Unit 22.6 Competing Function Model ValidationCalculator required

Three regressions are fit to original synthetic data. Which model is best supported by the residual summaries?

Residual summaries by increasing input
Model	Residuals
Linear	$2.4,0.8,-0.9,-1.5,-0.8,2.1$
Quadratic	$-0.7,0.9,0.6,-0.8,-0.5,0.7$
Exponential	$-1.6,-0.5,0.8,1.2,0.4,-1.3$

Linear, because its first and last residuals are positive
Quadratic, because its residuals are small and show no sustained pattern
Exponential, because its residuals change sign
No model, because every residual must equal zero

Answer and derivation

Correct answer: B

The quadratic residuals fluctuate near zero without a visible pattern.

Residuals need not be exactly zero.
The linear and exponential sequences show structured runs.
The quadratic residuals are smaller and alternate irregularly around zero.

Why the other options fail

A: Positive endpoints form part of a curved residual pattern, not evidence of fit.
B: Correct residual-plot criterion.
C: Changing sign is insufficient when a pattern remains.
D: Real data normally have nonzero residuals.

Common misconception: A high fit statistic or some sign changes do not replace checking residual pattern and context.

China/AP bridge: China-course bridge: regression is often calculator-led in Chinese classrooms; AP newly emphasizes residual evidence and justification; unlike choosing only the largest $R^2$ , inspect pattern and context; avoid demanding all residuals be zero.