Discussion
Let 8x - 10y + 66 = 0 and 40x - 18y = 214 be the lines of regression of Y on X and X on Y respectively. What is the correlation coefficient between X and Y ?
Answer: C
The correlation coefficient is 0.6.
Write each line in its regression form. Y on X: \(10y = 8x + 66\), so \(y = 0.8x + 6.6\) and \(b_{yx} = 0.8\). X on Y: \(40x = 18y + 214\), so \(x = 0.45y + 5.35\) and \(b_{xy} = 0.45\).
\[ \begin{aligned} r^{2} &= b_{yx} \times b_{xy} \\ &= 0.8 \times 0.45 = 0.36 \\ r &= 0.6 \end{aligned} \]
r takes the sign of the regression coefficients, which are both positive, so r = +0.6, not -0.6. 0.5 and -0.5 do not follow from 0.36. The check \(r^{2} \le 1\) also confirms the lines were labelled correctly.
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