How can I use the expression for a in this problem

[Faiq]

Homework Statement

A random sample of size $n$ from a bivariate distribution is denoted by $(x_r,y_r), r=1,2,3,...,n$. Show that if the regression line of $y$ on $x$ passes through the origin of its scatter diagram then[/B]
$$\bar y \sum^n_{r=1} x_r^2=\bar x\sum^n_{r=1} x_r y_r$$ where $(\bar x,\bar y)$ is the mean point of the sample.

I don't really know how to begin. I am aware the line equation is $$b=\frac{y}{x}=\frac{\sum xy-\frac{\sum x\sum y}{n}}{\sum x^2-\frac{(\sum x)^2}{n}}$$

Not sure what to do next.

 

[Ray Vickson]

Homework Statement

A random sample of size $n$ from a bivariate distribution is denoted by $(x_r,y_r), r=1,2,3,...,n$. Show that if the regression line of $y$ on $x$ passes through the origin of its scatter diagram then[/B]
$$\bar y\sum^n_{r=1} x_r^2=\bar x\sum^n_{r=1} x_r y_r$$ where $(\bar x,\ bar y)$ is the mean point of the sample.

I don't really know how to begin. I am aware the line equation is $$b=\frac{y}{x}=\frac{\sum xy-\frac{\sum x\sum y}{n}}{\sum x^2-\frac{(\sum x)^2}{n}}$$

Not sure what to do next.

(1) Do not write $\xbar$, write $\bar{x}$. Right-click on the formula to see its TeX commands.
Mod note: Fixed the TeX in the original post and above.
(2) What can you say about the data if the least-squares line has intercept $a$ equal to zero?

 

[Faiq]

Y is proportional to X

 

[Ray Vickson]

Y is proportional to X

That answer is not useful. Take the formula for $a$, in terms of $(x_i,y_i)$, and set it to zero. What do you get?

 

[SammyS]

Homework Statement

[/B]
A random sample of size $n$ from a bi-variate distribution is denoted by $(x_r,y_r), r=1,2,3,...,n$. Show that if the regression line of $y$ on $x$ passes through the origin of its scatter diagram then
$$\bar y \sum^n_{r=1} x_r^2=\bar x\sum^n_{r=1} x_r y_r$$ where $(\bar x,\bar y)$ is the mean point of the sample.

The Attempt at a Solution

I don't really know how to begin. I am aware the line equation is $$b=\frac{y}{x}=\frac{\sum xy-\frac{\sum x\sum y}{n}}{\sum x^2-\frac{(\sum x)^2}{n}}$$ Not sure what to do next.

I suppose the that the linear model you are working with is: $\ \displaystyle y=a+bx\$.

You have the correct expression for finding the linear coefficient, $\ b\,.\$( Leave out the $\ \displaystyle \frac yx\$ ).

It seems to me that you must also consider the expression for $\ a\,.\$ Then show that if $\ a=0\,,\$ then you obtain the desired result:

$\displaystyle \bar y \sum^n_{r=1} x_r^2=\bar x\sum^n_{r=1} x_r y_r$​

.