# Expected value of given x for simple linear regression?

I understand statisticians call the predicted y in a simple linear regression the “mean of y.” Let’s assume we only have 3 pairs of x and y values: (1,1), (2,3), (3,1). Whatever the regression line ends up being, for a value of x equal to 4 the regression line would output something representing the “mean of y given x = 4.” In what sense is this predicted y from the regression line at x = 4 the “mean of y?” It must be a guess of the mean of y at x = 4? It can’t be an “actual” mean calculated from our data. Do you see my confusion? I don’t understand why every y value on the regression line is the mean of y given whatever x is chosen.

Correct. It is an estimate of $$E(Y \vert X = x)$$.