AP Statistics · Unit 5 · Topic 5.3

Linear Regression Models Study Lab

A linear regression model, ŷ = a + bx, turns an x-value into a predicted y-value. Adjust the slope and intercept, choose where to predict, and see the difference between safe interpolation inside the data and risky extrapolation beyond it. The story explains every number in context.

1. Predict with a line

A linear regression model uses a line to predict the response from the explanatory variable. The hat on ŷ means "predicted." Adjust the line, choose an x-value, and decide whether the prediction is trustworthy.

ŷ = a + bx a = y-intercept · b = slope ŷ = predicted value of y
Contexts

Model inside the dataModel beyond the dataPredictionRange of observed x
The slope slider turns the line around the middle of the data, so the intercept changes too.
The green band shows the x-values in the data. Outside it, predictions are extrapolations.
x
Predicted ŷ
Type of prediction
Squared misses: your line vs. best

Show your work

2. Key ideas

Model

ŷ = a + bx

A regression line predicts the response y from the explanatory variable x. a is the y-intercept and b is the slope. Some texts write ŷ = b₀ + b₁x; it is the same model.

Notation

y vs. ŷ

y is an actual (observed) value. ŷ ("y-hat") is a predicted value from the line. Always write ŷ, or "predicted y," in the equation; writing y alone loses credit.

Slope

Change in ŷ per unit of x

The slope b tells how much the predicted response changes when x increases by 1 unit. Its sign matches the direction of the association.

Intercept

Predicted y when x = 0

The y-intercept a is the predicted response when x = 0. If x = 0 is far outside the data, or impossible, the intercept has no practical meaning.

Interpolation

Predicting inside the data

Using the line for x-values within the observed range is interpolation. These predictions are reasonable when the relationship is linear.

Extrapolation

Predicting beyond the data

Using the line for x-values outside the observed range is extrapolation. The pattern may change, so these predictions are often unreliable and can be impossible.

3. Interpretation templates

Fill in the context. Graders look for the word "predicted" (or "on average") and units.

QuantityTemplateExample (ŷ = 52 + 4.5x, hours → score)
Slope bFor each additional 1 [unit of x], the predicted [y] increases (decreases) by |b| [units of y].For each additional hour studied, the predicted exam score increases by 4.5 points.
Intercept aWhen [x] = 0, the predicted [y] is a [units].A student who studies 0 hours has a predicted score of 52 points.
Prediction ŷSubstitute x into the model and report ŷ with units.ŷ = 52 + 4.5(6) = 79 points.
Extrapolationx is outside the observed x-values, so the prediction may be unreliable.Predicting for 15 hours gives 119.5, more than the maximum score.

4. On the FRQ

How to earn the points

  • Write the equation with ŷ and variable names, for example: predicted score = 52 + 4.5(hours).
  • Interpret the slope as a change in the predicted response per one-unit increase in x, with units and context.
  • Show the substitution when you predict: "ŷ = 52 + 4.5(6) = 79 points."
  • When asked if a prediction is reliable, compare the x-value with the observed range and use the word extrapolation if it is outside.
  • Say whether the intercept makes sense in context. Often x = 0 is outside the data.

Common mistakes

  • Writing y instead of ŷ in the equation, or interpreting the slope without "predicted."
  • Saying "y increases by b" as if the line gives exact values for every individual.
  • Interpreting the intercept when x = 0 is impossible or far outside the data, without a warning.
  • Trusting extrapolated predictions because the line fits the data well.
  • Mixing up slope and intercept when reading an equation like ŷ = 3.2x + 15.

5. Practice questions

AP-style multiple choice. Pick an answer to see the explanation.

0 / 0 correct

6. Key terms

The vocabulary for topic 5.3. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 5 terminology guide.