Question

Regression of savings (s) of a family on income y may be expressed as$s = a + \frac{y}{m}$, 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 coefficient is found to be 0.4, the value of m is

• A. 2
• B. 5
• C. 8
• D. None of these

Answer 1

Answer: (B)

5

Answer 2

We have, $s = a + \frac{y}{m}$, $\sigma_{s}^{2} = \frac{1}{4}\sigma_{y}^{2}$ and $r = 0.4$

Now, $s = a + \frac{y}{m}$ ⇒ $\bar{s} = a + \frac{\bar{y}}{m}$⇒$s - \bar{s} = \frac{1}{m}(y - \bar{y})$.....(i)

⇒ $\sum(s - \bar{s})(y - \bar{y}) = \frac{1}{m}\sum(y - \bar{y})^{2}$

⇒ $Cov(s,y) = \frac{1}{m}\sigma_{y}^{2}$ .....(ii)

Now, $r = \frac{Cov(s,y)}{\sigma_{s}\sigma_{y}}$ ⇒ $0.4 = \frac{\frac{1}{m}\sigma_{y}^{2}}{\sigma_{s}\sigma_{y}}$

⇒ $0.4 = \frac{1}{m}\frac{\sigma_{y}}{\sigma_{s}}$ ⇒ $0.2 = \frac{1}{m}$, $\left\lbrack \because\mspace{6mu}\mspace{6mu}\mspace{6mu}\sigma_{s} = \frac{1}{2}\sigma_{y},(\text{given}) \right\rbrack$

s ⇒ $m = 5$.