Simple Linear Regression Example

In this lesson, we apply regression analysis to some fictitious data, and we show how to interpret the results of our analysis.

Note: Regression computations are usually handled by a software package or agraphing calculator. For this example, however, we will do the computations "manually", since the gory details have educational value.

Problem Statement

Last year, five randomly selected students took a math aptitude test before they began their statistics course. The Statistics Department has three questions.

• What linear regression equation best predicts statistics performance, based on math aptitude scores?
• If a student made an 80 on the aptitude test, what grade would we expect her to make in statistics?
• How well does the regression equation fit the data?

How to Find the Regression Equation

In the table below, the x_i column shows scores on the aptitude test. Similarly, the y_i column shows statistics grades. The last two rows show sums and mean scores that we will use to conduct the regression analysis.

The regression equation is a linear equation of the form: ŷ = b₀ + b₁x . To conduct a regression analysis, we need to solve for b₀ and b₁. Computations are shown below.

therefore, the regression equation is: ŷ = 26.768 + 0.644x .

How to Use the Regression Equation

Once you have the regression equation, using it is a snap. Choose a value for the independent variable (x), perform the computation, and you have an estimated value (ŷ) for the dependent variable.

In our example, the independent variable is the student's score on the aptitude test. The dependent variable is the student's statistics grade. If a student made an 80 on the aptitude test, the estimated statistics grade would be:

ŷ = 26.768 + 0.644x = 26.768 + 0.644 * 80 = 26.768 + 51.52 = 78.288

Warning: When you use a regression equation, do not use values for the independent variable that are outside the range of values used to create the equation. That is called extrapolation, and it can produce unreasonable estimates.

In this example, the aptitude test scores used to create the regression equation ranged from 60 to 95. Therefore, only use values inside that range to estimate statistics grades. Using values outside that range (less than 60 or greater than 95) is problematic.