Simple Linear Regression for a Numerical Explanatory Variable

Recall the concepts of algebra that the equation of a line is $𝑦 = 𝑎 + 𝑏 ⋅ 𝑥$. (Note that the ⋅ symbol is equivalent to the * “multiply by” mathematical symbol. We’ll use the ⋅ symbol in the rest of this course as it’s more succinct.) It’s defined by two coefficients $𝑎$ and $𝑏$. The intercept coefficient $𝑎$ is the value of $𝑦$ when $x$ = 0. The slope coefficient $𝑏$ for $𝑥$ is the increase in $𝑦$ for every increase of one in $𝑥$. This is also called the rise over run.

However, when defining a regression line, we use a slightly different notation, i.e., the equation of the regression line is $\hat y = b_0 + b_1 \cdot x$ ...