Q.The two lines of regression are $8x-10y+66=0$ and $40x-18y-214=0$. Find

The means are the intersection of the two lines; the product test decides which line is $Y$ on $X$.

(i) Means = point of intersection. Both regression lines pass through $(\bar x,\bar y)$, so solve them simultaneously:

Multiply (1) by $5$: $40x-50y=-330$. Subtract from (2):

Substitute in (1): $8x-10(17)=-66\ \Rightarrow\ 8x-170=-66\ \Rightarrow\ 8x=104\ \Rightarrow\ x=13.$ So $\bar x=13,\ \bar y=17$.

(ii) Which line is $Y$ on $X$? Use the product test $b_{yx}\cdot b_{xy}\le1$. Assume the first line is $Y$ on $X$: solve $8x-10y+66=0$ for $y$:

Then the second line is $X$ on $Y$: solve $40x-18y-214=0$ for $x$:

$40x=18y+214\ \Rightarrow\ x=0.45y+5.35\ \Rightarrow\ b_{xy}=0.45.$ …

To identify the lines you may instead check the other assignment: taking the first line as $X$ on $Y$ gives $b_{xy}=1.25$ and the second as $Y$ on $X$ gives $b_{yx}=40/18=2.22$, …

Guessing which line is $Y$ on $X$ without the product test. The only valid assignment is the one whose product of coefficients …