In order to plot a better understandable graph, I changed the number of years to intervals of five. That way, the graph will make more mathematical sense and show a more obvious trend. For this investigation, I use the software GeoGebra to plot all my graphs because it is high definition and clear, which shows the trend easily.
The above graph is all the plotted data points given of the growth population of China. As seen on the graph, there is a possibility of a linear trend because it has a general rising slope. But this slope can also be an exponential function because there is a immediate decrease after 1990. Therefore, in order to find a more suitable function, I used the graphic calculator (GDC) and input all the data points into STAT LIST and then calculated the points with a linear function (LinReg) and an exponential function (ExpReg) . To see which function suits the model best, I check the residue of each function.
For the linear function, the residue (r) is:
While the residue of the exponential function is:
Hence, this shows the most suitable function will be linear:
To test out this function, I have to choose two points from the data to find out the gradient of the function,
First Point: (0, 554.8)
Second Point: (5, 609)
To find the gradient, I use the formula
Therefore it is shown as:
m =
The next step is to find b in the function, I can find this value by inputting a data point into the function like this:
Thus, generating the complete linear function:
I then input this function into the graph to test the accuracy of the hand-calculated function.
As seen from the graph above, the line does not show the best fit as it does not touch upon all the lines, suggesting that the hand-calculated function may not be entirely reliable.
I then used my GDC to find the value for m and b, which is:
I then plot it into GeoGebra and it fits the line perfectly as shown as below:
To find a better fit line for the graph, I used a [FitLine] function from the GeoGebra to find a more accurate function, which looks like this:
This shows that the GDC and the GeoGebra are much more reliable resources than hand-calculated.
A researcher suggests that the population, P at time t can be modeled by
,
where K, L and M are parameters.
To find the value of K, L, M, I first input this function into GeoGebra to see how they fit into the data points, after a several trail and errors, I found the best fit for the line which looks like the following:
I named the x-axis as t and the y-axis as P to represent the time and the population respectively. The value for K, L, M are:
K: 1601
L: 2
M: 0.04
This can be interpreted back into the equation like this:
This model fits the data more than the linear one because it is almost impossible for the population to grow infinitely. Therefore, this model makes more sense, considering factors economically and logically.
To further explore the comparison of both models, I will calculate the expected population in 2100 for the linear model and the researcher’s model. Since the graph is using intervals of five for x-value, I convert the year 2100 to 150 for the x-value.
I plot the data point on the x-value of 150 for the linear model which is shown below:
This model states that by year 2100, the population in will reach up to 2851 million people in China. We can also calculate this by inputting the data point into the model:
As it is impossible to have 0.7 of a person, it is rounded off to 2851. This model is possible but not logical, referring to the diagram above, it is stating that as the time passes, the population will increase constantly. Although China is a large country and an increasing wealth, due to economical factors, it is almost impossible for China to have a infinite increasing population growth. It is possible to increase the population, but the growth difference between each year should become closer and closer.
To test the second model which is the researcher’s model, I input the data point on the x-value of 150 and the point is shown below:
This model states that by the year of 2100, the population in China will be
Similarly, it is impossible to have 0.035 of a human, thus, it is rounded off to approximately 1593 million of people in China, and this the correct degree of accuracy. This number is a more realistic and logical answer as oppose to the first model. Rather than increasing infinitely, this model shows a very steady increase in the beginning which eventually leads to no difference as the country has reached its maximum population.
To further compare the two models, there are additional data on population trends in China from the 2008 World Economic Outlook, published by the International Monetary Fund (IMF) which is shown on the table below:
The above data is individually tested on each of the models. For the first model, which is also the linear model, has only three points touching the model, including year 1983, 1992 and 1997; the rest of the data points were lower than the model. The second model is then tested, and for this researcher’s model, all the points are extremely close are touching the model, meaning the modified model should be based on the researcher’s model as it is more reliable than the linear model.
Hence, this model is established:
This model fit all the IMF data; hence it applies to all the given data from year 1950 to 2008 where the maximum points can touch on this model as shown below:
The modified model shows that the general data points are on the line, meaning this is the best that fits all the data since 1950 until 2008.
