Question at position 22 How is the residual calculated in a linear regression model?By multiplying the regression coefficient with the predictor variableBy subtracting the predicted value of the dependent variable from its actual valueBy adding the intercept and the product of the slope and the predictor variableBy observing the error.BlackTom题目解析

Question

BlackTom AI · Accepted Answer

To begin, let's lay out what a residual is in linear regression and how it is defined in common notation. The residual for an observation is the difference between the observed value of the dependent variable and the value predicted by the regression model.
Option 1: 'By multiplying the regression coefficient with the predictor variable' — This describes the component used to compute a predicted value (the slope times the predictor), not the residual itself. It does not account for the actual observed outcome, so it cannot be the residual.
Option 2: 'By subtracting the predicted value of the dependent variable from its actual value' — This matches the standard residual formula: residual = y − ŷ, where y is the actual value and ŷ is the predicted value. It directly expresses the difference between observation and model prediction.
Option 3: 'By adding the intercept and the product of the slope and the predictor variable' — This is the calculation of the predicted value ŷ (the regression line), not the residual. It lacks the actual observed value and thus does not yield the residual.
Option 4: 'By observing the error.' — This is vague and informal. While residuals are the errors of prediction, saying 'observing the error' without the explicit formula y − ŷ fails to define the residual precisely and can be misleading in a mathematical context.
In summary, only Option 2 directly expresses the precise residual definition as the difference between the observed outcome and its predicted value, aligning with the standard residual concept in linear regression.

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