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.

b. in what sense is the regression line the straight line that best fits the points in a scatterplot?

the regression line has the property that the sum / sum of squares of the residuals is the minimum / maximum possible sum.

Explanation:

Part a

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

• Option A: Residuals involve error (they measure the difference between observed and predicted values), 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 of a residual.
• Option C: A residual is not a "point with strong...", it's a numerical value, so this is incorrect.
• Option D: The description here refers to a slope or rate of change, not a residual, so this is incorrect.

The regression line is the "best - fit" line because it minimizes the sum of the squared residuals. Also, the sum of the residuals (without squaring) is zero (the smallest possible sum for the sum of residuals, as positive and negative residuals cancel out). So, the first blank should be "sum" and the second blank should be "smallest" (or "least", but in the context of the sum of residuals, the sum is zero which is the smallest possible sum).

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 \( y \).

Part b