- Year Observed Population Year Observed Population 1790 3 929 000 1910 91 972 000 1800 5 308 000 1920 105 711 000 1810 7 1 (161.27 KiB) Viewed 20 times
- Year Observed Population Year Observed Population 1790 3 929 000 1910 91 972 000 1800 5 308 000 1920 105 711 000 1810 7 2 (66.89 KiB) Viewed 20 times
Year Observed population Year Observed population 1790 3,929,000 1910 91,972,000 1800 5,308,000 1920 105,711,000 1810 7,240,000 1930 122,755,000 1820 9,638,000 1940 131,669,000 1830 12,866,000 1950 150,697,000 1840 17,069,000 1960 179,323,000 1850 23,192,000 1970 203,212,000 1860 31,443,000 1980 226,505,000 1870 38,558,000 1990 248,710,000 1880 50,156,000 2000 281,416,000 1890 62,948,000 2010 308,746,000 1900 75,995,000 Let P(t) be the population of the US in the year t (you can set t=0 for the year 1790). The exponential growth model is of the form P(t) = beat The logistic growth model is of the form P(t) M 1+e-(at+6) or in the form we use to fit to the data In P M-P E at +6 where M, a, b are parameters. M is the maximum value of the population.
1. a. First, estimate M using the data set. There are two methods you can use to estimate M: • The first one is to try different values of M, plot y = In M versus t to see if the points are on a straight line (approximately), you can also compute the SSE of the linear regression line y = at +b and the transformed) data set to see which SSE is minimum. • The second method is to estimate M/2 using the following observation. We have P'(t) = r(M - P)P (from the discussion on the logistic growth model); The second derivative of P(t) is P"(t) =r(-P')P+r(M – P)P'=rP'(M – 2P) thus, P" = 0 when P = M/2. This means that when the population Preaches half the limiting population M, the growth dP/dt is most rapid and then starts to diminish toward zero. The maximum rate of growth occurs at P = M/2. You can use this information to estimate M. Plot the points (t, P(t)) and use the plot to estimate M. b. When you have a reasonable value for M, find a, b using the least squares criterion.