Question

observation with smallest residual =
observation with largest residual =

Explanation:

Step1: Understand Residual

Residual is the vertical distance between a data point and the regression line. A smaller residual means the point is closer to the line; a larger residual means it's farther.

Step2: Analyze Each Point

- For the point at \( x = 6 \) (the one on the regression line), its residual is \( 0 \) (since it lies on the line), which is the smallest possible residual.
- For the point with the largest residual, we look for the point farthest from the line. The topmost blue dot (above the line) or the bottommost? Wait, the topmost dot (around \( y \approx 27 \)) is far above the line, but wait, no—wait the \( x = 6 \) has a point on the line. Wait, let's check coordinates. Wait the x-axis: let's assume x is maybe "something", but the key is the point on the line (x=6, y=19ish? Wait no, the line at x=6 has a point (the blue dot on the line). Then the other points: the one at x=5 (wait x-axis labels: 2,4,6,8,10. Wait the x-axis starts at, maybe x=0? Wait the first vertical line is at x=0? Wait no, the x-axis has ticks at 2,4,6,8,10. Wait the point at x=6: the blue dot on the line (residual 0). Then the point at x=10: y=10, which is on the line? Wait no, the line at x=10 is y=10? Wait the line goes from top-left to bottom-right. Wait the point at x=6 is on the line (residual 0). Then the point with the largest residual: the topmost blue dot (the highest one, above the line) or the lowest? Wait the lowest blue dot is at x=5 (y=11) and the line at x=5: let's see the line's equation. Wait, but visually, the point on the line (x=6) has residual 0. The point with the largest residual is the one farthest from the line. The topmost blue dot (around y=27) is far above the line, but wait, no—wait the x=6 has a point on the line. Wait, maybe the x=6 point (on the line) has smallest residual (0). Then the largest residual: the topmost blue dot (the highest one) or the bottommost? Wait the bottommost is at x=5 (y=11) and the line at x=5: let's estimate the line's y at x=5. The line at x=6 is y=19 (wait no, the line at x=6: the blue dot is on the line, so y there is, say, 19. Then at x=5, the line's y would be higher than 19? Wait no, the line is decreasing. So as x increases, y decreases. So at x=0, y is high; at x=10, y is 10. So the line equation: let's say at x=10, y=10; at x=6, y=19? No, wait the line at x=6 has a blue dot (on the line), so that point is (6, y_line). Then the other points: the one at x=6 (on the line) has residual 0. The point with the largest residual is the one farthest from the line. The topmost blue dot (the one above the line, highest) is far from the line, but wait, maybe the x=6 point (on the line) has smallest residual. Then the largest residual: the topmost blue dot (the highest one) or the bottommost? Wait the bottommost is at x=5 (y=11) and the line at x=5: let's see, the line at x=6 is y=19 (no, wait the line is decreasing, so at x=5, the line's y should be higher than at x=6. Wait, maybe I got the x-axis wrong. Wait the x-axis is labeled with 2,4,6,8,10, so the first vertical line is x=0? No, maybe x starts at 0, and the ticks are at 2,4,6,8,10. So the point at x=6 is on the line (residual 0). Then the point with the largest residual: the topmost blue dot (the one with y≈27) is far above the line, so its residual is large (positive). The bottommost blue dot (y≈11) is below the line, but is it farther than the topmost? Wait the topmost is much higher. Wait, but maybe the x=6 point (on the line) has smallest residual (0). Then the largest residual: the topmost blue dot (the highest one). Wait, but the question is "Observation with smallest residual" and "largest residual". So the smallest residual is 0 (the point on the line, at x=6). The largest residual is the point farthest from the line, which is the topmost blue dot (or maybe the bottommost? Wait no, the topmost is higher. Wait, let's check coordinates. Let's assume the line is \( y = -mx + b \). At x=6, y is the line's y (the point on the line). So that point has residual 0. So smallest residual is the point at x=6 (on the line). For largest residual, the point with the maximum vertical distance from the line. The topmost blue dot (above the line) is the farthest, so its residual is the largest (positive or negative, but magnitude).

Answer:

Observation with smallest residual = The point at \( x = 6 \) (on the regression line)
Observation with largest residual = The top - most blue dot (farthest from the regression line)