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.
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The least-squares line must pass through at least one data point.
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A. It is false.
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Example. Find the least-squares line for the data
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Solution. Here, we have 
and
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The least-squares line for the data is given by linear equation
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where the constants m and b satisfy the normal equations
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Then, we obtain the normal equations
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Solving them, we found 
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Therefore, the required least-squares line is 
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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.
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B. It is true
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Suppose that we are given two data points
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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 amountsd1,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
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be made as small as possible. But it is possible only in case when least-squares line passes through at least one data point.
Answer: A
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