In chapter 9 we proposed the correlation coefficient as a measure of the degree to which two random variables may be linearly related. In the present chapter we will show how information about one variable which is easily measured or well-understood can be exploited to improve our knowledge about a less easily measured or less familiar variable. To introduce the idea of a linear model, which is crucial for this chapter, we will begin with a simple non-linguistic example.

Suppose the manager of a shop is paid entirely on a commission basis and he receives at the end of each month an amount equal to 2% of the total value of sales made in that month. The problem, and the model for its solution, can be expressed mathematically. Let Y be the commission the manager ought to receive for the month just ended. Let X be the total value of the sales in that month. Then:

The model cart be represented graphically as in figure 13.1 by a straight line passing through the origin of the graph. When the value of X, the month's total sales, is known, then the corresponding value of Y, the commission, can be read off from the graph as shown in the figure. Note that for every £1 increase in X, the commission increases by 2p or £0.02. We would say that the slope or gradient of the line is 0.02.