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




































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:
If we let log y = Y and log a = A, then we recognize that this is the form of a linear equation
and we can use log x, log y as our ordered pairs to graph the points. The data we now have is


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
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.


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)  X1Mean  SQRED  Y1Mean  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  
YINTERCEPT  
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.