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

SECTION 16.1 | Introduction to Linear Equations and Regression 533
When drawing a graph
of a linear equation, it
is wise to compute and
plot at least three points
to be certain you have
not made a mistake.
Next, these two points are plotted on the graph: one point at X = 3 and Y = 80, the other
point at X = 8 and Y = 155. Because two points completely determine a straight line, we
simply drew the line so that it passed through these two points.
■ Regression
Because a straight line can be extremely useful for describing a relationship between two
variables, a statistical technique has been developed that provides a standardized method
for determining the best-fitting straight line for any set of data. The statistical procedure is
regression, and the resulting straight line is called the regression line.
DEFINITION
The statistical technique for finding the best-fitting straight line for a set of data is
called regression, and the resulting straight line is called the regression line.
The goal for regression is to find the best-fitting straight line for a set of data. To accomplish
this goal, however, it is first necessary to define precisely what is meant by “best
fit.” For any particular set of data, it is possible to draw lots of different straight lines that
all appear to pass through the center of the data points. Each of these lines can be defined
by a linear equation of the form Y = bX + a where b and a are constants that determine
the slope and Y-intercept of the line, respectively. Each individual line has its own unique
values for b and a. The problem is to find the specific line that provides the best fit to the
actual data points.
■ The Least-Squares Solution
To determine how well a line fits the data points, the first step is to define mathematically
the distance between the line and each data point. For every X value in the data, the linear
equation determines a Y value on the line. This value is the predicted Y and is called Ŷ
(“Y hat”). The distance between this predicted value and the actual Y value in the data is
determined by
distance = Y – Ŷ
Note that we simply are measuring the vertical distance between the actual data point (Y)
and the predicted point on the line. This distance measures the error between the line and
the actual data (Figure 16.3).
Because some of these distances will be positive and some will be negative, the next step
is to square each distance to obtain a uniformly positive measure of error. Finally, to determine
the total error between the line and the data, we add the squared errors for all of the
data points. The result is a measure of overall squared error between the line and the data:
total squared error = ∑(Y – Ŷ ) 2
Now we can define the best-fitting line as the one that has the smallest total squared
error. For obvious reasons, the resulting line is commonly called the least-squared-error
solution. In symbols, we are looking for a linear equation of the form
Ŷ = bX + a
For each value of X in the data, this equation determines the point on the line (Ŷ) that gives
the best prediction of Y. The problem is to find the specific values for a and b that make this
the best-fitting line.