Answer to Question #202278 in Statistics and Probability for RAGHAV SOOD

Question

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

$Question:-\\ For\space 25\space army\space personnels\\ \space line\space of\space regression\space of\space weight\space of\space kidneys\space (Y)\space on\space weight\space of\space heart\space (X)\space is\\ Y=(0.399)X+6.934\\ line\space of\space regression\space of\space weight\space of\space heart\space (X)\space on\space weight\space of\space kidney\space (Y)\space is\\ X=(1.212)Y\space -2.461\\ Find\space \\ (1)the\space correlation\space coefficient\space between\space X\space and\space Y\space \\ and\\ \space (2)their\space mean\space values \\------------------------------\\ (1)Correlation\space Coefficient\\ \space Y=0.399X+6.934\space is\space regression\space equation\space of\space y\space on\space x\\ \space ∴b_{yx}=0.399\space ........[b_{yx}\space is\space coefficient\space of\space X\space in\space regression\space equation\space of\space y\space on\space x]\\ X=1.212Y-2.461\space is\space regression\space equation\space of\space x\space on\space y\\ ∴bxy=1.212........[b_{xy}\space is\space coefficient\space of\space Y\space in\space regression\space equation\space of\space x\space on\space y]\\ Correlation\space Coefficient\space \space formula\\ r=\sqrt{byx⋅bxy}\\ r=\sqrt{0.399*1.212}\\ r=\sqrt{0.4836}\\ r=0.6954\\ ----------------------------------------- \\ (2)Mean\space of\space x\space and\space y\\ solve\space both\space equation\space and\space we\space will\space get\space mean\space of\space x\space and\space y\\ y=0.399x+6.934\\ ∴y-0.399x=6.934\\ and\\ \space x-1.212y+2.461=0\\ ∴x-1.21y=-2.46\\ -0.399x+y=6.934→equation(1)\\ x-1.212y=-2.461→equation(2)\\ equation(1)×1⇒-0.399x+y=6.934\\ equation(2)×0.399⇒0.399x-0.4836y=-0.9819\\ Adding\space both⇒0.5164y=5.9521\\ ⇒y=\frac{5.9521}{\space 0.5164}\\ ⇒y=11.5258\\ Putting\space y=11.5258\space in\space equation\space (1),\space we\space have\\ -0.399x+11.5258=6.934\\ ⇒-0.399x=6.934-11.5258\\ ⇒-0.399x=-4.5918\\ ⇒x=\frac{4.5918}{\space 0.399}\\ \space ⇒x=11.5083\\ ∴\bar{x}=11.5083\space and\space \bar{y}=11.5258\\$