Discussion
In a simple linear regression analysis the fitted line using least squares theory will :
Answer: A
The fitted line will definitely pass through \((\bar{x}, \bar{y})\) when the intercept is present.
With an intercept, the first normal equation is \(\sum(y_{i} - a - bx_{i}) = 0\), which gives \(a = \bar{y} - b\bar{x}\). So \(\hat{y} = \bar{y} + b(x - \bar{x})\), and at \(x = \bar{x}\) the line gives \(\bar{y}\), whatever the sign of the slope.
Without an intercept (regression through the origin) there is no such normal equation, and the line generally misses \((\bar{x}, \bar{y})\), so (B) is too strong. 'Never' is wrong, and the slope's sign is irrelevant.
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