# The two lines of regression are 8x - 10y = 66 and 40x - 18y = 214, and variance of x series is 9. What is the standard deviation of y series?

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The two lines of regression are 8x - 10y = 66 and 40x - 18y = 214, and variance of x series is 9. What is the standard deviation of y series?
1. 3
2. 4
3. 6
4. 8

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Correct Answer - Option 2 : 4

Concept:

Coefficient of correlation = $\rm r= \sqrt {b_{xy}\times b_{yx}}$

Where, byx and bxy are regression coefficients

or the slopes of the equation y on x and x on y are denoted as byx and bxy

Standard deviation = $\rm \sqrt {Variance }$

$\rm \frac{\sigma_y}{\sigma_x}=\frac{b_{yx}}{r}$$\rm \sigma_y$ and $\rm \sigma_x$are the standard deviation of y and x series respectively.

Calculation:

Here, The two lines of regression are 8x - 10y = 66 and 40x - 18y = 214

⇒ 10y = 8x - 66

⇒ byx = 8/10 = 4/5

And,

40x - 18y = 214

⇒ 40x =18y + 214

⇒ bxy = 18/40 = 9/20

Now, Coefficient of correlation = $\rm r= \sqrt {b_{xy}\times b_{yx}}$

$\pm\sqrt {\frac{4}{5}\times\frac{9}{20}}=\pm\sqrt {\frac{9}{25}}=\pm\frac 3 5$

bxy  > 0 and byx > 0.

So, r = 3/5

Here, variance of x series is 9

⇒ Standard deviation of x series is

$\rm \sigma_x$= √9 = 3

We know, $\rm \frac{\sigma_y}{\sigma_x}=\frac{b_{yx}}{r}$

So, $\rm {\sigma_y}=\frac{b_{yx}}{r}\times ​​\sigma_x$

$=\frac{\frac 4 5}{\frac 3 5}\times 3$

= 4

Hence, option (2) is correct.