Question

For a correlation of - .64 between X and Y, each 1- SD change in zx corresponds to a predicted change of __________ SD in zY. Why?

Answer 1

To solve this let's made some assumptions.

Let's assume that linear equation is:

Y = a + b*X

Mean of X = Mx

Mean of Y = My

Zx = z score of X at X = x

Zy = z score of Y at Y = y

SDx = Standard deviation of X

Correlation coefficient is given as, r = -0.64

Now Z score for X at X = x is given as:

Zx = (x - Mx) / SDx

By standradized regression formula we can say that

Zy = r * Zx

So, Zy = -0.64 * Zx ---------------(1)

Now we are given each 1-SD change in Zx. So, new Zx:

Zxnew = Zx + (1 - SDx)  

So, Zynew = r * Zxnew

Zynew = -0.64 * [Zx + (1 -SDx)}

Zynew = -0.64*Zx - 0.64 + 0.64 * SDx

Using equation (1), replacing -0.64 * Zx with Zy,

Zynew = Zy - 0.64 * (1 - SDx)

So from above equation it's clear that for each 1-SDx change in Zx, there is a change of

- 0.64 * (1 - SDx) in Zy

Note: I considered change to be positive i.e by adding the value in Zx. If you subtract (1 - SDx) from Zx then change in Zy will be positive i.e 0.64 * (1 - SDx)