In the context of residuals analysis, what does it mean if a data point has a residual value of $1.3$ with respect to the line of best fit for a data set?

• A. The residual is not related to the difference between observed and predicted values.
• B. The observed value is $1.3$ units above the value predicted by the line of best fit.
• C. The observed value is $1.3$ units below the value predicted by the line of best fit.
• D. The predicted value is $1.3$ times the observed value.

Answer: (B)

Verified step by step guidance

1. Understand that a residual in regression analysis is defined as the difference between the observed value and the predicted value from the line of best fit. Mathematically, it is expressed as: \[\text{Residual} = \text{Observed value} - \text{Predicted value}\]
2. Interpret the sign and magnitude of the residual: a positive residual means the observed value is greater than the predicted value, while a negative residual means the observed value is less than the predicted value.
3. Given a residual value of 1.3, recognize that this means the observed data point lies 1.3 units above the predicted value on the line of best fit.
4. Note that the residual is a measure of vertical distance from the data point to the regression line, not a ratio or multiple of the observed or predicted values.
5. Conclude that the correct interpretation is: the observed value is 1.3 units above the predicted value given by the line of best fit.