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:

Environmental Quality

The condition of the environment, considering factors like air and water purity, habitat health, and the presence of contaminants.

Richer People

Individuals or groups who have a significant amount of wealth or assets compared to the average population.

Environmental Degradation

The deterioration of the environment through depletion of resources, destruction of ecosystems, and pollution.

Property Rights

The legal rights to use, control, and derive benefit from property, whether it is tangible or intangible.