Question

serena says that if a regression line is drawn correctly in a scatterplot, there will always be the same number of points above the line as there are below the line. which of these must be a counterexample to her claim? a scatterplot with an odd number of points in which the correct regression line passes directly through 0 of the points a scatterplot with an even number of points in which the correct regression line passes directly through 0 of the points a scatterplot with an odd number of points in which the correct regression line passes directly through 1 of the points a scatterplot with an even number of points in which the correct regression line passes directly through 2 of the points

Explanation:

Step1: Analyze the nature of odd - numbered points

If there is an odd number of points (\(n = 2k + 1,k\in\mathbb{Z}\)) and the regression line passes through \(0\) of the points.
Let the number of points above the line be \(x\) and the number of points below the line be \(y\). Then \(x + y=2k + 1\). Since \(x + y\) is odd, \(x
eq y\).

Step2: Analyze the nature of even - numbered points

If there is an even number of points (\(n = 2k,k\in\mathbb{Z}\)) and the regression line passes through \(0\) of the points, \(x + y=2k\) (where \(x\) is the number of points above the line and \(y\) is the number of points below the line), and \(x=y = k\) is a possibility.
If there is an odd number of points (\(n=2k + 1,k\in\mathbb{Z}\)) and the regression line passes through \(1\) of the points, then \(x + y=2k\), and \(x=y = k\) is a possibility.
If there is an even number of points (\(n = 2k,k\in\mathbb{Z}\)) and the regression line passes through \(2\) of the points, then \(x + y=2k-2\), and \(x=y=k - 1\) is a possibility.

Answer:

a scatterplot with an odd number of points in which the correct regression line passes directly through \(0\) of the points