Civil Engineering (CE) Exam > Civil Engineering (CE) Questions > Given the regression lines X + 2Y - 5 = 0, 2X...
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Given the regression lines X + 2Y - 5 = 0, 2X + 3Y - 8 = 0 and Var(X) = 12, the value of Var(Y) is
- a)3/4
- b)4/3
- c)16
- d)4
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Given the regression lines X + 2Y - 5 = 0, 2X + 3Y - 8 = 0 and Var(X) ...
Given
Regression lines
x + 2y - 5 = 0
2x + 3y - 8 = 0
var(x)= σx = 12
Calculation
x + 2y - 5 = 0 ------(i)
Let y = - x/2 + 5/2 be the regression line of y on x [ from equation 1]
2x + 3y - 8
x = -(3/2)y + 8/2 be the regressiopn line of x on y
⇒ bxy = -1/2 and byx = -3/2
bxy = Regression line of y on x
byx = Regression line of x on y
We know that regression coefficient = r = √(byx × bxy)
⇒ r = √(-1/2 × -3/2)
∴ r = √3/2 < 1
σx = 12 = 2√3
We know that byx = r (σy/σx)
⇒ -1/2 = √3/2 (σy/2√3)
⇒ σy = - 2
∴ var(y) = variance of y =(-2)2 = 4
Most Upvoted Answer
Given the regression lines X + 2Y - 5 = 0, 2X + 3Y - 8 = 0 and Var(X) ...
Given Information:
Regression lines:
1) X + 2Y - 5 = 0
2) 2X + 3Y - 8 = 0
Var(X) = 12
Solution:
To find the value of Var(Y), we need to determine the slope of the regression line and use it to calculate the variance.
Finding Slope:
We can rewrite the given regression lines in the slope-intercept form (y = mx + c) by rearranging the equations:
1) X + 2Y - 5 = 0
2Y = -X + 5
Y = -0.5X + 2.5
2) 2X + 3Y - 8 = 0
3Y = -2X + 8
Y = -0.67X + 2.67
Comparing the equations with the slope-intercept form, we can determine the slopes of the regression lines:
1) Slope of Line 1 = -0.5
2) Slope of Line 2 = -0.67
Calculating Var(Y):
The variance of a random variable Y in a regression analysis is given by the formula:
Var(Y) = Var(X) * (1 - R^2)
where R^2 is the coefficient of determination, which is the square of the correlation coefficient (r) between X and Y.
To find Var(Y), we need to calculate R^2 first. The correlation coefficient (r) can be determined by multiplying the slopes of the regression lines:
r = (-0.5) * (-0.67)
r = 0.335
R^2 = r^2 = 0.335^2 = 0.112
Substituting the values, we can calculate Var(Y):
Var(Y) = 12 * (1 - 0.112)
Var(Y) = 12 * 0.888
Var(Y) = 10.656
Therefore, the value of Var(Y) is approximately 10.656, which is not given as an option in the provided choices. The correct answer may be a typo, and the actual answer might be 10.656 instead of 4.
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Given the regression lines X + 2Y - 5 = 0, 2X + 3Y - 8 = 0 and Var(X) = 12, the value of Var(Y) isa)3/4b)4/3c)16d)4Correct answer is option 'D'. Can you explain this answer?
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