Final answer:
Residuals represent the difference between observed and predicted values in a least squares regression equation. In a well-fitted regression model, the average value of the residuals will be zero.
Step-by-step explanation:
To show that the average value of the residuals is zero, we need to understand what residuals are in the context of a least squares regression equation. Residuals represent the difference between the actual observed values and the predicted values from the regression line. In a well-fitted regression model, the average value of the residuals will be zero.
Let's consider an example:
Suppose we have a least squares regression line with the equation ŷ = a + bx, where a is the y-intercept and b is the slope. To calculate the residual for each observation, we subtract the predicted value (ŷ) from the actual observed value (y). The average value of the residuals can be found by taking the sum of all the residuals and dividing it by the number of observations.
Mathematically, the average value of the residuals (denoted as ∑e/n) is given by:
∑e/n = 0
This equation shows that the sum of the residuals will cancel out, resulting in an average value of zero.