Answer to Question #279920 in Statistics and Probability for Siva

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Section: 1 of 1 Question: 3 of 7 Marks forthls Question: 10 a

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An analysis of the data of the partially destroyed dairy are

variance x=8 and the least square regression lines for the

random variables are 3x 4y = 20, 12x — 8y = 62. Find

the following basis of above infonnation:

(i) Mean value of x and y.

(ii) Correlation co-efficient between x and y.

(iii) SD of x and y.

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Answer 1

$i)$

Since two regression lines always intersect at a point $(\bar{x},\bar{y})$, representing mean values of the values x and y as shown below:

$3x+4y=20$

$12x-8y=62$

Multiplying the first equation by 4 and subtracting from the second,

$24y=18\implies\bar{y}=0.75$

From the first equation

$\bar{x}={(20-4\bar{y})\over3}={17\over3}=5.67$

Therefore, the mean value of x and y are 5.67 and 0.75 respectively.

$ii)$

To find the given regression equations in such a way that the coefficient of dependent variable is less than one at least in one equation. 

So, $3x+4y=20\implies4y=20-3x\implies y=5-{3\over4}x$

That is, $|m_{yx}|=|-{3\over4}|={3\over4}$

and

$12x-8y=62\implies12x=62+8y\implies x={62\over12}+{8\over12}y$

Thus, $m_{xy}={8\over 12}$

Hence coefficient of correlation $r$ between $x$ and $y$ is given by: 

$r=\sqrt{m_{xy}*m_{yx}}=\sqrt{{3\over4}*{8\over12}}=\sqrt{1\over2}=0.7071(4dp)$  

$iii)$

 Since the variance of x is 8, therefore, its standard deviation is,

$\sigma_x=\sqrt{8}=2.8284(4dp)$

To determine the standard deviation of y, consider the formula

$\sigma_y={m_{yx}\sigma_x\over r}=({0.75*2.8284\over0.7071})=3$

Therefore, the standard deviation for x and y are 2.8284 and 3 respectively.