Chapter 14

The Simple Linear Regression Model

I. Introduction

• We want to develop a model that hopes to successfully explain the relationship between one variable (x) and another (y).

• This statistical procedure is called regression analysis.

• Independent variables (x) are used to explain the dependent variable (y).

In what kind of x,y relationships are managers interested?

• Does advertising affect sales?• Do wages affect productivity?• Does price affect sales?• Will more fertilizer affect crop yield?• Does beer affect intelligence?

II. The Simple Linear Regression Model

For now, we will only use one independent variable in the model.

For example, while we’re fairly sure there exist many factors that affect your starting salary (y), we might include just your college grade point average (x).

A. Regression Model and Regression Equation

• The ß0 and ß1 are parameters that would describe the simple linear relationship between x and y.

• The is an error term that accounts for the variability in y that can’t be explained by the simple linear relationship above.

xy 10

• By taking the expected value of y, we get the simple linear regression equation.

• E(y) is the mean of y for a given value of x.

• ß0 is the y-intercept.

• ß1 is the slope of the equation.

xyE 10)(

3 Possibilities for ß1.

y

x

ß1>0

ß1 =0

ß1 <0

ß0

B. Estimated Regression Equation

• If ß0 and ß1 were both known, we could just use the regression equation to calculate E(y) for every value of x.

• But how much fun would that be?

• We will statistically estimate ß0 and ß1 by gathering sample data from the population and calculating b0 and b1.

• is the estimated value of y given a value of x.

• The method used for estimating b0 and b1 is called the least-squares method.

ii xbby 10ˆ

iy

III. Least Squares Method

A. An Example

Let’s develop a model between the age of a car (xi) and the repair cost (yi) at a local mechanic’s garage.

• So xi is the age of the ith auto, in years. is the estimated repair cost for the ith car.

Age Repair Cost1 135$ 2 175$ 3 320$ 4 300$ 5 450$

iy

ii xbby 10ˆ What is your expectation on the sign of b1? Positive, negative, or zero?

A scatter plot of our data.

Repair Cost and Age

$-

$100

$200

$300

$400

$500

0 1 2 3 4 5 6

Age (years)

Co

st

It looks like we have a pretty good positive linear relationship here!