Let the correlation coefficient between X and Y be 0.6, random variable Z and W are defined as Z=X+5 and `W=(y)/(3)` What is the correlation coefficient between Z and W

Options

A. 0.1
B. 0.2
C. 0.36
D. 0.6

Answer

To find the correlation coefficient between the random variables Z and W, we can follow these steps:

### Step 1: Understand the relationship between the variables
We are given that the correlation coefficient between X and Y is 0.6. The random variables Z and W are defined as:
- \( Z = X + 5 \)
- \( W = \frac{Y}{3} \)

### Step 2: Analyze the effect of transformations on correlation
The correlation coefficient measures the strength and direction of a linear relationship between two variables. Importantly, adding a constant to a variable (like adding 5 to X to get Z) does not change the correlation coefficient. Similarly, multiplying a variable by a constant (like dividing Y by 3 to get W) also does not change the correlation coefficient.

### Step 3: Apply the transformations
Since:
- \( Z = X + 5 \) (adding a constant)
- \( W = \frac{Y}{3} \) (multiplying by a constant)

The correlation coefficient between Z and W will remain the same as the correlation coefficient between X and Y.

### Step 4: Conclusion
Thus, the correlation coefficient between Z and W is the same as the correlation coefficient between X and Y, which is 0.6.

### Final Answer
The correlation coefficient between Z and W is **0.6**.

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