## Twovariable Regression Analysis Some Basic Ideas

In Chapter 1 we discussed the concept of regression in broad terms. In this chapter we approach the subject somewhat formally. Specifically, this and the following two chapters introduce the reader to the theory underlying the simplest possible regression analysis, namely, the bivariate, or two-variable, regression in which the dependent variable (the regressand) is related to a single explanatory variable (the regressor). This case is considered first, not because of its practical adequacy, but because it presents the fundamental ideas of regression analysis as simply as possible and some of these ideas can be illustrated with the aid of two-dimensional graphs. Moreover, as we shall see, the more general multiple regression analysis in which the regressand is related to one or more regressors is in many ways a logical extension of the two-variable case.

### 2.1 A HYPOTHETICAL EXAMPLE1

As noted in Section 1.2, regression analysis is largely concerned with estimating and/or predicting the (population) mean value of the dependent variable on the basis of the known or fixed values of the explanatory variable^).2 To understand this, consider the data given in Table 2.1. The data

1The reader whose statistical knowledge has become somewhat rusty may want to freshen it up by reading the statistical appendix, App. A, before reading this chapter.

2The expected value, or expectation, or population mean of a random variable Y is denoted by the symbol E(Y). On the other hand, the mean value computed from a sample of values from the Y population is denoted as Y, read as Y bar.

38 PART ONE: SINGLE-EQUATION REGRESSION MODELS

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