# 4.8 Fitting exponential models to data  (Page 5/12)

 Page 5 / 12
$y=\frac{c}{1+a{e}^{-bx}}$

Recall that:

• $\frac{c}{1+a}\text{\hspace{0.17em}}$ is the initial value of the model.
• when $\text{\hspace{0.17em}}b>0,$ the model increases rapidly at first until it reaches its point of maximum growth rate, $\text{\hspace{0.17em}}\left(\frac{\mathrm{ln}\left(a\right)}{b},\frac{c}{2}\right).\text{\hspace{0.17em}}$ At that point, growth steadily slows and the function becomes asymptotic to the upper bound $\text{\hspace{0.17em}}y=c.$
• $c\text{\hspace{0.17em}}$ 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

$y=\frac{c}{1+a{e}^{-bx}}$

Note that

• The initial value of the model is $\text{\hspace{0.17em}}\frac{c}{1+a}.$
• Output values for the model grow closer and closer to $\text{\hspace{0.17em}}y=c\text{\hspace{0.17em}}$ as time increases.

Given a set of data, perform logistic regression using a graphing utility.

1. Use the STAT then EDIT menu to enter given data.
1. Clear any existing data from the lists.
2. List the input values in the L1 column.
3. List the output values in the L2 column.
2. Graph and observe a scatter plot of the data using the STATPLOT feature.
1. Use ZOOM [9] to adjust axes to fit the data.
2. Verify the data follow a logistic pattern.
3. Find the equation that models the data.
1. Select “Logistic” from the STAT then CALC menu.
2. Use the values returned for $\text{\hspace{0.17em}}a,$ $\text{\hspace{0.17em}}b,$ and $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ to record the model, $\text{\hspace{0.17em}}y=\frac{c}{1+a{e}^{-bx}}.$
4. Graph the model in the same window as the scatterplot to verify it is a good fit for the data.

## Using logistic regression to fit a model to data

Mobile telephone service has increased rapidly in America since the mid 1990s. Today, almost all residents have cellular service. [link] shows the percentage of Americans with cellular service between the years 1995 and 2012 Source: The World Bank, 2013 .

Year 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
1. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ represent time in years starting with $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ for the year 1995. Let $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ represent the corresponding percentage of residents with cellular service. Use logistic regression to fit a model to these data.
2. Use the model to calculate the percentage of Americans with cell service in the year 2013. Round to the nearest tenth of a percent.
3. Discuss the value returned for the upper limit, $\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$ What does this tell you about the model? What would the limiting value be if the model were exact?
1. Using the 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 [link] :

Use the “Logistic” command from the STAT then CALC menu to obtain the logistic model,

$y=\frac{105.7379526}{1+6.88328979{e}^{-0.2595440013x}}$

Next, graph the model in the same window as shown in [link] the scatterplot to verify it is a good fit:

2. To approximate the percentage of Americans with cellular service in the year 2013, substitute $\text{\hspace{0.17em}}x=18\text{\hspace{0.17em}}$ for the in the model and solve for $\text{\hspace{0.17em}}y:$

According to the model, about 98.8% of Americans had cellular service in 2013.

3. The model gives a limiting value of about 105. This means that the maximum possible percentage of Americans with cellular service would be 105%, which is impossible. (How could over 100% of a population have cellular service?) If the model were exact, the limiting value would be $\text{\hspace{0.17em}}c=100\text{\hspace{0.17em}}$ and the model’s outputs would get very close to, but never actually reach 100%. After all, there will always be someone out there without cellular service!

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this