Which of the following statements are not true regarding the normal equations $n b _ { 0 } + \left( \sum x _ { i } \right) b _ { 1 } = \sum y _ { i } \text { and } \left( \sum x _ { i } \right) b _ { 0 } + \left( \sum x _ { i } ^ { 2 } \right) b _ { 1 } = \sum x _ { i } y _ { i }$ ?
A) The normal equations are linear in the unknowns $b _ { 0 } \text { and } b _ { 1 }$
)
B) The least squares estimates are always the unique solution to the system of normal equations.
C) Provided that at least two of the $x _ {i }$
Values are different, the least squares estimates are the unique solution to the system of normal equations.
D) The quantity $\sum y _ { i } ^ { 2 }$
Is not needed to solve the system of normal equations.
E) All of the above statements are true.

Question

Quizplus · Accepted Answer

The least squares estimates are unique only when the $X$ matrix (in this case, derived from the $x_i$ values) has full rank, meaning all columns are linearly independent. This is guaranteed when at least two of the $x_i$ values are different, as stated in option C, but not always, making B not universally true.

Which of the Following Statements Are Not True Regarding the Normal