Section 9.4
Multiple Regression
Section 9.4 Objectives
• Use a multiple regression equation to predict y-values
Multiple Regression Equation
• In many instances, a better prediction can be found
for a dependent (response) variable by using more
than one independent (explanatory) variable.
• For example, a more accurate prediction for the
carbon dioxide emissions discussed in previous
sections might be made by considering the number of
cars as well as the gross domestic product.
Multiple Regression Equation
Multiple regression equation
•
•
•
•
𝑦 = 𝑚𝑥 + 𝑏
ŷ = b + m1x1 + m2x2 + m3x3 + … + mkxk
x1, x2, x3,…, xk are independent variables
b is the y-intercept
y is the dependent variable
* Because the mathematics associated with this concept is
complicated, technology is generally used to calculate the
multiple regression equation.
Predicting y-Values
• After finding the equation of the multiple regression
line, you can use the equation to predict y-values over
the range of the data.
• To predict y-values, substitute the given value for
each independent variable into the equation, then
calculate ŷ.
Example: Finding a Multiple Regression
Equation
A researcher wants to determine how employee salaries
at a certain company are related to the length of
employment, previous experience, and education. The
researcher selects eight employees from the company
and obtains the data shown on the next slide.
Example: Finding a Multiple Regression
Equation
Employee Salary, y
A
57,310
B
57,380
C
54,135
D
56,985
E
58,715
F
60,620
G
59,200
H
60,320
Employment Experience Education
(yrs), x1
(yrs), x2
(yrs), x3
10
2
16
5
6
16
3
1
12
6
5
14
8
8
16
20
0
12
8
4
18
14
6
17
Example: Predicting y-Values
Use the regression equation
ŷ = 49,764 + 364x1 + 228x2 + 267x3
to predict an employee’s salary given 12 years of
current employment, 5 years of experience, and 16
years of education.
Solution:
ŷ = 49,764 + 364(12) + 228(5) + 267(16)
= 59,544
The employee’s predicted salary is $59,544.
Section 9.4 Summary
• Used a multiple regression equation to predict yvalues