The equations of two regression lines are \(8x-10y+66=0\) and \(40x-18y=214\). Find(a) The mean values of \(X\) and \(Y\)(b) Correlation coefficient between \(X\) and \(Y\)

Answer: (a) \(\bar{x}=13,\ \bar{y}=17\); (b) \(r=0.6\)

(a) The regression lines pass through \((\bar{x},\bar{y})\), so solve \(8x-10y=-66\) ... (1) and \(40x-18y=214\) ... (2).
(1) \(\times 5\): \(40x-50y=-330\) ... (3). (2) \(-\) (3): \(32y=544\Rightarrow y=17\). Then \(8x=170-66=104\Rightarrow x=13\).
\(\therefore\bar{x}=13,\ \bar{y}=17\).

(b) Let \(8x-10y+66=0\) be the regression of \(Y\) on \(X\): \(y=\frac{8}{10}x+\frac{66}{10}\), so \(b_{yx}=0.8\).
Let \(40x-18y=214\) be the regression of \(X\) on \(Y\): \(x=\frac{18}{40}y+\frac{214}{40}\), so \(b_{xy}=0.45\).
\(b_{yx}\cdot b_{xy}=0.36\le 1\), so the assumption is correct.

\(r^2=0.36\Rightarrow r=\pm 0.6\). Both regression coefficients are positive, so \(r=0.6\).