Which of the following are normal equations to fit a straight line trend \(y = a + bx\) by the method of least squares?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\sum y = na + b\sum x,\quad \sum xy = a\sum x + b\sum x^2\)
By the method of least squares, a and b make the sum of squared errors as small as possible:
\( S = \sum (y - a - bx)^2 \)
Setting the partial derivatives to zero:
\( \displaystyle \frac{\partial S}{\partial a} = -2\sum (y - a - bx) = 0 \)
\( \sum y = na + b\sum x \)
\( \displaystyle \frac{\partial S}{\partial b} = -2\sum x(y - a - bx) = 0 \)
\( \sum xy = a\sum x + b\sum x^2 \)
These are the two normal equations. They are the same as summing \(y = a + bx\) over all n points, and summing it again after multiplying by x.
CBSE Applied Mathematics (Class XII):
Computation of Straight-line Trend by Using Method of Least Squares
