Cassandra Evje, Ajla Demic, Nerrissa Seuberth, Avery Barger
MODELING UTAH POPULATION DATA
Math 1010 Intermediate Algebra Group Project
According to data from the U.S. Census Bureau, Population Division, the population of Utah
appears to have increased linearly over the years from 1980 to 2008. The following table shows
the population in 100,000’s living in Utah according to year. In this project, you will use the
data in the table to find a linear function f (x) that represents the data, reflecting the change in
population in Utah.
Estimates of Utah Resident Population, in 100,000's
Year
1981
1989
1993
1999
2005
2008
1
9
13
19
25
28
15.2
17.1
19
22
25
27.4
x
Population, y
Source: U.S. Census Bureau, Population Division
1.
2.
3.
Using the graph paper on the last page to plot the data given in the table as ordered pairs.
Label the x and y axes with words to indicate what the variables represent.
Use a straight edge to draw on your graph what appears to be the line that “best fits” the
data you plotted. You will only have one line drawn, rather than several pieces of lines.
Estimate the coordinates of two points that fall on your best-fitting line. Write these
points below.
( 27 , 28 ), ( 1 , 14 )
Use the points that you wrote down to find a linear function f(x) for the line. Show your
work!
F(x) = mx+b
Slope
𝟏𝟒−𝟐𝟖
𝟏−𝟐𝟕
=
𝟕
𝟏𝟑
For b we use one of the y-int points
f(x) =
7
13
𝑥 + 12.5
Cassandra Evje, Ajla Demic, Nerrissa Seuberth, Avery Barger
4.
What is the slope of your line?
m=
7
= .54
13
It might make sense to write your slope as a decimal. Interpret its meaning using a phrase
like “for any one unit increase in x we expect an m unit increase in y. Does it make sense
in the context of this situation? Please use complete sentences to respond to these
questions.
For any one unit increase in x, there is a unit increase of .54 in y.
5.
Find the value of f (45) using your function from part 3. Show your work, then write your
result in the blank below.
𝒇(𝟒𝟓) =
𝟕
(𝟒𝟓) + 𝟏𝟐. 𝟓
𝟏𝟑
= 24.3 + 12.5 = 36.8
f(45) = 36.8
Write a sentence interpreting the meaning of f (45) in the context of this project. What year does
the 45 represent?
In the equation f (45) =36.8, it is telling us that 45 years after 1980, the population is
estimated to be 3.68 million. The 45 represents the year 2025.
6.
Use your function from part 3 to approximate in what year the residential population of
Utah reached 2,000,000. Show your work.
2,000,000=20(100,000)
20 = .54x + 12.5
-12.5
-12.5
𝟕.𝟓 = .𝟓𝟒𝒙
.𝟓𝟒 .𝟓𝟒
x = 13.9
𝒇(𝟏𝟒) = . 𝟓𝟒(𝟏𝟒) + 𝟏𝟐. 𝟓 = 𝟐𝟎. 𝟎𝟒
14 + 1980 = 1994
Cassandra Evje, Ajla Demic, Nerrissa Seuberth, Avery Barger
7.
Compare your linear function with that of another student or group.
Comparison function: f(x) =
𝟓
𝟏𝟏
𝐱+
𝟏𝟓𝟓
𝟏𝟏
Is the comparison function the same as the function you wrote down for part 3?
No
If they are different, explain why.
The two functions are different because both groups chose to use different data points.
Therefore the function will have a different slope. The line is an estimation.
In actuality, using a linear growth model for population is not common. Most models are
exponential models, due to the fact that most populations experience relative growth, i.e. 2%
growth per year (What does this mean? You might have to look this up). Linear models for
nonlinear relationships like population work only within a small time frame valid close to the
time of the data modeled.
8.
Discuss some of the false conclusions you might reach if you use your linear model for
times far from 1980-2008.
When estimating population growth there are many variables that we cannot predict. The
linear function gives a false conclusion of a consistent level of growth. Each year’s
population growth will be greater than the previous year’s increase, thus resulting in more
of an exponential growth. For data points far beyond the time frame of 1980-2008
inaccuracies will occur. For example if we go back to 1950 with the linear model the
population will be a negative number.
Cassandra Evje, Ajla Demic, Nerrissa Seuberth, Avery Barger
9.
Determining a line of best fit using our method is sufficient for our project but leads to
subjectivity from one student to the next. Thus a more appropriate method is to perform
a linear regression in which every point is taken in to account to find the best line that fits
the given data. This process is algebraically intensive but is performed with ease using
technology. There are many free programs online that will do a linear regression for you
or you can create one in MS Excel. On our CANVAS page you will see a link to step by
step instructions to create a linear regression and produce a nice graphic that can be copy
and pasted into any word document. Here is a link that I found on YouTube that may
also be helpful
10.
Reflective Writing: This project is meant to connect our topic with lines to a bigger
picture where linear equations are used in the real world. Please a small paragraph to
discuss the following topics.
a.
Did this project change the way you think about application of mathematics?
If so why? If not then how did it support your current beliefs?
b. Discuss some other possible applications of Linear Regression
c.
There will be another part of this assignment that I will post on CANVAS
Cassandra Evje, Ajla Demic, Nerrissa Seuberth, Avery Barger
a) Yes, this project was a good way to finally see math put to use in a real world
application. Most people believe that math is useless and isn’t applicable in
everyday life, but this assignment takes the equations we have been learning in class
and gives meaning/understanding to them.
b) We can use linear Regression to estimate future sales of a company. If you know
previous and current sales you can use that data to predict company growth or
decline.