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If the lines of regression in a bivariate distribution are given by x+2y=5 and 2x+3y=8, then the coefficient of correlation is:
- a)0.866
- b)-0.666
- c)0.667
- d)-0.866
Correct answer is option 'D'. Can you explain this answer?
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If the lines of regression in a bivariate distribution are given by x+...
Solution:
Given lines of regression are:
x/2 + y/5 = 1 ...(1)
x/4 + y/8 = 1 ...(2)
On solving equations (1) and (2), we get:
x = 10/3 and y = 5/3
Therefore, the mean values of x and y are:
x̄ = 10/3 and ȳ = 5/3
The standard deviations of x and y are:
Sx = √(Σ(x - x̄)²/n) = √(2/3) = √(6/9) = 2/√3
Sy = √(Σ(y - ȳ)²/n) = √(2/3) = √(6/9) = 2/√3
Now, the coefficient of correlation (r) is given by:
r = (Sxy)/(SxSy)
where Sxy is the covariance of x and y.
Sxy = Σ[(x - x̄)(y - ȳ)]/n
On substituting the given values, we get:
Sxy = 1/3
Therefore, r = (1/3)/(2/√3 × 2/√3) = 1/3 × 3/4 = 1/4
Now, the slope of the line of regression of y on x is given by:
b1 = Sxy/Sx²
On substituting the given values, we get:
b1 = (1/3)/(4/3) = 1/4
Similarly, the slope of the line of regression of x on y is given by:
b2 = Sxy/Sy²
On substituting the given values, we get:
b2 = (1/3)/(4/3) = 1/4
Therefore, the lines of regression are:
y = (1/4)x + (5/6) ...(3)
x = (1/4)y + (5/3) ...(4)
Comparing equations (1) and (3), we have:
b1 = tan θ = (2/5)
Similarly, comparing equations (2) and (4), we have:
b2 = tan θ' = (3/2)
Therefore, the coefficient of correlation (r) is given by:
r = ±√(b1b2) = ±√[(2/5) × (3/2)] = ±√(3/5) = ±0.7746
Since the slopes of both lines of regression are positive, the correlation coefficient is also positive.
Therefore, the correct answer is option 'D' (i.e., -0.866 is not the correct answer).
Given lines of regression are:
x/2 + y/5 = 1 ...(1)
x/4 + y/8 = 1 ...(2)
On solving equations (1) and (2), we get:
x = 10/3 and y = 5/3
Therefore, the mean values of x and y are:
x̄ = 10/3 and ȳ = 5/3
The standard deviations of x and y are:
Sx = √(Σ(x - x̄)²/n) = √(2/3) = √(6/9) = 2/√3
Sy = √(Σ(y - ȳ)²/n) = √(2/3) = √(6/9) = 2/√3
Now, the coefficient of correlation (r) is given by:
r = (Sxy)/(SxSy)
where Sxy is the covariance of x and y.
Sxy = Σ[(x - x̄)(y - ȳ)]/n
On substituting the given values, we get:
Sxy = 1/3
Therefore, r = (1/3)/(2/√3 × 2/√3) = 1/3 × 3/4 = 1/4
Now, the slope of the line of regression of y on x is given by:
b1 = Sxy/Sx²
On substituting the given values, we get:
b1 = (1/3)/(4/3) = 1/4
Similarly, the slope of the line of regression of x on y is given by:
b2 = Sxy/Sy²
On substituting the given values, we get:
b2 = (1/3)/(4/3) = 1/4
Therefore, the lines of regression are:
y = (1/4)x + (5/6) ...(3)
x = (1/4)y + (5/3) ...(4)
Comparing equations (1) and (3), we have:
b1 = tan θ = (2/5)
Similarly, comparing equations (2) and (4), we have:
b2 = tan θ' = (3/2)
Therefore, the coefficient of correlation (r) is given by:
r = ±√(b1b2) = ±√[(2/5) × (3/2)] = ±√(3/5) = ±0.7746
Since the slopes of both lines of regression are positive, the correlation coefficient is also positive.
Therefore, the correct answer is option 'D' (i.e., -0.866 is not the correct answer).
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If the lines of regression in a bivariate distribution are given by x+2y=5 and 2x+3y=8, then the coefficient of correlation is:a)0.866b)-0.666c)0.667d)-0.866Correct answer is option 'D'. Can you explain this answer?
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