In the context of regression analysis, what is a residual, and under what condition is a residual considered positive?

• A. A residual is the sum of the observed and predicted values; it is positive when both values are positive.
• B. A residual is the difference between the observed value and the predicted value; it is positive when $y>{ŷ}_i$, that is, when the observed value is greater than the predicted value.
• C. A residual is the product of the observed and predicted values; it is positive when both values are greater than zero.
• D. A residual is the difference between the predicted value and the observed value; it is positive when ${ŷ}_i>y$, that is, when the predicted value is greater than the observed value.

Answer: (B)

Verified step by step guidance

1. Understand that in regression analysis, a residual represents the error or difference between what was actually observed and what the model predicted.
2. Express the residual mathematically as \(\text{Residual} = \text{Observed value} - \text{Predicted value}\).
3. Recognize that the sign of the residual depends on the relative sizes of the observed and predicted values.
4. A residual is positive when the observed value is greater than the predicted value, meaning the model underestimates the actual observation.
5. Conversely, a residual is negative when the observed value is less than the predicted value, indicating the model overestimates the actual observation.