[link] shows the population, in thousands, of harbor seals in the Wadden Sea over the years 1997 to 2012.
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
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ represent time in years starting with
$\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ for the year 1997. Let
$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ 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?
The logistic regression model that fits these data is
$\text{\hspace{0.17em}}y=\frac{25.65665979}{1+6.113686306{e}^{-0.3852149008x}}.$
If the population continues to grow at this rate, there will be about
$\text{\hspace{0.17em}}\text{25,634}\text{\hspace{0.17em}}$ seals in 2020.
To the nearest whole number, the carrying capacity is 25,657.
Visit
this website for additional practice questions from Learningpod.
Key concepts
Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero.
We use the command “ExpReg” on a graphing utility to fit function of the form
$\text{\hspace{0.17em}}y=a{b}^{x}\text{\hspace{0.17em}}$ to a set of data points. See
[link] .
Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time.
We use the command “LnReg” on a graphing utility to fit a function of the form
$\text{\hspace{0.17em}}y=a+b\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$ to a set of data points. See
[link] .
Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit.
We use the command “Logistic” on a graphing utility to fit a function of the form
$\text{\hspace{0.17em}}y=\frac{c}{1+a{e}^{-bx}}\text{\hspace{0.17em}}$ to a set of data points. See
[link] .
Section exercises
Verbal
What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.
Logistic models are best used for situations that have limited values. For example, populations cannot grow indefinitely since resources such as food, water, and space are limited, so a logistic model best describes populations.
What is regression analysis? Describe the process of performing regression analysis on a graphing utility.
Regression analysis is the process of finding an equation that best fits a given set of data points. To perform a regression analysis on a graphing utility, first list the given points using the STAT then EDIT menu. Next graph the scatter plot using the STAT PLOT feature. The shape of the data points on the scatter graph can help determine which regression feature to use. Once this is determined, select the appropriate regression analysis command from the STAT then CALC menu.