In the context of regression analysis, what is a residual, and what does it indicate when a residual is positive ( > )?
A
A residual is the sum of all prediction errors; a positive residual means the model fits the data perfectly.
B
A residual is the predicted value minus the observed value (); a positive residual means the predicted value is greater than the observed value.
C
A residual is the difference between the observed value and the predicted value (); a positive residual means the observed value is greater than the predicted value.
D
A residual is the average of observed values; a positive residual means the data are normally distributed.
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1
Understand that in regression analysis, a residual represents the difference between the observed value and the predicted value from the regression model. Mathematically, it is expressed as \(\text{Residual} = \text{Observed value} - \text{Predicted value}\).
Recognize that the residual measures the error or deviation of the prediction from the actual data point, indicating how well the model fits that particular observation.
Interpret the sign of the residual: a positive residual means that the observed value is greater than the predicted value, implying the model underestimates the actual data point.
Conversely, a negative residual means the observed value is less than the predicted value, indicating the model overestimates the data point.
Note that residuals are used to assess the overall fit of the regression model and to check assumptions such as homoscedasticity and normality of errors.