# Statistics: given total sum of squares, find R²

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1. Oct 4, 2014

### 939

1. The problem statement, all variables and given/known data

Given:
Σ(xi - x̄)² = 500
Σ(yi - ybar)² = 800 (total sum of squares, SST))
Σ(ŷ - ybar)² = 400 (total sum of estimators, SSE)
Σ(xi - x̄)²(yi) = 200
Σ(xi - x̄)²(εi) = 0
n = 1000
s² = 4

Find (or explain why you cannot find):
β1
β0
variance of β

2. Relevant equations

Σ(xi - x̄)² = 500
Σ(yi - ybar)² = 800 (total sum of squares, SST))
Σ(ŷ - ybar)² = 400 (total sum of estimators, SSE)
Σ(xi - x̄)²(yi) = 200
Σ(xi - x̄)²(εi) = 0
n = 1000
s² = 4

3. The attempt at a solution

R² = SSE/SST = 400/800 = 200

But to be honest, I have no idea how to find β1, β0, or the variance of β... Can anyone help?

2. Oct 4, 2014

### RUber

Normally, in a regression equation like this, $\beta_0 = \mu$ which is the overall sample mean. I don't see any immediately discernible information for finding those parameters, but it maybe in there with some algebra.
Your $R^2$ equation looks right, but that is not equal to 200. $R^2$ is always between 0 and 1.

3. Oct 4, 2014

### SteamKing

Staff Emeritus
Since when is 400 / 800 = 200? Is this the New Math everyone keeps talking about?

4. Oct 4, 2014

### 939

lol yea stupid error, 0.5, sorry :(

5. Oct 4, 2014

### WWGD

Most statistical packages will spit out all the estimators if you input relatively little data.

This Wiki page has explicit formulas for $\beta$ and $\beta_0$:

http://en.wikipedia.org/wiki/Simple_linear_regression

Last edited: Oct 4, 2014