Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$.

Question

Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$. Then, which of the following is the best answer?

1. $r_1 = r_2$.
2. $r_1 = 10r_2$
3. $r_1 = -10r_2$
4. $r_1 = -r_2$.
5. None of the above is correct.

I think Choice No.3 is the best answer because of the coef. of $Y$ is also $-10$. Isn't it?

Answer 1

$\newcommand{\Var}{\operatorname{Var}}\newcommand{\Corr}{\operatorname{Corr}}\newcommand{\Cov}{\operatorname{Cov}}$ Using the known properties of covariance gives \begin{align*} \Corr(X,Z) &=\frac{\Cov(X,Z)}{\sqrt{\Var(X)\Var(Z)}}\\ &=\frac{\Cov(-10Y+10,Z)}{\sqrt{\Var(-10Y+10)\Var(z)}}\\ &=\frac{-10\Cov(Y,Z)}{\sqrt{100\Var(Y)\Var(Z)}}\\ &=\frac{-10\Cov(Y,Z)}{10\sqrt{\Var(Y)\Var(Z)}}\\ &=-\Corr(Y,Z). \end{align*} Thus $$r_1 = -r_2,$$ which is option 4.

Answer 2

If in doubt, compute. We have $$\text{Cov}(X,Z)=\text{Cov}(-10Y+10,Z)=-10\text{Cov}(Y,Z).$$ Also, $\text{Var}(X)=100\text{Var}(Y)$. It follows that $$\rho(X,Z)=\frac{\text{Cov}(X,Z)}{\sigma_X\sigma_Z}=\frac{-10\text{Cov}(Y,Z)}{10\sigma_Y\sigma_Z},$$ and therefore $r_1=-r_2$.

Informally, scaling by some positive factor does not change the degree of linear relationship. Neither does a shift. Correlation does not depend on the scale of measurement.