• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

In this Internal Assessment, functions that best model the population of China from 1950-1995 were thoroughly investigated.

Extracts from this document...



In this Internal Assessment, functions that best model the population of China from 1950-1995 were thoroughly investigated.

The following table shows the population of China from 1950 to 1995:













in Millions











To begin, instead of using 1950, 1955, 1960, 1965, 1970, 1975, 1980, 1985, 1990, and 1995 for the Year values, 0, 5, 10, 15, 20, 25, 30, 35, 40, and 45 will be used during the investigation to facilitate analysis. Through the use of a Graphic Display Calculator (GDC), the following data points were plotted on a graph.


For easier interpretation and scrutinizing of the data points, Graphical Analysis 3 was used to provide a more understandable and vivid graph for analysis. image01.png

The graph seems to show trends also exhibited by linear graphs, quadratic graphs, and cubic graphs:

Linear Fit:
From the graph, it is apparent that from 20, or 1970 to 45, or 1995, the graph increases in a linear manner in approximately the same rate of change (slope). Therefore, a linear model could provide an efficient fit for the graph of the data points. Now, assuming a linear fit for the plot of the data points, the function would be in the form y=Mx + B. The parameters, in this case, will be M and B, while the variable will be x. Parameter M

...read more.


image03.png = image11.pngimage11.png = image02.pngimage02.png = 19.42 ≈ 19.4

Slope = image03.pngimage03.png = image12.pngimage12.png = image13.pngimage13.png = 14.22 ≈ 19.2

Slope = image03.pngimage03.png = image15.pngimage15.png = image13.pngimage13.png = 14.22 ≈ 14.2

Slope = image03.pngimage03.png = image16.pngimage16.png = image17.pngimage17.png = 17.06 ≈ 17.1

Slope = image03.pngimage03.png = image18.pngimage18.png = image19.pngimage19.png = 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) = image23.pngimage23.png  , where K, L

...read more.


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) = image43.pngimage43.png

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) = image43.pngimage43.png  shows the population trends in China with an efficient amount of accuracy and precision.

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Math IA -Modelling Population Growth in China.

    The researchers function of P(t) = (K)/((1+(Le)^(-Mt))) where K, L, and M are parameters This is a basic logistic function. The basic logistic function is: In order to find the function that the researchers suggested I will run a Logistic Regression on the TI 84 plus calculator.

  2. Maths Portfolio - Population trends in China

    One of the limitations that were noticed in the first linear model function was that although it approached the points close, it wasn't accurate enough, and it does not represent the population statistics as we would imagine, because of its tendency to portray results just in the short run.

  1. A logistic model

    run r ? ?1? 10?5 u ? 1.6 n n ? un?1 ? (?1? 10?5 u n ? 1.6)un ? (?1?10?5 )(u )(u ) ? 1.6u The logistic function model for un+1 is: n n n ? (?1?10?5 )(u 2 )

  2. An investigation of different functions that best model the population of China.

    Exponential Logarithmic Linear Power of x Polynomial (6th degree) Indeed the rate of this increase constantly changes with it getting faster and slower. Therefore, it would not make sense to use an exponential, or indeed a linear model to best fit the data.

  1. Investigating Quadratic functions

    However, you could see that the vertices of both equations are different. You could see from the graph that the vertex of the function y=x� is on the origin (0,0) and the vertex of y=(x-2)�+3 is (2,3). Therefore the positions of the two parabolas are different.

  2. IB Math Methods SL: Internal Assessment on Gold Medal Heights

    * - Although there are values for 1940 and 1944 (8 and 12 years elapsed respectively); as the original data set does not have such data for those years, we reject the two h-values. Let us now compare the table of values to the other tables wrought from the original

  1. The purpose of this investigation is to create and model a dice-based casino game ...

    With respects to fairly rational-minded casino patrons, having a large entry fee and higher expected earnings (or lower expected loss) would be a more attractive prospect in the long run. Thus by setting the payoff at 130% to 135% of the payment to play, and by setting the payment at an above-average value (e.g.

  2. This assignments purpose is to investigate how translation and enlargement of data affects statistical ...

    I would first have to multiply 5 to each height which is shown in the table below: Then repeating steps I used on the first question using my TI-84 (STAT: CALC: 1-Var Stats: ENTER: List: 2ND: 2: ENTER: Calculate: ENTER).

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work