Question

Find the lines of regression from the following data:

$x$	$y$
$2$	$6$
$4$	$5$
$6$	$4$
$8$	$3$
$10$	$2$

Answer 1

$x$	$y$	$xy$	$x^2$	$y^2$
$2$	$6$	$12$	$4$	$36$
$4$	$5$	$20$	$16$	$25$
$6$	$4$	$24$	$36$	$16$
$8$	$3$	$24$	$64$	$9$
$10$	$2$	$20$	$100$	$4$
$\sum x=30$	$\sum y=20$	$\sum xy=100$	$\sum x^2=220$	$\sum y^2=90$

$\overline{x}=\frac{30}{5}=6,\overline{y}=\frac{20}{5}=4$
$\overline{x}=6,\overline{y}=4$
$b_{xy}=\frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x{)}^2}=\frac{5\left(100\right)-30\times 20}{5\left(220\right)-900}$
$b_{yx}=\frac{500-600}{1100-900}=\frac{-100}{200}=\frac{-1}{2}$
Regression line $y$ on $x$
$y-\overline{y}=b_{yx}(x-\overline{x})$
$(y-4)=\frac{-1}{2}(x-6)$
$2y-8=-x+6$
$x+2y=14$
$b_{xy}=\frac{n\sum xy-\sum x\sum y}{n\sum y^2-(\sum y{)}^2}=\frac{5\left(100\right)-\left(30\right)\left(20\right)}{5\left(90\right)-\left(400\right)}$
$b_{xy}=\frac{500-600}{450-400}$
$b_{xy}=\frac{-100}{50}$
$=-2$
Regression line $x$ on $y$
$(x-\overline{x})=b_{xy}(y-\overline{y})$
$(x-6)=-2(y-4)$
$x-6=-2y+8$
$x+2y=14$
$x=-2y+14$.