Question

Suppose that ? is a set of bivariate data ? = {(?, ?) ∶ ? ∈ ?, ? ∈ ?} such that the standard deviations ?? and ?? are equal to one another (and non-zero). If ? corresponds to the slope for the line of best fit for ?, prove that it is impossible that |?| > 1.

Answer 1

The formula of the slope of fitted line Y on X is as follow:

Where r is the correlation coefficient between x and y

So for the given condition s_x = s_y, we get

Since r lies between -1 and 1, therefore | r | <= 1

this implies that

The formula of the slope of fitted line X on Y is as follow:

So for the given condition s_x = s_y, we get

this implies that

So for any fitted line on S the slope | m | <= 1 .