Question

From the given data:

Variable	x	y
Mean	$6$	$8$
Standard Deviation	$4$	$6$

and correlation coefficient: $\frac{2}{3}$. Find Most likely value of x when $y=14$.

Answer 1

To find $x$ when $y=14$ we need to first find regression line of $x$ on $y.$

Given: correlation coefficient $:r=\frac{2}{3}$

Mean of $x$ $:\overline{x}=6$, Mean of $y$ $:\overline{y}=8$,

Standard deviation of $x$ $:S_x=4$, Standard deviation of $y$ $:S_y=6$

Regression line $x$ on $y:$ $x=a+by$

where $a=\overline{x}-b\overline{y}$ and $b=b_{xy}=\frac{r}{{S_y}^2}$

$b_{xy}=\frac{r}{{S_y}^2}=\frac{2/3}{6^2}=\frac{1}{54}$

$a=6-\frac{1}{54}\times 8=6-\frac{4}{27}=\frac{158}{27}$

Regression line $:x=\frac{158}{27}+\frac{1}{54}y$

$\therefore$ when $y=14\to$ Estimated value(Most likely value) of $x=\frac{158}{27}+\frac{1}{54}\times 14=\frac{55}{9}$