The slope is an average change, predicted

$$\hat{y} = b_0 + b_1 x,$$

where $b_0$ is the y-intercept, $b_1$ the slope, and $\hat{y}$ the predicted value of the response at a given $x$. The hat is not decoration: $y$ is what an individual actually did, and $\hat{y}$ is what the model expects on average at that $x$. They differ by the residual, which is Topic 5.4.

For the eight students' study hours and exam scores,

$$\widehat{\text{score}} = 58.5 + 3.5(\text{hours}).$$

Writing the equation with the variable names rather than $x$ and $y$ is worth the extra characters: it makes the units visible and the interpretations almost automatic.

"For each additional hour of weekly study, the predicted exam score increases by 3.5 points, on average."

Four elements, and each is a place to lose credit:

1. Predicted. The model predicts; it does not determine. "Each extra hour makes a student score 3.5 points higher" is a causal, deterministic claim the line does not support.
2. On average. Individuals vary around the line. Two students with the same study hours will not have the same score.
3. Units on both variables. Points per hour. A slope with no units, or with the units flipped, has lost the interpretation.
4. Per one-unit change. Per additional hour, not per hour studied in total.

The direction follows the sign: a negative slope means the predicted response decreases as $x$ increases, and it is read with the same four elements.

"A student who studies 0 hours per week is predicted to score 58.5 points."

That reading is available here because $x = 0$ is close to the observed range, which ran from 1 to 8 hours. Often it is not:

1. A regression of weight on length for wolves might give an intercept of $-14$ kg, the predicted weight of a wolf of length zero. That is not a defect in the model; it is an extrapolation far outside the data, where the line was never meant to apply.
2. A regression of blood pressure on age with data from adults gives an intercept describing newborns, which the study never observed.

So the intercept is interpreted with the same predicted language, and accompanied by a note when $x = 0$ lies outside the data: the value is what the line says, and the line has no support there. Declaring a model wrong because its intercept is impossible confuses the arithmetic of a fitted line with a claim about the world.

To predict, substitute: at 6 hours, $\widehat{\text{score}} = 58.5 + 3.5(6) = 79.5$ points. The reading is "a student who studies 6 hours per week is predicted to score about 79.5 points", not "will score 79.5".

Extrapolation is predicting outside the range of observed $x$-values, and it is not supported by the data. At 20 hours the equation returns $58.5 + 3.5(20) = 128.5$ points, which exceeds the maximum possible score. The arithmetic is fine and the prediction is worthless: nothing in a data set covering 1 to 8 hours says the relationship stays linear at 20.

Two habits follow. State the range when reporting a model, so a reader knows where it applies. And when asked for a prediction outside that range, give the value if asked and say plainly that it is an extrapolation and unreliable.