Question 8
The following scatterplot shows a linear association between two numerical variables.
Choose the best description for the direction and strength of the association.
Not quite. Strong is right, but read the plot from left to right: $y$ starts near 28 and ends near 4. As $x$ increases, $y$ decreases, so the direction is negative.
That’s it. Step through the hints below to see why.
Not quite. Both words miss. The points fall from left to right, so the direction is negative, and they stay in a narrow band along a straight line, so the association is strong.
Not quite. Negative is right, but the points aren’t scattered widely. Every point sits close to one straight line from top left to bottom right. A little wobble is normal in real data and still counts as strong.
Scramble the points to weaken the association, or flip the plot to change its direction. Watch what each change does to $r$.
This is the plot from the paper. With the line of best fit on, the blue gaps from each point to the line are all short compared with how far the line drops.
Direction first. Read the plot from left to right, the way $x$ increases.
At $x=1$ the point is up near $y=28$. By $x=10$ it is down near $y=4$. As $x$ increases, $y$ decreases, so the association is negative. That rules out (A) and (C).
Strength is how closely the points follow a straight line. It has nothing to do with how steep the line is. Strength is not steepness.
Picture a thin band drawn around the points. Here it hugs a straight line from the top left to the bottom right, and no point wanders far outside it. A narrow band means a strong association.
The paper gives no data, so you judge by eye. If you did type in the points as drawn, the correlation coefficient would be .
The usual convention calls $|r|$ from 0.75 to 1 strong, 0.5 to 0.75 moderate, 0.25 to 0.5 weak and below 0.25 no association. These cut-offs are a teaching convention, not part of the syllabus. This plot is well inside strong. Slide the scramble to 60% on the left to see what a weak negative plot looks like.
Direction: as $x$ increases, $y$ decreases, so it is negative.
Strength: the points stay in a narrow band around a straight line, so it is strong.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2020 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2020, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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