# standard error of estimate

• Jun 7th 2008, 10:48 PM
Judi
standard error of estimate
standard error of estimate=
square root of sum of squares of difference of y and y hat divided by n-2

how does this become--------->
square root of sum of y square minus a times sum of y minus b times sum of xy and divided by 2

The text book doesn't show progress and just stated they are equivalent.

Would you please show me how to get there?

Judi
• Jun 7th 2008, 11:09 PM
mr fantastic
Quote:

Originally Posted by Judi
standard error of estimate=
square root of sum of squares of difference of y and y hat divided by n-2

how does this become--------->
square root of sum of y square minus b times sum of y minus b times sum of xy and divided by 2

The text book doesn't show progress and just stated they are equivalent.

Would you please show me how to get there?

Judi

It would help the common masses if you said that this was in the context of linear regression.

I can't be bothered working it out ..... Is $\displaystyle \hat{y}_i = ax_i + b$ or is it the other way around? And you have a formula for a in terms of the data $\displaystyle (x_i, ~ y_i)$ ....?
• Jun 8th 2008, 12:36 PM
Judi
yes, it is the least squares line yhat=a+bx.
I don't know what you mean by 'a formula for a in terms of the data ....?'
I have just given those(from the first thread)that two formulas are equal to each other. I want to see how one can become the other.
thanks,
• Jun 8th 2008, 07:23 PM
mr fantastic
Quote:

Originally Posted by Judi
yes, it is the least squares line yhat=a+bx.
I don't know what you mean by 'a formula for a in terms of the data ....?'
I have just given those(from the first thread)that two formulas are equal to each other. I want to see how one can become the other.
thanks,

The algebra is quite boring. Consider:

$\displaystyle \sum_{i=1}^{n} (Y_i - \hat{Y}_i)^2$

$\displaystyle = \sum_{i=1}^{n} (Y_i - [a + bX_i])^2$

$\displaystyle = \sum_{i=1}^{n} Y_i^2 - 2Y_i (a + bX_i) + (a + bX_i)^2$

$\displaystyle = .......... zzzzzzzzzzzz$ uh? uh? ughhh .....

$\displaystyle = \sum_{i=1}^{n} Y_i^2 - 2a \sum_{i=1}^{n}Y_i - 2b \sum_{i=1}^{n}Y_i X_i + \sum_{i=1}^{n} a^2 + 2ab \sum_{i=1}^{n} X_i + b^2 \sum_{i=1}^{n} X_i^2$

$\displaystyle = \sum_{i=1}^{n} Y_i^2 - 2a \sum_{i=1}^{n}Y_i - 2b \sum_{i=1}^{n}Y_i X_i + n a^2 + 2ab \sum_{i=1}^{n} X_i + b^2 \sum_{i=1}^{n} X_i^2$

$\displaystyle = \sum_{i=1}^{n} Y_i^2 - 2a \sum_{i=1}^{n}Y_i - 2b \sum_{i=1}^{n}Y_i X_i + n a^2 + b \left(2a \sum_{i=1}^{n} X_i + b X_i^2 \right)$

$\displaystyle = .......... zzzzzzzzzzzz$ uh? uh? ughhhh .....

$\displaystyle = \sum_{i=1}^{n} Y_i^2 - 2a \sum_{i=1}^{n}Y_i - 2b \sum_{i=1}^{n}Y_i X_i + n a^2 + b \left(a \sum_{i=1}^{n} X_i + {\color{red}a \sum_{i=1}^{n} X_i + b X_i^2} \right)$

$\displaystyle = \sum_{i=1}^{n} Y_i^2 - 2a \sum_{i=1}^{n}Y_i - 2b \sum_{i=1}^{n}Y_i X_i + n a^2 + b \left(a \sum_{i=1}^{n} X_i + {\color{red}\sum_{i=1}^{n} X_i Y_i} \right)$

using the normal equations for linear regression

$\displaystyle = \sum_{i=1}^{n} Y_i^2 - 2a \sum_{i=1}^{n}Y_i - b \sum_{i=1}^{n}Y_i X_i + n a^2 + b a \sum_{i=1}^{n} X_i$

$\displaystyle = \sum_{i=1}^{n} Y_i^2 - 2a \sum_{i=1}^{n}Y_i - b \sum_{i=1}^{n}Y_i X_i + n a^2 + a {\color{red}b\sum_{i=1}^{n} X_i}$

$\displaystyle = .......... zzzzzzzzzzzz$ uh? uh? ughhh .....

$\displaystyle = \sum_{i=1}^{n} Y_i^2 - 2a \sum_{i=1}^{n}Y_i - b \sum_{i=1}^{n}Y_i X_i + n a^2 + a \left( {\color{red}\sum_{i=1}^{n} Y_i - na} \right)$

using the normal equations for linear regression again

$\displaystyle = \sum_{i=1}^{n} Y_i^2 - 2a \sum_{i=1}^{n}Y_i - b \sum_{i=1}^{n}Y_i X_i + n a^2 + a \sum_{i=1}^{n} Y_i - na^2$

$\displaystyle = \sum_{i=1}^{n} Y_i^2 - 2a \sum_{i=1}^{n}Y_i - b \sum_{i=1}^{n}Y_i X_i + a \sum_{i=1}^{n} Y_i$

$\displaystyle = .......... zzzzzzzzzzzz$ uh? uh? .....

$\displaystyle = \sum_{i=1}^{n} Y_i^2 - a \sum_{i=1}^{n}Y_i - b \sum_{i=1}^{n}Y_i X_i$.

zzzzzzzzzzzzzzzzzzzzzzzzzzzzz ......... (Dreaming now. Star Wars: .... "Help me Mr Fantastic, you're my only hope. Help me Mr Fantastic, you're my only hope, everyone else hates statistics ......)
• Jun 8th 2008, 08:10 PM
TheEmptySet
Quote:

Originally Posted by mr fantastic
zzzzzzzzzzzzzzzzzzzzzzzzzzzzz ......... (Dreaming now. Star Wars: .... "Help me Mr Fantastic, you're my only hope. Help me Mr Fantastic, you're my only hope, everyone else hates statistics ......)

This is too funny!!(Clapping)
• Jun 9th 2008, 07:28 PM
Judi
Wow~
That's a lot of work. Thank you, Mr fantastic, for spending that much time on this problem
thank you
Judi
• Jun 9th 2008, 07:36 PM
mr fantastic
Quote:

Originally Posted by Judi
Wow~
That's a lot of work. Thank you, Mr fantastic, for spending that much time on this problem
thank you
Judi

Just wait till you get my invoice ...... (Rofl)

(Always a pleasure helping someone with such lovely manners).