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Math Help - Centroid point proof

  1. #1
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    Centroid 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!
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  2. #2
    MHF Contributor matheagle's Avatar
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    Just plug in \bar x in for the x and see if you get \bar y.
    You saw my other proof earlier today, use that post.
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  3. #3
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    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
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  4. #4
    MHF Contributor matheagle's Avatar
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    It looks right.
    But again I do need to know the model.
    However I think this is true for any model.

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

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

    giving us

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

    NOW setting x=\bar x in the line \hat y =\bar y+\hat\beta_1(x- \bar x) completes this.
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  5. #5
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    Matheagle,

    thank you very very much for all your help!
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