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If the two regression lines are as under :
Y = a + bX
X = c + dY
What is the correlation coefficient between variables X and Y?
- a)√bc
- b)√ac
- c)√ad
- d)√bd
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
If the two regression lines are as under :Y = a + bXX = c + dYWhat is ...
Y = a + bX
Where, a and b are constants
b = it is called the regression coefficient of Y on X and is denoted by byx. It measures the change in Y corresponding to a unit change in X.
Thus, byx = Slope of the line of regression of Y on X = b
X = c + dY
where, c and d are constants
d = it is called the regression coefficient of X on Y and is denoted by bxy. It measures the change in X corresponding to a unit change in Y.
Thus, bxy = Slope of the line of regression of X on Y = d
Correlation coefficient = √byx bxy
⇒ √bd
Most Upvoted Answer
If the two regression lines are as under :Y = a + bXX = c + dYWhat is ...
The correlation coefficient between variables X and Y can be calculated using the formula:
r = √(bd)
Since the regression lines are given as:
Y = a + bX
X = c + dY
We can rearrange the second equation to solve for Y:
Y = (X - c)/d
Substituting this value of Y in the first equation, we get:
Y = a + b((X - c)/d)
This can be simplified as:
Y = (ad + bc - bd)/d
Now, we can equate this expression to the original equation for Y:
(ad + bc - bd)/d = a + bX
Rearranging the equation, we get:
ad + bc - bd = ad + bd + b^2X
Simplifying further, we have:
bc = b^2X
Dividing both sides by b, we get:
c = bX
Now, substituting this value of c in the equation X = c + dY, we get:
X = bX + dY
Rearranging the equation, we have:
Y = (X - bX)/d
Simplifying further, we get:
Y = (1 - b)/d
Now, we can substitute these values of X and Y in the correlation coefficient formula:
r = √(bd)
Substituting the values of b and d, we get:
r = √(b(1 - b))
Therefore, the correlation coefficient between variables X and Y is √(b(1 - b)).
r = √(bd)
Since the regression lines are given as:
Y = a + bX
X = c + dY
We can rearrange the second equation to solve for Y:
Y = (X - c)/d
Substituting this value of Y in the first equation, we get:
Y = a + b((X - c)/d)
This can be simplified as:
Y = (ad + bc - bd)/d
Now, we can equate this expression to the original equation for Y:
(ad + bc - bd)/d = a + bX
Rearranging the equation, we get:
ad + bc - bd = ad + bd + b^2X
Simplifying further, we have:
bc = b^2X
Dividing both sides by b, we get:
c = bX
Now, substituting this value of c in the equation X = c + dY, we get:
X = bX + dY
Rearranging the equation, we have:
Y = (X - bX)/d
Simplifying further, we get:
Y = (1 - b)/d
Now, we can substitute these values of X and Y in the correlation coefficient formula:
r = √(bd)
Substituting the values of b and d, we get:
r = √(b(1 - b))
Therefore, the correlation coefficient between variables X and Y is √(b(1 - b)).
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