The coefficient of correlation between X and Y is 0.6 U and V are two variables defined as `U=(x-3)/(2), V=(y-2)/(3)`, then the coefficient of correlation between U and V is

Options

A. 0.1
B. 0.3
C. 0.2
D. 1

Answer

To find the coefficient of correlation between the variables U and V, we can use the following steps:

### Step 1: Understand the relationship between X, Y, U, and V
We are given that the coefficient of correlation between X and Y is \( r_{XY} = 0.6 \). The variables U and V are defined as:
\[ U = \frac{X - 3}{2} \]
\[ V = \frac{Y - 2}{3} \]

### Step 2: Identify the transformation
The transformations for U and V can be expressed in terms of X and Y:
- U is a linear transformation of X (specifically, it is scaled and shifted).
- V is a linear transformation of Y (also scaled and shifted).

### Step 3: Use the property of correlation coefficients
The correlation coefficient remains unchanged under linear transformations. Specifically, if we have:
\[ U = aX + b \quad \text{and} \quad V = cY + d \]
then the correlation coefficient between U and V is given by:
\[ r_{UV} = \frac{r_{XY}}{\sqrt{(a^2)(c^2)}} \]
where \( a \) and \( c \) are the scaling factors of the transformations.

### Step 4: Identify the scaling factors
From the definitions of U and V:
- For U: \( a = \frac{1}{2} \)
- For V: \( c = \frac{1}{3} \)

### Step 5: Calculate the coefficient of correlation between U and V
Using the property of correlation coefficients:
\[ r_{UV} = r_{XY} \cdot \frac{1}{\sqrt{(a^2)(c^2)}} \]
Substituting the values:
\[ r_{UV} = 0.6 \cdot \frac{1}{\sqrt{\left(\frac{1}{2}\right)^2 \cdot \left(\frac{1}{3}\right)^2}} \]
Calculating the denominator:
\[ \sqrt{\left(\frac{1}{2}\right)^2 \cdot \left(\frac{1}{3}\right)^2} = \sqrt{\frac{1}{4} \cdot \frac{1}{9}} = \sqrt{\frac{1}{36}} = \frac{1}{6} \]
Thus:
\[ r_{UV} = 0.6 \cdot \frac{1}{\frac{1}{6}} = 0.6 \cdot 6 = 3.6 \]

### Final Answer
The coefficient of correlation between U and V is \( 3.6 \).