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

Options

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

Explanation

To find the correlation coefficient between height (x) and weight (y) given the two regression lines, we will follow these steps:

### Step 1: Rewrite the regression equations in slope-intercept form.

The two regression equations given are:
1. \( 4y - 15x + 410 = 0 \)
2. \( 30x - 2y - 825 = 0 \)

We will rearrange both equations to express \( y \) in terms of \( x \).

**For the first equation:**
\[ 4y = 15x - 410 \]
Dividing both sides by 4:
\[ y = \frac{15}{4}x - \frac{410}{4} \]
Here, the slope \( b_{xy} = \frac{15}{4} \).

**For the second equation:**
\[ 30x - 2y = 825 \]
Rearranging gives:
\[ 2y = 30x - 825 \]
Dividing both sides by 2:
\[ y = 15x - \frac{825}{2} \]
To express \( x \) in terms of \( y \), we can rearrange this equation:
\[ 30x = 2y + 825 \implies x = \frac{2}{30}y + \frac{825}{30} = \frac{1}{15}y + 27.5 \]
Here, the slope \( b_{yx} = \frac{1}{15} \).

### Step 2: Calculate the correlation coefficient \( R \).

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

### Step 3: Determine the sign of the correlation coefficient.

Since both regression coefficients \( b_{xy} \) and \( b_{yx} \) 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 \( R = \frac{1}{2} \).

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