Question 1149659
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Part A


Scatter Plot and Regression Curves
<img src = "https://i.imgur.com/kS2gNb3.png">
Image generated by <a href = "https://www.geogebra.org/">GeoGebra</a> (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
<a href="http://pehs.psd202.org/documents/tfrey/1506536379.pdf">http://pehs.psd202.org/documents/tfrey/1506536379.pdf</a>
provides a guide on how to use a TI83 or TI84 calculator to do exponential regression. 


This link
<a href="https://education.ti.com/html/t3_free_courses/calculus84_online/mod05/mod05_lesson2.html">https://education.ti.com/html/t3_free_courses/calculus84_online/mod05/mod05_lesson2.html</a>
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
<table border = "1" cellpadding = "10">
<tr><td>x</td><td>y</td><td>Point</td></tr>
<tr><td>4</td><td>1</td><td>A</td></tr>
<tr><td>5</td><td>6</td><td>B</td></tr>
<tr><td>6</td><td>12</td><td>C</td></tr>
<tr><td>7</td><td>58</td><td>D</td></tr>
<tr><td>8</td><td>145</td><td>E</td></tr>
<tr><td>9</td><td>360</td><td>F</td></tr>
<tr><td>10</td><td>608</td><td>G</td></tr>
<tr><td>11</td><td>845</td><td>H</td></tr>
<tr><td>12</td><td>1056</td><td>I</td></tr>
<tr><td>13</td><td>1230</td><td>J</td></tr>
<tr><td>14</td><td>1440</td><td>K</td></tr>
<tr><td>15</td><td>1710</td><td>L</td></tr>
<tr><td>16</td><td>2000</td><td>M</td></tr>
</table>


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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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