Determine Whether the Statement Is True or False

Multiple Choice

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. ​
If the data consist of two distinct points, then the least-squares line is just the line that passes through the two points.
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A) It is true.
Suppose that we are given two data points
If we try to fit a straight line to these data points, the line will miss the first and the second data points by the amounts d1, d2, respectively.
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The principle of least squares states that the straight line L that fits the data points best is the one chosen by requiring that the sum of the squares
Be made as small as possible. In this case the smallest sum is zero. , when and .
Therefore, the required least-squares line is just the line that passes through the two points.
B) It is false.
Example: Find the least-squares line for the data Solution: Here, we have n = 2 and
The least-squares line for the data is given by linear equation y = f(x) = mx + b
Where the constants m and b satisfy the normal equations Then, we obtain the normal equations Solving them, we found Therefore, the required least-squares line is y = x + 0.4. The scatter diagram and the least-squares line are shown in the figure. We can see that the line does not pass through any data point.