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##### (Solution) - An alternative to the regression formula y

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An alternative to the regression formula ?y = ? + ?x expresses each y value, rather than the mean of the y values, in terms of x. This approach models an observation on y as
y = mean + error = ? + ?x + ?,
Where the mean ?y = ? + ?x and the error = ?.
The error term denoted by ? (the Greek letter epsilon) represents the deviation of the observation from the mean, that is, ? = error = y - mean.
a. If an observation has ? > 0, explain why the observation falls above the mean.
b. What does ? equal when the observation falls exactly at the mean? The ? term represents the error that results from using the mean value (? + ?x) of y at a certain value of x for the prediction of the individual observation on y.
c. For the sample data and their prediction equation  = ? + ?x, explain why an analogous equation to the population equation y = ? + ?x + ?, is y = a + bx + e, where e is the residual, e = y - . (The residual e estimates ?. We can interpret e as a sample residual and ? as a population residual.)
d. Explain why it does not make sense to use the simpler model, y = ? + ?x, which does not have the error term.

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