SLR - simple linear regression (in R, but about the math behind, not the programming)

Question

So I have some problems understanding simple linear regression. I did read a lot, so I have the basic ideas in mind, but I cannot quite follow when we do one. So I have this equation:

yi = a + bxi + ei

Okay so I do realize this is the equation for a straight line, even though I do wonder about the "ei" as I cannot find it on the internet, but my professor keeps using it.

So I want to find a and b, so I can find a straight line which I hope isn't to far away from my data (is that right?). I know I can calculate that, but this is not my question.

I hope it is alright if I add my example here, so I can explain what I'm doing: data set

x        y
8        6.4
8        6.8
3        1.7
2        2.3
2        3.8
1        2.3
1        5.0
1        4.0
1        3.4
0        2.3

calculation everything that's needed, I get: b = 0.4599 a = 2.55827

(and doing the lm with R shows me it is right). Now if I draw this straight line abline(2.55827,0.4599) (entering the intercept first??), it shows me that this is just not a good line and looking at the table I would totally agree. But do I understand right? If the x|y points arrange like they do through the given values (meaning without a specific pattern), there's just no good line to find, so I can only find a rather good one.

Can someone maybe help me out here?

Answer 1

Okay so I do realize this is the equation for a straight line, even though I do wonder about the "ei" as I cannot find it on the internet, but my professor keeps using it.

It's not the equation for a line. y_i = a + bx_i is the equation for a line. That e_i is the error between this straight line given by a and b and your measurements. In other words, e_i = y_i - (a + bx_i).

What linear regression does is to find the values for a and b that minimizes the sum of the squares of those error terms. This fit is not necessarily a good one; it's just the best possible (in a least squares sense). The size of the residual gives you an idea of how good the fit was.

To be able to make sense of whether the fit is good or bad, you need to know not just the residuals but also the errors in the individual measurements.