Hello everyone!

Does anybody know how to prove the fact that the centroid point (x_bar, y_bar) lies on the regression line?

Thank you!

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- May 30th 2009, 08:34 PMStillRussianCentroid point proof
Hello everyone!

Does anybody know how to prove the fact that the centroid point (x_bar, y_bar) lies on the regression line?

Thank you! - May 30th 2009, 09:48 PMmatheagle
Just plug in $\displaystyle \bar x $ in for the x and see if you get $\displaystyle \bar y$.

You saw my other proof earlier today, use that post. - May 31st 2009, 10:10 AMStillRussian
Right, so I have

β0_hat + β1x_bar = y_bar – β1_hat*x_bar + β1_hat*x_bar =y_bar

It just seemed too good to be true (Smile) - May 31st 2009, 08:38 PMmatheagle
It looks right.

But again I do need to know the model.

However I think this is true for any model.

$\displaystyle \hat y=\hat\beta_0+\hat\beta_1 x$

is the fitted line, with $\displaystyle \hat\beta_0=\bar y-\hat\beta_1 \bar x$

giving us

$\displaystyle \hat y =\bar y-\hat\beta_1 \bar x +\hat\beta_1 x =\bar y+\hat\beta_1(x- \bar x)$.

NOW setting $\displaystyle x=\bar x$ in the line $\displaystyle \hat y =\bar y+\hat\beta_1(x- \bar x)$ completes this. - Jun 1st 2009, 04:50 AMStillRussian
Matheagle,

thank you very very much for all your help!