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

534 CHAPTER 16 | Introduction to Regression
Y ˆ = bX + a
F I G U R E 16.3
The distance between the actual
data point (Y) and the predicted
point on the line (Ŷ) is defined as
Y – Ŷ . The goal of regression is to
find the equation for the line that
minimizes these distances.
Y values
X values
X, Y
data point
Distance = Y – Ŷ
The calculations that are needed to find this equation require calculus and some sophisticated
algebra, so we will not present the details of the solution. The results, however, are
relatively straightforward, and the solutions for b and a are as follows:
b 5 SP
SS X
(16.2)
where SP is the sum of products and SS X
is the sum of squares for the X scores.
A commonly used alternative formula for the slope is based on the standard deviations
for X and Y. The alternative formula is
b 5 r s Y
s X
(16.3)
where s Y
is the standard deviation for the Y scores, s X
is the standard deviation for the
X scores, and r is the Pearson correlation for X and Y. After the value of b is computed,
the value of the constant a in the equation is determined by
a = M Y
– bM X
(16.4)
Note that these formulas determine the linear equation that provides the best prediction of
Y values. This equation is called the regression equation for Y.
DEFINITION
The regression equation for Y is the linear equation
Ŷ = bX + a (16.5)
where the constant b is determined by Equation 16.2 or 16.3, and the constant a
is determined by Equation 16.4. This equation results in the least squared error
between the data points and the line.