Question 1149659: Assessment Tool
Data for the number of facebook users can be readily found on the internet. Below is listed facebook user numbers from the years 2004 to 2016. Answer the following questions.
YEAR 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016
1 6 12 58 145 360 608 845 1056 1230 1440 1710 2000
USERS (in millions)
Enter this data into your calculator and view the scatterplot. Use years since 2000 as your independent variable. You can use your calculator to find the answers to any of these questions.
Exponential Model
Logistic Model
a)
Fit both an exponential model and a logistic model to the data; sketch the graphs of those models on the scatter plots above.
b)
Which of the two models above best fits the data? Write the equation of that model.
c)
Based on this model, what will be the predicted number of Facebook users in the year 2025? Include units.
d)
is the value in part (c) a reasonable or unreasonable prediction for the number of Facebook users in 2025? Explain
why or why not.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Part A
Scatter Plot and Regression Curves
Image generated by GeoGebra (free graphing software).
I used the FitExp and FitLogistic functions in GeoGebra to get the exponential curve (green) and the logistic curve (blue) respectively.
This link
http://pehs.psd202.org/documents/tfrey/1506536379.pdf
provides a guide on how to use a TI83 or TI84 calculator to do exponential regression.
This link
https://education.ti.com/html/t3_free_courses/calculus84_online/mod05/mod05_lesson2.html
provides a guide on how to use a TI83 or TI84 calculator to do logistic regression.
I'll stick with using GeoGebra since it offers more dynamic options.
Green curve = exponential curve = f(x) = 0.55e^(0.59x)
Blue curve = logistic curve = g(x) = 2210.36/(1 + 477.27e^(-0.5x))
Decimal values listed for each function are approximate values.
Table of Values
x = number of years since 2000
y = number of users in millions
| x | y | Point |
| 4 | 1 | A |
| 5 | 6 | B |
| 6 | 12 | C |
| 7 | 58 | D |
| 8 | 145 | E |
| 9 | 360 | F |
| 10 | 608 | G |
| 11 | 845 | H |
| 12 | 1056 | I |
| 13 | 1230 | J |
| 14 | 1440 | K |
| 15 | 1710 | L |
| 16 | 2000 | M |
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Part B
The blue logistic curve is a much better fit as this curve is closer to all of the data points.
A logistic curve is a better fit for populations because populations do not grow forever (which is what exponential functions do). Rather there is some limiting factors that cap the max population. This could be based on habitat size, amout of food/water, etc.
The logistic regression function found was
g(x) = 2210.36/(1 + 477.27e^(-0.5x))
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Part C
Plug x = 25 into the g(x) function.
g(x) = 2210.36/(1 + 477.27e^(-0.5x))
g(25) = 2210.36/(1 + 477.27e^(-0.5*25))
g(25) = 2206.44
g(25) = 2206
I rounded to the nearest whole number to stay consistent with the other y values being whole numbers as well.
We estimate roughly 2206 million users in the year 2025
note: 2206 million = 2.206 billion = 2,206,000,000
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Part D
This is a reasonable projection considering that there are roughly 7.7 billion people on earth in 2019 (based on estimates).
It's likely the population will grow beyond this value because of more efficient uses of farming, food distribution, housing improvements, etc.
Plus the company is likely to aim for growth in users along with revenue/profit growth.
Moreover, in the year 2016, we're told that 2 billion people are users (2000 million = 2 billion).
This is not far off compared to 2.206 billion people.
So in short, it's not unreasonable to estimate that there are 2.206 billion Facebook users in the year 2025.
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