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

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Introduction

MATHEMATICS SL
INTERNAL ASSESSMENT TYPE II
POPULATION TRENDS IN CHINA

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:

Year

1950

1955

1960

1965

1970

1975

1980

1985

1990

1995

Population

in Millions

554.8

609.0

657.5

729.2

830.7

927.8

998.9

1070.0

1155.3

1220.5

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.

image00.png

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

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Middle

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.

image20.png

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.

image21.pngimage22.png

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

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Conclusion

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:

image42.png

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:

image44.png

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.

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