Question 22
A person used a fitness tracker to monitor their hours of sleep and the distance they travelled each day.
| Sleep (hours) | 7 | 7.5 | 8 | 6.5 | 9.5 | 6 | 8.5 |
|---|---|---|---|---|---|---|---|
| Distance (km) | 5 | 7 | 5 | 7 | 4 | 8 | 6 |
Using distance as the response variable, display the data in a scatterplot with labelled axes.
[3 marks]Note: If you make a mistake in the scatterplot, cancel it by ruling a single diagonal line through your work and use the additional response space at the back of this question and response book.
Identify the direction of the association between hours of sleep and distance travelled.
[1 mark]Distance goes up the side, and each column becomes one dot
Watch the response variable claim the vertical axis, the scales go on at equal steps, then the table turn into seven points, one column at a time.
The response variable always goes on the vertical axis. With an equal scale on each axis, the seven columns land as seven dots that drift downward.
Set up the axes on QCAA’s blank grid, then plot all seven points
The grid below is the one in the paper, 31 squares by 22, with the axes placed a few squares in to leave room for the scale. Choose what goes where and how the scale runs. Once the axes are right, plot each column of the table and press Check. Part b) is at the bottom.
Read your scatterplot from left to right. As sleep increases, does distance tend to go up, go down, or neither?
Check the ends of the plot. The least sleep, 6 hours, goes with the longest distance, 8 km. The most sleep, 9.5 hours, goes with the shortest, 4 km. Reading the table in its given order can make the numbers look random. Read the scatterplot instead.
That’s it. As hours of sleep increase, distance travelled tends to decrease, so the association is negative. “Downward” is also accepted. If you found the correlation coefficient on your calculator, it is about , and the sign agrees. Part b) doesn’t need it.
The points are not on a line, but they still drift down from left to right. Direction asks which way the points trend, and how tightly they cluster is the strength. Here the trend is downward, so the direction is negative.
Three marks for the plot, one word for the direction
Try each step yourself before you reveal it. Part a) gives a mark for the axis choice, a mark for scaling and labelling, and a mark for plotting every point. Part b) is one mark.
The question names the response variable for you. The response variable always goes on the vertical axis, and the explanatory variable goes across.
Vertical axis: distance (km). Horizontal axis: sleep (hours).
The guide notes that this mark may be implied by subsequent working. You don’t have to write “response variable” anywhere. A plot with distance up the side earns it.
Marker: correctly identifies the axis for the response variable
The grid is 31 squares wide and 22 high, with nothing on it. Leave a few squares for the scale, then choose equal steps that fit. Sleep goes up in half hours, so give each hour two squares.
$$\text{Sleep: }1\text{ hour}=2\text{ squares}\qquad\text{Distance: }1\text{ km}=2\text{ squares}$$
From 0 to 10 then takes 20 squares on each axis, so both fit, and every half hour lands on a gridline. Write the scale values at equal steps and label the axes with the table’s headings, Sleep (hours) and Distance (km). The guide’s sample labels sleep every hour and distance every 2 km. Any equally spaced scale that fits the data is fine.
Follow-through applies, so if the variables went on the wrong axes in Step 1, a well-scaled, labelled plot can still earn this mark.
Marker: formats scatterplot with appropriate scaling and labelling for both axes
Each column of the table is one point, written as (sleep, distance).
$$(7,\,5),\ (7.5,\,7),\ (8,\,5),\ (6.5,\,7),\ (9.5,\,4),\ (6,\,8),\ (8.5,\,6)$$
Make each point a clear dot and leave them unjoined. Watch the pairs that share a distance: (7, 5) and (8, 5) sit level, and so do (7.5, 7) and (6.5, 7).
Marker: accurately plots all points
Read your scatterplot from left to right. As hours of sleep increase, the distance travelled tends to decrease.
The guide accepts similar statements, such as “downward”. One word is enough here, because the question says identify. The correlation coefficient, $r\approx-0.78$, has the same sign, but you don’t need it.
Marker: identifies direction
Each phrase of the question gave you something
Turning a table of two numerical variables into a scatterplot is routine Unit 3 work, and the question even tells you which variable is the response. Part b) is a one-word answer, and the 2023 subject report listed identifying the direction of an association among the things students did well.
If I had to guess where you might lose marks, it would be the scale. The grid comes blank, so the scaling is yours to choose, and half-hour sleep values punish an uneven one. The 2023 subject report recommended writing scale values at equally spaced increments on each axis so that points can be plotted accurately. Give each hour two squares and every half hour lands on a line. Then label both axes with the variable and its unit, and check each dot against its column before you move on.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and marking-guide steps are from the 2023 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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