Simple Regression, Linear and Curvilinear
Determining the way one variable changes when another changes: (2) according to the straight-line function
There are several ways by which a straight line can be determined to show the functional relation between two variables. One way would be simply to place a ruler over the chart along the several group averages, or to stretch a black thread over them, and draw the line in by eye so as to fall as nearly as possible along them. Although no two persons would draw their lines exactly the same, still this method might give fairly satisfactory results where only a rough measure was wanted. However, it would often be advantageous to determine a particular straight line that could be exactly duplicated by other persons and that would qualify in some sense as the best possible straight line for expressing the relationship in this particular set of data. We shall therefore use the exact regression method of determining the straight line. But first we must consider the meaning of a straight line.
The determination of what this line will be consists in finding the constants for the equation of the line. Just as we have already seen (Chapter 3) that the curve showing the relation between the distance a body has to fall and the time it takes can be expressed by the relation,