theory proofs

#1
Hi, I am doing a study guide for my stat test coming up, and got stuck on 3 proofs. They seem a bit confusing to me, and I just keep going in circles and not reaching the end product. If you can help or show me how this is supposed to go, in case it ends up on the test so I can learn it in time, that would be great.

1. Prove that
n
SSR = b1^2∑(xi-xbar)^2
i=1

2. Prove that

F=MSR/MSE=(b1/se(b1))^2=t^2

3. Prove that

R^2 = (corr(y,x))^2

sorry for the lack of skills to write in proper math code so it looks nice, but I haven't caught on to that yet. Thank you in advance for any and all help with any of these three proofs!!
 

Dragan

Super Moderator
#2
Hi, I am doing a study guide for my stat test coming up, and got stuck on 3 proofs. They seem a bit confusing to me, and I just keep going in circles and not reaching the end product. If you can help or show me how this is supposed to go, in case it ends up on the test so I can learn it in time, that would be great.

1. Prove that
n
SSR = b1^2∑(xi-xbar)^2
i=1

2. Prove that

F=MSR/MSE=(b1/se(b1))^2=t^2

3. Prove that

R^2 = (corr(y,x))^2

sorry for the lack of skills to write in proper math code so it looks nice, but I haven't caught on to that yet. Thank you in advance for any and all help with any of these three proofs!!

I will help you with the first one - that should get you started.


\(SS_{Reg}=r^{2}SS_{Y}\)

and so it follows that:

\( SS_{Reg}=r^{2}\frac{Var_{Y}}{Var_{X}}Var_{X}\left ( N-1 \right ) \).

Now, it can be shown that:

\( b_{1}^{2}=r^{2}\frac{Var_{Y}}{Var_{X}} \)

and thus you have your desired result:

\( SS_{Reg}=b_{1}^{2}SS_{X} \).