Question

You are given that the Variance of x = 36. The regression equations are 60x-27y = 321 and 12x – 15y + 99 = 0.

(i) Find the correlation coefficient between two variables.
(ii) Find the standard deviation of Y.
(iii) Find the average values of r and y.​

Answer 1

1. The correlation coefficient between variables is 0.6.

2. Standard deviation of y = 8.

3. The average value of x and y is 13 and 17.

Given that,

The Variance of x = 36. The equations are 60x-27y = 321 and 12x – 15y + 99 = 0.

We know that,

1. We have to find the correlation coefficient between two variables.

Take equation

12x – 15y + 99 = 0 ⇒ y = $\frac{12}{15}$x + $\frac{90}{15}$    --------> equation(1)

60x-27y = 321 ⇒ x = $\frac{27}{60}$y + $\frac{321}{60}$ ----------->equation(2)

The equation (1) is of the regression from y on x

$b_{yx}$ = $\frac{12}{15}$

The equation (2) is of the regression from x on y

$b_{xy}$ = $\frac{27}{60}$

The coefficient of correlation $r_{xy} = \pm\sqrt{b_{xy}b_{yx}}$

$r_{xy} = \pm\sqrt{\frac{12}{15} \times \frac{27}{60} }$ = $\frac{3}{5}$ = 0.6

Therefore, the correlation coefficient between two variables = 0.6.

2. We have to find the standard deviation of y.

$b_{xy} = r_{xy}\frac{\sigma_x}{\sigma_y}$

Here,

$r_{xy}$ = 0.6

$b_{xy}$ = 0.45

$\sigma_x$ = √36 = 6

We get

0.45 = 0.6×$\frac{6}{\sigma_y}$

0.75 = $\frac{6}{\sigma_y}$

$\sigma_y$ = 8

Therefore, standard deviation of y = 8

3. We have to find the average values of r and y.

Solve the given equation,

12x – 15y + 99 = 0 ⇒ x = $\frac{1}{12}$(15y-99)

Substitute in the other equation

60x-27y = 321

20x - 9y = 107

20($\frac{1}{12}$(15y-99)) - 9y = 107

25y - 165 - 9y = 107

16y = 272

y = 17

Then

x = $\frac{1}{12}$(15y-99)

x = $\frac{1}{12}$(15(17)-99)

x = 13

Therefore, The average value of x and y is 13 and 17.

To know more, visit:

https://brainly.in/question/38799865

https://brainly.in/question/14032691

#SPJ1