A least-squares line is only useful if you can say what its numbers mean for the situation. That means two sentences, one for the slope and one for the intercept, each with units and the word "predicted."
Idea clave
Slope $b$: for each one-unit increase in x, the predicted y changes by $b$ (in y's units). Intercept $a$: the predicted y when $x = 0$, which has a real meaning only if $x = 0$ makes sense and is near the data.
Por qué funciona
The line describes the trend of many individuals, not a rule each one follows. Real points scatter above and below it, so the slope is the change in the average (predicted) y, not a guaranteed change for any one individual. The intercept is just where the line crosses $x = 0$; if no data come from near $x = 0$, it is only an anchor for the line.
Ejemplo resuelto: Reading a slope
For used cars of one model, $\widehat{\text{price}} = 24{,}000 - 1{,}800 \cdot \text{age}$ (dollars, years). Interpret the slope and the intercept.
- Slope $-1{,}800$: for each additional year of age, the predicted price drops by \$1,800.
- Intercept $24{,}000$: a brand-new car (age 0) is predicted to cost \$24,000. New cars exist and the data include young cars, so this makes sense.
Respuesta: Each year of age lowers the predicted price by \$1,800; a new one is predicted at \$24,000.
Ejemplo resuelto: Comparing two individuals
Two cars of this model differ in age by 3 years. How do their predicted prices compare?
- Each year changes the prediction by $-1{,}800$.
- For 3 years: $3 \times (-1{,}800) = -5{,}400$.
Respuesta: The older car is predicted to sell for \$5,400 less.
Ejemplo resuelto: An intercept with no meaning
For adults, $\widehat{\text{weight}} = -180 + 5 \cdot \text{height}$ (pounds, inches). Interpret the intercept.
- At height 0 inches the line predicts $-180$ pounds.
- No adult is 0 inches tall, and a negative weight is impossible: $x = 0$ is far outside the data.
- The intercept only positions the line; it has no meaning here.
Respuesta: It has no real meaning; it just anchors the line.
Error común
Stating a predicted change as an exact change for every individual: "every car loses exactly \$1,800 a year." The line describes the average trend; individual cars sit above or below it. Check your sentence for the word "predicted" or "on average."
Slopes always carry units: dollars per year, points per hour. If you cannot name the units, you have not yet said what the slope means. In econometrics, the slope is the start of every "how much does y change with x?" question.