FIGURE 2.5 Sample and population regression lines.
the PRF based on the SRF is at best an approximate one. This approximation is shown diagrammatically in Figure 2.5.
For X = Xi, we have one (sample) observation Y = Yi. In terms of the SRF, the observed Yi can be expressed as
and in terms of the PRF, it can be expressed as
Now obviously in Figure 2.5 Y overestimates the true E(Y | Xi) for the Xi shown therein. By the same token, for any Xi to the left of the point A, the SRF will underestimate the true PRF. But the reader can readily see that such over- and underestimation is inevitable because of sampling fluctuations.
The critical question now is: Granted that the SRF is but an approximation of the PRF, can we devise a rule or a method that will make this approximation as "close" as possible? In other words, how should the SRF be constructed so that fi\ is as "close" as possible to the true Pi and /32 is as "close" as possible to the true p2 even though we will never know the true P and P2?
CHAPTER TWO: TWO-VARIABLE REGRESSION ANALYSIS: SOME BASIC IDEAS 51
The answer to this question will occupy much of our attention in Chapter 3. We note here that we can develop procedures that tell us how to construct the SRF to mirror the PRF as faithfully as possible. It is fascinating to consider that this can be done even though we never actually determine the PRF itself.
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