Least squares regression
Everyone draws a slightly different line through the same scatter plot. Which rule picks one line that everybody agrees on, and what can that line be trusted to say?
Where you are 5 inside
What this area is about
A line drawn by eye through a scatter plot is a guess, and no two guesses match. Least-squares regression replaces the guess with a rule: choose the line \hat{y} = a + bx that makes the sum of the squared vertical misses as small as possible. As long as the x-values are not all equal, that rule gives exactly one line, which can then be read, used for predictions, and judged.
How these fit together
The definition. The least squares criterion states the rule and shows why it picks a single line.
A result. Slope from r and the spreads writes that line from five summary numbers, slope b = r\,\dfrac{s_y}{s_x}, and explains regression to the mean.
Procedures. Fitting models with technology gets the line (or an exponential curve) from a calculator or spreadsheet. Interpreting the slope and intercept says what the numbers mean in context, and predicting with a regression line uses it for cases nobody measured, with a warning about extrapolation.
A path through it
Take the criterion first, then the slope from r and the spreads. Fitting with technology, interpreting the slope and predicting can follow in that order.
Next, residuals show what the line missed, influential points show what can drag it, and the sampling distribution of the slope leads to inference.