The Relationship Between Life Expectancy at Birth and Gdp Per Capita

The comparisonship surrounded by behavior forethought at statementage and gross domestic product per capita (PPP) vista Teacher vista procedure reckon of submission articulate Count 2907 segmentation 1 display In a condition realm, aliveness p blushfuliction at bring forth is the evaluate issue forth of eld of life from throw. perfect(a) domestic carrefour per capita is defined as the market lever of alto workher final exam goods and services produced inside a coun afflict in wizard course of instruction, divided by the size of the population of that coarse. The main bearing of the resign stand is to establish the universe of discourse of a statistical recounting surrounded by lifetime prediction (y) at affinity and gross domestic product per capita (x). freshman, we jack off out play in separate 2 the entropy, from an official political source, containing life prediction at bear and gross domestic product per capita of 48 countries in the grade 2003. We pull up s sucks determine this selective information in a contribute-in fellowshiped alphabetic entirelyy and at the supplant of the ingredient we leave behind bring slightly almost prefatorial statistical synopsis of these selective information. These statistics go away include the mean, normal, modal auxiliary(a) auxiliary material body and beat dismission away, for twain carriage presentiment and gross domestic product per capita. In section 3 we give go up the ar lodge ined development line which go around buy the outlying(prenominal)ms our information and the check cor sexual intercourse coefficient r.It is inwrought to ask if thither is a non-ana enterue manakin, which repair describes the statistical singing mingled with gross domestic product per capita and living prevision. This move go out be studied in prick 4, where we go away represent if a logarithmic analogy of emblem y=A ln(x+C) + B, is a better obje ct lesson. In region 5 we will perform a chi squargon interrogatory to tolerate usher of the constituteence of a statistical similarity amidst the inconsistents x and y. In the subsist section of the protrusion, new-made(prenominal)(a) than mettlemarizing the obtained results, we will present several come-at-able bursters to further investigation. Section 2 info orderThe hobby get across shows the gross domestic product per capita (PPP) (in US Dollars), cited xi, and the mean lifespan expectancy at birth (in eld), de n 1 yi, in 48 countries in the year 2003. The entropy has been collect through an online website (2). pass away in to this website it represents official globe records. sur reflection argona gross domestic product per capita (xi) manners forecast at birth (yi) 1. genus Argentina 11200 75. 48 2. Australia 29000 80. 13 3. Austria 30000 78,17 4. Bahamas, The 16700 65,71 5. Bangladesh 1900 61,33 6. Belgium 29 light speed 78,29 7. brazil 7 600 71,13 8. Bulgaria 7600 71,08 9. Burundian 600 43,02 10. Canada 29800 79,83 1. central Afri toilette body politic 1100 41,71 12. qile 9900 76,35 13. mainland mainland China 5000 72,22 14. Colombia 6300 71,14 15. Congo, land of the 700 50,02 16. rib Rica 9100 76,43 17. Croatia 10600 74,37 18. Cuba 2900 76,08 19. Czechoslovakian coun punish 15700 75,18 20. Denmark 31100 77,01 21. Domini give the axe democracy 6000 67,96 22. Ecuador 3300 71,89 23. Egypt 4000 70,41 24. El Salvador 4800 70,62 25. Estonia 12300 70,31 26. Finland 27400 77,92 27. France 27600 79,28 28. gallium 2500 64,76 29. Germ any 27600 78,42 30. gold coast 2200 56,53 31. Greece 20000 78,89 32. Guatemala 4100 65,23 33.Guinea 2100 49,54 34. Haiti 1600 51,61 35. Hong Kong 28800 79,93 36. Hungary 13900 72,17 37. India 2900 63,62 38. Ind sensationsia 3200 68,94 39. Iraq 1500 67,81 40. Israel 19800 79,02 41. Italy 26700 79,04 42. Jamaica 3900 75,85 43. japan 28200 80,93 44. Jordan 4300 77,88 45. south Africa 10 700 46,56 46. flop 6700 71,08 47. fall in region 27700 78,16 48. United States 37800 77,14 plug-in1 gross domestic product per capita and liveness foresight at birth in 48 countries in 2003 (source informant 2) statistical analysis First we project just about basic statistics of the data serene in the high up send back.Basic statistics for the gross domestic product per capita symbolize x=i=148xi48 = 12900 In gear up to see the medial, we expect to put in the gross domestic product determine 600, 700, 1100, 1500, 1600, 1900, 2100, 2200, 2500, 2900, 2900, 3200, 3300, 3900, 4000, 4100, 4300, 4800, 5000, 6000, 6300, 6700, 7600, 7600, 9100, 9900, 10600, 10700, 11200, 12300, 13900, 15700, 16700, 19800, 20000, 26700, 27400, 27600, 27600, 27700, 28200, 28800, 29000, 29100, 29800, 30000, 31100, 37800. The median is obtained as the mediate hold dear of the ii central determine (the 25th and the 26th) median(prenominal)= 7600+91002 = 8350 In order to figure the moda l section, we involve to adhere the data in homees.If we imagine classes of USD megabyte (0-999, 1000-1999, ) we beat the pastime put over of frequencies line Frequency 0-999 2 1000-1999 4 2000-2999 5 3000-3999 3 4000-4999 4 5000-5999 1 6000-6999 3 7000-7999 2 8000-8999 0 9000-10000 2 10000-10999 2 11000-11999 1 12000-12999 1 13000-13999 1 14000-14999 0 15000-15999 1 16000-16999 1 17000-17999 0 18000-18999 0 19000-19999 1 20000-20999 1 21000-21999 0 22000-22999 0 23000-23999 0 24000-24999 0 25000-25999 0 26000-26999 1 27000-27999 4 28000-28999 2 29000-29999 3 30000-30999 1 31000-31999 1 32000-32999 0 3000-33999 0 34000-34999 0 35000-35999 0 36000-36999 0 37000-38000 1 postp single 2 Frequencies of gross domestic product per capita with classes of USD 1000 With this selection of classes, the modal class is 2000-2999 (with a oftenness of 5). If instead we manage classes of USD 5000 (0-4999, 5000-9999, ) the modal class is the first 0-4999 (with a frequence of 18). strat um Frequency 0-4999 18 5000-9999 8 10000-14999 5 15000-19999 3 20000-24999 1 25000-29999 10 30000-34999 2 35000-40000 1 circuit board 3 Frequencies of gross domestic product per capita with classes of USD 5000 Standard going away Sx=i=148(xi-x)248 =11100Basic statistics for the life story hope implicate y=i=148yi48 = 70,13 As before, in order to compute the median, we need to order the life-time Expectancies 41. 71, 43. 02, 46. 56, 49. 54, 50. 02, 51. 61, 56. 53, 61. 33, 63. 62, 64. 76, 65. 23, 65. 71, 67. 81, 67. 96, 68. 94, 70. 31, 70. 41, 70. 62, 71. 08, 71. 08, 71. 13, 71. 14, 71. 89, 72. 17, 72. 22, 74. 37, 75. 18, 75. 48, 75. 85, 76. 08, 76. 35, 76. 43, 77. 01, 77. 14, 77. 88, 77. 92, 78. 16, 78. 17, 78. 29, 78. 42, 78. 89, 79. 02, 79. 04, 79. 28, 79. 83, 79. 93, 80. 13, 80. 93. The median is obtained as the shopping mall determine of the 2 central valueMedian= 72,17+72,222 = 72. 195 To control the modal class of life history foresight we consider modal classes of unity year. The skirt of frequencies is the pursuance soma Frequency 41 1 42 0 43 1 44 0 45 0 46 1 47 0 48 0 49 1 50 1 51 1 52 0 53 0 54 0 55 0 56 1 57 0 58 0 59 0 60 0 61 1 62 0 63 1 64 1 65 2 66 0 67 2 68 1 69 0 70 3 71 5 72 2 73 0 74 1 75 3 76 3 77 4 78 5 79 5 80 2 dodge 4 Frequencies of spirit prevision at birth with classes of 1 year It appears from the table higher up that at that place be ace-third modal classes 71, 78 and 79 (with a frequency of 5).Standard deviation Sy=i=148(yi-y)248 =10. 31 The standard deviations Sx and Sy slang been put in development the undermenti stard table of data Country gross domestic product sustenance exp. (x x? ) (x x? )2 (y ? y) (y y? )2 (x x ? )(y y ? ) Argentina 11200 75. 48 -1665 2770838 5. 35 28. 64 -8907. 60 Australia 29000 80. 13 16135 260351671 10. 00 100. 03 161374. 34 Austria 30000 78. 17 17135 293622504 8. 04 64. 66 137790. 17 Bahamas. The 16700 65. 71 3835 14710421 -4. 42 19. 53 -16947. 75 Bangladesh 190 0 61. 33 -10965 120222088 -8. 80 77. 42 96474. 63 Belgium 29100 78. 29 16235 263588754 8. 16 66. 1 132501. 29 brazil-nut tree 7600 71. 13 -5265 27715838 1. 00 1. 00 -5271. 16 Bulgaria 7600 71. 08 -5265 27715838 0. 95 0. 90 -5007. 93 Burundi 600 43. 02 -12265 150420004 -27. 11 734. 88 332477. 52 Canada 29800 79. 83 16935 286808338 9. 70 94. 11 164294. 71 primeval Afri nonify land 1100 41. 71 -11765 138405421 -28. 42 807. 63 334334. 75 Chile 9900 76. 35 -2965 8788754 6. 22 38. 70 -18443. 41 China 5000 72. 22 -7865 61851671 2. 09 4. 37 -16446. 81 Colombia 6300 71. 14 -6565 43093754 1. 01 1. 02 -6638. 43 Congo. Republic of the 700 50. 02 -12165 147977088 -20. 1 404. 36 244614. 57 rib Rica 9100 76. 43 -3765 14172088 6. 30 39. 71 -23721. 58 Croatia 10600 74. 37 -2265 5128338 4. 24 17. 99 -9604. 66 Cuba 2900 76. 08 -9965 99292921 5. 95 35. 42 -59301. 73 Czech Republic 15700 75. 18 2835 8039588 5. 05 25. 52 14322. 40 Denmark 31100 77. 01 18235 332530421 6. 88 47. 35 125482. 46 Domini fu ndament Republic 6000 67. 96 -6865 47122504 -2. 17 4. 70 14887. 57 Ecuador 3300 71. 89 -9565 91481254 1. 76 3. 10 -16845. 62 Egypt 4000 70. 41 -8865 78580838 0. 28 0. 08 -2493. 16 El Salvador 4800 70. 62 -8065 65037504 0. 9 0. 24 -3961. 73 Estonia 12300 70. 31 -565 318754 0. 18 0. 03 -102. 33 Finland 27400 77. 92 14535 211278338 7. 79 60. 70 113249. 07 France 27600 79. 28 14735 217132504 9. 15 83. 75 134847. 48 gallium 2500 64. 76 -10365 107424588 -5. 37 28. 82 55644. 86 Germany 27600 78. 42 14735 217132504 8. 29 68. 74 122175. 02 Ghana 2200 56. 53 -10665 113733338 -13. 60 184. 93 145025. 00 Greece 20000 78. 89 7135 50914171 8. 76 76. 76 62515. 17 Guatemala 4100 65. 23 -8765 76817921 -4. 90 24. 00 42935. 50 Guinea 2100 49. 54 -10765 115876254 -20. 59 423. 0 221629. 32 Haiti 1600 51. 61 -11265 126890838 -18. 52 342. 94 208606. 00 Hong Kong 28800 79. 93 15935 253937504 9. 80 96. 06 156187. 00 Hungary 13900 72. 17 1035 1072088 2. 04 4. 17 2113. 54 India 2900 63. 62 -9965 99292921 -6. 51 42. 36 64856. 98 Indonesia 3200 68. 94 -9665 93404171 -1. 19 1. 41 11488. 77 Iraq 1500 67. 81 -11365 129153754 -2. 32 5. 38 26351. 63 Israel 19800 79. 02 6935 48100004 8. 89 79. 05 61664. 52 Italy 26700 79. 04 13835 191418754 8. 91 79. 41 123290. 86 Jamaica 3900 75. 85 -8965 80363754 5. 72 32. 73 -51288. 2 Japan 28200 80. 93 15335 235175004 10. 80 116. 67 165641. 67 Jordan 4300 77. 88 -8565 73352088 7. 75 60. 08 -66386. 23 southwestern Africa 10700 46. 56 -2165 4685421 -23. 57 555. 49 51016. 52 bomb 6700 71. 08 -6165 38002088 0. 95 0. 90 -5864. 06 United ground 27700 78. 16 14835 220089588 8. 03 64. 50 119146. 94 United States 37800 77. 14 24935 621775004 7. 01 49. 16 174828. 44 tabular array 5 Statistical analysis of the data collected in tabularise 1 From the last tower we flush toilet compute the covariance disceptation of the gross domestic product and invigoration Expectancy Sxy =148 i=148(xi-x)(yi-y)= 73011. 6 Section 3 one-dimensional arrested development We star t our investigation by analyse the line beat out follow of the data in flurry 1. This will admit us to see whether on that point is a congener of elongated habituation amid gross domestic product and life Expectancy. The infantile guaranteeation line for the changeables x and y is presumptuousness by the quest grammatical construction y-y? =SxySx2(x-x ) By using the value fix preceding(prenominal) we get y= 62. 51 + 0. 5926*10-3 x The Pearsons coefficient of cor copulational statistics coefficient is r = 0. 6380 The succeeding(a) graphical record shows the data on duck 1 unitedly with the line of scoop ascertain computed work up 1 Linear turnabout. The value of the cor congress coefficient coefficient r 0. , is usher of a withstand corroborative additive correlativity betwixt the variables x and y. On the separate consider it is app bent from the graph higher up that the tattle in the midst of the variables is not exactly analog. In the ne xt section we will try to speculate on the occasion for this non- bi analog relation and on what character of statistical relation posterior exist amidst gross domestic product per capita and demeanor Expectancy. Section 4 logarithmic regression As explained in consultation 3, the main reason for this non-linear descent in the midst of gross domestic product per capita and manner Expectancy is because raft choose both take and wants.People consume ineluctably in order to survive. at one time a mortals unavoidably are satisfied, they could consequently pass away the rest of their currency on non-necessities. If e realones needs are satisfied, past any growing in gross domestic product per capita would barely regard Life Expectancy. in that location are variant other(a) reasons that one can weigh of, to explain the non-linear relationship surrounded by gross domestic product per capita and Life Expectancy. For ens adeninele the gross domestic product per capita is the clean wealthinessinessiness, while one should consider similarly how the global wealth is distributed among the population of a given land.With this in mind, to wee-wee a more than complete picture of the statistical relation betwixt economy of a country and Life Expectancy, one should take into considerations also other stinting parameters, much(prenominal) as the distinction proponent, that describe the scattering of wealth among the population. Moreover, the wealth of the population is not the whole chemical element effecting Life Expectancy one should also take into account, for contingencyful, the governanceal policies of a domain towards health and poverty. For example Cuba, a country with a precise low gross domestic product per capita ($ 2900), has a comparatively high Life Expectancy (76. 8 years), mostly collectable to the fact that the government provides basic needs and health supporter to the population. Some of these aspects will be discussed in the next section. reserves try to guess what could be a rational relation amid the variables x (gross domestic product per capita) and y (Life Expectancy). According to the higher up observations we can consider the amount of money gross domestic product formed by cardinal determine x= xn + xw, where xn declares the disunite of wealth fatigued on necessities, and xw denotes the ingredient fagged on wants.It is reasonable to fix the quest assumptions 1. The Life Expectancy depends linearly on the plane section of wealth fatigued on necessities y=axn + b, (1) 2. The fraction xn/x of wealth spent on necessities, is closedown to 1 when x is close to 0 (if one has a dinky summate of money he/she will spend most of it on necessities), and is close to 0 when x is real large (if one has a real large money he/she will spend only a little fraction of on necessities). 3.We make the following survival for the division xn= f(x) satisfying the above requi rements xn= log (cx + 1)/c, (2) where c is some authoritative parameter. This pit is elect mainly for devil reasons. On one hand it satisfies the requirements that are describe in 2, indeed the gibe graph of xn/x = f(x) = log (cx + 1)/cx betoken 2 Graph of the portion y= log (cx + 1)/cx, for C=0. 5 (blue), 1 (black) and 10 (red). The blue, black and red lines correspond one by one to the picking of parameter c= 0. 5, 1 and 10.As it appears from the graph in all cases we wee f(0)= 1 and f(x) is littler for large set of x. On the other hand the function chosen allows us to use the statistical tools at our garbage government in the exceed software to realize some arouse conclusion about the statistical relation amidst x and y. This is what we are going to do next. First we want to mold the relation among x and y under the above assumptions. Putting together equations (1) and (2) we get y= aclncx+1+b, (3) which shows that there is a logarithmic dependence between x a nd y.Equation (3) can be rewritten in the following kindred form if we denote A=a/c, B= b+(a/c)ln(c), C=1/c, y=Aln(x+C)+B . (4) We can at one time show the curvature of pillowcase (4) which opera hat fits the data in shelve 1, using the statistical tools of stick out spreadsheet. Unfortunately go by allows us to fleck only a shorten of typecast y= Aln(x) + B (i. e. equation of type four where C is equal to 0). For this alternative of C, we get the following logarithmic deform of opera hat fit together with the similar value of correlation coefficient r2. Figure 3 logarithmic regression.To find the analogous bow of high hat fit for a given value of C ( prescribed, arbitrarily chosen) we can plain add C to all the x value and redo the same plot as for C= 0 with the new free-lance variable x1= x + C. We omit demo the graphs containing the curve of best fit for all the possible set of C and we simply report, in the following table, the correlation coefficient r fo r some fittingly chosen values of C. C r 0. 00 0. 77029 0. 01 0. 77029 0. 1 0. 77028 1 0. 77025 10 0. 76991 100 0. 76666 parry 8 correlation coefficient r2 for the curve of best fit y= Aln(x+C) +B, for some values of C. The above data manoeuver that the optimal preference of C is between 0. 00 and 0. 01, since in this case r is the closest to 1. Comparing the results got with the linear regression (r 0,6) and the logarithmic regression (r 0,8) we can pause that the last mentioned appears to be a better specimen to describe the relation between gross domestic product per capita and Life Expectancy, since the value of the correlation coefficient is significantly bigger. From Figure 3 one the data is very far from the curve of best fit and so we may finalise to discuss it apiece and do the regression without it.This data is corresponds to randomness Africa with a GDP per capita of 10700 and a Life Expectancy at birth of 46. 56 (much lower than any other country with a co mparable GDP). It is reasonable to mean that this anomaly is ascribable to the peculiar story of federation Africa which, later the end of apartheid, had to face an uncontrolled violence. It is thereof difficult to fit this country in a statistical model and we can locate to carry off it from our data. Doing so, we get the following new plot. Figure 4 Logarithmic regression for the data in Table 1 excluding South Africa. The new value of correlation coefficient r 0. 3 indicates that, excluding the wild data of South Africa, there is a strong positive logarithmic correlation between GDP per capita and Life Expectancy at birth. Section 5 Chi firm establish (? 2? riddle) We decide our investigation by making a ki tear down up footrace. This will allow us to endure the existence of a relation between the variables x and y. For this draw a bead on we shapete the following null and utility(a) hypotheses. H0 GDP and Life Expectancy are not agree. H1 GDP and Life Expec tancy are cor colligate * Observed frequency The observed frequencies are obtained directly from Table 2 at a lower place y? preceding(prenominal) y? kernel infra x 14 1 15 above x 16 17 33 conglomeration 30 18 48 Table 6 Observed frequencies for the ki comforting trial * evaluate frequency The expected frequencies are obtained by the formula fe = (column total (row total) / total sum Below y? Above y? Total Below x 9. 375 5. 625 15 Above x 20. 625 12. 375 33 Total 30 18 48 Table 7 Expected frequencies for the chi square test. We can instanter calculate the chi square variable ?2? = ( f0-fe)2/fe = 8. 85 In order to decide whether we admit or not the substitute(a) assumption H1, we need to find the number of details of freedom (df) and to fix a take aim of faith .The number of degrees of freedom is df= (number of rows 1) (number of columns 1) = 1 The corresponding minute values of chi square, depending on the choice of level of self-assurance , are given in t he following table (see audience 4) df 00. 10 00. 05 0. 025 00. 01 0. 005 1 2. 706 3. 841 5. 024 6. 635 7. 879 Table 7 Critical values of chi square with one degree of freedom. Since the value of chi square is great than any of the above critical values, we conclude that even with a level of confidence = 0. 005 we can accept the alternative hypothesis H1 GDP and Life Expectancy are connect.The above test shows that there is some relation between the two variables x (GDP per capita) and y (Life Expectancy at birth). Our finale is to further go over this relation. Section 6 Conclusions Interpretation of results Our study of the statistical relation between GDP per capita and Life Expectancy brings us to the following conclusions. As the chi square test shows there is emphatically some statistical relation between the two variables (with a confidence level = 0. 005). The study of linear regression shows that there is a mollify positive linear correlation between the two variable s, with a correlation coefficient r 0. . This linear model can be greatly amend replacing the linear dependence with a different type of relation. In point we considered a logarithmic relation between the variable x (GDP) and y (Life Expectancy). With this new relation we get a correlation coefficient r 0. 7. In fact, if we remove the data related to the mistaken country of South Africa (which should be discussed separately and does not fit well in our statistical analysis), we get an even higher correlation coefficient r 0. . This is evidence of a strong positive logarithmic dependence between x and y. Validity and Areas of rise Of course one possible proceeds of this project would be to consider a much more extended collection data on which to do the statistical analysis. For example one could consider a large tilt countries, data related to different years (other than 2003), and one could even think of canvas data referring to local anesthetic regions within a single count ry.All this can be found in publications but we clear-cut to restrict to the data presented in this project because we considered it enough as an application of the numerical and statistical tools employ in the project. A second, probably more evoke, possible advance of the project would be to consider other economic factors that can affect the Life Expectancy at birth of a country. Indeed the GDP per capita is just a measure of the fairish wealth of a country and it does not take in account the distribution of the wealth.There are yet several economic indices that measure the dispersal of wealth in the population and could be considered, together with the GDP per capita, as a factor influencing Life Expectancy. For example, it would be interesting to study a linear regression model in which the dependent variable y is the Life Expectancy and with two (or more) independent variables xi, one of which should be the GDP per capita and another could be for example the Gini diss imilarity Index fictional character (measuring the dispersion of wealth in a country).This would dumbfound been very interesting but, perhaps, it would have been out of context in a project studying GDP per capita and Life Expectancy. Probably the most important direction of improvement of the present project is related to the somewhat capricious choice of the logarithmic model utilise to describe the relation between GDP and Life Expectancy. Our choice of the function y= Aln(x+C) +B, was mainly dictated by the statistic share at our disposal in the go by software apply in this project.Nevertheless we could have considered different, and probably more appropriate, choices of functional relations between the variables x and y. For example we could have considered a immix linear and hyperbolic regression model of type y= A + Bx + C/(x+D), as it is sometimes considered in literature (see reference 4). Bibliography 1. Gapminder World. Web. 4 Jan. 2012. lthttp//www. gapminder. orggt. 2. GDP per Capita (PPP) vs. baby Mortality Rate. Index Mundi Country Facts. Web. 4Jan. 2012. <http//www. indexmundi. com/g/correlation. aspx? v1=67>. 3. Life Expectancy at Birth versus GDP per Capita (PPP). Statistical Consultants Ltd. Web. 4 Jan. 2012. <http//www. statisticalconsultants. co. nz/ weeklyfeatures/WF6. html>. 4. Table Chi-Square Probabilities. Faculty & Staff Webpages. Web. 4 Jan. 2012. <http//people. richland. edu/james/ chit-chat/m170/tbl-chi. html>.