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Statistical Inference: A Short Course by Michael J. Panik

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12.4 Decomposition of the Sample Variation in Y

Let us express the variance of the random variable Y as

(12.10) equation

We noted above that there were two components to Yi—a systematic component reflecting the linear influence of Xi and a random component εi due to chance factors. A moment's reflection reveals that if most of the variation in Y is due to random factors, then the estimated least squares line is probably worthless as far as predictions are concerned. However, if most of the variation in Y is due to the linear influence of X, then we can obtain a meaningful regression equation. So the question arises—“How much of the variation in Y is due to the linear influence of X and how much can be attributed to random factors? We can answer this question with the aid of Fig. 12.7.

Figure 12.7 Decomposition of imginto imgand ei.

img

Since our goal is to explain the variation in Y, let's start with the numerator of Equation (12.10). That is, from Fig. 12.7,

(12.11) equation

where is attributed to the linear ...

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