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

Options

A. `-0.5`
B. `+0.5`
C. `-0.6`
D. `+0.6`

Answer

To find the correlation coefficient between the transformed variables \( U \) and \( V \), we will follow these steps:

### Step 1: Understand the transformations
The transformations given are:
- \( U = 2 - 3X \)
- \( V = 4Y + 1 \)

### Step 2: Recall the properties of correlation
The correlation coefficient is unaffected by linear transformations of the variables. Specifically, if you transform a variable \( X \) to \( aX + b \) (where \( a \) and \( b \) are constants), the correlation coefficient remains the same.

### Step 3: Apply the properties to the transformations
In our case:
- For \( U = 2 - 3X \), the transformation involves multiplying \( X \) by \(-3\) (which is a linear transformation). The constant \( 2 \) does not affect the correlation.
- For \( V = 4Y + 1 \), the transformation involves multiplying \( Y \) by \( 4\) (which is also a linear transformation). The constant \( 1 \) does not affect the correlation.

### Step 4: Calculate the new correlation coefficient
Given that the original correlation coefficient between \( X \) and \( Y \) is \( 0.6 \), we can apply the property of correlation:
- The correlation coefficient between \( U \) and \( V \) will be:
\[ \rho_{U,V} = \rho_{X,Y} \cdot (-3) \cdot 4 \]
However, since correlation is a measure of the strength and direction of a linear relationship, the sign of the correlation will change due to the negative multiplication by \(-3\).

Thus, the correlation coefficient between \( U \) and \( V \) will be:
\[ \rho_{U,V} = -0.6 \]

### Final Answer:
The correlation coefficient between the transformed variables \( U \) and \( V \) is \( -0.6 \).

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