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In a study on the relationship between investment $(X)$ and profit $(Y)$, the following two regression equations were beained hased on the data on $X$ and $Y$ $3 X+Y-12=0$
$X+2 Y-14=0$
What is the mean $\bar{X}$ ?
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$X+2 Y-14=0$
What is the mean $\bar{X}$ ?
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Verified Answer
The correct answer is:
2
Given lines of regression are $3 X+Y-12=0$ and $X+2 Y-14=0$
Since, lines of regression passes through $(\bar{X}, \bar{Y})$.
therefore $(\bar{X}, \bar{Y})$ satisfies the given equations.
$\therefore 3 \bar{X}+\bar{Y}-12=0$
and $\bar{X}+2 \bar{Y}-14=0$
Multiply equation (ii) by 3 and subtract from (i), we get
$(3 \bar{X}+\bar{Y}-12)-(3 \bar{X}+6 \bar{Y}-42)=0$
$\Rightarrow 5 \bar{Y}+30=0 \Rightarrow \bar{Y}=6$
Thus, $\bar{X}=14-2 \bar{Y}=14-12=2$
Hence, mean, $\mathrm{X}=2$
Since, lines of regression passes through $(\bar{X}, \bar{Y})$.
therefore $(\bar{X}, \bar{Y})$ satisfies the given equations.
$\therefore 3 \bar{X}+\bar{Y}-12=0$
and $\bar{X}+2 \bar{Y}-14=0$
Multiply equation (ii) by 3 and subtract from (i), we get
$(3 \bar{X}+\bar{Y}-12)-(3 \bar{X}+6 \bar{Y}-42)=0$
$\Rightarrow 5 \bar{Y}+30=0 \Rightarrow \bar{Y}=6$
Thus, $\bar{X}=14-2 \bar{Y}=14-12=2$
Hence, mean, $\mathrm{X}=2$
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