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pattern is no longer valid.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 Ôdoubles every ten yearsÕ is not a good model.