In this assignment you will be working with a very small data set. You need to build a regression from the ground up. I want you to focus on your understanding of the regression error term. Specifically, can you have an error term without first having a regression line?

Read pages 9-25 in the Lewis-Beck book Applied Regression for help.

Raw Data

X	Y
2	1
4	5
6	6
8	7
10	11

Our regression model is: \(y = b_0 + b_1 x + e\)

The variance/covariance of our data is:

var(x): 10
var(y): 13
cov(x,y): 11

Answer the following:

Calculate \(b_1\) using the knowledge the slope can be calculated as cov(x,y) / var(x).

ANSWER:

Interpret the coefficient \(b_1\) in plain English.

ANSWER:

Calculate \(b_0\). Recall that the regression line always passes through the mean of X and Y:

\(\bar{y} = b_0 + b_1 \bar{x}\)

ANSWER:

What is the predicted value of Y when X has a value of 14?

ANSWER:

Calculate the sum of the squared errors (see page Lewis-Beck, p14).

ANSWER:

Calculating the Residual

Replace each a in the table with the correct value.

X	Y	\(\hat{Y}\)	e	\(e^2\)
2	1	a	a	a
4	5	a	a	a
6	6	a	a	a
8	7	a	a	a
10	11	a	a	a

Calculate the regression sum of squares (Lewis-Beck p21). You can check your results against the ANOVA table below.

ANSWER:

Calculate the \(R^2\) using the sum of squares in the table.

ANSWER:

Analysis of Variance (ANOVA) Table:

library( dplyr )
library( knitr )
options( digits=2 )
m <- lm( y ~ x )
anova( m ) %>% pander()

Analysis of Variance Table
	Df	Sum Sq	Mean Sq	F value	Pr(>F)
x	1	48.4	48.4	40.33	0.007898
Residuals	3	3.6	1.2	NA	NA

Submission Instructions

After you have completed your lab, knit your RMD file. Login to Canvas at http://canvas.asu.edu and navigate to the assignments tab in the course repository. Upload your RMD and your HTML files to the appropriate lab submission link.

You can create a new RMarkdown file, or download the lab template: RMD Template

Remember to:

name your files according to the convention: Lab-##-LastName.Rmd
show your solution, include your code.
do not print excessive output (like a full data set).
follow appropriate style guidelines (spaces between arguments, etc.).

Lab 01 - Regression Review

Raw Data

Answer the following:

Calculating the Residual

Analysis of Variance (ANOVA) Table:

Submission Instructions