Simple Linear Regression and Inference on the Slope

The simple linear model writes each response as an intercept plus a slope times the predictor plus random error. The error term collects everything the line cannot explain, and the model assumes those errors centre on zero with steady spread and no pattern. Least squares picks the line that minimises the sum of squared residuals. The slope predicts how much the response changes for a one unit step in x.

To test whether the slope differs from zero you divide the slope estimate by its standard error and compare the ratio to a t distribution. A tiny p value means the line explains more than chance would, and the matching interval bounds the slope. A large p value never proves the slope is zero; it only means the data do not rule zero out.