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Statistical Inference: A Short Course by Michael J. Panik

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12.9 The Prediction of a Particular Value of Y given X

We now turn to the task of predicting a specific Y value from a given level of X. In particular, for X = Xo, let us predict or forecast the value of the random variable Yo = β0 + β1Xo + εo. If the true population regression line were known, the predictor of Yo would be E(Yo|Xo)= β0 + β1Xo, a point on the population regression line corresponding to Xo. Since E(Yo|Xo) is unknown, and must be estimated from sample data, let us use the estimator img, a point on the sample regression line. If img is used to estimate Yo, then the forecast error is the random variable img its mean is img that is, img is an unbiased estimator for Yo since img. Moreover, it can be demonstrated that the estimated standard deviation of the forecast error is

(12.26) equation

If we next assume ...

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