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MODELING REAL REALTIONS

 

By Dario Gonzalez Martinez

 

 

 

The idea now is to draw a mathematic model to represent and predict the behavior of a quantity that is related to another.To make this real, we will base our discussion on real data.We will consider the data based on the first class letter postage for the US Mail from 1919 to 2008, and we will model this problem to predict future rates.

 

Here is the data

 

Year

Rate (in cents)

1919

2

1932

3

1958

4

1963

5

1968

6

1971

8

1974

10

1975

13

1978

15

1981

20

1985

22

1988

25

1991

29

1994

32

1997

33

1999

34

2002

37

2004

39

2006

41

2008

42

 

We can elaborate a graph to show the variation of the rate through the years.Figure 1 below shows this relation:

 

Figure 1

 

Using a graph always can help us to determine a curve that could fit for our problem.In other words, the idea is to find a function that relates years and rates such that the relation expressed by the function describes acceptably the real relation between years and rates.

 

For our analysis, consider the following table:

 

Year

Rate (in cents)

Rateís average Increment per year

1919

2

0.076923077

1932

3

0.038461538

1958

4

0.2

1963

5

0.2

1968

6

0.666666667

1971

8

0.666666667

1974

10

3

1975

13

0.666666667

1978

15

1.666666667

1981

20

0.5

1985

22

1

1988

25

1.333333333

1991

29

1

1994

32

0.333333333

1997

33

0.5

1999

34

1

2002

37

1

2004

39

1

2006

41

0.5

2008

42

 

The third column that I added represents the rateís average increment per year, that is, each of these cells is the rateís average increment per year from the year in the same row to the year in the next row.For example, the result of the first cell in the third column can be obtained by

 

 

If we observe this column, we can appreciate that the rate increase relatively fast from 1919 to 1988 with a great explosion about 1958.It is possible appreciate that the population in the United State start increasing from 1956 to 1959 by looking the following resource on the web:

 

http://www.census.gov/popest/archives/1990s/popclockest.txt

 

This could explain why the first class letter postage cost increase fast from 1958.Since the population was increasing, the demand for letter postage increases too, which in turn could have increased the rate.

 

 

DEDUCTION OF A SUITABLE MODEL

 

Although it seems highly improbable the first class letter postage cost increase indefinitely since nobody will pay a high price to send a letter, the rate is still relatively small, we can use an exponential model to predict the cost in a near future.

 

An exponential model is:

 

 

Where a and b are the parameters that we need to find.To this end, we will take natural logarithm in both sides of the above relation.

 

 

We just need to find the values of a and ln(b) in a linear regression, and then we will obtain the values for the parameters of our exponential model.By using the following statistic formulas

 

 

The following table made in Excel did the work to calculate these relations for us:

 

VALUES OF PARAMTERS

 

Thus, our exponential model will be

 

 

Where x represents the number of years from 1919, and y is the rates of the first class letter postage (theoretical).Letís see how well our model fit.Figure 2 shows the real graph and our theoretical model to predict the rate:

 

Figure 2

 

This model could be a good approximation to the real relation between years and rates.

 

Suppose that we want to know when the cost will be 1 dollar (100 cents).According our model, we will have:

 

 

This result suggests that first class letter postage will cost 1 dollar about 2028.

 

 

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