3.2: Fitting Linear Models to Data (2024)

To create a scatter plot on Desmos

1. Open your Desmos app, or go to https://www.desmos.com/calculator.
2. Above the equation list, click the + icon.
3. Click "table."
4. Enter the x-values in the first column and the y-values in the second column.
5. Click the magnifying glass to automatically adjust the viewing window to see the points. You can also manually adjust the window as necessary.

Example \(\PageIndex{1}\): Using a Scatter Plot to Investigate Cricket Chirps

The table below shows the number of cricket chirps in 15 seconds, for several different air temperatures, in degrees Fahrenheit. Plot this data and determine whether the data appears to be linearly related.

Solution

Plotting this data, as depicted in Figure \(\PageIndex{2}\) suggests that there may be a trend. We can see from the trend in the data that the number of chirps increases as the temperature increases. The trend appears to be roughly linear, though certainly not perfectly so.

Example \(\PageIndex{2}\): "Eyeballing" a Line of Best Fit

Find a linear function that fits the data in Table \(\PageIndex{1}\) by “eyeballing” a line that seems to fit.

Solution

On a graph, we could try sketching a line.

Using the starting and ending points of our hand drawn line, points \((0, 30)\) and \((50, 90)\), this graph has a slope of

\[m=\dfrac{60}{50}=1.2\nonumber\]

and a y-intercept at 30. This gives an equation of

\[T(c)=1.2c+30\nonumber\]

where \(c\) is the number of chirps in 15 seconds, and \(T(c)\) is the temperature in degrees Fahrenheit. The resulting equation is represented in Figure \(\PageIndex{3}\).

Analysis

This linear equation can then be used to approximate answers to various questions we might ask about the trend.

Exercise \(\PageIndex{1}\)

According to the data from Table \(\PageIndex{1}\), what temperature can we predict it is if we counted 29 chirps in 15 seconds?

Solution

According to the line the temperature would be somewhere in the low 60s, approximately 63°F.

To find a line of best fit using Desmos

1. Open your Desmos app, or go to https://www.desmos.com/calculator.
2. Create a scatter plot, as listed above.
3. After the xy table, in the space for the next equation, write \(y_1\)~\(mx_1+b\). This tells the program to use the values of \(x_1\) and \(y_1\) from the table above.
   • The ~ sign tells the program to approximate the equation, since a linear equation won't fit the data precisely.
   • The \(x_1\) and \(y_1\) tell the program to use the values of \(x_1\) and \(y_1\) from the table above.
   • The \(m\) and \(b\) represent slope and y-intercept of the line, respectively.
4. Press Enter.
5. The resulting information will display the appropriate slope (\(m\)) and y-intercept (\(b\)) values of the line of best fit.
   • The information will also display statistics of \(r^2\) and \(r\). An r-value of 1 will indicate an exact fit. The closer the r-value is to 1, the closer the data fits the line of best fit.
6. For a video explanation of this process, click HERE. The first 53 seconds show how to enter a scatter plot. The line of best fit begins at the 0:53 mark.

Example \(\PageIndex{3}\): Finding a Linear Regression Equation

Find the line of best fit using the cricket-chirp data in Table \(\PageIndex{1}\).

Solution

Enter the input (chirps) in \(x_1\). Enter the output (temperature) in \(y_1\).

Enter \(y_1~mx_1+b\) in the next equation space.

Note the m and b values for the data.

Therefore we obtain the equation:

\[y=1.143x+30.281 \nonumber \]

or

\[T=1.143c+30.281 \nonumber \]

Example \(\PageIndex{4}\): Using a Regression Line to Make Predictions

Gasoline consumption in the United States has been steadily increasing. Consumption data from 1994 to 2004 is shown in Table \(\PageIndex{3}\).

a. Graph the data and determine whether the trend is linear. If so, find a model for the data.

b. Use the model to predict the consumption in 2008.

c. Use the model to predict when oil consumption would reach 160 billion gallons.

a. The scatter plot of the data, including the least squares regression line, is shown in Figure \(\PageIndex{5}\).

We can introduce new input variable, \(t\),representing years since 1994.

The linear regression equation is:

\[C=113.318+2.209t \nonumber \]

b. Using this to predict consumption in 2008, set \(t=14\).

\[\begin{align*} C&=113.318+2.209(14) \\ &=144.244 \end{align*} \]

The model predicts 144.244 billion gallons of gasoline consumption in 2008.

c. Using the model to predict time when consumption is 160 billion gallons, set \(C=160\).

\[\begin{align*} 160&=113.318+2.209(t) \\ 46.682&=2.209(t) \\ 21.133 &=t \end{align*} \]

21 years after 1994 is the year 2015, so oil consumption is expected to reach 160 billion gallons in 2015.

Exercise \(\PageIndex{2}\)

Use the model we created using technology in Example \(\PageIndex{4}\) to predict the gas consumption in 2011.

Answer

150.871 billion gallons