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 $\hat y = b_0 + b_1 \cdot x$. The intercept coefficient is $b_0$, so $b_0$ is the value of $\hat y$ when 𝑥 = 0. The slope coefficient for $𝑥$ is $b_1$, i.e., the increase in $\hat y$ for every increase of one in $𝑥$. Why do we put a hat on top of the $y$? It’s a form of notation commonly used in regression to indicate that $\hat y$ is a fitted value, representing the value of $𝑦$ on the regression line for a given $𝑥$ value.

Recall that the regression line has a positive slope $b_1$ corresponding to our explanatory $𝑥$ variable bty_avg. This is because instructors tend to have higher bty_avg scores, they also tend to have higher teaching evaluation scores. However, what is the numerical value of the slope $b_1$? What about the intercept $b_0$?

We can obtain the values of the intercept $b_0$ and the slope for bty_avg $b_1$ by outputting a linear regression table. This is done in two steps:

1. We first fit the linear regression model using the lm() function and save it in score_model.

2. We get the regression table by applying the get_regression_table() function from the moderndive package to score_model