Question

Using the following regression equations, the correlation coefficient between two survey quantities x and y will be __________________ (round off to 2 decimal places). \[ 2x - 5y + 98 = 0 \] \[ 6x - 7y + 114 = 0 \]

Hint:

The correlation coefficient can be determined using the product of the slopes of the two regression lines.

Answer 1

Correct Answer: 0.68

The regression equations are:
\[ 2x - 5y + 98 = 0 \quad \text{and} \quad 6x - 7y + 114 = 0. \] From the equations, we can extract the values of \( b_{xy} \) (slope of the regression of \( y \) on \( x \)) and \( b_{yx} \) (slope of the regression of \( x \) on \( y \)):
- \( b_{xy} = \frac{-2}{5} = -0.4 \)
- \( b_{yx} = \frac{-6}{7} \approx -0.8571 \) The correlation coefficient \( r \) can be calculated using the formula:
\[ r = \sqrt{b_{xy} \times b_{yx}}. \] Substituting the values:
\[ r = \sqrt{(-0.4) \times (-0.8571)} = \sqrt{0.34284} \approx 0.585. \] Thus, the correlation coefficient between \( x \) and \( y \) is approximately \( 0.69 \) when rounded to 2 decimal places.