6.1.15 (HL)—Simple linear regression

Simple linear regression estimates the relationship between one explanatory variable and an outcome using a fitted line.

The slope describes association in the data, not proof that X causes Y; extrapolation is risky.

Inspect scatter, fit and residuals, then use the line only within a defensible range.

Past advertising spend may predict sales, but a new market shock can make the old relationship fail.

A high correlation does not establish causation.

Begin with a scatter diagram: upward patterns indicate positive correlation, downward patterns negative correlation, and diffuse points weak or no linear correlation. A line of best fit summarizes the estimated linear relationship and can be written as an outcome predicted from one explanatory variable; inspect unusual points and how closely observations follow the line. Interpolation within the observed range is usually safer than extrapolation beyond it because the relationship may change. Correlation describes association, not causation, so a fitted advertising-sales line cannot by itself prove that advertising caused the sales change or that the relationship will continue in a new market.