Question 13·Easy·Two-Variable Data: Models and Scatterplots
A beach stand owner recorded the afternoon temperature (in °C) and the number of ice-cream cones sold on eight different days.
| (°C) | 18 | 20 | 22 | 24 | 26 | 28 | 30 | 32 |
|---|---|---|---|---|---|---|---|---|
| (cones) | 53 | 60 | 70 | 79 | 88 | 96 | 104 | 110 |
The owner creates a scatterplot of the data and wants to choose a model to predict future sales. Which type of model is most appropriate for these data?
For scatterplot and data-model questions, first decide the direction of the relationship (do the values increase or decrease as increases?), then look at the shape: are the points roughly on a straight line or do they show clear curvature or rapid change that levels off? Quickly check whether the differences between consecutive values are roughly constant (suggesting linear), clearly growing or shrinking in a curved way (suggesting quadratic), or if the ratios are about constant (suggesting exponential). Once you match direction and shape, eliminate model types that do not fit those features and select the remaining option.
Hints
Look at the direction of change
As the temperature values increase from 18°C to 32°C, do the sales numbers tend to go up or go down? Think about what that tells you about the overall direction of the relationship.
Compare the size of the increases
Check how much the number of cones sold increases when the temperature goes up by 2°C each time. Are these increases roughly similar, getting steadily bigger, or doing something else?
Connect the pattern to common model shapes
Recall that straight-line models have a roughly constant rate of change, exponential decay models quickly decrease and then level off, and upward-opening quadratics curve upward more and more. Which of these general shapes best matches the pattern you see in the data?
Desmos Guide
Enter the data as a table
In Desmos, add a table. In the first column (say ), enter the temperatures 18, 20, 22, 24, 26, 28, 30, 32. In the second column (), enter the corresponding sales values 53, 60, 70, 79, 88, 96, 104, 110. Look at the scatterplot that appears.
Fit a straight-line model and inspect the trend
Below the table, type y1 ~ m x1 + b to perform a linear regression. Desmos will draw the best-fit line and give values for and . Focus on whether the line slopes upward or downward and how well it follows the points; this tells you both the direction and whether a straight-line model is reasonable.
Compare with curved models if you want to check
Optionally, you can try typing y1 ~ a x1^2 + b x1 + c for a quadratic fit or y1 ~ a b^(x1) for an exponential fit. Compare how well each curve follows the plotted points and what general shape it has (increasing, decreasing, curved, straight). Choose the model type whose shape and direction best match the scatter of points.
Step-by-step Explanation
Determine whether the relationship is positive or negative
Look at how (cones sold) changes as (temperature) increases from 18°C to 32°C.
- At 18°C, ; at 20°C, ; at 22°C, ; and so on, up to 110 cones at 32°C.
- Each time increases, also increases.
This means the association between and is positive, not negative.
Check whether the pattern is roughly linear or clearly curved
Compute how much increases each time goes up by 2°C:
These increases are all in the same ballpark (around 7 to 10 cones), so the rate of change is roughly constant, which is what you expect from a straight line. There is no strong sign of the increases getting dramatically larger and larger (which would suggest an upward-opening curve) or of any sudden sharp drop and leveling off (which would suggest decay).
Match the observed pattern to the model type
The data show:
- A positive relationship (more temperature, more sales).
- An approximately constant rate of increase, which corresponds to a straight-line trend rather than a curved or decaying one.
A model that is decreasing with temperature cannot work, and models that are strongly curved (quadratic upward or exponential decay) do not match the nearly constant step-by-step increases. Therefore, the most appropriate choice is D) A positive linear model.