Final
Assignment – Part 2.C
Stamp Data
by
Victor L. BrunaudVega

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
writeup and use your analysis to answer the questions a new: á
When will the cost of a first class postage stamp reach $1.00? á
When will the cost be 74 cents? á
How soon should we expect the next increase? á
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seem to argue that pattern is no longer valid.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? 

My first attempt was graph the data. Then I thought that the curve might be close to a power or
an exponential function. But just
watching the scatter plot of the data, I cannot say if the curve follows either
a power or an exponential function.
However, using the coefficient of correlation (R^{2}) between
the variables maybe I can find how well the variables are related and then
realize what curve is the best fit for the data.
The coefficient of correlation values in both graphics are pretty
close: 0.923 for exponential curve, and 0.933 for power curve. As the difference is not big enough, I
think I can choose either one for predicting other values because both will be
equally good predictors.
LetŐs see how close are the outcomes using both equations in the
following table:
x=Year 
y=Price (rate in cents) 
Exponential
Function
y=2E33e^0.0394 
Power
Function
y=4E253*x^76.917 

Year Prediction 
Price Prediction

Year Prediction 
Price Prediction 

1919 
2 
1919 
1.4 
1929 
1.3 
1932 
3 
1929 
2.3 
1939 
2.2 
1958 
4 
1936 
6.4 
1947 
6.3 
1963 
5 
1942 
7.8 
1952 
7.6 
1968 
6 
1947 
9.5 
1957 
9.3 
1971 
8 
1954 
10.6 
1964 
10.5 
1974 
10 
1959 
12.0 
1970 
11.8 
1975 
13 
1966 
12.5 
1977 
12.2 
1978 
15 
1970 
14.0 
1980 
13.7 
1981 
20 
1977 
15.8 
1988 
15.4 
1985 
22 
1979 
18.5 
1990 
18.0 
1988 
25 
1983 
20.8 
1993 
20.2 
1991 
29 
1986 
23.4 
1997 
22.7 
1994 
32 
1989 
26.3 
2000 
25.5 
1997 
33 
1990 
29.7 
2001 
28.7 
1999 
34 
1990 
32.1 
2001 
31.0 
2002 
37 
1993 
36.1 
2004 
34.7 
2006 
39 
1994 
42.3 
2005 
40.5 
2022 
74 
2010 
79.4 
2022 
74.6 
2029 
100 
2018 
104.6 
2030 
97.3 
I chose the power curve as predictor because is a little
bigger. The equation for
this curve is y = 4^{253} x^{76.917}, where x represents the
year and y represents the price (rate in cents).
To know when the price will reach 1.00 dollar, we can replace y by 100 cents and
then find the new value of x.
So, now 100 = 4^{253} ● x^{76.917}
x = (100 / 4^{253}) ^{76.917}
and x = 2030. This means that the stamp will cost a
dollar the year 2030. Of course,
this is an approximate value.
The same can be done to find the year in which the stamp will cost 74 cents. So, now 74 = 4^{253} ● x^{76.917}
x = (74 / 4^{253}) ^{76.917}
and x = 2022. This means that the stamp will cost 74
cents the year 2022.
Remember that these values are approximate.
Are there significant differences if I use power function or
exponential function? We can see
in trhe next chart 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 Ôdoubles every ten
yearsŐ is not a good model.
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