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Linear Regression Analysis of Automobile Data: Predicting US Car Numbers from 1959 to 2000, Study Guides, Projects, Research of Mathematics

A project for math 130 students to analyze the relationship between the number of automobiles in the us and time using linear regression. Students are asked to create graphs, calculate slopes, and make predictions based on the data provided. The goal is to determine if a linear model is an appropriate fit for the data and to compare the results with excel's trendline.

Typology: Study Guides, Projects, Research

Pre 2010

Uploaded on 08/19/2009

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Math 130 Spring 2009
Math 130 - QProject 1: Linear Functions
Background: Many scientists are interested in the subject of global warming.
The basic issue concerns whether human activities are influencing the climate of
the entire planet. One widely held theory says that the amount of carbon dioxide
(CO2) in the atmosphere is increasing as a result of burning various kinds of fuel.
According to this theory, the atmosphere will heat up as a result of the increased
carbon dioxide levels. How much? And how soon? These questions are studied
by developing models and making predictions. The models are very involved,
and consider many variables. In this project, we will look at just one of the
variables: the number of automobiles in the U.S. This variable is used to predict
how much gasoline is burned in the U.S., and that leads to predictions about the
amount of carbon dioxide added to the atmosphere. In the table below are data
on the number of automobiles in the U.S1. The figures are in units of one million,
so that in 1959 there were roughly 59,500,000 automobiles (rounded to the
nearest hundred thousand).
You should use Excel to complete the following explorations. The explorations
will enable you to write a well-written report describing in detail your results.
Include graphs as appropriate in the body of your report. It is this report that will
be graded. (You should not turn in this sheet as part of your final report but you
may wish to turn it in with a rough/preliminary draft.)
Year Number of Automobiles
1959 59.5
1964 72.0
1969 86.9
1975 106.7
1979 118.4
1. Consider the year 1959 to be our starting year, or time zero. Fill in the
remaining times in the table below.
Time Number of Automobiles
0 59.5
72.0
86.9
106.7
20 118.4
1
QP1 Page 1
Note the
change in
years is
not the
same
each time
pf3
pf4

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Math 130 - QProject 1: Linear Functions

Background: Many scientists are interested in the subject of global warming. The basic issue concerns whether human activities are influencing the climate of the entire planet. One widely held theory says that the amount of carbon dioxide (CO 2 ) in the atmosphere is increasing as a result of burning various kinds of fuel. According to this theory, the atmosphere will heat up as a result of the increased carbon dioxide levels. How much? And how soon? These questions are studied by developing models and making predictions. The models are very involved, and consider many variables. In this project, we will look at just one of the variables: the number of automobiles in the U.S. This variable is used to predict how much gasoline is burned in the U.S., and that leads to predictions about the amount of carbon dioxide added to the atmosphere. In the table below are data on the number of automobiles in the U.S^1. The figures are in units of one million, so that in 1959 there were roughly 59,500,000 automobiles (rounded to the nearest hundred thousand). You should use Excel to complete the following explorations. The explorations will enable you to write a well-written report describing in detail your results. Include graphs as appropriate in the body of your report. It is this report that will be graded. (You should not turn in this sheet as part of your final report but you may wish to turn it in with a rough/preliminary draft.) Year Number of Automobiles 1959 59. 1964 72. 1969 86. 1975 106. 1979 118.

  1. Consider the year 1959 to be our starting year, or time zero. Fill in the remaining times in the table below. Time Number of Automobiles 0 59. 72. 86.

20 118. 1 Note the change in years is not the same each time

  1. Use Excel to make a graph of the data with Time (0-20) on the horizontal axis and number of automobiles (in millions) on the vertical axis. Adjust your vertical axis so that it displays only the values from 55 to 140 million. Does the graph appear to be reasonably linear?
  2. A further check on whether this graph is really linear is to compute the slope from each point to the next. If these slopes are all identical, then they must line up exactly. If the slopes are approximately equal, a linear model might be appropriate. Compute these slopes and fill them in the table below: Time Number of Automobiles Slopes 0 59.5 --- 72.0 2. 86.

20 118.

  1. Are these slopes all reasonably similar? What value might we use as a 'common' slope for these data? Explain.
  2. Using the common slope you mentioned in #4, and the intercept shown in the table (0, 59.5), find the slope-intercept equation of a linear model for this data:
  3. Use your model to predict the number of automobiles in the years 1959, 1964, 1969, 1975 and 1979 and fill them in the table below. Then add the graph of your model to the graph of your data from #2. Does this linear model appear to be a good fit? Year Time Number of Automobiles Model Predictions 1959 0 59. 1964 72. 1969 86. 1975 106. 1979 20 118.
  4. There is a theoretical way (called linear regression) to choose the 'best' line to fit a set of data. Excel can find the equation of that line for you as follows: Right click on the curve that displays the original data. Select " Add Trendline " Under the “ Trendline Options " -- select " linear " by checking that button. Then put a check in the button next to " Display equation on chart " (Second line from the bottom).

purple box down until that box includes the value 36. These two points should be added to your graph. Ask for help if you have any trouble with this. Use this altered graph to explain why both of our models did such a poor job of predicting the number of automobiles after 1979.

  1. Write a report summarizing your work in this project. Include tables and graphs (in the body of the report*) as necessary. Use complete sentences and make it interesting! *You can copy any Excel graph and paste it into Word, then resize it as you like. The same is true of tables. (^1) http://www.fhwa.dot.gov/ohim/summary95/mv200.xlw QP1 Page 4 Drag these 4 cells to the graph Drag these 4 cells to the graph