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:

Analyze Serena's claim

Using the Scatterplot Analysis and Line of Best Fit knowledge points

Serena claims that a correctly drawn regression line (line of best fit) in a scatterplot will always have the same number of points above the line as below the line. Let \(N\) be the total number of points in the scatterplot, \(A\) be the number of points above the line, \(B\) be the number of points below the line, and \(O\) be the number of points directly on the line. Thus, the total number of points is:

$$ N = A + B + O $$

Serena's claim is that \(A = B\) must always hold true. A counterexample must be a scenario where \(A
eq B\) is guaranteed to occur.

Evaluate an odd number of points with zero points on the line

Using the Scatterplot Analysis knowledge point

Let's test the first option: a scatterplot with an odd number of points (\(N\) is odd) where the regression line passes through 0 points (\(O = 0\)).

$$ N = A + B + 0 \implies N = A + B $$

Since \(N\) is an odd integer, the sum of two integers \(A\) and \(B\) is odd. An odd number cannot be divided into two equal integers:

$$ A eq B $$

Therefore, it is mathematically impossible for the number of points above the line to equal the number of points below the line. This scenario must be a counterexample to Serena's claim.

Evaluate other options for consistency

Using the Scatterplot Analysis knowledge point

Let's check why the other options do not guarantee a counterexample:

• Even points, 0 on the line: \(N\) is even, \(O = 0\). We can have \(A + B = N\). Since \(N\) is even, \(A = B = \frac{N}{2}\) is possible.
• Odd points, 1 on the line: \(N\) is odd, \(O = 1\). We have \(A + B = N - 1\). Since \(N\) is odd, \(N - 1\) is even, so \(A = B = \frac{N-1}{2}\) is possible.
• Even points, 2 on the line: \(N\) is even, \(O = 2\). We have \(A + B = N - 2\). Since \(N\) is even, \(N - 2\) is even, so \(A = B = \frac{N-2}{2}\) is possible.

Answer:

• (A) a scatterplot with an odd number of points in which the correct regression line passes directly through 0 of the points (Correct answer)
• (B) a scatterplot with an even number of points in which the correct regression line passes directly through 0 of the points
• (C) a scatterplot with an odd number of points in which the correct regression line passes directly through 1 of the points
• (D) a scatterplot with an even number of points in which the correct regression line passes directly through 2 of the points