Quick questions on Bivariate data: scatter plots, Pearson correlation and least-squares regression for HSC Maths Advanced

A scatter plot displays paired data $(x_i, y_i)$ as points in the plane, one dot per observation, and is the first thing you draw because your eyes catch structure that a single number hides. By convention the independent (explanatory) variable goes on the $x$-axis and the dependent (response) variable on the $y$-axis. Never join the dots: a scatter plot is a cloud of separate observations, not a quantity changing over time, and joining them turns it into a line graph.

The Pearson correlation coefficient $r$ measures the strength and direction of the linear relationship between two variables. It is defined by

The least-squares regression line of $y$ on $x$ is the line $y = a + b x$ that minimises the sum of squared vertical residuals $\sum (y_i - (a + b x_i))^2$. The residuals are the vertical gaps from each point to the line; squaring them before adding penalises big misses heavily and refuses to let the line drift, so the result is the single straight line that fits the cloud best overall. The solution is