The correlation coefficient between two variables X and Y is found to be 0.6. All the observations on X and Y are transformed using the transformations U = 2 – 3X and V = 4Y + 1. The correlation coefficient between the transformed variables U and V will be

Concept:

• Corr(a + bX, c +dY) = Corr(bX, dY), Where, var (X) = b & var (Y) = d
• Corr(aX, bY) = (ab)Corr (X, Y) where a, b, c, and d are constants
• \(\begin{aligned} \operatorname{Corr}(X, Y) &=\frac{\operatorname{Corr} (X,Y)}{\sqrt{\operatorname{Var}(X) \operatorname{Var}(Y)}} \\ \end{aligned}\)

Calculation:

Given:

\(\begin{aligned} \operatorname{Corr}(X, Y) &=0.6 \\ \operatorname{Corr}(U, V) &=\frac{\operatorname{Corr} (U,V)}{\sqrt{\operatorname{Var}(U) \operatorname{Var}(V)}} \\ &=\frac{\operatorname{Corr}(2-3 X, 4Y+1)}{\sqrt{(-3)^{2}} \sqrt{4^{2}}} \\ &=\frac{\operatorname{Corr} (-3 X, 4Y)}{12} \\ \\ &=\frac{(-3\times 4)\operatorname{Corr} ( X, Y)}{12} \\\\ &=\frac{(-12)\operatorname{Corr} ( X, Y)}{12} \\&=\frac{-12}{12}\times{0.6} \\ &=-0.6 \end{aligned}\)

Hence, option (3) is correct.