I will now list the values of ‘x’ and ‘y’.
Modelling the population of China
Using technology[1] I graphed the given data to analyse the shape of the trend, shown in Graph 1.
Looking at the graph one can see a gradual increase in population between the years 1950 to 1995 at intervals of five years. At first, it seems as if it following a linear trend as it represents the overall population increase. However, it is not an accurate depiction of the data given to us. Below I have used three different functions to decide which model will best fit the data.
Linear
Quadratic
Cubic
Cubic equation is:
In this case function is:
As shown above, the graph with the best fit line is graph 4, the Cubic function. I believe that this is because polynomials can be integrated to have a more precision and so I have decided that this will best model the population growth of China. I have not used Quartic or Quintic equations due to time and technological constraints. I have also not used logarithmic and power of x functions as I believe they wouldn’t be as accurate.
Finding Equation analytically
After further analysis I decided to try and acquire a different cubic equation using matrices. Using TI-84, I put the data into Matrix form, using 1950 as 0. After putting in the data I found the equation by putting the matrix in reduced row echelon form.[2]
Matrix:
Input looks like this:
I will now use the rref() function on my TI-84 to solve the matrix and get a single equation:
On pressing I receive:
The output is:
Equation is:
Therefore the equation I get is:
The Graph below shows the equation (the red line is the data set):
Since the equation I developed is not representing the data set well enough (Graph 5) I will use the cubic equation I had got earlier via technology.
Researcher’s equation
A researcher suggests that
This is a logistic function where K,L and M are the parameters.
Using Logger Pro, I input the equation and use the ‘Best Fit’ function. I receive:
In this case:
The graph of the equation is shown below:
In graph 7 I will compare the researcher’s function against the original data:
As we can see the researcher’s model does not fit the data as well as the cubic function.
Implications
I will now see what the 2 models predict the future populations will be:
Cubic model
Researcher’s model
As we can see, the cubic model (graph 7) is declining which is not possible for a country’s population. Thus the cubic equation cannot be relied upon. The Researcher’s model (graph 8) shows exponential growth which is more realistic.
Additional data
I will now compare the two models with the new information I have received and see how well they have predicted the values.
Cubic model
Researcher’s model
The cubic model (although it is closest to the new data) is not a realistic representation therefore I will continue with the Researcher’s model. The Researcher’s model shows a constant rise in the population.
Final model
Original Data
I now graph the new set of data:
The new model and new Researcher’s equation:
After putting in the new data into Logger pro, I once again find the best fit line:
The new equation is:
Its future implications:
Conclusion
With the new data the Researcher’s Equation is much better. The modified model forms an ‘S’ shaped logistic function which is best suited to show population growth. Some of the points in the beginning of the graph are a little off and the last few points show a decrease in the rate of population growth. However, it does model the general increase and the levelling off of the population and gives a decent estimate of the population increase for future years.
According to me is the best function to forecast China’s population.
Bibliography
TI-84; Solving Linear Equations with Row Reductions to Echelon Form on Augmented Matrices method learnt from:
[Accessed on 21 February 2011]
Technology used:
Texas Instruments 84 Silver Edition calculator
Microsoft Paint (Version 6.1)
MathType (Version 6.7)
Microsoft Word 2010
Vernier Logger Pro
Emulator Used: Virtual TI-84 (Wabbitemu by Revolution Software)
Graph 1
Graph 2
Graph 3
Graph 4
Graph 5
Graph 7
Graph 6
Graph 8
Graph 9
Graph 10
Graph 11
Graph 12
Graph 14
[1] Logger pro
[2] Method learnt from:
[Accessed on 21 February 2011]
by
ishaan463 (student)
