Logisitc growth Model

S-shaped curves, often referred to as sigmoidal curves, arise in various applications, including bioassay, signal detection theory, engineering, and economics. Various types of growth data often conform to sigmoidal curves. Logistic growth model is a nonlinear regression model that has been widely used in such situations. A logistic growth model involves three parameters 1: By and Ba. and has the form B1 Vi =1+ Bye -Bax + i where yi is the population size, and x; is time; for equally spaced observations, it is conventional to take x; = i - 1, and so x = 0, 1, 2, .. The parameters are interpreted in terms of the underlying growth process: 1 represents the limiting value of the response past which the output cannot grow (that is, the asymptote that bounds the function and therefore specifies the level at which the growth process saturates), By is the location parameter (it shifts the function in time, but it does not affect the function's shape), and Ba denotes the rate of growth (Ba affects the steepness of the curve, and so, as a increases, the curve approaches the asymptote , more rapidly). Another quantity of interest is the point of inflection, Lossz, at go . at which the growth rate reaches a maximum. Consider the decennial population data for the United States for the period from 1790 to 2000. We want to fit the logistic growth model to these data. Population of the United States, in Millions, 1790-2000. Year Population (yt) Year Xi Population (7) 1790 3.929 1900 11 75.995 1800 5.308 1910 12 91.972 1810 7.240 1920 13 105.711 1820 9.638 1930 14 122.775 1830 12.866 1940 15 131.669 1840 17.069 1950 16 150.697 1850 23.192 1960 17 179.323 1860 31.443 1970 18 203.302 1870 39.818 1980 19 226.542 1880 50.156 1990 20 248.718 1890 10 62.948 2000 21 281.425 a) Fit the logistic growth model to the decennial population data for the United States. Find an estimate of the point of inflection. b) Test the hypotheses Ho: Ba = 0 against H1: Ba # 0. c) Plot y against x, and show a plot of the estimated regression function on the same graph