5.4 Interpreting Coefficients in Linear Regression

Slope in context

The slope of a least-squares regression line predicts the change in the response variable for every one-unit increase in the explanatory variable. Always state it in context: 'For each additional [unit of x], the predicted [y] changes by [slope] [units].' The sign of the slope tells you the direction of the association.

Y-intercept in context

The y-intercept is the predicted value of y when x = 0. It's only meaningful if x = 0 is realistic or within (or near) the observed range of data. If x = 0 is impossible or far outside the data (extrapolation), the intercept has no real-world interpretation — it's just a mathematical anchor for the line.

Units matter

Both slope and intercept carry the units of y (intercept) or units of y per unit of x (slope). Misreading units is a common error — always double check what each variable represents and its measurement units before interpreting coefficients on an exam.

Key terms

How do you interpret the slope of a regression line?

For each one-unit increase in x, the predicted y changes by the slope amount (in the units of y).

How do you interpret the y-intercept?

The predicted value of y when x = 0 — only meaningful if x = 0 is realistic or within the data's range.

Units of the slope

Units of y per unit of x (e.g., kg per cm).

Extrapolation

Using a regression line to predict y for x-values outside the range of observed data; can produce unreliable or meaningless results.

Why interpret coefficients 'in context'?

Numbers alone aren't enough — you must reference the actual variables and units to make the interpretation meaningful and complete.

Try an AP-style question

Which interprets the slope correctly?

1. Studying causes a 3.5-point increase per hour
2. For each additional hour studied, the predicted score increases by 3.5 points
3. A student who studies 0 hours scores 3.5
4. For each additional hour, every student's score increases by 3.5