Determine Whether the Statement Is True or False

question 37

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.

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Definitions:

Ex Post Facto

A study or analysis conducted after an event has occurred, used to discover causes even though variables are not manipulated by the researcher.

Pretest Posttest

A research design that measures the effects of an intervention by comparing subject scores before and after the intervention.

Experimental Design

The blueprint for conducting a scientific study to ensure that it is valid and reliable; it includes planning, conducting, and analyzing an experiment.

Two-by-two

A 2x2 matrix or table used to categorize or compare data points based on two binary criteria, often used in research for analysis or decision-making.