# Questions on linear regression

I had a few true / false questions on a practice test that I would like to discuss if possible.

• A value $R^2$ close to 1 indicates the linear regression is a good fit to data

Yes, but I am not sure how to reason it other than intuitively. More precisely, why do we decide to define $R^2$ as 1-(sum of residuals/total sum of squares)?

• The estimate of the error variance $s^2$ is a random variable

I believe it is because how the error varies is normally distributed if I remember correctly. If anyone could elaborate it would be much appreciated.

• $∑(Y_{1,i}−Y_{2,i})^2 = 0$, where $Y_1$ is the predicted value and $Y_2$ is the actual value.

Since the sum of residuals is zero and since this is equivalent to the sum of residuals it must be zero.

• Welcome to the site, @user26091. I took the liberty of editing your question with $\LaTeX$. Please make sure it still says what you want it to. – gung - Reinstate Monica May 25 '13 at 1:11
• Thanks! It does not quite say what I want it to however but close. – user26091 May 25 '13 at 1:36
• I thought there was something wrong with c, but I couldn't figure out what it was supposed to be. – gung - Reinstate Monica May 25 '13 at 1:39
• Since it's for the purpose of study, please add the self-study tag. I took a stab at improving "c" - if that's wrong, please clarify. – Glen_b -Reinstate Monica May 25 '13 at 1:45
• @PeterFlom, $\LaTeX$ is "\$\LaTeX\$". – gung - Reinstate Monica May 25 '13 at 13:41

a) $R^2$ is the proportion of explained variation on the whole variation. The formula is just a different representation of $R^2$.