Stamp Data

By: Lucia Zapata


Consider the Stamp Problem in Assignment 12. Update the data to include the price increases for a first class letter through January 2006 -- when the price will become 39 cents. (Recent increases were 33 cents in 1997, 34 cents in 1999, and 37 cents in 2002.) Prepare a write-up and use your analysis to answer the questions anew:

When will the cost of a first class postage stamp reach $1.00?

When will the cost be 74 cents?

How soon should we expect the next increase?

In 1996, the analysis of stamp data historically seemed to show that the postage doubled every 10 years approximately. The cost in 2006 would seem to argue that pattern is no longer valid. Is there evidence to show a change in the growth pattern? Or, was the 'doubles every ten years' just a bad model?


If we see the scatter plot of the data, we cannot decide, just watching, if the curve follows either a power or an exponential function.  However we are able to use the R-squared coefficient to decide what curve is the best fit for the data (remember that the R-square is the coefficient of correlation between the variables; it explains how well the variables are related) 

 

The R-squared values in both graphics are close: 0.9229 for exponential curve and 0.9328 for power curve. Because the difference is not big enough, we are able to choose either one for predicting other values and both will equally be good predictors.

 

Since power R-square for power curve is a little bigger, let us choose this curve as our predictor. A power curve has the form: . The equation for this curve is which is calculated automatically using excel; however you can also use algebra to find the values of c and n.  Try yourself here.  In our equation x-axe represents year and y-axe represents price. 

 

If we want to know when the price will reach 1.00 dollar, we only need to substitute y by 100 cents and to find the new value for x.

 

So,  where x=2030.  This means that 2030 will be the year in which the stamp will cost a dollar.  Table 1 shows this fact.

 

Something similar we must do for 74 cents.  We substitute 74 in y and calculate the value of x.  So,  where x=2022.  This means that 2022 will be the year in which the stamp will cost 74 cents.

 

Remember that these values are approximate.

 

Next chart shows the difference when we use power function or exponential function.  We can see that the difference in price is very small between them (some cents),  and some years were predicted exactly similar.  The cells highlighted are the cells where there was any difference.

 

Table 1

Year

Price

Exponential Function

Power Function

Prediction price using equation 

Predictions year using equation

Prediction price using equation

Prediction year using equation

1919

2

1.37

1929

1.34

1929

1932

3

2.29

1939

2.25

1939

1958

4

6.38

1946

6.29

1947

1963

5

7.77

1952

7.65

1952

1968

6

9.46

1956

9.30

1957

1971

8

10.65

1964

10.46

1964

1974

10

11.98

1969

11.76

1970

1975

13

12.46

1976

12.22

1977

1978

15

14.03

1980

13.74

1980

1981

20

15.79

1987

15.44

1988

1985

22

18.48

1989

18.03

1990

1988

25

20.80

1993

20.25

1993

1991

29

23.41

1996

22.74

1997

1994

32

26.35

1999

25.53

2000

1997

33

29.66

2000

28.66

2001

1999

34

32.09

2000

30.95

2001

2002

37

36.11

2003

34.74

2004

2006

39

42.28

2004

40.50

2005

2022

74

79.41

2020

74.62

2022

2029

100

104.63

2028

97.34

2030


In 1996, the analysis of stamp data historically seemed to show that the postage doubled every 10 years approximately. The cost in 2006 would seem to argue that pattern is no longer valid. Is there evidence to show a change in the growth pattern? Or, was the 'doubles every ten years' just a bad model?

If we observe in the table 1,  the price in 1981 was 20 and 10 years later  in 1991 was 29.  This fact suggests that the evidence does not show us that the postage price doubled every ten years.  As a result the ‘doubles every ten years’ is not a good model.

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