Utilizing the data set of the first class letter postage for the US Mail from 1933 to 1996 do:
a.) plot the data and develop a prediction function for the growing price of stamps
b.) Answer the questions: when will the cost of a first class postage stamp reach $1.00? when will the cost be 64 cents? how soon should we expect the next 3 cent increase?
c.) Calculate a measure of error between your model and the observed data by taking the square of the difference for each time, sum the squares, and divide by the number of data points. You can use this statistic to guide refinement of your function to model the data.
Let us begin with the data set for the first class letter postage from 1933 to 1996.
Now using Excel we will construct a plot of the data.
Next using our data set, we will have Excel compute two lines of best fit, a 3-degree polynomial and an logarithmic function. Then using the lines of best fit we can hypothesize: when the cost of a first class postage stamp will reach $1.00, when the cost will be 64 cents, and how soon we can expect the next 3 cent increase.
Asking Excel to compute two lines of best fit, we get the following lines with each defined on the graph.
Next, using the lines of best fit we can compute the predictions of the next 3 cent increase, when the stamp will reach 64 cents, and when the stamp will reach $1.00. Looking at the table below we can see that the polynomial and logarithmic function predict two different years for each prediction. The polynomial function predicts the stamp will increase by 3 cents in 2021, while the exponential function predicts it has already happened in 2001. The prediction year for the cost of stamp reaching 64 cents using the polynomial function is 2485, while the logarithmic function is 2012. In this instance it appears the logarithmic function is intuitively a 'better' prediction. Lastly, the prediction of the stamp costing $1.00, the polynomial function predicts 5231, while the logarithmic function predicts 2022. Once again, the logarithmic function seems 'better'.
Next to determine which of our best fit lines, polynomial or logarithmic function, better predict the rising cost of stamps we will compute the error using Excel. To compute the error between the actual data points versus our polynomial and logarithmic function predictions we begin by taking the square of the difference for each time, sum the squares, and then divide by the number of data points. This error will act as our measure in determining our predictions. As illustrated in the above table, the error for each prediction function is in bold red with the polynomial error as 43.335 and the logarithmic error as 40.155. As you can see our logarithmic function has a smaller error, however the error is still quite large. Although the error is large, we can still hypothesize the increase in the cost of first class postage stamps. Using our logarithmic function predictions we predict that the next 3 cent increase in the price of stamps is just around the corner, the cost of a first class postage stamp reaching 64 cents will be in 2012 and the first class postage stamp will be $1.00 in 2022!