1. What is a residual? Explain when a residual is positive, negative, and zero.

Verified step by step guidance

1. Understand that a residual is the difference between an observed value and the predicted value from a regression model. Mathematically, it is expressed as $e=y-y$, where $y$ is the observed value and $y$ is the predicted value.
2. Recognize that a residual measures the error or deviation of the prediction from the actual data point, helping to assess the accuracy of the regression model.
3. A residual is positive when the observed value is greater than the predicted value, meaning the model underestimates the actual data point.
4. A residual is negative when the observed value is less than the predicted value, indicating the model overestimates the actual data point.
5. A residual is zero when the observed value exactly equals the predicted value, showing a perfect prediction for that data point.

Key Concepts

Residual

A residual is the difference between an observed value and the predicted value from a regression model. It measures the error or deviation of the prediction from the actual data point, indicating how well the model fits the data.

Positive Residual

A residual is positive when the observed value is greater than the predicted value. This means the model underestimates the actual data point, and the error is above the regression line.

Negative and Zero Residuals

A residual is negative when the observed value is less than the predicted value, indicating the model overestimates the data point. A residual is zero when the observed and predicted values are equal, meaning the model perfectly predicts that data point.