# Question: An alternative to the regression formula y

An alternative to the regression formula μy = α + βx 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 = α + βx + ɛ,

Where the mean μy = α + βx and the error = ɛ.

The error term denoted by ɛ (the Greek letter epsilon) represents the deviation of the observation from the mean, that is, ɛ = error = y - mean.

a. If an observation has ɛ > 0, explain why the observation falls above the mean.

b. What does ɛ equal when the observation falls exactly at the mean? The ɛ term represents the error that results from using the mean value (α + βx) 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 = α + βx, explain why an analogous equation to the population equation y = α + βx + ɛ, is y = a + bx + e, where e is the residual, e = y - . (The residual e estimates ɛ. We can interpret e as a sample residual and ɛ as a population residual.)

d. Explain why it does not make sense to use the simpler model, y = α + βx, which does not have the error term.

y = mean + error = α + βx + ɛ,

Where the mean μy = α + βx and the error = ɛ.

The error term denoted by ɛ (the Greek letter epsilon) represents the deviation of the observation from the mean, that is, ɛ = error = y - mean.

a. If an observation has ɛ > 0, explain why the observation falls above the mean.

b. What does ɛ equal when the observation falls exactly at the mean? The ɛ term represents the error that results from using the mean value (α + βx) 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 = α + βx, explain why an analogous equation to the population equation y = α + βx + ɛ, is y = a + bx + e, where e is the residual, e = y - . (The residual e estimates ɛ. We can interpret e as a sample residual and ɛ as a population residual.)

d. Explain why it does not make sense to use the simpler model, y = α + βx, which does not have the error term.

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