# 1.3: Models and Data

**At Grade**Created by: CK-12

## Learning Objectives

A student will be able to:

- Fit data to linear models.
- Fit data to quadratic models.
- Fit data to trigonometric models.
- Fit data to exponential growth and decay models.

## Introduction

In our last lesson we examined functions and learned how to classify and sketch functions. In this lesson we will use some classic functions to model data. The lesson will be a set of examples of each of the models. For each, we will make extensive use of the graphing calculator.

Let’s do a quick review of how to model data on the graphing calculator.

*Enter Data in Lists*

Press **[STAT]** and then **[EDIT]** to access the lists, **L1 - L6**.

*View a Scatter Plot*

Press **2nd [STAT PLOT]** and choose accordingly.

Then press **[WINDOW]** to set the limits of the axes.

*Compute the Regression Equation*

Press **[STAT]** then choose **[CALC]** to access the regression equation menu. Choose the appropriate regression equation (Linear, Quad, Cubic, Exponential, Sine).

*Graph the Regression Equation Over Your Scatter Plot*

Go to **Y**=> **[MENU]** and clear equations. Press **[VARS]**, then enter \begin{align*}5\end{align*} and **EQ** and press **[ENTER]** (This series of entries will copy the regression equation to your **Y =** screen.) Press **[GRAPH]** to view the regression equation over your scatter plot

*Plotting and Regression in Excel*

You can also do regression in an Excel spreadsheet. To start, copy and paste the table of data into Excel. With the two columns highlighted, including the column headings, click on the **Chart** icon and select **XY scatter**. Accept the defaults until a graph appears. Select the graph, then click **Chart**, then **Add Trendline**. From the choices of trendlines choose **Linear**.

Now let’s begin our survey of the various modeling situations.

*Linear Models*

For these kinds of situations, the data will be modeled by the classic linear equation \begin{align*}y = mx + b.\end{align*} Our task will be to find appropriate values of \begin{align*}m\end{align*} and \begin{align*}b\end{align*} for given data.

**Example 1:**

It is said that the height of a person is equal to his or her wingspan (the measurement from fingertip to fingertip when your arms are stretched horizontally). If this is true, we should be able to take a table of measurements, graph the measurements in an \begin{align*}x-y\end{align*} coordinate system, and verify this relationship. What kind of graph would you expect to see? ** (Answer: You would expect to see the points on the line** \begin{align*}y = x\end{align*}.)

Suppose you measure the height and wingspans of nine of your classmates and gather the following data. Use your graphing calculator to see if the following measurements fit this linear model (the line \begin{align*}y = x\end{align*}).

Height (inches) |
Wingspan (inches) |
---|---|

\begin{align*}67\end{align*} | \begin{align*}65\end{align*} |

\begin{align*}64\end{align*} | \begin{align*}63\end{align*} |

\begin{align*}56\end{align*} | \begin{align*}57\end{align*} |

\begin{align*}60\end{align*} | \begin{align*}61\end{align*} |

\begin{align*}62\end{align*} | \begin{align*}63\end{align*} |

\begin{align*}71\end{align*} | \begin{align*}70\end{align*} |

\begin{align*}72\end{align*} | \begin{align*}69\end{align*} |

\begin{align*}68\end{align*} | \begin{align*}67\end{align*} |

\begin{align*}65\end{align*} | \begin{align*}65\end{align*} |

We observe that only one of the measurements has the condition that they are equal. Why aren’t more of the measurements equal to each other? *(Answer: The data do not always conform to exact specifications of the model. For example, measurements tend to be loosely documented so there may be an error arising in the way that measurements were taken.)*

We enter the data in our calculator in **L1** and **L2**. We then view a scatter plot. (Caution: note that the data ranges exceed the viewing window range of \begin{align*}[-10, 10].\end{align*} Change the window ranges accordingly to include all of the data, say \begin{align*}[40, 80].\end{align*})

Here is the scatter plot:

Now let us compute the regression equation. Since we expect the data to be linear, we will choose the **linear regression** option from the menu. We get the equation \begin{align*}y = .76x + 14.\end{align*}

In general we will always wish to graph the regression equation over our data to see the goodness of fit. Doing so yields the following graph, which was drawn with Excel:

Since our calculator will also allow for a variety of non-linear functions to be used as models, we can therefore examine quite a few real life situations. We will first consider an example of quadratic modeling.

*Quadratic Models*

**Example 2:**

The following table lists the number of Food Stamp recipients (in millions) for each year after 1990.

years after 1990 |
Participants |
---|---|

\begin{align*}1\end{align*} | \begin{align*}22.6\end{align*} |

\begin{align*}2\end{align*} | \begin{align*}25.4\end{align*} |

\begin{align*}3\end{align*} | \begin{align*}27.0\end{align*} |

\begin{align*}4\end{align*} | \begin{align*}27.5\end{align*} |

\begin{align*}5\end{align*} | \begin{align*}26.6\end{align*} |

\begin{align*}6\end{align*} | \begin{align*}25.5\end{align*} |

\begin{align*}7\end{align*} | \begin{align*}22.5\end{align*} |

\begin{align*}8\end{align*} | \begin{align*}19.8\end{align*} |

\begin{align*}9\end{align*} | \begin{align*}18.2\end{align*} |

\begin{align*}10\end{align*} | \begin{align*}17.2\end{align*} |

We enter the data in our calculator in **L3** and **L4** (that enables us to save the last example’s data). We then will view a scatter plot. Change the window ranges accordingly to include all of the data. Use \begin{align*}[-2, 10]\end{align*} for \begin{align*}x\end{align*} and \begin{align*}[-2, 30]\end{align*} for \begin{align*}y.\end{align*}

Here is the scatter plot:

Now let us compute the regression equation. Since our scatter plot suggests a quadratic model for the data, we will choose **Quadratic Regression** from the menu. We get the equation:

\begin{align*}y = -0.30x^2 + 2.38x + 21.67.\end{align*}

Let’s graph the equation over our data. We see the following graph:

*Trigonometric Models*

The following example shows how a trigonometric function can be used to model data.

**Example 3:**

With the skyrocketing cost of gasoline, more people have looked to mass transit as an option for getting around. The following table uses data from the *American Public Transportation Association* to show the number of mass transit trips (in billions) between 1992 and 2000.

year |
Trips (billions) |
---|---|

1992 | \begin{align*}8.5\end{align*} |

1993 | \begin{align*}8.2\end{align*} |

1994 | \begin{align*}7.93\end{align*} |

1995 | \begin{align*}7.8\end{align*} |

1996 | \begin{align*}7.87\end{align*} |

1997 | \begin{align*}8.23\end{align*} |

1998 | \begin{align*}8.6\end{align*} |

1999 | \begin{align*}9.08\end{align*} |

2000 | \begin{align*}9.4\end{align*} |

We enter the data in our calculator in **L5** and **L6**, starting in L5 with the number one for 1992 (the first year). We then will view a scatter plot. Change the window ranges accordingly to include all of the data. Use \begin{align*}[-2, 10]\end{align*} for both \begin{align*}x\end{align*} and \begin{align*}y\end{align*} ranges.

Here is the scatter plot:

Now let us compute the regression equation. Since our scatter plot suggests a sine model for the data, we will choose **Sine Regression** from the menu. We get the equation:

\begin{align*}y = .9327 * \sin(.4681x + 2.8734) + 8.7358.\end{align*}

Let us graph the equation over our data. We see the following graph:

This example suggests that the sine over time \begin{align*}t\end{align*} is a function that is used in a variety of modeling situations.

**Caution:** Although the fit to the data appears quite good, do we really expect the number of trips to continue to go up and down in the future? Probably not. Here is what the graph looks like when projected an additional ten years:

*Exponential Models*

Our last class of models involves exponential functions. Exponential models can be used to model growth and decay situations. Consider the following data about the declining number of farms for the years 1980 - 2005.

**Example 4:**

The number of dairy farms has been declining over the past \begin{align*}20+\end{align*} years. The following table charts the decline:

Year |
Farms (thousands) |
---|---|

1980 | \begin{align*}334\end{align*} |

1985 | \begin{align*}269\end{align*} |

1990 | \begin{align*}193\end{align*} |

1995 | \begin{align*}140\end{align*} |

2000 | \begin{align*}105\end{align*} |

2005 | \begin{align*}67\end{align*} |

We enter the data in our calculator in L5 (entering the year 1980 as 1, the year 1985 as 2, etc.) and L6. We then will view a scatter plot. Change the window ranges accordingly to include all of the data. For the large \begin{align*}y-\end{align*}values, choose the range \begin{align*}[-50, 350]\end{align*} with a scale of \begin{align*}25.\end{align*}

Here is the scatter plot:

Now let us compute the regression equation. Since our scatter plot suggests an exponential model for the data, we will choose **Exponential Regression** from the menu. We get the equation: \begin{align*}y = 490.6317 * .7266^x\end{align*}

Let’s graph the equation over our data. We see the following graph:

In the homework we will practice using our calculator extensively to model data.

## Lesson Summary

- Fit data to linear models.
- Fit data to quadratic models.
- Fit data to trigonometric models.
- Fit data to exponential growth and decay models.

## Review Questions

- Consider the following table of measurements of circular objects:
- Make a scatter plot of the data.
- Based on your plot, which type of regression will you use?
- Find the line of best fit.
- Comment on the values of \begin{align*}m\end{align*} and \begin{align*}b\end{align*} in the equation.

Object |
Diameter (cm) |
Circumference (cm) |
---|---|---|

Glass | \begin{align*}8.3\end{align*} | \begin{align*}26.5\end{align*} |

Flashlight | \begin{align*}5.2\end{align*} | \begin{align*}16.7\end{align*} |

Aztec calendar | \begin{align*}20.2\end{align*} | \begin{align*}61.6\end{align*} |

Tylenol bottle | \begin{align*}3.4\end{align*} | \begin{align*}11.6\end{align*} |

Popcorn can | \begin{align*}13\end{align*} | \begin{align*}41.4\end{align*} |

Salt shaker | \begin{align*}6.3\end{align*} | \begin{align*}20.1\end{align*} |

Coffee canister | \begin{align*}11.3\end{align*} | \begin{align*}35.8\end{align*} |

Cat food bucket | \begin{align*}33.5\end{align*} | \begin{align*}106.5\end{align*} |

Dinner plate | \begin{align*}27.3\end{align*} | \begin{align*}85.6\end{align*} |

Ritz cracker | \begin{align*}4.9\end{align*} | \begin{align*}15.5\end{align*} |

- Manatees are large, gentle sea creatures that live along the Florida coast. Many manatees are killed or injured by power boats. Here are data on powerboat registrations (in thousands) and the number of manatees killed by boats in Florida from 1987 - 1997.
- Make a scatter plot of the data.
- Use linear regression to find the line of best fit.
- Suppose in the year 2000, powerboat registrations increase to \begin{align*}700,000\end{align*}. Predict how many manatees will be killed.

Year |
Boats |
Manatees killed |
---|---|---|

1987 | \begin{align*}447\end{align*} | \begin{align*}13\end{align*} |

1988 | \begin{align*}460\end{align*} | \begin{align*}21\end{align*} |

1989 | \begin{align*}480\end{align*} | \begin{align*}24\end{align*} |

1990 | \begin{align*}497\end{align*} | \begin{align*}16\end{align*} |

1991 | \begin{align*}512\end{align*} | \begin{align*}24\end{align*} |

1992 | \begin{align*}513\end{align*} | \begin{align*}21\end{align*} |

1993 | \begin{align*}526\end{align*} | \begin{align*}15\end{align*} |

1994 | \begin{align*}557\end{align*} | \begin{align*}33\end{align*} |

1995 | \begin{align*}585\end{align*} | \begin{align*}34\end{align*} |

1996 | \begin{align*}614\end{align*} | \begin{align*}34\end{align*} |

1997 | \begin{align*}645\end{align*} | \begin{align*}39\end{align*} |

- A passage in
states that the measurement of “Twice around the wrist is once around the neck.” The table below contains the wrist and neck measurements of \begin{align*}10\end{align*}people.*Gulliver’s Travels*- Make a scatter plot of the data.
- Find the line of best fit and comment on the accuracy of the quote from the book.
- Predict the distance around the neck of Gulliver if the distance around his wrist is found to be \begin{align*}52 \;\mathrm{cm}\end{align*}.

Wrist (cm) |
Neck (cm) |
---|---|

\begin{align*}17.9\end{align*} | \begin{align*}39.5\end{align*} |

\begin{align*}16\end{align*} | \begin{align*}32.5\end{align*} |

\begin{align*}16.5\end{align*} | \begin{align*}34.7\end{align*} |

\begin{align*}15.9\end{align*} | \begin{align*}32\end{align*} |

\begin{align*}17\end{align*} | \begin{align*}33.3\end{align*} |

\begin{align*}17.3\end{align*} | \begin{align*}32.6\end{align*} |

\begin{align*}16.8\end{align*} | \begin{align*}33\end{align*} |

\begin{align*}17.3\end{align*} | \begin{align*}31.6\end{align*} |

\begin{align*}17.7\end{align*} | \begin{align*}35\end{align*} |

\begin{align*}16.9\end{align*} | \begin{align*}34\end{align*} |

- The following table gives women’s average percentage of men’s salaries for the same jobs for each \begin{align*}5\end{align*}-year period from 1960 - 2005.
- Make a scatter plot of the data.
- Based on your sketch, should you use a linear or quadratic model for the data?
- Find a model for the data.
- Can you explain why the data seems to dip at first and then grow?

Year |
Percentage |
---|---|

1960 | \begin{align*}42\end{align*} |

1965 | \begin{align*}36\end{align*} |

1970 | \begin{align*}30\end{align*} |

1975 | \begin{align*}37\end{align*} |

1980 | \begin{align*}41\end{align*} |

1985 | \begin{align*}42\end{align*} |

1990 | \begin{align*}48\end{align*} |

1995 | \begin{align*}55\end{align*} |

2000 | \begin{align*}58\end{align*} |

2005 | \begin{align*}60\end{align*} |

- Based on the model for the previous problem, when will women make as much as men? Is your answer a realistic prediction?
- The average price of a gallon of gas for selected years from 1975 - 2008 is given in the following table:
- Make a scatter plot of the data.
- Based on your sketch, should you use a linear, quadratic, or cubic model for the data?
- Find a model for the data.
- If gas continues to rise at this rate, predict the price of gas in the year 2012.

Year |
Cost |
---|---|

1975 | \begin{align*}1\end{align*} |

1976 | \begin{align*}1.75\end{align*} |

1981 | \begin{align*}2\end{align*} |

1985 | \begin{align*}2.57\end{align*} |

1995 | \begin{align*}2.45\end{align*} |

2005 | \begin{align*}2.75\end{align*} |

2008 | \begin{align*}3.45\end{align*} |

- For the previous problem, use a linear model to analyze the situation. Does the linear method provide a better estimate for the predicted cost for the year 2012? Why or why not?
- Suppose that you place \begin{align*}\$1,000\end{align*}in a bank account where it grows exponentially and is compounded annually over the course of six years. The table below shows the amount of money you have at the end of each year.
- Find the exponential model.
- In what year will you triple your original amount?

Year |
Amount |
---|---|

\begin{align*}1000\end{align*} | |

\begin{align*}1\end{align*} | \begin{align*}1127.50\end{align*} |

\begin{align*}2\end{align*} | \begin{align*}1271.24\end{align*} |

\begin{align*}3\end{align*} | \begin{align*}1433.33\end{align*} |

\begin{align*}4\end{align*} | \begin{align*}1616.07\end{align*} |

\begin{align*}5\end{align*} | \begin{align*}1822.11\end{align*} |

\begin{align*}6\end{align*} | \begin{align*}2054.43\end{align*} |

- Suppose that in the previous problem, you started with \begin{align*}\$3,000\end{align*}but maintained the same interest rate.
- Give a formula for the exponential model. (Hint: note the coefficient in the previous answer!)
- How long will it take for the initial amount, \begin{align*}\$3,000\end{align*}, to triple? Explain your answer.

- The following table gives the average daily temperature for Indianapolis, Indiana for each month of the year:
- Construct a scatter plot of the data.
- Find the sine model for the data.

Month |
Avg Temp (F) |
---|---|

Jan | \begin{align*}22\end{align*} |

Feb | \begin{align*}26.3\end{align*} |

March | \begin{align*}37.8\end{align*} |

April | \begin{align*}51\end{align*} |

May | \begin{align*}61.7\end{align*} |

June | \begin{align*}75.3\end{align*} |

July | \begin{align*}78.5\end{align*} |

Aug | \begin{align*}84.3\end{align*} |

Sept | \begin{align*}68.5\end{align*} |

Oct | \begin{align*}53.2\end{align*} |

Nov | \begin{align*}38.7\end{align*} |

Dec | \begin{align*}26.6\end{align*} |

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