Question

make a residual plot for the linear model and the data set it represents. how well does the model fit the data set?
y = -0.04x + 12.6

x	y
10	10
15	13
20	14
25	13
30	9

(there are also some graphs on the right side and a text box with the following: the points in the residual plot dropdown to be randomly distributed on either side of the x - axis, and they dropdown clustered fairly close to the x - axis. the model dropdown)

Explanation:

Step1: Calculate Residuals

Residual \( r = y - \hat{y} \), where \( \hat{y} = -0.04x + 12.6 \).

• For \( x = 10 \): \( \hat{y} = -0.04(10) + 12.6 = 12.2 \), \( r = 10 - 12.2 = -2.2 \)
• For \( x = 15 \): \( \hat{y} = -0.04(15) + 12.6 = 12.0 \), \( r = 13 - 12.0 = 1.0 \)
• For \( x = 20 \): \( \hat{y} = -0.04(20) + 12.6 = 11.8 \), \( r = 14 - 11.8 = 2.2 \)
• For \( x = 25 \): \( \hat{y} = -0.04(25) + 12.6 = 11.6 \), \( r = 13 - 11.6 = 1.4 \)
• For \( x = 30 \): \( \hat{y} = -0.04(30) + 12.6 = 11.4 \), \( r = 9 - 11.4 = -2.4 \)

Step2: Identify Residual Plot

Residuals: \( (-2.2, 1.0, 2.2, 1.4, -2.4) \) at \( x = 10,15,20,25,30 \).
Plot A has points matching these residuals (e.g., \( x=10 \) residual ~-2.2, \( x=20 \) ~2.2, etc.).

Step3: Analyze Fit

Residuals are randomly distributed (no pattern) and close to x-axis (small magnitude). So the model fits reasonably well.

Answer:

Residual plot: Plot A (top-left).
Fit analysis: The points in the residual plot appear to be randomly distributed on either side of the x - axis, and they are clustered fairly close to the x - axis. The model fits the data reasonably well.