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 ß 0 and ß 1 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.

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. ß 0 is the y-intercept. ß 1 is the slope of the equation.

3 Possibilities for ß 1. y x ß 1 >0 ß 1 =0 ß 1 <0 ß0ß0

B. Estimated Regression Equation If ß 0 and ß 1 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 ß 0 and ß 1 by gathering sample data from the population and calculating b 0 and b 1.

is the estimated value of y given a value of x. The method used for estimating b 0 and b 1 is called the least-squares method.

III. Least Squares Method A. An Example Let’s develop a model between the age of a car (x i ) and the repair cost (y i ) at a local mechanic’s garage. So x i is the age of the i th auto, in years. is the estimated repair cost for the i th car. What is your expectation on the sign of b 1 ? Positive, negative, or zero?

A scatter plot of our data. It looks like we have a pretty good positive linear relationship here!