Maria Trujillo
Project: Linear Modeling (QLP)
Mat 115:Prof. Rudy Meangru
Project#1
In the article Public University Tuition Rises Sharply Again for “04 in the October
20, 2004 New York Times newspaper it was stated that tuition at public
institutions has seen its sharpest increase in decades. It was noted that it was
the first time that the average tuition at postsecondary institutions has “surpassed
$20,000 for a private college, $5,000 for a public university and $2,000 for a
community college. According to the College Board survey, more and more
students are becoming dependent on loans to pay for their college education.
The pursuit of a higher education should be a path to freedom and intellectual. It
should not lead to a downfall to indebtedness. The rising cost of tuition has
become a burden for low to middle income students.
In this activity students will use their algebraic skills to develop a liner model of
the cost of tuition from the data complied by the College Board.
Step 1: Briefly write why do we need to study Linear Function.
We often use a linear function to model data that are generally line up on a
straight line. There are many natural and economic situations in which
bivariate data tends to follow a linear pattern.
We can use a Linear Function in our everyday life. With the help of the
calculation a Linear Function we can predict our earnings in few years or
what will be the cost of certain products. If we want to do that, the years will
go on the x-axis and the money amounts on the y-axis.
Example:
The cost of a Mustang en 1999 was $35,000.00 suppose the cost of the car
depreciates to $28,000.00 in 2004. Assume the cost is a linear function. Let C
represents the cost and x the age of the car.
a) Write a linear function, C(x)
b) What is the cost of the car in 2009?
c) When will it worth $20,000.00?
Age
x=0
x=5
1999
2004
m=
28000  3500
50
m=
7000
5
Cost
35,000
28,000
m = - 1400
a) C(x) = mx + b
C(x) = m(0) + 35,000
C(x) = 0 + 35,000
C(x) = -1,400x + 3500
Ans. = The linear function is C(x) = -1,400x + 3500
b) 2009
 x = 10
C(10) = 1400 (10) + 35,000
C(10) = -14,000 + 35,000
C(10) = $21,000
Ans. = The cost of the car in 2009 will be $21,000.
c) -1,400x + 35,000 = 20,000
- 1,400x = -15,000
x = 10.7
Ans. = The car will be worth $20,000.00 in around 10 years.
Step 2: Define what a linear function is and list some of its properties.
Use the short description below to guide you.
A function f is a linear function if
f(x) = ax + b
Where a and b are real numbers. The domain of f is the set of all real
number and the range is the set of all real number.
One of the characteristics a linear function is that the graph of a linear
function is a straight line that is neither horizontal nor vertical. List the
other characteristics of a linear function. The rate of change of a linear
function f is constant over every interval of f. Show algebraically why the
slope of f is the constant a.
The average rate of f is the constant a which also represent the slope of
the line. To show that the average rate of change of f is a, consider the
two points (x1,f(x1)) and (x2, f(x2)).
The average rate of change of f is given by
f ( x 2 )  f ( x1 ) ax2  b  ( ax1  b)

x 2  x1
x2  x1
ax  b  ax1  b
= 2
x2  x1
=
ax 2  ax1
x 2  x1
=
a ( x 2  x1 )
x 2  x1
=a
Step 3:
(x2  x1 )
Inquiry Learning Exercise.
Many real-life situations can be modeled by using a linear function. In
statistics the method of least squares is used to fine the best-fit line for a
data set. Research this method and give a brief explanation of the idea
behind this approach of finding such line.
Carefully define all terms associated with this method. Use a scatter graph
to illustrate your point. {You may use a graph paper to draw your graph by
hand}
Least squares may be interpreted as a method of fitting data. The best fit,
between modeled data and observed data, in its least-squares sense, is
an instance of the model for which the sum of squared residuals has its
least value, where a residual is the difference between an observed value
and the value provided by the model.
Least Square-Regression is a method utilized to find a line that
summarizes the relationship between two variables, at least within the
domain of the explanatory variable, x. The least-squares regression line
(LSRL) is a mathematical model for the data.
Regression Line: A regression line is a line drawn through a scatter plot
two variables. The line is chosen so that it comes as close to the points as
possible. A straight line that describes how a response variable y
changes as an explanatory variable x changes. It can sometimes be used
to predict the value of y for a given value of x.
This graph is an example of scatter plot you can notice that a line is drawn where
there are more points.
Step 4:
Using an actual data set to fit a linear model.
The cost of a college education is an important factor when a student is
deciding whether to attend college or not and what college to attend. The
College Board puts out an annual report that discuses the trend in college
tuition and fees.
Read the report from the College Board and write a brief summary of it.
Read the introduction, write a brief summary
http://www.collegeboard.com/press/cost02/html/CBTrendsPricing02.pd
f
In the Introduction the College Board is explaining how it was able to
obtain the tuition college data. In the present year’s report new information has
been added as race and gender. The report has been calculated taking in
consideration financial aid, which most of the students are able to obtain. Other
expenses as room, board and transportation are also included as non – fixed
components.
The sample’s data for this analysis is obtained from surveys that are send
to different colleges (private and public). The analysis includes only colleges
from which the College Board has two consecutive years of data and college’s
data from which the College Board has sufficient information to justify an
average.
Weighted data could be more accurate because fixed charges are included
in its calculation. The results of the future trend in the cost of tuition are
somewhat over estimated in purpose. This is because the College Board wants
college students and their parents to be prepared for any unusual change in the
future.
Using table 6a of the report on Tuition and Fees perm the following tasks.
(a) Identify the costs for a 4-year public education for the period 1992
through 2003.
(b) Provide a table of the data.
1994
3122
1995
3239
1996
3277
1997
3372
1998
3464
1999
3557
2000
3581
2001
3584
(c) Sketch a scatter plot of the data neatly on a graph paper.
Tuition and Fee 4-yr Colleges
(d)
4500
4000
Tuition
1993
2949
3500
3000
2500
2000
Series1
1500
1000
500
0
1992
1994
1996
1998
Years
2000
2002
2004
2002
3765
2003
4081
Tuition and Fee 4-yr Colleges
Tuition
4500
4000
3500
3000
2500
2000
Series1
1500
1000
500
0
1992
1994
1996
1998
2000
2002
2004
Years
(d) Describe observable pattern/trend of the data.
The trend of the data observed in this graph allows us to notice that
tuition keeps increasing over the years. Between 1999 and 2001 the
tuition was steady, but after 2001 the increasing trend came back
again. After the year 2001 we can notice how tuition began to increase
once more.
Tuition and Fee 4-yr Colleges
Tuition
4500
4000
3500
3000
2500
2000
y = 91.455x - 179272
R2 = 0.9358
1500
1000
500
0
1992
1994
1996
1998
2000
Series1
Linear (Series1)
2002
2004
Years
(e) Assume this trend is linear; draw a possible line through the points.
(f) Using two points on this line compute the equation of it.
Let choose the year 1993 and 2001. Also Let the year 1993 be
represent by x = 0.
So the points are (0,2949) and (8,3584)
Y = mx + b
m
y 2  y1 4081  3584 497


 62.125
x2  x1
80
8
Y = 62.125x + 2949
The predicted tuition for 2010s computed as follow.
For 2007, the value of x is 17
Y = 62.125 (17) + 2949
= $4.005.125
With the help of a technological tool:
(a) Obtain a scatter plot of the data
(b) Find a linear model that fits the data.
(c) Use the model to estimate the cost of tuition for 2010?
Tuition and Fee 4-yr Colleges
Tuition
4500
4000
3500
3000
2500
2000
y = 91.455x - 179272
R2 = 0.9358
1500
1000
500
0
1992
1994
1996
1998
Years
2000
Series1
Linear (Series1)
2002
2004
m= 91.455
y = 91.455x - 179272
y = 91.455(2010) - 179272
y = $4552.55
Compare the two approaches and write a short summary.
Step 5:
In this activity I was able to calculate predicted tuition for different future
years. The construction of the graphs was very helpful because only by looking
at them I can notice the increasing trend of college tuition. Some of the
challenges that I face while doing this activity were some of the construction of
the graphs and the long reading in order to do the summaries. Well I think I was
able to overcome the challenges by having patience and keep going. The most
interesting part in this activity was to find out that a linear equation could be
useful in real life stuff. I didn’t find any part that I could call the least interesting,
for me all the outcomes that I obtained were interesting. The procedure to obtain
the outcomes was confusing sometimes. Well after finishing this activity I am
glad to be in college now because I found out that tuition keeps increasing while
years pass by. I am glad that I didn’t wait more time because tuition could have
been expensive.