**Property 1**:

Proof: The first and last equations are trivial. We now show the second. By the first equation:

Taking the sum of both sides of the equation over all values of *i* and then squaring both sides, the desired result follows provided

This follows by substituting ŷ* _{i}* by

*bx*and simplifying.

_{i}+ a_{i}**Property 2**:

Proof: We give the proof for the case where both *x* and y have mean 0 and standard deviation 1. The general case follows via some tedious algebra.

In the case where *x *and y have mean 0 and standard deviation 1, by Theorem 1 of Method of Least Squares and Property 1 of Method of Least Squares, we know that for all* i*

it follows that

and so *n* – 1 = *SS _{T}*, which means that

**Property 3**:

Proof: From Property 2, solving for *r ^{2}*, we have the following by Property 1:

**Property 4**:

Proof: The first assertion is trivial. The second is a consequence of Property 1and 2 since

**Property 5**:

a) The sums of the y values is equal to the sum of the ŷ values; i.e. =

b) The mean of the y values and ŷ values are equal; i.e. ȳ = the mean of the ŷ_{i}

c) The sums of the error terms is 0; i.e. = 0

d) The correlation coefficient of *x* with ŷ is sign(b); i.e. *r*_{xŷ} = sign(*r*_{xy})

e) The correlation coefficient of y with ŷ is the absolute value of the correlation coefficient of *x* with y; i.e. = ||

f) The coefficient of determination of y with ŷ is the same as the correlation coefficient of *x* with y; i.e. =

Proof:

a) By Theorem 1 of Method of Least Squares

b) That the mean of the ŷ* _{i}* is ȳ follows from (a) since the mean of ŷ = ŷ

*/*

_{i}*n*= y

_{i}/

*n*= ȳ.

c) This property follows from (a) since

d) First note that by Property 3 of Expectation

Now by Property A of Correlation

Thus, *r* = 1 if *b* > 0 and* r* = -1 if *b* < 0. If *b* = 0, *r* is undefined since there is division by 0. By Property 1 of Method of Least Squares, *r* = *sign*(*b*) = *sign*(*r*_{xy}).

e) Using property (b), the correlation of y with ŷ is

By Theorem 1 of Method of Least Squares

As we saw in the proof of (d),

and so putting all the pieces together, we get

By Property 1 of Method of Least Squares, *r* = *sign*(*b*) = *sign*(*r*_{xy}), and so

f) This follows from (e)