The data for the price of stamps from the years 1933 to 1994 is given below:
| 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 |
I will let t = the number of years since 1900. For example, in 1919, t = 19.
This table is given below:
| t (in years) | Rate (in cents) |
| 19 | 2 |
| 32 | 3 |
| 58 | 4 |
| 63 | 5 |
| 68 | 6 |
| 71 | 8 |
| 74 | 10 |
| 75 | 13 |
| 78 | 15 |
| 81 | 20 |
| 85 | 22 |
| 88 | 25 |
| 91 | 29 |
| 94 | 32 |
Excel was used to create the following scatter plot of the data:
The stamp prices appear to increase exponentially, so Excel can be used to find the exponential function that fits this data.
To find the exponential function, highlight the scatter plot, and then click Chart > Add Trendline... Under the "Type" tab, highlight "Exponential." Click the "Options" tab, and then check "Display equation on chart."
Excel will display the trendline on top of the scatter plot, and it will give the equation of the trendline. The equation of the trendline for this data is
As you can see, the trendline falls below the data points for each year after 1974. Notice that the first two stamp prices (years 1919 and 1932) are extremely low. I think they are pulling down the trendline and making it less accurate than it could be. If the first two data points are ignored, a more accurate prediction function can be created.
The equation of the trendline for this data when the first two data points are ignored is
The trendline
fits the data more closely than the first trendline, so it will be used to make predictions.
To find the year when the price stamps will reach $1.00, we can set the price to 100 and solve for t.
Thus, the price of stamps will be $1.00 when t = 109.933, or in the year 2010.
To find the year when the price of stamps will be 64 cents (twice the cost in 1994), we can set the price to 64 and solve for t.
Thus, the price of stamps will be $0.64 when t = 102.826, or in the year 2003.
This estimate for the price of stamps in 2003 seems to be very high. Currently the year is 2001, and the price of stamps is only 34 cents.
The next 3 cent increase (from the year 1994) will occur when the price = 32 + 3 = 35. To find out when this will occur, we can set the price to 35 and solve for t.
Thus, the price of stamps will be 35 cents when t = 93.2156, or in the year 1994. This estimate, of course, is not accurate. The price of stamps in 1994 was only 32 cents. This error is caused by the fact that the trendline does not exactly fit the data. In the year 1994, the trendline is slightly above the data.
Click here for the Excel file that was created and used for this exploration.