Assume that we are given a collection of n points \{ (x_ i, y_ i) \} _{i=1}^{n}. Which one of the following choices, on its own, provides a sufficient condition under which a unique least-squares estimator (\hat{a}, \hat{b}) exists?
There are at least two total observations; i.e. (n \geq 2).
There are at least two distinct x's in the collection; i.e. x_1 \neq x_2.
There are at least two distinct y's in the collection; i.e. y_1 \neq y_2.
The correct choice that provides a sufficient condition under which a unique least-squares estimator exists is:
There are at least two distinct x's in the collection; i.e. x_1 ≠ x_2.
This is because when there are at least two distinct x-values, it ensures that there is enough variation in the independent variable to estimate the slope of the line accurately. Without distinct x-values, there would be no variation in the independent variable, making it impossible to estimate the slope.
The correct choice that provides a sufficient condition for a unique least-squares estimator (\hat{a}, \hat{b}) to exist is:
There are at least two distinct x's in the collection; i.e. x_1 ≠ x_2.
This condition ensures that there is sufficient variation in the x-values, which allows for a unique estimation of the parameters a and b in the linear equation y = ax + b. Without distinct x-values, the equation may become degenerate and not allow for a unique solution. The condition of having at least two total observations (n ≥ 2) or at least two distinct y-values (y_1 ≠ y_2) does not guarantee a unique least-squares estimator.