Note: This particular item has several instructional components and was developed only to be used on the pilot form of the LOCUS Assessments.
The scatterplots below show the number of wins plotted against the number of goals scored by a soccer team and the number of wins plotted against the number of goals allowed by a soccer team.
(a) Compare the associations displayed in the scatterplots. In which scatterplot does the association appear to be the stronger? Explain.
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This part of the question is intended for students that are unfamiliar with the QCR (quadrant count ratio). If you are familiar with the QCR, please proceed to the text just below this box.
One measure of the strength of an association is the QCR (quadrant count ratio). To calculate the QCR, divide the scatterplot into 4 quadrants by drawing a vertical line at the mean value for the explanatory variable (either Goals Scored or Goals Allowed), and by drawing a horizontal line a the mean value of the response variable (number of wins). Then, count the number of points in each quadrant and calculate:
For example, here is the scatterplot for the number of wins plotted against the number of goals scored by a team, along with a vertical line at the mean number of goals scored (38.1) and a horizontal line at the mean number of wins (10.8).
The QCR for the association between Wins and Goals Scored is:
A QCR close to +1 suggests a strong positive association. A positive QCR close to 0 suggests a weak positive association. A QCR close to -1 suggests a strong negative association. A negative QCR close to 0 suggests a weak negative association.
Thus, a QCR of 0.2 is a positive number, and suggests a weak positive association between wins and goals scored.
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The scatterplot below shows the number of wins plotted against the number of goals allowed. The mean number of goals allowed is 38.1 and the mean number of wins is 10.4.
(b) Show that the QCR for the relationship between wins and goals allowed is -0.73.
(c) The QCR between wins and goals scored is 0.2 and the QCR between wins and goals allowed is -0.73. Do these values confirm or contradict your response to part (a)? Explain.