Population Trends in China Essay Sample free essay sample

Using the Chinese population informations from 1950 to 1995. allow us build a graph utilizing engineering. Before charting the information though. we must foremost find the relevant variables. which are. the twelvemonth and the population ( in 1000000s ) of each co-occuring twelvemonth. The parametric quantities are purely confined to the information for the old ages 1950 and 1995 in the sense that the informations can non fall below the population figure for the twelvemonth 1950 and can non fall above the information for the twelvemonth 1995. Upon reexamining the graph. we notice that the information appears to increase. but non exactly in a consecutive line. Traveling through the tendency lines on excel. we find that the tendency line that fits the best is the multinomial tendency line. which is displayed in the graph down at a lower place. If we were to analytically develop one theoretical account map to find if the multinomial tendency line is so the most accurate tantrum. We will write a custom essay sample on Population Trends in China Essay Sample or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page I would suggest making a system of equations. Before leaping to far in front. we need to do it clear the equation we are traveling to be analysing. We will utilize the equation given to us by the multinomial tendency line which is: y= ax2 + bx +c and the ground that we are utilizing this equation is because of the fact that the R2 value is 0. 9955. The closer the R2 value is to 1 the better it will suit the graph. We will rearrange the equation y= ax2 + bx +c so that we can work out for the terra incognitas which are. the letters a. B and c. To make this. we need to add informations to the equation and make three matrices. In order to go on on. allow us foremost add the known values to the equation. Given y= ax2 + bx +c. we know that the y-values are China’s population in 1000000s and the x-values are the old ages at which the population is measured. We will utilize three points to work out this. one from the beginning. one from the center. and one from the terminal in order to make the matrices that will so be used to happen values for a. b. and degree Celsius. So we take the equation y= ax2 + bx +c and stopper 1950 ( first twelvemonth ) in as the x-value and 554. 8 as the y-value and we will make the same with the following two points that will be used.y= ax2 + bx +c554. 8=a ( 19502 ) + B ( 1950 ) + degree Celsius830. 7=a ( 19702 ) + B ( 1970 ) + degree Celsius1120. 5=a ( 19952 ) + B ( 1995 ) + degree CelsiusNow that we have created our system of equations we can divide them up into matrices. Looking at the equations we have come up with. we notice that we have a column of y-values ( bold ) . a column of x-values ( blue ) . and a column of terra incognitas ( purple ) 554. 8=a ( 19502 ) + B ( 1950 ) + degree Celsius830. 7=a ( 19702 ) + B ( 1970 ) + degree Celsius1120. 5=a ( 19952 ) + B ( 1995 ) + degree Celsius We can name the y-values matrix ‘B’ or [ B ] which is traveling to be a 3?1 matrix. the x-values matrix ‘A’ or [ A ] which will be a 3?3 matrix. and the unknowns matrix ‘C’ or [ C ] which is another 3?1 matrix. [ A ] = [ B ] = [ C ] = So so we get an equation that looks like this: [ B ] = [ A ] * [ C ] . We can split [ A ] from one side of the equation to insulate [ C ] so. [ A ] -1* [ B ] = [ C ] . We multiply by the opposite of [ A ] . Multiplying [ A ] -1* [ B ] we get the [ C ] to be. Year | 1950| 1955| 1960| 1965| 1970| 1975| 1980| 1985| 1990| 1995| New Population in Millions| 554. 8| 620. 8| 688. 8| 758. 7| 830. 7|904. 6| 980. 6| 1058. 6| 1138. 5| 1220. 5| Having plugged the original old ages into the found map we receive the information in the informations tabular array above and acquire a graph that looks like the information is close to the original graph. We are following shown an equation where the population at clip is modeled by: P ( T ) = Using a reckoner we can run a logistical trial on the original informations where we get K to be 1950. L to equal 4. 34. and M to be. 0333. We get the informations tabular array below. Year | 1950| 1955| 1960| 1965| 1970| 1975| 1980| 1985| 1990| 1995| New Population in Millions| 537. 8| 605. 0| 676. 5| 751. 8| 830. 0| 910. 0| 991. 0| 1071. 7| 1151. 1| 1228. 1| The graph created by the logistic map is a really close tantrum. it about covers up the original population graph. Even looking at the informations tabular array for the map. we see that the information is really near to the original set of informations. We know that in multinomial maps. the independent and dependent variables are straight related to each other. In this survey the form of the original informations indicates a uninterrupted gradual slope. so as the old ages increase so will the population. The logistic map is different in the sense that while it will go on to increase. after awhile it will non increase as much and will get down to decelerate down until it stops increasing and degrees off. The graphs below are to assistance in the apprehension of the construct. We are following given a set of informations on the population trends in China from the 2008 World Economic Outlook published from the International Monetary Fund ( IMF ) Year | 1983| 1992| 1997| 2000| 2003| 2005| 2008|New Population in Millions| 1030. 1| 1171. 7| 1236. 3| 1267. 4| 1292. 3| 1307. 6| 1327. 7| We will get down with the multinomial map and once more choose three different points for the above information. So we will acquire: y= ax2 + bx +c 1030. 1=a ( 19832 ) + B ( 1983 ) + degree Celsius1267. 4=a ( 20002 ) + B ( 2000 ) + degree Celsius1327. 7=a ( 20082 ) + B ( 2008 ) + degree Celsius We can name the y-values matrix ‘B’ or [ B ] which is traveling to be a 3?1 matrix. the x-values matrix ‘A’ or [ A ] which will be a 3?3 matrix. and the unknowns matrix ‘C’ or [ C ] which is another 3?1 matrix. [ A ] = [ B ] = [ C ] =We can split [ A ] from one side of the equation to insulate [ C ] so. [ A ] -1* [ B ] = [ C ] . We multiply by the opposite of [ A ] . Multiplying [ A ] -1* [ B ] we get the [ C ] to be. Graphing this we get the graph displayed below which we see looks indistinguishable to the original to the point where we do non even see the original graph. Now we run the 2nd map trial to see how that one besides fits the new informations. Again we have the equation P ( T ) = Using a reckoner we can run a logistical trial on the original informations where we get K to be 1436. L to equal 1. 36. and M to be. 0633. We get the informations tabular array below. Year | 1983| 1992| 1997| 2000| 2003| 2005| 2008|New Population in Millions| 1030. 1| 1171. 7| 1236. 3| 1267. 4| 1292. 3| 1307. 6| 1327. 7| We so take this found information and see how good it fits the original informations that we graphed earlier. As we see the logistic equation fits the original informations absolutely. Now let us see how both graphs expression together when consolidated into one graph. As we see here. the information from both the original graphs and their logistic maps line up rather good about making one uninterrupted line. The theoretical accounts fit each other with no outliers and making one solid line on the graph. The same can be seen with the 2nd graph of the original informations and the original IMF informations and their several multinomial maps. In decision to this mathematical survey. we have found tendency lines to chart informations. created system of equations for that informations and successfully graphed them against their original graphs. We learned how logistic graphs and multinomial graphs work and how they differ from each other. To make another challenge it would be interesting to seek and prove more maps. It would be interesting to prove parametric quantities and see what would go on if we tested outside of them. It would be a challenge to larn how the logistic map works mathematically to bring forth the Numberss it produces on the reckoner.