Question

One of the normal equations for fitting a straight line \(y=a+bx\) is

• A. \(\sum y=a\sum x+b\sum x^2\)
• B. \(\sum y=na+b\sum x\)
• C. \(\sum xy=na+b\sum x^2\)
• D. \(\sum x=\sum y+b\sum x\)

Hint:

Normal equations for the regression line \[ \boxed{ y=a+bx } \] are \[ \boxed{ \begin{aligned} \sum y &= na+b\sum x,\\[2mm] \sum xy &= a\sum x+b\sum x^2. \end{aligned} } \]

Answer 1

The Correct Option is B

Step 1: Set up the least-squares error.
For the line $y=a+bx$ fitted to points $(x_i,y_i)$, the total squared error is $S=\sum(y_i-a-bx_i)^2$.
Step 2: Minimize with respect to a.
Setting $\dfrac{\partial S}{\partial a}=0$ gives $-2\sum(y_i-a-bx_i)=0$, which simplifies to $\sum y=na+b\sum x$.
Step 3: Match this against the options.
This is exactly one of the two normal equations, and it appears among the choices.
\[ \boxed{\sum y=na+b\sum x} \]