Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

552 CHAPTER 16 | Introduction to Regression
churches and crime, the partial correlation showed that the actual relationship was zero
after the population size was controlled. Multiple regression produces the same conclusion.
A multiple-regression analysis of these data using churches and population to
predict crime, produced a beta value of β = 0.961 for the population variable and
β = 0 for the church variable. Thus, there is a positive correlation between population
and crime but the underlying correlation between churches and crime is zero (not positive
and not negative).
LEARNING CHECK
1. A multiple regression equation with two predictor variables is calculated for a set of
scores. If the constant values in the equation are b 1
= 2, b 2
= –3, and a = 7, then
what Y value would be predicted for an individual with X 1
= 2 and X 2
= 4?
a. 1
b. –1
c. 17
d. 41
2. A multiple regression equation with two predictor variables produces R 2 = 0.10.
What portion of the variability for the Y scores is predicted by the equation?
a. 1%
b. 10%
c. 90%
d. 99%
3. The multiple regression with two predictor variables is computed for a sample of
n = 25 participants. What is the value for degrees of freedom for the predicted
portion of the Y-score variance, MS regression
?
a. 2
b. 3
c. 22
d. 23
ANSWERS
1. B, 2. B, 3. A
SUMMARY
1. When there is a general linear relationship between
two variables, X and Y, it is possible to construct a
linear equation that allows you to predict the Y value
corresponding to any known value of X:
predicted Y value = Ŷ = bX + a
The technique for determining this equation is
called regression. By using a least-squares method to
minimize the error between the predicted Y values and
the actual Y values, the best-fitting line is achieved
when the linear equation has
s Y
b 5 SP 5 r and a 5 M
SS X
s Y
2 bM X
X
2. The linear equation generated by regression (called
the regression equation) can be used to compute a
predicted Y value for any value of X. However, the
prediction is not perfect, so for each Y value, there is