Population Trends in China - Math Problem Example

?Population Trends in China Introduction China has been documented as the world’s most populous country, with a population of about 3 billion, which is about a fifth of the world’s population. The rapid growth in population has become a concern to the government, such that it has introduced various measures to curb the rate of growth. China therefore is a good case study in population growth trends, its analysis and prediction. This paper concentrates on population trends in China between 1950 and 2008, and outlines various models that could be used to represent the data. A variable can be defined as a quantity or attribute that changes according to different situations in a certain process. In our case, the time (t) in years is a variable because it moves from 1950 to 1995. The second variable is population (P) of the people of China in millions. A parameter is a constant that varies from one group of equations to another. The parameters are m, the gradient and c the y-intercept. The graph below shows the population trends in China between 1950 and 1995. Trends Seen in the Graph From the graph, it can be noted that population in China has been increasing over the years. Between 1950 and 1975, the population increases at an increasing rate whereas, between 1975 and 1995, it increases steadily. This difference could be attributed to the one child policy that was introduced by the Chinese government in 1978, to curb population increase. The points in the above graph assume a linear pattern and if a line of best fit is drawn, it has a gradient of approximately 15.49 (calculated using technology). This means that China’s population grows by about 15.49 million people each year. The increase in population has remained steady probably because, there has been a decrease in the infant mortality rate and national deaths due to improved healthcare. Another factor could be immigration by people from other countries. The correlation coefficient (R2) is 0.994, translating to a 99.4% fit, which is a good fit. Functions that could Model the Behaviour of the Graph Some of the functions that could possibly be used to model the behaviour of this graph are, exponential, linear, logarithmic and polynomial functions. Starting off with the exponential function, it takes the form, The growth rate of an exponential function is proportional to its value. For example, if the rate of population growth is proportional to its size, then the population after t years will be Fig 2: Graph showing an exponential model of China’s population. The correlation coefficient in the above graph is 0.990 which translates to a 99.0% fit of the data points to the curve. This is a relatively good fit. However, the data points for 1970 up to 1980 are overestimated while that of 1995 is underrated. A linear function is one that can be written in the form, where m is the gradient and c is the y-intercept. A linear function often implies uniformity. Fig 3: A graph showing a linear model of China’s population. The correlation coefficient (R2) is 0.994 translating to a 99.4% fit of the data points to the line. This is a much better fit than the exponential model. In this model, the points in 1960 and 1965 are underestimated while that of 1950 is overrated. A logarithmic function can be defined as the inverse of an exponential function. It can be expressed by the following identity, Fig 4: A graph showing logarithmic model of China’s population. The correlation coefficient of the above graph is 0.994 translating to a 99.4% fit. It is similar to the linear function fit, but also better than the exponential model fit. The above three models can be used because they all have a 99.0% and above fit, which is an excellent choice as we cannot have real data with a 100% fit. Developing a Model Function that Fits the Data In this case, I choose to use a polynomial function. A polynomial function can be defined as a mathematical function consisting of several terms added together. This includes a linear function, which is discussed above. A polynomial function has the general form, Taking a polynomial of degree 3 and 4 distinct points A(10, 657.5), B(20, 830.7), C(25, 927.8), D(35, 1070.0) with p, q, r and s as the parameters, we come up with the following system: The above system may be solved using matrices, X=Y-1 Z where Y is the coefficient matrix and Z is a vector representing solutions to the systems. This will be done as shown below. After solving the system, we get the following equation: Using the above equation, we can plot a graph showing a polynomial model. The correlation coefficient in the polynomial model function is 0.998 which translates to a 99.8% fit of the data points. All the data points in this graph appear to lie perfectly on the curve. It is evident from the above graph that the polynomial model function is the best model to use as compared to the previous three models. This is because it has an almost perfect fit. It is therefore unnecessary to revise the above model. Researcher’s Model The researcher’s model: can be solved using technology, i.e. Texas Instruments 84SE Calculator, we find that the values of the parameters K, L and M are: K= 1946, L=2.62 and M=0.0333. Thus, the researcher’s model is now: This model represents logistic growth, as population increases increasingly at first, but then after some time, grows at a constant rate. This can be explained by the fact that population growth is subject to the resources available. At some point, the population reaches a level known as transport capacity, where the factors that promote population growth e.g. births are countered by factors that diminish population, such as mortality. This can be shown in the equation as follows: As t approaches infinity (t >?), e-Mt approaches zero (e-Mt >0) and therefore, where K is the transport capacity. If K was greater, this would imply that population at time t will be more, and vice versa. In case L was increased, the population at time t would then decrease, while increasing M would have the same effect as when t approaches infinity. Fig 6: A graph representing the researcher’s model of China’s population As shown above, the researcher’s model data points appear to fall on the line, apart from a slight overestimation of the points between 1965 and 1975, while the point in 1995 is slightly underrated. The correlation coefficient is 0.995 which translates to a 99.5% fit, making it the second best model discussed here. It is therefore also a good choice. Future Implications of Each Model The exponential model suggests that the population will continue to increase at an increasing rate, which is not possible in reality. This is because, according to logistic model, population reaches transport capacity as time t increases, thus population cannot grow further due to limited resources. The linear model implies that population increases at a constant rate throughout the years, which is again not true, because it does not put into account the transport capacity, natural catastrophes such as earthquakes and diseases. It gives an impression of uniformity in population growth. This is not practical, because fertility rate could fluctuate depending on situations such as the one child policy. This is also the case with the logarithmic model. The polynomial model gives the most accurate impression on population growth in China. It portrays increasing rate of population growth at the beginning, which then changes after some time and population growth decelerates. In reality, when population increase gets out of hand, the government introduces policies to control the increase. Also, natural factors such as mortality eventually balance out with births thus reducing the rate at which population grows. Population Trends in China by the IMF Fig 7: Population trends in China from the 2008 World Economic Outlook published by the IMF This linear model fits the points relatively well, though the points between 1998 and 2003 are slightly overestimated, while the point in 2008 is slightly underrated. The correlation coefficient is 0.974 which implies a 97.4% fit of the points on the line. Fig 8: A graph of population trends in China using exponential model Four points in the graph are off the line, which is half the number of points. This is not one of the best models to adopt for this data, as it does not represent majority of the data points. Its correlation is also lower than the previous model, being 0.962 which translates to a 96.2% fit. It would be unwise to choose this model as it does not portray the real world situation. Fig 8: A graph showing logarithmic model for IMF data. In this case, only the last point clearly stands out, i.e. the 2008 data point is completely off the line as compared to the others. The correlation coefficient (0.975) implies that the fit is 97.5% perfect. It can therefore be safely deduced that, the logarithmic model is more efficient than the linear and exponential models. Fig 9: A graph showing the researcher’s model for IMF population data. The points in the researcher’s model assume a constant growth rate in population. Its correlation is 0.998 which translates to 99.8% fit. It is better than the logarithmic, linear and exponential models thus may be chosen over the three. Fig 10: Graph showing the polynomial model using the IMF data. The polynomial model fits perfectly on all data points with a correlation coefficient of 0.999, which implies a 99.9% perfect fit. It is by far the best model that could be used for this data. Modifying Model to Apply to all Data from 1950 to 2008 Fig 11: A graph combining initial data with IMF data to show China’s population from 1950 to 2008. The polynomial model, which best fits the previous data, in this case gives a correlation of 0.994, which is a 99.4% fit. This is a relatively good model, as 11 out of 17 points lie on the line. Thus, it leaves room for error, which is inevitable in reality as we cannot have a perfect model. Conclusion Despite having various models for this data, we found out that the models differ in efficiency. In our case, the polynomial model turned out to be the best choice as it displays an accurate picture of the real situation concerning population growth in China over the years. It is evident from the polynomial model that in years to come, population growth rate in China will decrease, and most probably it will no longer be the world’s most populous country. Works Cited Szecsei, Denise. Homework Helpers Pre-Calculus. 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