Determine Whether the Statement Is True or False

question 73

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. The least-squares line must pass through at least one data point.

• A) It is false. Example. Find the least-squares line for the data Solution. Here, we have n = 5 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 m = 1, b = 0.4 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.
• B) 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. But it is possible only in case when least-squares line passes through at least one data point.

---

Definitions:

Difference

The quantitative distance or discrepancy between two values or results.

Test-Retest Reliability

The consistency of a measure or test over time, assessed by administering the same test to the same group on two different occasions.

Scores

Represents numerical expressions of performance or achievement often used in assessments and research.

Time 1

Refers to the initial point or first measurement time in a longitudinal study or time-series analysis.