The intercept

There are numerous combulated equations to calculate the intercept of the linear regression model. Here I want us to use a procedure that you probably learned in 6th grade.

If you look at the linear regression model equation,

\[\begin{equation} Y = mX + b \end{equation}\]

You could isolate the intercept, \(b\), as:

\[\begin{equation} b = Y - mX \end{equation}\]

If you recall, the line of the least-squares regression line has to pass by the coordinates defined by the mean value of X and the mean value of X. So to estimate the intercept all you have to do is to replace the \(X\) and \(Y\) parameters of the linear regression equation by the mean of X, \(\bar{x}\), and the mean of Y, \(\bar{y}\). Like this:

\[\begin{equation} b = \bar{y} - m\bar{x} \end{equation}\]

Because you also know the slope from the prior section. Then you have all parameters of the equation above to stimate the intercept.

Let’s estimate the intercept for the relationship between time studying and grades. From that data, the mean of X, \(\bar{x}\), was 2.6, and the mean of Y, \(\bar{y}\), was 74.2. So, the intercept for that linear regression is:

\[\begin{equation} Intercept = b =74.2 - 2.6*9.586 \end{equation}\]

\[\begin{equation} Intercept = b =49.277 \end{equation}\]

So the Y-Intercept of the least-square line for the relationship between grades and time studying is 49.28. You can interpret that value as the grade that you can expect in my class if you do not study a single hour of the week. Hmm, not good.