COST OF A STAMP

The following data set is based on the first class letter postage for the US Mail from 1933 to 1996.

 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

By plotting the points, we can look at the increase as shown on the graph.

Can we find a function to fit the graph of the data shown? Is it a power function, exponential, logarithmic, polynomial? It appears to be exponential, but we need to look at some other graphs.

A power function is of the form y = ax^b

We begin with the form for the power function and take the logarithm of both sides:

log y = log(ax^b)

If we let log y = Y and log a = A, then we recognize that this is the form of a linear equation

Y = A + bx

and we can use log x, log y as our ordered pairs to graph the points. The data we now have is

 log x log y 3.28307497473547 0.301029995663981 3.28600712207947 0.477121254719662 3.29181268746712 0.602059991327962 3.29292029960001 0.698970004336019 3.29402509409532 0.778151250383644 3.29468662427944 0.903089986991943 3.29534714833362 1 3.29556709996248 1.11394335230684 3.29622628726116 1.17609125905568 3.29688447553855 1.30102999566398 3.29776051109913 1.34242268082221 3.29841638006129 1.39794000867204 3.29907126002741 1.46239799789896 3.29972515397564 1.50514997831991 3.3003780648707 1.51851393987789 3.30081279411812 1.53147891704226 3.3014640731433 1.56820172406699

The graph below shows this data with the trend line added. The correlation coefficient is given as 0.96. The closer this correlation coefficient is to 1, the better it fits the function.

Now we can look at the exponential function along with the correlation coefficient.

We begin by using the form y = ab^x. If we again take the log of both sides, we have

log y = log (ab^x)

or

log y = log a + x log b

If we let log y = Y, log a = A, and log b = B, then we have Y = A + Bx. So now the equation again is linear. We will use x and log y to graph the data.

 log y x (year) 0.301029995663981 1919 0.477121254719662 1932 0.602059991327962 1958 0.698970004336019 1963 0.778151250383644 1968 0.903089986991943 1971 1 1974 1.11394335230684 1975 1.17609125905568 1978 1.30102999566398 1981 1.34242268082221 1985 1.39794000867204 1988 1.46239799789896 1991 1.50514997831991 1994 1.51851393987789 1997 1.53147891704226 1999 1.56820172406699 2002

The correlation using the trend line is 0.9598. This is slightly less than the correlation coefficient given by the power function, but both are very close to 1 and both will give a good approximation of the trend.

So let's use the power function to predict the year that stamps will double in value? If they cost 0.37 in 2002, what year will they cost .74? When will they cost \$1.00?

 year rate(in cents) X1-Mean SQRED Y1-Mean SQRED 1919 2 -56 3136 -15.52 240.8704 1932 3 -43 1849 -14.52 210.8304 1958 4 -17 289 -13.52 182.7904 1963 5 -12 144 -12.52 156.7504 1968 6 -7 49 -11.52 132.7104 1971 8 -4 16 -9.52 90.6304 1974 10 -1 1 -7.52 56.5504 1975 13 0 0 -4.52 20.4304 1978 15 3 9 -2.52 6.3504 1981 20 6 36 2.48 6.1504 1985 22 10 100 4.48 20.0704 1988 25 13 169 7.48 55.9504 1991 29 16 256 11.48 131.7904 1994 32 19 361 14.48 209.6704 1997 33 22 484 15.48 239.6304 1999 34 24 576 16.48 271.5904 2002 37 27 729 19.48 379.4704 Total Sum Squared 8204 2412.2368 Divide by Sample 512.75 150.7648 Sx 22.6439837484485 Sy 12.2786318456089 Sx/Sy 1.84417808377784 SLOPE POWER 1.77041096042673 EXPON. 1.77004212480997 Y-INTERCEPT POWER 1943.98239997332 EXPON. 1943.98886197333

Using the information found in the above chart, we know that the equation y = 1.770x + 1943.98 will give us the value of stamps in any year.