Question

8(b) For 25 army personnels, line of regression of weight of kidneys (Y) on weight of 

heart (X ) is Y = .0 399X + .6 934 and the line of regression of weight of heart on 

weight of kidney is X − .1 212Y + .2 461= .0 Find the correlation coefficient between 

X and Y and their mean values.

Answer 1

Rewrite the regression equations uniformly

$Y = .0 399X + .6 934$

$X = .1 212Y - .2 461$

Denote mean values as $\overline{x}$ and $\overline{y}$, correlation coefficient as $r$, standard deviations as $s_x$ and $s_y$. Then the basic formulas are

$Y=r\dfrac{s_y}{s_x}X-r\dfrac{s_y}{s_x}\overline{x}+\overline{y}$

$X=r\dfrac{s_x}{s_y}Y-r\dfrac{s_x}{s_y}\overline{y}+\overline{x}$

So find correlation coefficient from following

$\left\{\begin{array}{rr}r\dfrac{s_y}{s_x}=.0 399\\ \\ r\dfrac{s_x}{s_y}= .1 212\end{array}\right.$

Multiply the equations

$r\dfrac{s_y}{s_x}\cdot r\dfrac{s_x}{s_y}=.0 399 \cdot .1 212$ $\Rightarrow$ $r^2=.00483588$

Thus

$r=\sqrt{.00483588}=.069540492$

Further, find mean values from

$-r\dfrac{s_y}{s_x}\overline{x}+\overline{y}= .6 934$

$-r\dfrac{s_x}{s_y}\overline{y}+\overline{x}=- .2 461$

or

$- .0 399\overline{x}+\overline{y}= .6 934$

$-.1 212\overline{y}+\overline{x}=- .2 461$

Construct the system of linear equations and solve it using Cramer's rule

$\left\{\begin{array}{rrr}- .0 399\ \overline{x}&+\ \overline{y}=& .6 934\\ \overline{x}&-.1 212\ \overline{y}=&- .2 461\end{array}\right.$

$\det(A)= \begin{vmatrix} - .0 399 & 1 \\ 1 & -.1 212 \end{vmatrix}=$ $-.0 399\cdot( -.1 212)-1\cdot1=$ $-.99516412$

$\det(A_x)= \begin{vmatrix} .6 934 & 1 \\ - .2 461 & -.1 212 \end{vmatrix}=$ $.6934\cdot( -.1 212)-1\cdot(-.2461)=$ $.16205992$

$\det(A_y)= \begin{vmatrix} - .0 399 & .6 934 \\ 1 & - .2 461 \end{vmatrix}=$ $-.0 399\cdot( -.2461)-.6934\cdot1=$ $-.68358061$

$\overline{x}=\dfrac{\det(A_x)}{\det(A)}=\dfrac{.16205992}{-.99516412}=-.162847431$

$\overline{y}=\dfrac{\det(A_y)}{\det(A)}=\dfrac{-.68358061}{-.99516412}=.686902388$

Answer

correlation coefficient

$r=.0695$

mean values

$\overline{x}=-.1628$

$\overline{y}=.6869$

Remark. Weight must be positive. Perhaps condition is incorrect.