Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- If a scatterplot has a linear form, a linear regression model can approximate the relationship.
- The model uses the explanatory variable x to predict the response variable y.
- ŷ (“y-hat”) is a predicted value. Real observed values, y, are scattered around it.
- Model sentence: “The model predicts that the kiosk will sell about 99 iced drinks on a day with a maximum temperature of 24 °C.”
Key relationships
| Idea | What to remember |
|---|---|
| Model | ŷ = a + bx |
| a | y-intercept: ŷ when x = 0 |
| b | slope: change in ŷ for each increase of 1 in x |
| Predicting | substitute x, multiply by b, add a; round at the end |
| Interpolation | x inside the interval from the smallest to the largest x-value in the data (end points included) |
| Extrapolation | x outside that interval; less reliable, and more so the further out |
Assumptions and conventions
- The scatterplot shows a linear form. A line is a poor model for a curved pattern, even when r is close to −1 or 1.
- In this topic a and b are given. In Topic 5.5 you find them with technology (least-squares regression).
- Write the model with the hat, and preferably with variable names: “predicted sales = −30.26 + 5.37 × (temperature)”.
Mistakes to avoid
- Dropping the hat: y = a + bx says every point is on the line.
- Treating ŷ as certain: it is a prediction; actual values vary.
- Mixing up a and b: b is the number multiplied by x.
- Substituting a y-value for x: the model predicts y from x, not the other way round.
- Calling an end-point prediction an extrapolation: the end points are inside the interval.
- Trusting a far extrapolation: it can give impossible values, such as negative heights.
- Rounding coefficients early, which can shift the prediction a lot.
- No context or units in the final answer.
Quick self-check
- ŷ = 4.5 + 2.2x. Predict y when x = 10. (ŷ = 4.5 + 22 = 26.5)
- ŷ = 120 − 3.5x was fitted to data with x from 2 to 15. Find ŷ at x = 8. Is this interpolation or extrapolation? (ŷ = 120 − 28 = 92; interpolation, since 8 is between 2 and 15)
- The same model is used at x = 40. Why should you be cautious? (40 is far outside 2 to 15, so it is an extrapolation; the linear pattern may not continue. Here ŷ = −20, which may be impossible in context.)
Next: practice questions.