Question

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

Answer 1

Correct Answer - Option 2 : 4

Concept:

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

Where, b_yx and b_xy are regression coefficients

or the slopes of the equation y on x and x on y are denoted as b_yx and b_xy

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.