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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 50 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 80 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.