If \( Y = \frac{x}{2} + 2 \) and \( X = \frac{y}{8} - 1 \) are regression lines of Y on X and X on Y respectively, then the correlation coefficient between X and Y is equal to:
1/4
1. Y on X: \( Y = \frac{1}{2}X + 2 \) ⇒ \( b_{yx} = \frac{1}{2} \)
2. X on Y: \( X = \frac{1}{8}Y - 1 \) ⇒ \( b_{xy} = \frac{1}{8} \)
The correlation coefficient (r) is given by:
\( r = \pm\sqrt{b_{yx} \times b_{xy}} \)
Substituting the values:
\( r = \pm\sqrt{\frac{1}{2} \times \frac{1}{8}} = \pm\sqrt{\frac{1}{16}} = \pm\frac{1}{4} \)
Since both regression coefficients are positive, we take the positive value:
\( r = \frac{1}{4} \)
Correct Answer: Option 3. \(\frac{1}{4}\)
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