## Use Data to Build a Logistic Model

Like exponential and logarithmic growth, logistic growth increases over time. One of the most notable differences with logistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or limiting value. Because of this, logistic regression is best for modeling phenomena where there are limits in expansion, such as availability of living space or nutrients.

It is worth pointing out that logistic functions actually model resource-limited exponential growth. There are many examples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and even stains in fabric. When performing logistic regression analysis, we use the form most commonly used on graphing utilities:

Recall that:

- is the initial value of the model.
- when , the model increases rapidly at first until it reaches its point of maximum growth rate, . At that point, growth steadily slows and the function becomes asymptotic to the upper bound .
- is the limiting value, sometimes called the carrying capacity, of the model.

#### Logistic Regression

Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows to an upper limit. We use the command "Logistic" on a graphing utility to fit a logistic function to a set of data points. This returns an equation of the form

- The initial value of the model is .
- Output values for the model grow closer and closer to as time increases.

##### How To

Given a set of data, perform logistic regression using a graphing utility.- Use the
**STAT**then**EDIT**menu to enter given data.- Clear any existing data from the lists.
- List the input values in the L1 column.
- List the output values in the L2 column.

- Graph and observe a scatter plot of the data using the
**STATPLOT**feature.- Use
**ZOOM [9]**to adjust axes to fit the data. - Verify the data follow a logistic pattern.

- Use
- Find the equation that models the data.
- Graph the model in the same window as the scatterplot to verify it is a good fit for the data.

##### Example 3

Using Logistic Regression to Fit a Model to DataYear | Americans with Cellular Service (%) | Year | Americans with Cellular Service (%) |
---|---|---|---|

1995 | 12.69 | 2004 | 62.852 |

1996 | 16.35 | 2005 | 68.63 |

1997 | 20.29 | 2006 | 76.64 |

1998 | 25.08 | 2007 | 82.47 |

1999 | 30.81 | 2008 | 85.68 |

2000 | 38.75 | 2009 | 89.14 |

2001 | 45.00 | 2010 | 91.86 |

2002 | 49.16 | 2011 | 95.28 |

2003 | 55.15 | 2012 | 98.17 |

ⓐ Let represent time in years starting with for the year 1995. Let represent the corresponding percentage of residents with cellular service. Use logistic regression to fit a model to these data.

ⓑ Use the model to calculate the percentage of Americans with cell service in the year 2013. Round to the nearest tenth of a percent.

ⓒ Discuss the value returned for the upper limit, . What does this tell you about the model? What would the limiting value be if the model were exact?

*Solution*

**STAT**then

**EDIT**menu on a graphing utility, list the years using values 0–15 in L1 and the corresponding percentage in L2. Then use the

**STATPLOT**feature to verify that the scatterplot follows a logistic pattern as shown in Figure 5:

**Logistic**" command from the

**STAT**then

**CALC**menu to obtain the logistic model,

##### Try It #3

Year | Seal Population (Thousands) | Year | Seal Population (Thousands) |
---|---|---|---|

1997 | 3.493 | 2005 | 19.590 |

1998 | 5.282 | 2006 | 21.955 |

1999 | 6.357 | 2007 | 22.862 |

2000 | 9.201 | 2008 | 23.869 |

2001 | 11.224 | 2009 | 24.243 |

2002 | 12.964 | 2010 | 24.344 |

2003 | 16.226 | 2011 | 24.919 |

2004 | 18.137 | 2012 | 25.108 |

ⓐ Let represent time in years starting with for the year 1997. Let represent the number of seals in thousands. Use logistic regression to fit a model to these data.

ⓑ Use the model to predict the seal population for the year 2020.

ⓒ To the nearest whole number, what is the limiting value of this model?

Source: Rice University, https://openstax.org/books/college-algebra/pages/6-8-fitting-exponential-models-to-data

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