5.2 Estimating the Slope and Intercept in a Regression Model

The Least Squares Regression Line

The least squares regression line (LSRL) minimizes the sum of squared residuals (observed − predicted y). Its equation, ŷ = a + bx (or ŷ = b₀ + b₁x), gives predicted y-values. This line passes through the point (x̄, ȳ), the mean of x and mean of y, guaranteeing the regression line always includes that point.

Computing Slope and Intercept

The slope b = r(sy/sx), where r is the correlation coefficient and sy, sx are the standard deviations of y and x. Slope represents the predicted change in y for a 1-unit increase in x. The intercept a = ȳ − b(x̄), found by solving for a using the point (x̄, ȳ). Software/calculator output usually reports both directly.

Interpreting Slope and Intercept in Context

Always interpret slope in context: 'for each additional [unit of x], the predicted [y] changes by [b] [units].' The intercept is the predicted y-value when x = 0 — but this interpretation is only meaningful if x = 0 is realistic and within the range of the data; otherwise, avoid interpreting it literally.

Key terms

Least squares regression line (LSRL)

The line ŷ = a + bx that minimizes the sum of squared residuals.

Slope formula

b = r(sy/sx); predicted change in y for a 1-unit increase in x.

Intercept formula

a = ȳ − b(x̄); the predicted y-value when x = 0.

Point (x̄, ȳ)

The point that the least squares regression line always passes through.

Interpreting the intercept

Only meaningful if x = 0 is realistic and within the range of the data; otherwise avoid literal interpretation.

Try an AP-style question

r = 0.6, s_y = 10, and s_x = 5. What is the slope of the least-squares line?

1. 12
2. 1.2
3. 3
4. 0.3