Let `5x+3y=55` and `7x+y=45` be two lines of regression for a bivariate data
Coefficient of correlation between x and y

Options

A. `sqrt((5)/(21))`
B. `-sqrt((5)/(21))`
C. `sqrt((21)/(5))`
D. `-sqrt((21)/(5))`

Answer

To find the coefficient of correlation \( r \) between \( x \) and \( y \) given the lines of regression \( 5x + 3y = 55 \) and \( 7x + y = 45 \), we can follow these steps:

### Step 1: Rearranging the equations to find the regression coefficients

1. **First Line of Regression**:
\[ 5x + 3y = 55 \]
Rearranging for \( y \):
\[ 3y = 55 - 5x \implies y = -\frac{5}{3}x + \frac{55}{3} \]
The coefficient of \( x \) here is \( b_{yx} = -\frac{5}{3} \).

2. **Second Line of Regression**:
\[ 7x + y = 45 \]
Rearranging for \( y \):
\[ y = 45 - 7x \]
The coefficient of \( x \) here is \( b_{xy} = -7 \).

### Step 2: Finding the product of the regression coefficients

Now, we can find the product of the regression coefficients:
\[ b_{yx} \cdot b_{xy} = \left(-\frac{5}{3}\right) \cdot (-7) = \frac{5 \cdot 7}{3} = \frac{35}{3} \]

### Step 3: Relating the product to the coefficient of correlation

The relationship between the regression coefficients and the coefficient of correlation \( r \) is given by:
\[ b_{yx} \cdot b_{xy} = r^2 \]
Thus,
\[ r^2 = \frac{35}{3} \]

### Step 4: Finding the value of \( r \)

To find \( r \), we take the square root of \( r^2 \):
\[ r = \sqrt{\frac{35}{3}} \]

### Step 5: Determining the sign of \( r \)

Since both regression coefficients \( b_{yx} \) and \( b_{xy} \) are negative, the correlation will be negative:
\[ r = -\sqrt{\frac{35}{3}} \]

### Final Answer

The coefficient of correlation \( r \) between \( x \) and \( y \) is:
\[ r = -\sqrt{\frac{35}{3}} \]

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