Consider the estimated regression model Y_hat = 0.5 + 1.2*X1. This model was estimated on a data set where X1 has a range of values between 8 and 34. Should we use the model to predict Y_hat=2.9 = 0.5 + 1.2*2 for X1=2?

Question

Consider the estimated regression model Y_hat = 0.5 + 1.2*X1. This model was estimated on a data set where X1 has a range of values between 8 and 34. Should we use the model to predict Y_hat=2.9  = 0.5 + 1.2*2   for X1=2?

BlackTom AI · Accepted Answer

Question restatement: The estimated regression model is Y_hat = 0.5 + 1.2*X1, estimated on data where X1 ranges from 8 to 34. The question asks whether we should use the model to predict Y_hat = 2.9 for X1 = 2, given the formula Y_hat = 0.5 + 1.2*2.

Option 1: True. Why this is not correct: The model was trained with X1 values only between 8 and 34, so predicting at X1 = 2 would be an extrapolation outside the observed data range. Predictions outside the training domain are unreliable because the relationship estimated may not hold beyond the range of X1 used to fit the model. Also, the stated calculation Y_hat = 2.9 using X1 = 2 ignores the warning about extrapolation and uses a value not supported by the data range.

Option 2: False. Why this is correct: It is inappropriate to use the model to predict at X1 = 2 since 2 lies well below the observed range (8 to 34). Predictions outside the training interval are prone to large errors and lack empirical justification. Even though the algebraic substitution yields Y_hat = 0.5 + 1.2*2 = 2.9, this predicted value is not trustworthy because it corresponds to an X1 value outside the domain where the model was estimated, so the model should not be used to make that prediction.

Additional clarifications to avoid a common pitfall: It is tempting to compute the numerical prediction as if the algebra were the only criterion, but the key methodological rule is about the data domain. The conclusion hinges on the domain of X1, not on the arithmetic of the equation itself. Since the question explicitly mentions the training range, the correct stance is to reject predictions at X1 = 2 as inappropriate.

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Consider the estimated regression model Y_hat = 0.5 + 1.2*X1. This model was estimated on a data set where X1 has a range of values between 8 and 34. Should we use the model to predict Y_hat=2.9 = 0.5 + 1.2*2 for X1=2?判断题