Question

a. what is a residual?
b. in what sense is the regression line the straight line that \best\ fits the points in a scatterplot?
a. what is a residual?
a. a residual is a value that is determined exactly, without any error.
b. a residual is a value of $y - \hat{y}$, which is the difference between an observed value of $y$ and a predicted value of $y$.
c. a residual is a point that has a strong effect on the regression equation.
d. a residual is the amount that one variable changes when the other variable changes by exactly one unit.

Explanation:

To determine the correct definition of a residual, we analyze each option:

• Option A: Residuals involve error (difference between observed and predicted), so this is incorrect.
• Option B: A residual is defined as \( y - \hat{y} \), where \( y \) is the observed value and \( \hat{y} \) is the predicted value from the regression model. This matches the definition.
• Option C: A point with strong effect on regression is an influential point, not a residual.
• Option D: The amount one variable changes with a unit change in another is the slope, not a residual.

Answer:

B. A residual is a value of \( y - \hat{y} \), which is the difference between an observed value of \( y \) and a predicted value of \( y \).