$\frac{c}{1+a}\text{\hspace{0.17em}}$ 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.
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.
Find the equation that models the data.
Select “Logistic” from the STAT then CALC menu.
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,
$\text{\hspace{0.17em}}y=\frac{c}{1+a{e}^{-bx}}.$
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
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.
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,
$\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$ What does this tell you about the model? What would the limiting value be if the model were exact?
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,
Next, graph the model in the same window as shown in
[link] the scatterplot to verify it is a good fit:
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.
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!
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Someone should please solve it for me
Add 2over ×+3 +y-4 over 5
simplify (×+a)with square root of two -×root 2 all over a
multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15
Second one, I got Root 2
Third one, I got 1/(y to the fourth power)
I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
graph the following linear equation using intercepts method.
2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b
you were already given the 'm' and 'b'.
so..
y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line.
where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2
2=3x
x=3/2
then .
y=3/2X-2
I think
Given
co ordinates for x
x=0,(-2,0)
x=1,(1,1)
x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
I've run into this:
x = r*cos(angle1 + angle2)
Which expands to:
x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2))
The r value confuses me here, because distributing it makes:
(r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1))
How does this make sense? Why does the r distribute once
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis
vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0