Question 26·Medium·Two-Variable Data: Models and Scatterplots
A biologist recorded the air temperature and the average number of cricket chirps per minute at several times during an evening. The results are shown below.
| Temperature (°F) | Chirps per minute |
|---|---|
| 60 | 80 |
| 65 | 88 |
| 70 | 95 |
| 75 | 100 |
| 80 | 110 |
Assuming a linear model best describes the relationship between temperature and chirping rate, what is the best estimate of the number of chirps per minute when the temperature is 68°F?
For questions with a roughly linear relationship in a table or scatterplot, quickly estimate the slope using two convenient points (often the endpoints), then write a simple linear equation using one data point. Substitute the given -value to predict , and always sanity-check your answer by making sure it falls between the known data values surrounding that in the table.
Hints
Locate 68°F in the table context
Notice that lies between and . The chirp rate at 68°F should therefore be between the chirp rates given for 65°F and 70°F.
Use a linear pattern
The problem tells you to assume a linear model. That means you can think of the data as lying roughly on a straight line. What is the overall rate of change in chirps per minute as temperature increases from 60°F to 80°F?
Create an equation
Once you know the slope (chirps per degree), use one of the data points to write an equation in the form , where is temperature and is chirps per minute.
Substitute 68°F into your model
After you have an equation, plug in to find the predicted chirps per minute, and then choose the answer choice closest to that value.
Desmos Guide
Enter the data into a table
Create a table in Desmos with as temperature and as chirps per minute, and input the five points: , , , , and .
Fit a linear model (line of best fit)
Below the table, type y1 ~ m x1 + b to let Desmos compute a linear regression. Desmos will display estimated values for (slope) and (y-intercept), giving you an equation of the form that models the data.
Use the model to estimate chirps at 68°F
In a new expression line, type m*68 + b using the and values from the regression. The numerical result is the predicted chirps per minute at ; compare this value to the answer choices.
Step-by-step Explanation
Recognize and model the linear relationship
The table shows that as temperature increases, the chirps per minute also increase in a roughly straight-line pattern. To create a linear model, pick two convenient points from the table, such as and , where the first number is temperature and the second is chirps per minute.
Find the slope (rate of change)
Use the two chosen points to find the slope of the line.
This means the model predicts that for each 1°F increase in temperature, the chirp rate increases by about chirps per minute.
Write an equation for the linear model
Use slope-intercept form and one of the points to find .
Using point :
So an equation that models the data is
where is the temperature in °F and is the chirps per minute.
Predict the chirp rate at 68°F and choose the answer
Substitute into the equation to estimate the chirp rate:
So the best estimate of the number of chirps per minute at is , which corresponds to choice B) 92.