Let us develop the following notation. The population regression line will be written as

(12.4)

and the sample regression line will appear as

where is the sample estimate of and is the sample estimate of . The sample regression line is illustrated in Fig. 12.4. Note that there are two values of Y that occur at each X_{i} value—the observed value of Y or Y_{i}, and the estimated value of Y or . Their difference, , will be termed the ith residual or ith deviation from the sample regression line. This residual is “observed” and serves as an estimate of ε_{i}, which is purely random and thus “unobserved.”

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