`u=(x-20)/(5) and v=(y-30)/4,` then `b_(xy) =`

Options

A. `4/5 b_(vu)`
B. `4/5 b_(uv)`
C. `5/4 b_(uv)`
D. `5/4 b_(vu)`

Answer

To solve the problem, we need to find the value of \( b_{xy} \) given the transformations of \( u \) and \( v \). Let's break down the steps systematically.

### Step-by-Step Solution:

1. **Understand the Transformations**:
We are given:
\[ u = \frac{x - 20}{5} \quad \text{and} \quad v = \frac{y - 30}{4} \]
From these equations, we can express \( x \) and \( y \) in terms of \( u \) and \( v \):
\[ x = 20 + 5u \quad \text{and} \quad y = 30 + 4v \]

2. **Identify the Relationship**:
The relationship between the regression coefficients \( b_{xy} \) and \( b_{uv} \) can be established using the correlation coefficient \( r \):
\[ b_{xy} = r_{xy} \frac{\sigma_x}{\sigma_y} \]
and
\[ b_{uv} = r_{uv} \frac{\sigma_u}{\sigma_v} \]

3. **Calculate Standard Deviations**:
The standard deviations of \( x \) and \( y \) in terms of \( u \) and \( v \) are:
\[ \sigma_x = 5 \sigma_u \quad \text{and} \quad \sigma_y = 4 \sigma_v \]

4. **Substituting into the Regression Coefficient Formula**:
We can express \( b_{xy} \) in terms of \( b_{uv} \):
\[ b_{xy} = r_{xy} \frac{\sigma_x}{\sigma_y} = r_{xy} \frac{5 \sigma_u}{4 \sigma_v} \]

5. **Relate \( r_{xy} \) and \( r_{uv} \)**:
Since the correlation coefficients are invariant under linear transformations, we have:
\[ r_{xy} = r_{uv} \]

6. **Final Expression**:
Substituting \( r_{uv} \) into the equation for \( b_{xy} \):
\[ b_{xy} = r_{uv} \frac{5 \sigma_u}{4 \sigma_v} \]
This can be rearranged to:
\[ b_{xy} = \frac{5}{4} b_{uv} \]

### Final Answer:
Thus, the value of \( b_{xy} \) is:
\[ b_{xy} = \frac{5}{4} b_{uv} \]