Two samples from bivariate populations have 15 observations each.… - Vidyadip

Question

Two samples from bivariate populations have $15$ observations each. The sample means of $X$ and $Y$ are $25$ and $18$ respectively. The corresponding sum of squares of deviations from means are $136$ and $148$ respectively. The sum of product of deviations from respective means is $122.$ Obtain the regression equation of $x$ on $y$

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Answer

$ \text { Given, } n =15, \bar{x}=25, \bar{y}=18,$
$\sum\left(x_{ i }-\bar{x}\right)^2=136, \sum\left(y_{ i }-\bar{y}\right)^2=148,$
$\sum\left(x_{ i }-\bar{x}\right)\left(y_{ i }-\bar{y}\right)=122$
$\text { Now, } b _{ xy }=\frac{\sum\left(x_{ i }-\bar{x}\right)\left(y_{ i }-\bar{y}\right)}{\sum\left(y_{ i }-\bar{y}\right)^2}$
$=\frac{122}{148}$
$=0.82 $
Also, $a ^{\prime}=\bar{x}- b _{x y} \bar{y}$
$ =25-0.82 \times 18$
$=25-14.76$
$=10.24 $
$\therefore$ The regression equation of $X$ on $Y$ is
$ X=a^{\prime}+b_{x y} Y$
$\therefore X=10.24+0.82 Y $

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