AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/2/2-6)
Unit 2 · Topic 2.6
2.6 Competing Function Model Validation
Several models can seem to fit the same data. Residuals, the gaps between actual and predicted values, show you which model is best and where a model overestimates or underestimates.
Key terms
- residual
- residual plot
- overestimate / underestimate
- model validation
Competing models
Data with a slowly changing rate of change could look linear, quadratic or exponential over a short stretch. To choose, look at the context and at how each model's errors behave.
Context clues matter. Money earning a fixed percent should be exponential. A quantity that can't go below zero shouldn't be modeled by a line that eventually goes negative.
Each function type has a signature in data with equally spaced inputs. A linear pattern has constant differences. A quadratic pattern has constant second differences (the differences of the differences). An exponential pattern has constant ratios. Real data rarely match any of these exactly, so you look for the closest match and then confirm with residuals.
Residuals
For each data point, the residual is the actual value minus the predicted value:
residual = actual y − predicted y
A positive residual means the actual value is above the model, so the model underestimated. A negative residual means the model overestimated.
The size of the error is the absolute value of the residual. A residual of −2.3 and one of +2.3 are equally large errors in opposite directions.
Residual plots
A residual plot puts the input variable on the horizontal axis and the residuals on the vertical axis.
A model is appropriate when its residual plot shows no pattern: the points scatter randomly above and below zero. A clear pattern, like a U shape or an upside-down U, means the model misses something about the data's shape, and a different function type would fit better.
Small residuals help too, but the pattern is the main test. A model with a curved residual pattern is systematically wrong, even if the errors are small.
A calculator regression finds the constants that fit the data as closely as it can for the function type you chose. It can't tell you whether that type was the right choice. The residual plot can.
When over or under is better
Sometimes the context tells you which kind of error is safer. A school ordering lunches would rather overestimate how many students will eat than run out. An engineer estimating how much weight a beam can hold would rather underestimate.
So the best model for a decision might be the one that errs in the safer direction on the interval that matters.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Computing and interpreting a residual
A model predicts 52.3 thousand visitors for a museum in 2022. The actual number was 50.0 thousand. Find the residual and say whether the model overestimates or underestimates.
Show the solutionHide the solution
- Step 1: residual = actual − predicted = 50.0 − 52.3 = −2.3.
- Step 2: Negative means the actual value is below the prediction.
- Step 3: Watch the order: predicted − actual would give +2.3 and lead to the wrong conclusion.
Answer: The residual is −2.3 thousand visitors; the model overestimates by 2,300 visitors for 2022.
- Example 2
Comparing residual plots
For the data (0, 3.1), (1, 4.4), (2, 6.5), (3, 9.4), (4, 13.8), (5, 20.1), a linear regression gives residuals 1.843, −0.174, −1.391, −1.809, −0.726, 2.257, and an exponential regression gives residuals 0.036, −0.060, 0.007, −0.052, 0.039, 0.067. Which model is more appropriate?
Show the solutionHide the solution
- Step 1: Linear residuals: positive, then four negatives, then positive. Plotted against x, they form a U shape: a clear pattern.
- Step 2: That pattern means the line is too high in the middle and too low at both ends, because the data curve upward.
- Step 3: Exponential residuals: small and mixed in sign with no visible pattern.
Answer: The exponential model, because its residual plot shows no pattern (and its residuals are much smaller), while the linear model's residuals form a U shape.
- Example 3
Choosing the safer error
A clinic plans flu-season staffing. On weeks 4 to 8, model A has residuals between 3 and 10 patients, and model B has residuals between −8 and −2 patients. Which model is safer for planning?
Show the solutionHide the solution
- Step 1: Model A's residuals are positive: actual > predicted, so A underestimates the number of patients.
- Step 2: Model B's residuals are negative: B overestimates.
- Step 3: Planning for too many patients is safer than planning for too few.
Answer: Model B, because it overestimates the number of patients on weeks 4 to 8, so the clinic won't be understaffed.
Common mistakes
- Computing residuals as predicted minus actual. The definition is actual minus predicted.
- Choosing the model with one small residual. Judge the whole residual plot, and look for a pattern first.
- Saying a positive residual means the model overestimated. Positive means the actual value is above the model, so it underestimated.
On the exam
- Expect to compute a residual, interpret its sign, or choose between models by comparing residual plots. Justify with “the residual plot for this model shows no pattern.”
- Calculator questions may ask you to run two regressions on the same data and decide which is more appropriate.
Connected topics
Videos
Check yourself
5 questions on 2.6 Competing Function Model Validation. Pick an answer to see if you got it, and why.
| x (days) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Actual mass (g) | 3.40 | 5.10 | 8.00 | 12.30 | 18.90 | 28.60 |
| M(x) (g) | 3.39 | 5.22 | 8.03 | 12.37 | 19.06 | 29.35 |
Invented data
What is the residual for x = 6?
Which of the following statements is supported by the table?
A student fits a linear model and an exponential model to the same data set. The residual plot for the linear model shows a clear U-shaped pattern: positive residuals at both ends and negative residuals in the middle. The residual plot for the exponential model shows residuals scattered above and below 0 with no clear pattern. Which conclusion is best supported?
| x (days) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Actual area (cm²) | 12.1 | 14.6 | 18.2 | 21.9 | 27.2 | 33.3 |
| L(x) | 10.70 | 15.00 | 19.30 | 23.60 | 27.90 | 32.20 |
| E(x) | 12.00 | 14.70 | 18.01 | 22.06 | 27.02 | 33.10 |
Invented data
For which value of x does L overestimate the actual area by the greatest amount?
Which of the following is best supported by the table?
0 of 5 answered