Cubic Fit:
From the graph, the data points also seem to show trends apparent in graphs of cubic functions, as the rate at which the population increases proliferates as the years go by. Assuming a quadratic fit for the plot of the data points, the function would be in the form y=Ax3+Bx2+Cx+ D. In order to define parameters clearly, the function y=Ax3+Bx2+Cx+ D can be represented as
y=A(x-H)3+K. Parameter A represents the horizontal stretch of the graph. Parameter H represents the horizontal translation of the function. Parameter K represents the vertical translation of the function. The coordinate (-H, K) represents the turning point of the function. Regarding constraints, the cubic function for the population data would be limited to the first quadrant. A cubic fit applied to the data points via Graphical Analysis 3 follows:
Next, a model function can be developed to fit the data points of the graph. In this investigation, a linear function will be created.
A linear function, as discussed above, is a function in the form y=Mx+B. Parameter B represents the y-intercept, while Parameter M represents the slope. The y-intercept of our data is 554.8, as that is the population of China (in millions), when x=0 (Year=1950). For the slope value, the slopes between several different data points can be averaged to find the most efficient slope for the entire function:
Slope = = = = 10.8
Slope = = = = 8.7
Slope = = = = 14.34 ≈ 13.3
Slope = = = = 20.3
Slope = = = = 19.42 ≈ 19.4
Slope = = = = 14.22 ≈ 19.2
Slope = = = = 14.22 ≈ 14.2
Slope = = = = 17.06 ≈ 17.1
Slope = = = = 13.04 ≈ 13.0
Now that we have gathered several slope values, we can find their average to acquire the optimal slope value. Using a GDC, the optimal slope was found.
Slope ≈ 14.7
Therefore, the analytically developed model function that fits the data points on the graph is:
y = 14.7x + 554.8
y=14.7x + 555
Note that the section of the linear function’s graph that applies to the population trends in China is constrained to the first quadrant.
The following graphs show the analytically developed model in collaboration with the plot of the data points. One was generated with a GDC, and the other through Graphical Analysis 3.
My model linear function fits the original data quite well. The first portion of the function is very close to fitting the original data points. The second portion of the function fits the original data points almost exactly. Therefore, the slope of this linear function is efficient enough to model the original data points, as only 2 out of the 10 data points are a somewhat significant distance away from the model linear function.
A researcher proposes that the population of China, P at time t can be modeled by the following function:
P(t) = , where K, L, and M are parameters.
First, in order to estimate and interpret the parameters K, L, and M, the type of function must be identified. This function resembles the form of a logistic function, such as y =. Therefore, it can be established that the researcher’s model is a logistic function.
Now, through Logistic Regression, the values of the parameters K, L, and M can be estimated. Performing a Logistic Regression on a GDC reveals the estimated values for parameters K, L, and M:
Parameter K represents the asymptote of the function. Regarding the population of China, it represents the carrying capacity, or maximum population China can sustain. The value of parameter K is approximately 1941.466886, or 1940. The following graphs demonstrate how K represents the asymptote. The first graph has K value 500; the second 1000; and the third 1500, while the L and M values are kept the same:
Parameter L affects the y-intercept of the function. As L becomes bigger, the curve gets closer to the x-axis. As L becomes smaller, the curve gets further away from the x-axis. The value of parameter L is approximately 2.609932064, or 2.61. The following graphs demonstrate how L affects the y-intercept and the curve’s proximity to the x-axis. The first graph has L value 2; the second 4; and the third 8, while the K and M values are kept the same:
Parameter M represents the rate at which the population of China grows. As M becomes bigger, the rate of change of the curve increases. As M becomes smaller, the rate of change of the curve decreases. The value of parameter L is approximately 0.0333536861, or 0.0333. The following graphs demonstrate how M represents the rate at of change of the logistic function. The first graph has M value 0.03; the second 0.09; and the third 0.18, while the K and L values are kept the same:
The values of parameters K, L, and M were stored in their non-rounded forms in the GDC as K, L, and M, respectively, to increase the accuracy of the investigation.
With the estimated values of the parameters, the researcher’s model can be constructed:
P(t) =
On the following graph the researcher’s model and the original data are plotted using the stored values of parameters K, L, and M in the GDC for the sake of accuracy.
The researcher’s model seems to fit the original data almost perfectly. The curve passes through or extremely near the data points. The researcher’s model is by far a better fit for the data points than the one that I had developed analytically (y=14.7x + 555). Note that the section of the researcher’s function’s graph that applies to the population trends in China is constrained to the first quadrant.
Regarding population growth for China in the future, the model that I developed analytically and the model developed by the researcher have different features.
First, let’s have a look at the function I developed analytically:
y=14.7x + 555
The function has no x or y limit. Therefore, according to this model, the population of China will increase at a steady rate to infinity. Hence, this model would not be a very accurate representation of population trends in China for the future, as if the population was to increase infinitely, it would exceed China’s carrying capacity.
Now, let’s have a look at the researcher’s logistic function:
P(t) =
The function has no x limit, but has a y limit of approximately 1940. Therefore, the population, according to the researcher’s model will not exceed 1940 million, or 1.94 billion people. In the researcher’s model, the rate at which the population grows decreases as it nears its carrying capacity of 1940, or the asymptote of the graph. In conclusion, the rate at which the population grew in the beginning stage of the graph is a significant amount higher than the rate in the later years, or higher values of t.
The following are additional data on population trends in China from the 2008 World Economic Outlook, published by the International Monetary Fund (IMF).
Again, for the purpose of analysis, instead of using 1983, 1992, 1997, 2000, 2003, 2005, and 2008 for the Year values, 33, 42, 47, 50, 53, 55, and 58 will be used as the starting year for the data in the investigation was 1950 (Year - 1950=time).
First, let’s examine how the model I developed analytically fits the IMF data for the years 1983-2008:
y=14.7x + 555
The graph of the model I developed seems to fit the first three data points in a reasonably genuine manner. However, starting at the 4th data point, or year 2000, the curve and the data points shift away from each other in different directions. The curve extends toward infinity at a steady rate of change while the data points slow down in their rate of change and near and asymptote.
Now, let’s examine how the researcher’s model fits the IMF data for the years 1983-2008.
P(t) =
The graph of the researcher’s model seems to fit only the first two data points in a reasonably genuine manner. Starting at the 3rd data point, or year 1997, the curve and the data points shift slightly away from each other in different directions. The curve extends towards an asymptote of approximately 1940 million, while the data points extend toward a different asymptote. However, even though the researcher’s model fits only two data points in a genuine manner, it has a better fit than the linear model function I developed analytically, as my model extends toward infinity, further away from the data points, while the researcher’s model shifts to an asymptote of 1940 million.
Now, as stated above, the researcher’s model fits the IMF data best in the long-run. Therefore, after combining the original data points from 1950-1995 with the IMF data points from 1983-2008, the Logistic Regression function of the GDC can be used to acquire new estimates for the parameters K, L, and M of the researcher’s model that would apply to the data from 1950-2008.
After combining the two data sets, Logistic Regression yields the following results:
Parameter K is approximately 1617.675865, or 1620.
Parameter L is approximately2.066947811, or 2.07.
Parameter M is approximately 0.0399161863, or 0.0399.
The values of parameters K, L, and M were stored in their non-rounded forms in the GDC as A, B, and C, respectively, to increase the accuracy of the investigation.
Therefore, the modified researcher’s model is:
P(t) =
The graph of this logistic function collaborated with the data points from 1950-2008 follows:
This modified version of the researcher’s model fits the data from 1950-2008 of China’s population perfectly. The line passed through or is extremely close to passing through all of the data points. The curve and the data points both have about the same rate of change, same y-intercept, and the same asymptote of approximately 1620 million. Additionally, it must again be noted that the portion of the function’s graph that applies to population trends in China is constrained to the first quadrant. Therefore, it can be concluded that the model P(t) = shows the population trends in China with an efficient amount of accuracy and precision.