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Mathematics sl IA 26.03.2012

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Mathematics sl IA 26.03.2012

  1. 1. Mathematics SL Margaux LOESCHE Internal AssessmentPOPULATION TRENDS IN CHINA 26/03/2012
  2. 2. Population Trends in China are common to observe since China has the largest population on Earthsince a number of years. We can observe its growth with the data collected from 1950 to 1995. The variable for this data is the growing populations. The data collected is not completelyaccuratethough since a lot of parameters influence this data in many ways. The health of the people in Chinaaffects the data. The new medicine and technology discovered throughout the years affected a lot theconditions of living in many countries such as China. The migration of the people throughout the world alsoaffects the set of data evidently. During this period of time there were a lot of geographical and naturaldisasters that affected the population of China by killing many in an unexpected situation. China is also aparticular country because it set a one child policy in the country; this is a factor that affected a lot thegrowth of the population. Year 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 Populations 554.8 609.0 657.5 729.2 830.7 927.8 998.9 1070.0 1155.3 1220.5 in Millions Here is a chart with the data and a graph showing its growth throughout the years. The X is the yearsin which the data was collected, the variable Y is the population that varies: 1400 Population in Millions 1300 1200 1100 1000 900 800 700 600 Year 500 1950 1960 1970 1980 1990 2000 2010 This is a clear graph obtained using Autograph by plotting in the data. We can see on this graph that the population is constantly increasing. However we can also identify asmall part of a curve which could by identified as an exponential function from the year 1950 up to the year1975. A quadratic or sine curve could be used to fit this data. The points following that year however follow asimple linear function. So the data could fit a simple linear function. I tried a linear function to see how well itcould fit.
  3. 3. The equation of a linear function is y=mx+p where m is the gradient and p is the y intercept M= M= M= M=15.05 To find p you substitute the coordinates of a point into the equation. Here I used x=1985 y=1070 Y=15.05x+p 1070=15.05×1985+p 1070=29874.25+p -P=29874.25-1070 -P=28804.25 P=-28804.25 Therefore the equation for the function would be: Y=15.05x-28804.25 1500 Population in Millions 1400 1300 1200 1100 1000 900 800 700 600 Year 500 1950 1960 1970 1980 1990 2000 We can see that this model does not correctly fit the data because it does not go through all thepoints.
  4. 4. Year Population Value of population in Difference in millions millions produced by the model (using technology)1950 554.8 543.25 554.8-543.25= 11.551955 609 618.5 609-618.5= -9.51960 657.5 693.75 657.5-693.75= -36.251965 729.2 729 729.5-729= .51970 830.7 844.25 830.7-844.25= -13.551975 927.8 919.5 927.8-919.5= 8.31980 998.9 994.75 998.9-994.75= 4.151985 1070 1070 1070-1070= 01990 1155.3 1145.25 1155.3-1145.25= 10.051995 1220.5 1220.5 1220.5-1220.5= 0 = =-2.475 The difference between the model and the data is quite high since the sum of all the differences foreach point and then divided by the number of points is equal to -2.475. The closer this value is to zero thebetter the model will fit the data. Another model that could be used to fit this equation could be a polynomial. Using the calculator wecan find the model that fits the best and that would be following this equation: y= + +ex+f Where a=0,0003161 b=-0,01663 c= 0,3948 d= -3,848 e=22,58 f=554,4 This gives us the following equation: y=0,0003161x⁵-0,01663x⁴+0,3948x³-3,848x²+22,58x+554,4 When this equation is graphed on the data points we get a very accurate model that fits thedata.
  5. 5. 1500 Population in millions 1400 1300 1200 1100 1000 900 800 700 600 Years 500 0 10 20 30 40 50Year Population Value of population in Difference in millions millions produced by the model (using technology)1950 554.8 554.4 554.8-554.4= .41955 609 611 609-611= -21960 657.5 653.3 657.5-653.3= 4.21965 729.2 732.3 729.5-732.3= -2.81970 830.7 832 830.7-832= -1.31975 927.8 924.5 927.8-924.5= 3.31980 998.9 999.9 998.9-999.9= -11985 1070 1071 1070-1071= -11990 1155.3 1154 1155.3-1154= 1.31995 1220.5 1221 1220.5-1221= -0.5 = =0.06 By doing a difference table here we managed to see that this model fits almost perfectly tothe data since the sum of all the differences between points over the number of points is only equalto 0.06 which is very close to 0. A researcher suggests that the population, P at time t can be modeled by : Where K, L and M are parameters.
  6. 6. P=PopulationT=TimeK=carrying capacityL=growth rateM=rate of change in growth rateUsing the GDC, the logistic we can easily find the values of these unknowns.K 1946L 2.619M 0.03332P(t)= 1500 Population in Millions 1400 1300 1200 1100 1000 900 800 700 600 Year 500 0 10 20 30 40 50
  7. 7. Year Population Value of population in Difference in millions millions produced by the model (using technology)1950 554.8 537.7 554.8-537.7= 17.11955 609 604.9 609-604.9= 4.11960 657.5 676.4 657.5-676.4= -18.91965 729.2 751.7 729.5-751.7= -22.21970 830.7 829.9 830.7-829.9= 0.81975 927.8 909.9 927.8-909.9= 17.91980 998.9 990.9 998.9-990.9= 81985 1070 1072 1070-1072= -21990 1155.3 1151 1155.3-1151= 4.31995 1220.5 1228 1220.5-1228= -7.5 = =0.16 Here is some additional data given:Year 1983 1992 1997 2000 2003 2005 2008Population in millions 1030.1 1171.1 1236.3 1267.4 1292.3 1307.6 1327.7 1500 Population in Millions 1400 1300 1200 1100 1000 900 800 700 600 Year 500 1950 1960 1970 1980 1990 2000 2010
  8. 8. 1500 Population in Millions14001300120011001000 900 800 700 600 Year 500 0 10 20 30 40 50 60 When we add the new data the researcher’s model does not fit as well as with the previous set ofdata .we see that the points that were added do not correspond or are not close to points from the modelespecially towards the end, those points were added points. These result also go for the linear function used in the beginning:
  9. 9. By “fiddling” with the numbers of the linear equation I found a function that corresponded more tothe new set of data although not quit the perfect fit. 1500 Population in Millions 1400 1300 1200 1100 1000 900 800 700 600 Year 500 1950 1960 1970 1980 1990 2000 2010

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