Decide what a fitted line can support · AP Statistics

Assess a fitted line with its original scatterplot, residuals and prediction domain. A compact equation can hide systematic error. Treat coefficients as descriptions of a fitted relation first; causal claims require an appropriate design beyond a strong numerical association.

Build the method

The intercept is the predicted response at x=0. If observed trip lengths range from twenty to fifty kilometers, zero is outside the observed domain. A six-liter intercept is an algebraic consequence of the fit, not direct evidence that a zero-length trip consumes six liters.

Worked example

For a simple least-squares line with intercept, R²=0.16 and a negative slope imply r=−0.4. Squaring removes direction, so the slope supplies the sign. A positive square root alone would contradict the direction visible in the scatterplot.

Compare the alternatives

Residuals 2,−1,−2,−1,2 at increasing x values form a curved pattern. Predictions are too low at both ends and too high in the middle. That systematic structure suggests the straight-line form misses part of the relationship, even if an overall correlation is fairly large.

Avoid a specific trap

A point far outside the usual x range has high leverage. Actual influence asks how much the fitted coefficients change when it is omitted. A remote point close to the existing line can have high leverage without greatly changing the slope. This distinction is a diagnostic enrichment, not a claim that every related regression method is assessed by the current exam.

Transfer check with explanation

Transfer: investigate a remote observation for measurement error and compare fitted results with and without it, documenting the reason for any exclusion. Do not silently delete it to obtain a desired line. Also avoid inferring independence or valid extrapolation merely because the residual signs alternate.

WORKED EXAMPLE

A fitted line predicts fuel use from trip length observed only between 20 and 50 km. Its intercept is 6 liters. Which statement about the intercept is justified?

Locate x=0 relative to the observed explanatory-variable range.
The intercept has a formal meaning, but zero km was not represented by the observed trips.
Why an alternative fails: Treat an unsupported extrapolated prediction as a certain observed outcome.

Answer: It is the algebraic prediction at zero km, outside the observed range

Compare the other answer choices

It proves a zero-distance trip consumes exactly 6 liters

Treat an unsupported extrapolated prediction as a certain observed outcome.

It is the increase in fuel use per additional kilometer

Confuse the intercept with the slope.

It is the average residual for the observed trips

The intercept is a model coefficient, not an average prediction error.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

AP Statistics Course and Exam Description, effective Fall 2026