Question at position 25 Suppose you have the following estimated regression model: y^i=2.671−0.592xi\hat{y}_i = 2.671 - 0.592 x_i A data point (xi,yi)(x_i, y_i) is (4.5, -1.3). What is the value of the residual for this data point? Do not round any calculations or your answer.AnswerSuppose you have the following estimated regression model: y^i=2.671−0.592xi\hat{y}_i = 2.671 - 0.592 x_i A data point (xi,yi)(x_i, y_i) is (4.5, -1.3). What is the value of the residual for this data point? Do not round any calculations or your answer.[input]BlackTom题目解析

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BlackTom AI · Accepted Answer

We start by identifying what a residual is in linear regression: it is the difference between the observed value y_i and the predicted value ŷ_i, i.e., residual = y_i − ŷ_i.

First, compute the predicted value ŷ for x_i = 4.5 using the model ŷ = 2.671 − 0.592 x.
- Multiply the slope by x: 0.592 × 4.5 = 2.664.
- Subtract from the intercept: ŷ = 2.671 − 2.664 = 0.007.

Next, use the observed y_i = −1.3 to obtain the residual:
- Residual = y_i − ŷ = −1.3 − 0.007 = −1.307.

Thus the residual for the data point (4.5, −1.3) is −1.307 (without rounding any intermediate steps).

If you compare this to the provided answer of −1.463, note that that value would imply a different ŷ (for example, a predicted value around −0.163), which doesn’t align with the given regression equation and calculation shown here. The step-by-step arithmetic above follows directly from the model specification and the definition of residuals.

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