Question #179627

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.


1
Expert's answer
2021-04-15T06:50:21-0400

Rewrite the regression equations uniformly


Y=0.0399X+0.6934Y = 0.0 399X + 0.6 934




X=0.1212Y0.2461X = 0 .1 212Y - 0.2 461


The basic formulas are


Y=rsysxXrsysxxˉ+yˉY = r\dfrac{s_y}{s_x}X -r\dfrac{s_y}{s_x}\bar{x}+\bar{y}

X=rsxsyYrsxsyyˉ+xˉX = r\dfrac{s_x}{s_y}Y -r\dfrac{s_x}{s_y}\bar{y}+\bar{x}

Then


rsysx=0.0399r\dfrac{s_y}{s_x}=0.0399

rsxsy=0.1212r\dfrac{s_x}{s_y}=0.1212

Hence


r2sysxsysx=0.0399(0.1212)r^2\dfrac{s_y}{s_x}\cdot\dfrac{s_y}{s_x}=0.0399(0.1212)

r2=0.00483588r^2=0.00483588

r=0.004835880.06954049r=\sqrt{0.00483588}\approx0.06954049

{0.0399xˉ+yˉ=0.69340.1212yˉ+xˉ=0.2461\begin{cases} -0.0399\bar{x}+\bar{y}=0.6934 \\ -0.1212\bar{y}+\bar{x}=-0.2461 \end{cases}

{yˉ=0.6934+0.0399xˉ0.1212(0.6934+0.0399xˉ)+xˉ=0.2461\begin{cases} \bar{y}=0.6934+0.0399\bar{x} \\ -0.1212(0.6934+0.0399\bar{x} )+\bar{x}=-0.2461 \end{cases}

{yˉ=0.6934+0.0399xˉxˉ=0.162059920.99516412\begin{cases} \bar{y}=0.6934+0.0399\bar{x} \\ \bar{x}=-\dfrac{0.16205992}{0.99516412} \end{cases}

{yˉ=0.6869xˉ=0.1628\begin{cases} \bar{y}=0.6869 \\ \bar{x}=-0.1628 \end{cases}

r=0.0695r=0.0695


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