Regression and the sum of residuals

Question

The answer to the last part provided is The sum of square of residuals is minimum for points lying on the regression line and so cannot be less than 8.8 for any other line.

Can somebody please explain what this means?

It is almost evident that these points doesn't lie on the given regression line. So if I were to provide a much more accurate regression line, won't the square of sum much more smaller?

Answer 1

The sum of square of residuals is minimum for points lying on the regression line and so cannot be less than $8.8$ for any other line.

This is misleadingly stated. It says "for points lying on the regression line". What it ought to say is that for the line whose slope was specified slope and intercept, the sum of squares of residuals is smaller than it is for any other slope or other intercept.

Note that only seven of the eight pairs are given. You are asked to find the eighth pair.

Answer 2

One way to look at the result line we get from a linear regression is that this is the line we get by minimizing the sum of squared residuals of the points (to visualize, it is the sum of squared vertical distance of points to the regression line).

Thus any line other than the regression line will not have a smaller sum of squared residues.

EDIT

I feel you might misunderstand what is a regression line - so a regression line is not a line that you give arbitrary $a$, $b$ parameters to it. Instead, it is a line that you calculate you parameter so that the sum of squared residual is the smallest out of all lines. Hope this would help.