Final Assignment – Part 2.C

Stamp Data

by

Victor L. Brunaud-Vega

 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 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?

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 (R2) 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=2E-33e^0.0394 Power        Function                           y=4E-253*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 x76.917, 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 x76.917

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 x76.917

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.