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Aim: in this task, you will investigate the different functions that best model the population of China from 1950 to 1995.

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Introduction

Steven Burnett        IB Year 1        SL Type II

Aim: in this task, you will investigate the different functions that best model the population of China from 1950 to 1995.

Data on 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

Variables and parameters:

Variables

Parameters

  • The population (in millions). In the data provided, China's population is increasing every 5 years.
  • The year is also changing (increasing every 5 years between 1950 and 1995).
  • Since the population data starts at 1950, the graph shouldn't start at year 0 (since there is no data).
  • The years and population can't be negative (they can only go down to 0).

image04.png

  • The graph above is the data for China's population (in millions) plotted as a scatter graph. The population is clearly increasing as the years increase, seemingly at a steady rate similar to that of a linear progression.

image05.png

Linear trend line:

  • The above graph now has a linear trend line (with the equation of f(x) = 15.5x – 29690.25).Although it is very close to each point, it isn't perfect (for example, the data for 1950 and 1965 hardly touch the line). Furthermore, this model predicts that China's population would simply increase at a constant rate – which is unlikely since it is expected that a population would increase exponentially, in an ideal world.
...read more.

Middle

Logarithmic trend line:

  • The logarithmic trend line (f(x) = 30561.48 In(x) – 230995.52) appears to fit the data almost identically to the linear line, with it missing the data points at 1950 and 1965. However, as it is a logarithmic line, it is likely that the curve will level out in the future – which isn't likely to actually happen in the real world.

Final model:

  • Since a linear model fitted the most amount of points, I modified the original line (new equation: y=15.5x-2.969E+004). Some points don't quite fit onto the line, but every point after 1975 fits perfectly; it also shows a steady increase in the population which could be accurately used to predict the population growth in the near future (although curved trend lines might be more appropriate for prediction far into the future).

image07.png

Researcher's model:

image08.png

  • The years (on the x-axis) have been replaced with smaller numbers: 1950 is 1, 1955 is 2 and so on.
  • The population in millions (on the y-axis) have been divided by 100.

I changed these values so that I could calculate the values for L, K and M – the scale of the graph needed to change. By using autograph's constant controller, I was able to estimate values for L, K and M:

L

4.5

K

0.7

M

-0.098

  • The model does fit most of the points, except for 5 (1970), 6 (1975) and 10 (1995). However, when compared to the linear model above, the points that aren't matched to the model are further away from the line
...read more.

Conclusion

Year

1983

1992

1997

2000

2003

2005

2008

Population (in millions)

1030.1

1171.1

1236.3

1267.4

1292.3

1307.6

1327.7

image10.png

Linear trend line:

image11.png

  • The linear trend line (f(x) = 11.94x – 22621.29) fits all of the IMF data points well, although at 1983 and 2008 it barely touches them. However, it might be more appropriate to use a curved trend line to fit all of the points exactly.

Polynomial trend line:

image12.png

  • This order 6 polynomial (y=0.001356x³-1.036x²-2296x+2.804E+006) fits all of the points on the IMF data almost exactly – although since it begins to slope downwards, I think that it would only be appropriate for predictions up to 2015 as the population is most likely going to rise; not fall.

Final model:

  • Since the polynomial trend line fitted all of the points, I modified it to fit all of the data:

image06.png

  • Although this model does slightly miss the points at 1960, 1965 and 1975, it matches all of the other points almost exactly. The equation for this model is y=-0.002044x³+3.266x²-1.247E+044x+1.213E+007. Furthermore, I think that this model would be accurate for predicting the population of China up to 2012, at which point the model takes an unexpected downwards slope.

...read more.

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