Question 23·Medium·Two-Variable Data: Models and Scatterplots
Eight students recorded the number of hours they slept the night before an exam () and their scores on the exam (). Their results are shown below.
| Hours of sleep | 4 | 5 | 6 | 6 | 7 | 8 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|
| Test score | 65 | 67 | 70 | 72 | 75 | 80 | 82 | 85 |
Based on the scatterplot these data would produce, which statement best describes the relationship between and ?
For scatterplot description questions, quickly analyze three things: direction, form, and strength. First, see whether tends to increase or decrease as increases (positive vs. negative). Next, decide if the points roughly follow a straight line (linear) or a curve (nonlinear). Finally, judge strength by how tightly the points cluster around that pattern. Then pick the answer choice that combines the correct direction, form, and strength without actually needing to draw the scatterplot.
Hints
Look at how the two variables move together
As you move from left to right across the hours-of-sleep row, what generally happens to the test scores in the row below—do they go up, go down, or stay random?
Consider how clear the pattern is
If you plotted these pairs as points, would they seem to follow a clear trend, or would they be scattered without any obvious pattern?
Think about the shape of the trend
Imagine drawing a smooth line that best follows the points. Would that line be mostly straight, or would it need to curve to fit the pattern?
Desmos Guide
Enter the data into a table
In Desmos, add a table and type the hours of sleep into the column: 4, 5, 6, 6, 7, 8, 8, 9. Then type the corresponding scores into the column: 65, 67, 70, 72, 75, 80, 82, 85. Desmos will automatically plot these as points.
Observe direction and strength
Look at how the plotted points move from left to right: do they tend to go upward or downward, and are they close to a single trend or widely scattered? Use this to decide whether the relationship is positive or negative, and whether it looks strong or weak.
Check the shape (linear vs. nonlinear)
Visually imagine the best-fitting smooth line through the points. Ask yourself: would a straight line fit them well, or would you need a curved line to follow the pattern? Based on this, choose the option that best describes both the direction, the strength, and whether the pattern is straight or curved.
Step-by-step Explanation
Visualize the scatterplot from the table
Each pair in the table would become a point on the scatterplot:
- and
- and
Imagine these as dots, where is on the horizontal axis (hours of sleep) and is on the vertical axis (test score).
Determine the direction of the relationship
Look at how changes as increases.
- When goes from to , goes from to .
- At every step, when hours of sleep go up, the test score also goes up (or at least does not go down).
So the pattern of points would go upward from left to right, which means the relationship is positive, not negative.
Judge how strong the relationship is
Check how regular and consistent the pattern is:
- As increases by hour (or sometimes hours), the score increases by only a few points each time (for example, to , to about –, then to , then to –, then to ).
- There are no points far away from the general upward pattern.
Because the points would cluster closely around a clear trend, the relationship is strong, not weak or “little to no relationship.”
Decide if the pattern is linear or nonlinear and match the choice
Ask whether the points would follow a straight line or a curved pattern:
- The amount the score increases for each extra hour of sleep is roughly similar across the table; it does not suddenly speed up or slow down in a curved way.
- That means the pattern would look like a rough straight line on the scatterplot, not a curve.
You have now identified a relationship that is positive, strong, and linear, which matches answer choice A) There is a strong positive linear relationship.