Cogito
AP Statistics · Chapter 5 · Lesson 1
Least-Squares Regression
The line, the slope, and what r² means.
11 problems · about 25 minutes · AP Statistics 2.5, 2.7, 2.8
What this lesson teaches
The student interprets the slope and intercept of a regression line and the value of r².
- The least-squares line minimises the sum of squared residuals.
- The slope is a predicted change per unit, and r² is the proportion of variation explained.
- A line says nothing outside the range of the data it was fitted to.
Warm Up
Straightforward practice. Get the method working first.
4 problemsr = 0.9. What is r²?
Answer 0.81
Why 0.81.
r = 0.5. What is r²?
Answer 0.25
Why Square it.
ŷ = 3x + 4. What is y predicted at x = 2?
Answer 10
Why 6 + 4.
r² = 0.9. Percent of variation explained?
Answer 90
Why As a percent.
Build It Up
The same ideas with more to keep track of.
3 problemsA residual is actual minus predicted. Actual 12, predicted 10. Residual?
Answer 2
Why 12 − 10.
Does a strong r² prove causation? 1 for yes, 0 for no.
Answer 0
Why Design decides causation, not fit.
r = −0.8. Is the slope negative? 1 for yes, 0 for no.
Answer 1
Why They share a sign.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsData covers x from 1 to 10. Is predicting at x = 50 sound? 1 for yes, 0 for no.
Answer 0
Why That is extrapolation.
Match each quantity to what it tells you.
Answer Slope → Predicted change in y per unit of x; r² → Proportion of variation explained; Residual → How far one point sits from the line
Why Only one of the three describes a single point.
From r to r²: r = 0.6. What is r²?
Answer 0.36
Why 0.36.
Unexplained: r² = 0.75. What proportion of the variation is not explained?
Answer 0.25
Why 0.25.