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 Four arbitrary point (x1,y1), (x2,y2), (x3,y3), (x4,y4), are given in the x, y – plane Using the method of least squares, if, regressing y upon x gives the fitted line y = ax + b; and regressing y upon x given the fitted line y = ax + b; and regressing x upon y gives the fitted line x = cy + d then  
  • a)
    The two fitted lines must coincide
  • b)
    The two fitted lines need not coincide  
  • c)
    It is possible that ac = 0  
  • d)
    A must be 1/c 
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Four arbitrary point (x1,y1), (x2,y2), (x3,y3), (x4,y4), are given in ...
y =ax+b − (i) and x = cy + d − (ii)
 
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Four arbitrary point (x1,y1), (x2,y2), (x3,y3), (x4,y4), are given in ...
Explanation:

The method of least squares is a statistical technique used to find the best-fitting line or curve that minimizes the sum of the squared differences between the observed and predicted values. In this case, we have two regression lines: one regressing y upon x and another regressing x upon y.

The two fitted lines need not coincide:
The fitted line y = ax + b obtained by regressing y upon x and the fitted line x = cy + d obtained by regressing x upon y need not coincide. In other words, they can be two separate lines with different slopes and intercepts.

It is possible that ac = 0:
The product of the slopes of the two lines, ac, can be zero. This means that either a or c can be zero, but not both. If a = 0, then the fitted line y = 0x + b simplifies to y = b, which is a horizontal line. If c = 0, then the fitted line x = 0y + d simplifies to x = d, which is a vertical line. So, it is possible for either line to be vertical or horizontal, but not both.

A must be 1/c:
Given that the two lines need not coincide, it is not necessary for a and c to be reciprocals of each other. Therefore, option D is incorrect.

Therefore, the correct answer is option C: It is possible that ac = 0.
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Four arbitrary point (x1,y1), (x2,y2), (x3,y3), (x4,y4), are given in the x, y – plane Using the method of least squares, if, regressing y upon x gives the fitted line y = ax + b; and regressing y upon x given the fitted line y = ax + b; and regressing x upon y gives the fitted line x = cy + d then a)The two fitted lines must coincideb)The two fitted lines need not coincide c)It is possible that ac = 0 d)A must be 1/cCorrect answer is option 'D'. Can you explain this answer?
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