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Question

# The regression of savings $\left(s\right)$ of a family on income $y$may be expressed as $s=a+\left(\frac{y}{m}\right)$, where $a$ and $m$ are constants. In a random sample of $100$ families, the variance of savings is one-quarter of the variance of incomes and the correlation is found to be $0.4$ the value of $m$ is

A

$2$

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B

$5$

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C

$8$

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D

None of these

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Solution

## The correct option is B $5$Explanation for the correct answer:Finding the value of $m$:Given that, $s=a+\left(\frac{y}{m}\right)$,${s}^{2}=\left(\frac{1}{4}\right){y}^{2}$ and $r=0.4$$\begin{array}{rcl}s& =& a+\left(\frac{y}{m}\right)\\ and\overline{)s}& =& a+\left(\frac{\overline{)y}}{m}\right)\end{array}$On subtracting $s-\overline{)s}$, we get$s-\overline{)s}=\left(\frac{1}{m}\right)\left(y-\overline{)y}\right)...\left(i\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\sum \left(s-\overline{)s}\right)\left(y-\overline{)y}\right)=\left(\frac{1}{m}\right)\sum {\left(y-\overline{)y}\right)}^{2}$Also, $Cov\left(s,y\right)=\left(\frac{1}{m}\right){y}^{2}...\left(ii\right)$Finding the value of $m.$$\begin{array}{rcl}r& =& \frac{cov\left(s,y\right)}{sy}\\ & ⇒& 0.4=\frac{\left(\frac{1}{m}\right){y}^{2}}{sy}\\ & ⇒& 0.4=\frac{\left(\frac{1}{m}\right)}{\left(\frac{y}{s}\right)}\\ & ⇒& 0.2=\frac{1}{m}\\ & ⇒& m=5\end{array}$Therefore, the correct answer is option (B).

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