1. In the lecture notes where we studied a Malthusian growth model for the U. S. population, there was an applet that allowed you to adjust the growth rate, r, and the range of years to try to fit a discrete Malthusian growth model. This problem examines a version of that applet in more detail. In this problem you will be given a couple ranges of years to try to fit with a Malthusian growth model, then you will find the least squares fit to the census data by adjusting the parameter, r. Finally, you will use your model to compare to other census data.
a. Take the applet given below and adjust the data range to go from 1800 to 1880. Next adjust the parameter, r, until you obtain the smallest value of the sum of squares error (the least squares fit to the data). Write this value of r and the sum of squares error in your lab report and give the discrete Malthusian growth model for this set of data. Note that the initial value, P0, of the data agrees with the population at 1800. Use Excel to graph the data and the solution to your model for the range from 1800 to 1900.
b. Use your model to predict the population in 1860, 1890, and 1900. Find the percent error between the actual census data and these predictions. Write a short paragraph describing how well this model works on the predictions for these dates, and briefly describe any discrepancies that you observe on the graph between the model and the data and put these errors in the context of what you know about U. S. history.
c. Repeat the process from Parts a and b, but use the range 1840 to 1920 with the initial population, P0, of the data agrees with the population at 1840. Use the model to predict the populations in 1870, 1930, and 1940. Which range of data provides the better information for predicting next two decades and why?