Explanation

In simple linear regression, the residual (also called error term) for an observation is defined as:

Residual = Observed value of Y - Predicted/Estimated value of Y

Mathematically: $e_i = Y_i - \hat{Y}_i$

Where:

• $Y_i$ = actual observed value of the dependent variable
• $\hat{Y}_i$ = predicted value from the regression equation

Why option C is correct:

• This is the standard definition of a residual in regression analysis
• Residuals represent the vertical distance between actual data points and the regression line
• They measure the unexplained portion of the variation after accounting for the relationship with the independent variable

Why other options are incorrect:

• Option A: This describes a ratio, not a residual. Residuals are differences, not ratios.
• Option B: This describes a ratio of unexplained to explained variation, which is related to R-squared concepts but not the definition of an individual residual.

Key Points:

• Residuals should sum to zero in ordinary least squares regression
• Analysis of residuals is crucial for checking regression assumptions (linearity, homoscedasticity, normality)
• Residuals represent the error or unexplained variation for each observation