Slope and intercept

A normal linear model,

μy  =  β0  +  β1x

σy  =  σ

involves 3 parameters, β0, β1 and σ. The model's slope, β1, and intercept, β0, can be interpreted in a similar way to the slope and intercept of a least squares line.

Slope
Increase in the mean response per unit increase in X.
Intercept
Mean response when X = 0.

Examples of interpretation

Context Interpretation of β1 Interpretation of β0
Y = Yield of wheat per acre
X = Fertiliser (kg per m2)
Increase in mean yield per acre for each additional kg/m2 of fertiliser Mean yield per acre if no fertiliser is used
Y = Exam mark
X = Hours of study by student before exam
Increase in expected mark for each additional hour of study Expected mark if there is no study
Y = Hospital stay (days)
X = Age of patient
Average extra days in hospital per extra year of age Average days in hospital at age 0. Not particularly meaningful here.