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.

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.

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.

Assume That a Certain Commodity's Demand Equation Has the Form

Multiple Choice

Correct Answer:VerifiedUnlock this answer nowGet Access to more Verified Answers free of chargeAccess For Free

Correct Answer:
Verified
Unlock this answer now
Get Access to more Verified Answers free of charge