If two regression lines between height (x) and weight (s) are 4y - 15 x +410=0 and 30x-2y- 825=0. then what will be the correlation coefficient between height and weight?

Options

A. `1/2`
B. `1/2`
C. `2/3`
D. `3/4`

Answer

To find the correlation coefficient between height (x) and weight (s) using the given regression equations, we will follow these steps:

### Step 1: Identify the regression equations
The two regression lines provided are:
1. \( 4y - 15x + 410 = 0 \)
2. \( 30x - 2y - 825 = 0 \)

### Step 2: Rearranging the first equation
We will rearrange the first equation to express \( y \) in terms of \( x \):
\[ 4y = 15x - 410 \]
\[ y = \frac{15}{4}x - \frac{410}{4} \]
From this equation, we can identify the slope \( b_{yx} \) (the regression coefficient of \( y \) on \( x \)):
\[ b_{yx} = \frac{15}{4} \]

### Step 3: Rearranging the second equation
Now, we will rearrange the second equation to express \( x \) in terms of \( y \):
\[ 30x = 2y + 825 \]
\[ x = \frac{2}{30}y + \frac{825}{30} \]
\[ x = \frac{1}{15}y + 27.5 \]
From this equation, we can identify the slope \( b_{xy} \) (the regression coefficient of \( x \) on \( y \)):
\[ b_{xy} = \frac{1}{15} \]

### Step 4: Calculate the correlation coefficient
The correlation coefficient \( r \) can be calculated using the formula:
\[ r = \sqrt{b_{yx} \cdot b_{xy}} \]
Substituting the values we found:
\[ r = \sqrt{\left(\frac{15}{4}\right) \cdot \left(\frac{1}{15}\right)} \]
\[ r = \sqrt{\frac{15}{4} \cdot \frac{1}{15}} = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

### Step 5: Determine the sign of the correlation coefficient
Since both regression coefficients \( b_{yx} \) and \( b_{xy} \) are positive, the correlation coefficient \( r \) will also be positive. Therefore, we take:
\[ r = \frac{1}{2} \]

### Final Answer
The correlation coefficient between height and weight is:
\[ \boxed{\frac{1}{2}} \]