"
##  [901] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X" "X" "X" "X"
##  [919] "D" "X" "E" "D" "X" "E" "X" "X" "E" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
##  [937] "X" "X" "X" "X" "X" "X" "X" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X" "X" "X" "X"
##  [955] "Z" "Z" "X" "X" "Z" "Z" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
##  [973] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
##  [991] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [1009] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "Z" "Z" "X" "X" "D"
## [1027] "D" "X" "Z" "Z" "X" "Z" "Z" "Z" "Z" "X" "X" "D" "D" "X" "X" "X" "X" "D"
## [1045] "D" "Z" "Z" "Z" "X" "X" "X" "X" "Z" "D" "X" "Z" "Z" "Z" "Z" "D" "X" "X"
## [1063] "X" "X" "X" "X" "X" "X" "X" "X" "X" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [1081] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "D" "Z" "Z" "Z" "Z"
## [1099] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "D" "Z" "D" "Z" "Z" "D" "Z" "Z" "Z" "D" "Z"
## [1117] "Z" "Z" "Z" "D" "Z" "Z" "D" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [1135] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [1153] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "D" "D" "Z" "Z" "Z" "Z"
## [1171] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [1189] "Z" "Z" "Z" "Z" "Z" "Z" "D" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "D" "Z"
## [1207] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [1225] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "D" "Z" "Z" "Z" "Z" "Z"
## [1243] "D" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X" "X" "X" "X" "X" "E" "X"
## [1261] "E" "X" "Z" "Z" "Z" "X" "Z" "Z" "X" "Z" "Z" "X" "X" "Z" "X" "Z" "X" "Z"
## [1279] "X" "Z" "Z" "Z" "Z" "Z" "X" "X" "X" "X" "Z" "X" "X" "Z" "Z" "X" "Z" "Z"
## [1297] "Z" "X" "X" "X" "X" "X" "X" "Z" "Z" "Z" "Z" "X" "X" "X" "X" "D" "X" "X"
## [1315] "X" "X" "E" "X" "X" "X" "E" "X" "X" "X" "X" "E" "X" "X" "X" "E" "X" "X"
## [1333] "E" "Z" "X" "X" "E" "X" "E" "Z" "X" "E" "X" "E" "Z" "X" "X" "X" "X" "X"
## [1351] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "E" "X" "E" "X" "E" "X" "E" "X"
## [1369] "X" "X" "X" "X" "X" "X" "E" "X" "X" "E" "X" "X" "E" "X" "X" "E" "X" "X"
## [1387] "X" "E" "X" "E" "X" "X" "Z" "X" "Z" "Z" "Z" "Z" "Z" "X" "X" "E" "X" "X"
## [1405] "E" "X" "X" "X" "X" "E" "X" "Z" "Z" "X" "X" "E" "X" "X" "X" "X" "X" "X"
## [1423] "X" "E" "X" "X" "E" "X" "X" "E" "X" "Z" "X" "X" "X" "X" "X" "X" "X" "X"
## [1441] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [1459] "X" "X" "X" "E" "X" "X" "E" "X" "X" "X" "E" "X" "E" "X" "X" "X" "E" "X"
## [1477] "E" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "E" "X" "E" "X" "E" "X"
## [1495] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "E" "X" "X"
## [1513] "X" "E" "X" "E" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [1531] "E" "X" "X" "X" "E" "X" "E" "X" "E" "Z" "X" "E" "X" "E" "Z" "X" "X" "E"
## [1549] "X" "X" "X" "X" "E" "X" "E" "Z" "Z" "X" "E" "X" "E" "Z" "Z" "X" "X" "X"
## [1567] "X" "X" "X" "X" "X" "X" "X" "Z" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [1585] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "Z" "X" "Z" "Z" "Z" "X" "X"
## [1603] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [1621] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X"
## [1639] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "E" "Z" "Z" "Z" "E" "Z" "Z"
## [1657] "E" "Z" "Z" "Z" "Z" "Z" "Z" "X" "X" "E" "X" "X" "X" "E" "X" "X" "X" "X"
## [1675] "X" "X" "X" "X" "X" "X" "X" "X" "D" "X" "E" "X" "Z" "X" "X" "X" "X" "X"
## [1693] "X" "X" "X" "X" "E" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [1711] "X" "X" "X" "X" "X" "X" "X" "Z" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [1729] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "Z" "Z" "Z" "Z"
## [1747] "Z" "Z" "Z" "Z" "E" "Z" "E" "Z" "Z" "Z" "Z" "E" "X" "X" "X" "Z" "Z" "Z"
## [1765] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X" "Z" "Z" "Z"
## [1783] "X" "X" "Z" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "Z" "Z"
## [1801] "Z" "Z" "Z" "Z" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [1819] "X" "X" "X" "X" "X" "X" "X" "X" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [1837] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "D" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [1855] "Z" "Z" "D" "Z" "Z" "Z" "Z" "Z" "D" "Z" "D" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [1873] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [1891] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [1909] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [1927] "X" "X" "X" "Z" "Z" "Z" "X" "X" "X" "X" "X" "X" "Z" "X" "X" "X" "X" "Z"
## [1945] "Z" "Z" "Z" "X" "X" "X" "X" "X" "X" "X" "X" "Z" "X" "X" "X" "Z" "Z" "X"
## [1963] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [1981] "X" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [1999] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [2017] "Z" "Z" "Z" "Z" "D" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [2035] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X" "X" "E" "X" "X" "X" "D" "D" "X" "X" "E"
## [2053] "X" "X" "D" "X" "E" "X" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X" "X" "X"
## [2071] "X" "X" "Z" "Z" "Z" "Z" "Z" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [2089] "X" "Z" "Z" "Z" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [2107] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [2125] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
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## [6895] "X" "E" "Z" "X" "X" "E" "X" "X" "E" "Z" "X" "X" "X" "X" "X" "X" "X" "X"
## [6913] "X" "X" "Z" "X" "X" "Z" "Z" "X" "X" "X" "X" "Z" "Z" "X" "X" "X" "X" "X"
## [6931] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [6949] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [6967] "X" "X" "X" "X" "X" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [6985] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X"
## [7003] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "Z"
## [7021] "Z" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "D" "X" "X" "X" "X" "X"
## [7039] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [7057] "Z" "X" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "D" "Z" "Z" "Z" "D" "Z" "Z" "Z"
## [7075] "Z" "Z" "D" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X" "X" "X" "Z" "Z" "Z"
## [7093] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X" "Z" "Z" "Z" "X" "X" "Z" "Z"
## [7111] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "Z" "Z" "Z" "Z" "Z"
## [7129] "Z" "Z" "Z" "Z" "Z" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [7147] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "Z" "Z" "Z" "Z" "Z"
## [7165] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [7183] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z"
## [7201] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "X" "X" "X" "X" "Z" "X"
## [7219] "X" "X" "Z" "Z" "Z" "Z" "X" "X" "X" "X" "X" "Z" "X" "Z" "X" "X" "X" "X"
## [7237] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "Z" "Z" "X" "X" "Z" "Z" "Z" "Z"
## [7255] "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "Z" "D"
## [7273] "X" "X" "X" "X" "D" "X" "X" "X" "X" "X" "X" "X" "X" "Z" "Z" "Z" "Z" "Z"
## [7291] "Z" "Z" "Z" "X" "Z" "Z" "Z" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [7309] "Z" "Z" "Z" "X" "X" "Z" "Z" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X"
## [7327] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "Z" "X" "X" "X" "X" "X" "X" "X"
## [7345] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "Z" "X"
## [7363] "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "X" "Z" "Z" "Z" "Z" "Z" "Z" "X"
## [7381] "Z" "Z" "Z" "Z" "Z"

Question 6: (5 points)

In your own words, describe how categorical variables with multiple levels are coded.

They are set equal to the function factor() that contains the variables in them and they have a parameter called “levels” that you set equal to a list of the levels you want them in, in order from least to greatest.

Question 7: (5 points)

Using the method you described above write out this coding.

RELEVEL

co2$Fuel.Type <- factor(co2$Fuel.Type, 
                        levels = c("X", "Z", "D", "E", "N"))

Question 8: (5 points)

Create a side-by-side boxplot to compare the distributions of carbon emissions across fuel types. Feel free to use color! What are your observations?

## SPACE FOR CODE

ggplot(data = co2)+
  geom_boxplot(aes(x = Fuel.Type, y = CO2.Emissions.g.km., color = Fuel.Type))

Question 9: (5 points)

Perform the appropriate hypothesis test whether there is a significant difference in the average carbon emissions across fuels types. Please state the following components:

  • Name of the hypothesis test: T-Test
  • Hypotheses (null and alternative): Null: slope = 0, alt: slope!=0
  • Provide the test statistic
  • Reference distribution:
  • P-value: <2.2e-16 for the fuel types that have signifigance

Then communicate your findings in sentence form (5 part summary).

We fail to reject the null hypothesis with p-value of 2.2e-16 High signifigance level. This is highly suggestive that the fuel type has a linear coorelation to the co2 emmisions.

## SPACE FOR CODE
m1=lm(CO2.Emissions.g.km.~Fuel.Type, data = co2)
summary(m1)
## 
## Call:
## lm(formula = CO2.Emissions.g.km. ~ Fuel.Type, data = co2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -147.092  -42.119   -8.043   35.881  255.957 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 235.1193     0.9335 251.855   <2e-16 ***
## Fuel.TypeZ   30.9241     1.3643  22.666   <2e-16 ***
## Fuel.TypeD    2.4292     4.3571   0.558    0.577    
## Fuel.TypeE   39.9726     3.0722  13.011   <2e-16 ***
## Fuel.TypeN  -22.1193    56.3078  -0.393    0.694    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 56.3 on 7380 degrees of freedom
## Multiple R-squared:  0.0747, Adjusted R-squared:  0.0742 
## F-statistic: 148.9 on 4 and 7380 DF,  p-value: < 2.2e-16
anova(m1)
## Analysis of Variance Table
## 
## Response: CO2.Emissions.g.km.
##             Df   Sum Sq Mean Sq F value    Pr(>F)    
## Fuel.Type    4  1888452  472113  148.95 < 2.2e-16 ***
## Residuals 7380 23392397    3170                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Part 3 (55 points): Fitting Models

Question 10: (5 points)

Create a scatter plot to look at the relationship between Fuel Consumption Comb (L/100km) and CO2 Emissions (g/km). Describe the scatterplot above (direction, form, strength, outliers).

There is a negative direction, with a curve, but the coorelation seems pretty strong and you can tell the difference between the city and hwy. This doesn’t make much sense to me as the more fuel consumed should lead to a higher co2 emission. There are not any outliers that are noticable.

## SPACE FOR CODE
ggplot(data = co2, aes(x =Fuel.Consumption.Comb..mpg., y= CO2.Emissions.g.km.))+
  geom_point()

MODEL A - Simple Linear Regression (SLR)

Question 11: (5 points)

Create a simple linear model for the relationship between Fuel Consumption Comb (L/100km) and CO2 Emissions (g/km). Write the equation for the estimated fitted model.

y = 452.35-7.34x

## SPACE FOR CODE

mod2 = lm(CO2.Emissions.g.km.~Fuel.Consumption.Comb..mpg., data = co2)
summary(mod2)
## 
## Call:
## lm(formula = CO2.Emissions.g.km. ~ Fuel.Consumption.Comb..mpg., 
##     data = co2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -148.147  -11.463   -5.727    7.666  165.092 
## 
## Coefficients:
##                              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                 452.35304    1.12443   402.3   <2e-16 ***
## Fuel.Consumption.Comb..mpg.  -7.34193    0.03957  -185.6   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 24.59 on 7383 degrees of freedom
## Multiple R-squared:  0.8234, Adjusted R-squared:  0.8234 
## F-statistic: 3.443e+04 on 1 and 7383 DF,  p-value: < 2.2e-16

Question 12: (5 points)

Perform a hypothesis test for the slope. Perform a test for the slope. State the reference distribution, degrees of freedom, the test statistic, and p-value in the form of a five-part conclusion in the context of the problem.

We fail to reject the null hypothesis with p-value of 2.2e-16 High signifigance level. This is highly suggestive that the fuel consumption combined has a linear coorelation to the co2 emmisions.

t-value:-185.6 DF:7384

## SPACE FOR CODE

Question 13: (5 points)

Should we trust the inference we made in the previous step? To assess this check the model diagnostics.

  • Create a residual plot - comment on mean zero assumption and homoscedasticity.
  • Create a qq plot - comment on normality
  • Create a leverage plot - comment on the presence of influential outliers
## SPACE FOR CODE
# QQ NORM Plot 
hist(mod2$residuals)

qqnorm(mod2$residuals)
qqline(mod2$residuals)

co2<-cbind(co2, 
            fit=mod2$fitted.values,
            residual=mod2$residuals)

ggplot(data=co2, aes(residual))+
  geom_histogram(bins=8)+
  ggtitle("Histogram of Residuals")+
  theme_bw()

# Residual Plot
ggplot(data=co2, aes(x = CO2.Emissions.g.km., y = residual))+
  geom_point()+
  ggtitle("Residual Plot")+
  xlab("CO2 emissions)")+
  ylab("Residuals")+
  theme_bw()+
  geom_hline(yintercept = 0,
             color="blue", lty=2, lwd=1)

Multiple Linear Regression:

Question 14: (5 points)

Create a scatter plot to look at the relationship between Fuel Consumption Comb (L/100km) and CO2 Emissions (g/km) but now color the points based on the Fuel Type. Discuss your observations.

It looks like the different fuel types would have different intercepts, but pretty similar slopes for their linear relationship to co2 emissions.

Diesel seems to have a higher average co2 emission.

## SPACE FOR CODE
ggplot(data = co2)+
  geom_point(aes(x = Fuel.Consumption.Comb..mpg., y = CO2.Emissions.g.km., color = Fuel.Type))

MODEL B - Parallel Lines

Question 14: (5 points)

Create a parallel lines model for CO2 Emissions (g/km) using Fuel Consumption Comb (L/100km) and Fuel Type. Write the equations for the estimated fitted models for the fuel types.

## SPACE FOR CODE. 

mod3 = lm(CO2.Emissions.g.km.~Fuel.Consumption.Comb..mpg.+Fuel.Type, data = co2)
summary(mod3)
## 
## Call:
## lm(formula = CO2.Emissions.g.km. ~ Fuel.Consumption.Comb..mpg. + 
##     Fuel.Type, data = co2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -93.406 -12.169  -7.082   4.947 175.246 
## 
## Coefficients:
##                              Estimate Std. Error  t value Pr(>|t|)    
## (Intercept)                 473.42045    1.16081  407.835  < 2e-16 ***
## Fuel.Consumption.Comb..mpg.  -8.00966    0.03724 -215.081  < 2e-16 ***
## Fuel.TypeZ                   -0.88193    0.52724   -1.673   0.0944 .  
## Fuel.TypeD                   28.35533    1.62065   17.496  < 2e-16 ***
## Fuel.TypeE                  -59.78310    1.23033  -48.591  < 2e-16 ***
## Fuel.TypeN                  -84.20795   20.88813   -4.031  5.6e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 20.88 on 7379 degrees of freedom
## Multiple R-squared:  0.8727, Adjusted R-squared:  0.8726 
## F-statistic: 1.012e+04 on 5 and 7379 DF,  p-value: < 2.2e-16
ggplot(co2, aes(x = Fuel.Consumption.Comb..mpg., y = CO2.Emissions.g.km.,color = Fuel.Type))+
  geom_point()+
  ggtitle("Scatterplot of Fuel consumption combined vs CO2 emission")+
  theme_bw()+
  geom_abline(slope = mod3$coefficients[2],intercept =mod3$coefficients[1])+
  geom_abline(slope = mod3$coefficients[2],intercept =mod3$coefficients[3]+mod3$coefficients[1])+
  geom_abline(slope = mod3$coefficients[2],intercept =mod3$coefficients[4]+mod3$coefficients[1])+
 geom_abline(slope = mod3$coefficients[2],intercept =mod3$coefficients[5]+mod3$coefficients[1])+
   geom_abline(slope = mod3$coefficients[2],intercept =mod3$coefficients[6]+mod3$coefficients[1])

Question 15: (5 points)

Create a graphic for your parallel lines model showing the fitted models for each type of fuel. Comment on which shifts of intercept are significant.

the three significant shifts are those of ethanol, diesel, and premium combined with regualr

## SPACE FOR CODE

MODEL C - Interactions (Unrelated) Lines

Question 16: (5 points)

Create an unrelated lines (interaction) model for CO2 Emissions (g/km) using Fuel Consumption Comb (L/100km) and Fuel Type. Write the equations for the estimated fitted models for the fuel types.

Note: The interaction for Fuel.Consumption.Comb..L.100.km.:Fuel.TypeN is not estimable due to the small sample size. Therefore the slope for this group will not be shifted from the reference group.

## SPACE FOR CODE
mod4 = lm(CO2.Emissions.g.km.~Fuel.Consumption.Comb..mpg.*Fuel.Type, data = co2)
summary(mod4)
## 
## Call:
## lm(formula = CO2.Emissions.g.km. ~ Fuel.Consumption.Comb..mpg. * 
##     Fuel.Type, data = co2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -51.752 -10.769  -4.650   4.679 140.338 
## 
## Coefficients: (1 not defined because of singularities)
##                                         Estimate Std. Error  t value Pr(>|t|)
## (Intercept)                            442.41325    1.17713  375.842  < 2e-16
## Fuel.Consumption.Comb..mpg.             -6.96746    0.03830 -181.913  < 2e-16
## Fuel.TypeZ                              91.08802    1.96594   46.333  < 2e-16
## Fuel.TypeD                              21.67081    7.62144    2.843  0.00448
## Fuel.TypeE                              78.71618    5.26997   14.937  < 2e-16
## Fuel.TypeN                             -76.12912   17.80540   -4.276 1.93e-05
## Fuel.Consumption.Comb..mpg.:Fuel.TypeZ  -3.40686    0.07105  -47.949  < 2e-16
## Fuel.Consumption.Comb..mpg.:Fuel.TypeD   0.10037    0.22783    0.441  0.65955
## Fuel.Consumption.Comb..mpg.:Fuel.TypeE  -7.25658    0.29461  -24.631  < 2e-16
## Fuel.Consumption.Comb..mpg.:Fuel.TypeN        NA         NA       NA       NA
##                                           
## (Intercept)                            ***
## Fuel.Consumption.Comb..mpg.            ***
## Fuel.TypeZ                             ***
## Fuel.TypeD                             ** 
## Fuel.TypeE                             ***
## Fuel.TypeN                             ***
## Fuel.Consumption.Comb..mpg.:Fuel.TypeZ ***
## Fuel.Consumption.Comb..mpg.:Fuel.TypeD    
## Fuel.Consumption.Comb..mpg.:Fuel.TypeE ***
## Fuel.Consumption.Comb..mpg.:Fuel.TypeN    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 17.8 on 7376 degrees of freedom
## Multiple R-squared:  0.9076, Adjusted R-squared:  0.9075 
## F-statistic:  9051 on 8 and 7376 DF,  p-value: < 2.2e-16
ggplot(co2, aes(x = Fuel.Consumption.Comb..mpg., y = CO2.Emissions.g.km.,color = Fuel.Type))+
  geom_point()+
  ggtitle("Scatterplot of Fuel consumption combined vs CO2 emission")+
  theme_bw()+
  geom_abline(slope = mod4$coefficients[2],intercept =mod4$coefficients[3]+mod4$coefficients[1])+
  geom_abline(slope = mod4$coefficients[2],intercept =mod4$coefficients[4]+mod4$coefficients[1])+
 geom_abline(slope = mod4$coefficients[2],intercept =mod4$coefficients[5]+mod4$coefficients[1])+
   geom_abline(slope = mod4$coefficients[2],intercept =mod4$coefficients[6]+mod4$coefficients[1])

Question 17: (5 points)

Create a graphic for your unrelated lines model showing the fitted models for each type of fuel. Comment on which shifts of intercept and/or slope are significant.

## SPACE FOR CODE

Model Comparison and Selection

Question 18: (5 points)

Compare Models A, B, and C by reporting their Adjusted R-squared values. Which model would you pick? Make an argument based on an assessment of the model assumptions, simplicity, and interpretability as well as model fit.

Model 1: .82

Model 2: .87

model 3: .9

There is a big enough difference to pick model 2 over model 1, but model 3 doesn’t show that big of a difference in the R-squared values shown above and the simplicity of model 2 over model 3 makes it more favorable. It is also just easier to understand the differences in the intercept differences in model 2, making it the most favorable one.

## SPACE FOR CODE

Prediction

Question 19: (5 points)

Professor Smalley drives a 2001 Toyota Rav4. She calculated that her car gets a combined fuel composition of 25 miles per gallon. You can convert this to L/100km with the following equation: \[1 mile/US gallon = 235.21 liters/100 km\] She puts regular gasoline in her car. Predict the CO2 Emissions (g/km) for her car, using the model you chose in Question 18.

about 120 (g/km)

is there a function for this?

## SPACE FOR CODE

Extra Credit: (5 points)

Create a 95% prediction interval for the CO2 Emissions (g/km) for Professor Smalley’s car, using the model you chose in Question 18.