Project Linear Modeling _QLP_ by xumiaomaio

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									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                      Cost
              1999           x=0                     35,000
              2004           x=5                     28,000

     28000  3500             7000
m=                      m=                     m = - 1400
        50                    5
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 ( x2 )  f ( x1 ) ax2  b  ( ax1  b)
                          
            x2  x1              x2  x1
                            ax  b  ax1  b
                          = 2
                                x2  x1

                         ax2  ax1
                     =
                          x2  x1

                         a ( x2  x1 )
                     =
                           x2  x1

                     =a                  (x2  x1 )

Step 3:      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.

1993   1994             1995     1996    1997     1998      1999       2000      2001   2002   2003
2949   3122             3239     3277    3372     3464      3557       3581      3584   3765   4081




        (c) Sketch a scatter plot of the data neatly on a graph paper.



                                 Tuition and Fee 4-yr Colleges

        (d)
                        4500
                        4000
                        3500
                        3000
              Tuition




                        2500
                                                                              Series1
                        2000
                        1500
                        1000
                        500
                          0
                          1992    1994   1996   1998     2000   2002    2004
                                                Years
                      Tuition and Fee 4-yr Colleges

             4500
             4000
             3500
             3000
   Tuition


             2500
                                                               Series1
             2000
             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

             4500
             4000
             3500
             3000
   Tuition




             2500
                             y = 91.455x - 179272       Series1
             2000
                                  R2 = 0.9358           Linear (Series1)
             1500
             1000
             500
               0
               1992   1994    1996    1998    2000   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


           y 2  y1 4081  3584 497
   m                              62.125
           x 2  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

                  4500
                  4000
                  3500
                  3000
        Tuition




                  2500
                                  y = 91.455x - 179272       Series1
                  2000
                                       R2 = 0.9358           Linear (Series1)
                  1500
                  1000
                  500
                    0
                    1992   1994    1996    1998    2000   2002   2004
                                          Years
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.

								
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