1

A linear regression gives us a correlation coefficient $r=0$.

  • What is the equation of the best fit line?

  • Give an example of data with $r=0$

  • What is the value of the correlation coefficient of data on a line parallel to the x-axis or y-axis?

I have no idea what the answers to any of these questions are. How can we answer this with just one piece of data?

FlacchusMaximus
  • 53
  • 3
  • If you are doing a linear regression, you have at least two (usually many more) data points $(x_1,y_1)$ and $(x_2,y_2)$, and you are trying to find a straight line $y= rx + b$ of slope $r$ such that $$\sum_{i=1}^n (y_i-rx_i-b)^2$$ ($n=2$ in the simplest case) is as small as possible. Here, "find" means you get to choose $r$ and $b$ so as to minimize the sum of squares above. You are now told that $r=0$. What does that tell you about the line? – Dilip Sarwate Mar 31 '13 at 14:11
  • @DilipSarwate It is a horizontal line of the form $y=a$? If true, I would only still not know what the value of r is on a line parallel to the y-axis – FlacchusMaximus Mar 31 '13 at 14:12
  • Well, $y=b$ if you follow the notation I used, but yes, the line is horizontal, and the average value of the $y_i$'s is $b$ (or $a$ or whatever you choose to call the height of the line above the $x$ axis). – Dilip Sarwate Mar 31 '13 at 14:16
  • @DilipSarwate Wait, what do you mean by the average value of the $y_i$'s is $b$? Isn't every value of $y_i = b$? (This of course would also mean the average value is $b$, but isn't that obvious). Also, what is the answer on the third question? If I try on my calculator with a horizontal line it only says $r = $ and a vertical line it says domain error – FlacchusMaximus Mar 31 '13 at 15:42
  • Try working out the values of $r$ and $b$ for $3$ points $(-1,-1), (1,0), (1,-1)$ so that you are minimizing $$\begin{align}(-1+r-b)^2+(1-b)^2+(-1-r-b)^2&=(r-(1+b))^2+(b-1)^2+(r+(1+b))^2\&=2r^2+2(b+1)^2+(b-1)^2\&=2r^2+3b^2+2b+3.\end{align}$$ Do you get $r=0, b=-1/3=(y_1+y_2+y_3)/3=(-1+1-1)/3$? – Dilip Sarwate Mar 31 '13 at 19:17

0 Answers0