The Relationship Between Life Expectancy at Birth and Gdp Per Capita

The descent amidst emotional state apprehension at birth and gross domestic product per capita (PPP) Candidate Teacher Candidate number pick up of submission Word Count 2907 contribution 1 Introduction In a disposed(p) state of matter, manners hope at birth is the evaluate number of years of life from birth. Gross domestic product per capita is out preeminenced as the market cling to of all(a) final goods and services produced inwardly a country in champion year, divided by the size of it of the population of that country. The main objective of the present intention is to establish the initiation of a statistical congener mingled with life history apprehension (y) at birth and gross domestic product per capita (x).First, we volition present in discussion section 2 the selective information, from an official political source, containing lifespan Expectancy at birth and gross domestic product per capita of 48 countries in the year 2003. We ordain put this data in a table ordered alphabetically and at the end of the character we will perform well-nigh grassroots statistical analysis of these data. These statistics will include the mean, median, modal(a) physique and standard deviation, for both tone Expectancy and gross domestic product per capita. In Section 3 we will find the turnabout toward the mean line which best equips our data and the check offing coefficient of cor sexual intercourse coefficient coefficient r.It is natural to command if at that place is a non- running(a) model, which offend describes the statistical relation in the midst of gross domestic product per capita and spirit Expectancy. This question will be studied in Section 4, where we will see if a enterarithmic relation of type y=A ln(x+C) + B, is a better model. In Section 5 we will perform a ki squ atomic number 18 test to grow assure of the humankind of a statistical relation amidst the covariants x and y. In the last section of the proj ect, other than summarizing the obtained results, we will present some(prenominal) thinkable directions to further investigation. Section 2 Data collectionThe chase table shows the gross domestic product per capita (PPP) (in US Dollars), pertaind xi, and the mean Life Expectancy at birth (in years), de none yi, in 48 countries in the year 2003. The data has been collected through and through an online website (2). According to this website it represents official world records. Country gross domestic product per capita (xi) Life Expectancy at birth (yi) 1. 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 29100 78,29 7. brazil 7600 71,13 8. Bulgaria 7600 71,08 9. Burundian 600 43,02 10. Canada 29800 79,83 1. underlying Afri discount state 1100 41,71 12. chili con carne 9900 76,35 13. China 5000 72,22 14. Colombia 6300 71,14 15. Congo, Republic of the 700 50,02 16. costa Rica 9100 76,43 17 . Croatia 10600 74,37 18. Cuba 2900 76,08 19. Czech Republic 15700 75,18 20. Denmark 31100 77,01 21. Domini give nonice Republic 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. tabun 2500 64,76 29. Germany 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 10700 46,56 46. Turkey 6700 71,08 47. United nation 27700 78,16 48. United States 37800 77,14 defer1 gross domestic product per capita and Life Expectancy at birth in 48 countries in 2003 (source reference 2) Statistical analysis First we compute some base statistics of the data collected in the in a higher place table.Basic statistics for the GDP per capita blind drunk x=i=148xi48 = 12900 In order to compute the median, we study to order the GDP set 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 middle grade of the devil aboriginal determine (the 25th and the 26th) Median= 7600+91002 = 8350 In order to compute the modal class, we need to split the data in classes.If we consider classes of USD 1000 (0-999, 1000-1999, ) we guard the following table of frequencies flesh 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 16 000-16999 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 dishearten 2 Frequencies of GDP per capita with classes of USD 1000 With this choice of classes, the modal class is 2000-2999 (with a relative frequency of 5). If instead we consider classes of USD 5000 (0-4999, 5000-9999, ) the modal class is the commencement ceremony 0-4999 (with a frequency of 18). Class 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 Table 3 Frequencies of GDP per capita with classes of USD 5000 Standard deviation Sx=i=148(xi-x)248 =11100Basic statistics for the Life Expectancy Mean y=i=148yi48 = 70,13 As before, in order to compute the median, we need to order the Life Expectancies 41. 71, 4 3. 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 middle value of the twain central valuesMedian= 72,17+72,222 = 72. 195 To find the modal class of Life Expectancy we consider modal classes of one year. The table of frequencies is the following Class 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 Table 4 Frequencies of Life Expectancy at birth with classes of 1 year It appears from the table in a higher place that there are three modal classes 71, 78 and 79 (with a frequ ency of 5).Standard deviation Sy=i=148(yi-y)248 =10. 31 The standard deviations Sx and Sy father been show using the following table of data Country GDP Life 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 1900 61. 33 -10965 120222088 -8. 80 77. 42 96474. 63 Belgium 29100 78. 29 16235 263588754 8. 16 66. 1 132501. 29 Brazil 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 Central African Republic 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 Costa 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 Dominican 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 Georgia 2500 64. 76 -10365 107424588 -5. 37 28. 82 55644. 86 Germany 27600 78. 42 14735 217132504 8. 29 68. 74 1 22175. 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 South Africa 10700 46. 56 -2165 4685421 -23. 57 555. 49 51016. 52 Turkey 6700 71. 08 -6165 3800208 8 0. 95 0. 90 -5864. 06 United Kingdom 27700 78. 16 14835 220089588 8. 03 64. 50 119146. 94 United States 37800 77. 14 24935 621775004 7. 01 49. 16 174828. 44 Table 5 Statistical analysis of the data collected in Table 1 From the last column we can compute the covariance argument of the GDP and Life Expectancy Sxy =148 i=148(xi-x)(yi-y)= 73011. 6 Section 3 running(a) atavism We start our investigation by studying the line best equalise of the data in Table 1. This will take into account us to see whether there is a relation of linear addiction between GDP and Life Expectancy. The regression line for the variables x and y is prone by the following formula y-y? =SxySx2(x-x ) By using the values order in a higher place we get y= 62. 51 + 0. 5926*10-3 x The Pearsons correlation coefficient is r = 0. 6380 The following graph shows the data on Table 1 unneurotic with the line of best fit computed simulacrum 1 Linear regression. The value of the correlation coefficient r 0. , is evidence of a moderate confirming linear correlation between the variables x and y. On the other manus it is apparent from the graph above that the relation between the variables is not merely linear. In the next section we will try to speculate on the reason for this non-linear relation and on what type of statistical relation can exist between GDP per capita and Life Expectancy. Section 4 logarithmic regression As explained in reference 3, the main reason for this non-linear transactionhip between GDP per capita and Life Expectancy is because people postulate both needs and wants.People consume needs in order to survive. Once a persons needs are satisfied, they could then spend the rest of their money on non-necessities. If e genuinelyones needs are satisfied, then any increase in GDP per capita would barely affect Life Expectancy. There are sundry(a) other reasons that one can think of, to explain the non-linear relationship between GDP per capita and Life Expectancy. F or exercise the GDP per capita is the average wealth, while one should consider also how the global wealth is distributed among the population of a given country.With this in mind, to have a much complete picture of the statistical relation between economy of a country and Life Expectancy, one should take into considerations also other economic parameters, such as the unlikeness Index, that describe the distribution of wealth among the population. Moreover, the wealth of the population is not the unaccompanied factor effecting Life Expectancy one should also take into account, for example, the governmental policies of a nation towards health and poverty. For example Cuba, a country with a very(prenominal) low GDP per capita ($ 2900), has a relatively high Life Expectancy (76. 8 years), mostly due to the fact that the government provides basic needs and health assistance to the population. Some of these aspects will be discussed in the next section. Lets try to guess what could be a commonsensical relation between the variables x (GDP per capita) and y (Life Expectancy). According to the above observations we can consider the total GDP formed by two values x= xn + xw, where xn denotes the per centum of wealth spent on necessities, and xw denotes the part spent on wants.It is reasonable to make the following assumptions 1. The Life Expectancy depends linearly on the part of wealth spent on necessities y=axn + b, (1) 2. The fraction xn/x of wealth spent on necessities, is pen up to 1 when x is close to 0 (if one has a little amount of money of money he/she will spend most of it on necessities), and is close to 0 when x is very large (if one has a very large money he/she will spend only a little fraction of on necessities). 3.We make the following choice for the amour xn= f(x) satisfying the above requirements xn= log (cx + 1)/c, (2) where c is some positive parameter. This business office is chosen mainly for two reasons. On one hand it satisfies the r equirements that are describe in 2, indeed the corresponding graph of xn/x = f(x) = log (cx + 1)/cx Figure 2 Graph of the function y= log (cx + 1)/cx, for C=0. 5 (blue), 1 (black) and 10 (red). The blue, black and red lines correspond respectively to the choice of parameter c= 0. 5, 1 and 10.As it appears from the graph in all cases we have f(0)= 1 and f(x) is small for large values of x. On the other hand the function chosen allows us to use the statistical tools at our disposal in the excel packet to derive some interesting conclusion about the statistical relation between x and y. This is what we are going to do next. First we want to find the relation between 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 habituation between x and y.Equation (3) can be rewritten in the following analogous form if we denote A=a/c, B= b+(a/c)ln(c), C=1/c, y=Aln(x+C)+B . (4) We can now study the mold of type (4) which best fits the data in Table 1, using the statistical tools of excel spreadsheet. Unfortunately excel allows us to plot only a worm of type y= Aln(x) + B (i. e. equation of type four-spot where C is equal to 0). For this choice of C, we get the following logarithmic curve of best fit together with the corresponding value of correlation coefficient r2. Figure 3 Logarithmic regression.To find the analogous curve of best fit for a given value of C (positive, arbitrarily chosen) we can merely add C to all the x values and redo the identical plot as for C= 0 with the new self-governing variable x1= x + C. We omit showing the graphs containing the curve of best fit for all the realizable values of C and we simply report, in the following table, the correlation coefficient r for some appropriately 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 Table 8 correlation coefficient r2 for the curve of best fit y= Aln( x+C) +B, for some values of C. The above data indicate that the optimal choice 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 conclude that the latter appears to be a better model to describe the relation between GDP 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 steady down to discuss it separately and do the regression without it.This data is corresponds to South 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 comparable GDP). It is reasonable to think that this anomaly is due to the peculiar history of South Africa which, afterwards the end of apartheid, had to face an uncontrolled violence. It is therefore difficult to fit this country in a statistical model and we can decide to despatch 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 anomalous data of South Africa, there is a strong positive logarithmic correlation between GDP per capita and Life Expectancy at birth. Section 5 Chi real test (? 2? test) We conclude our investigation by making a chi settle test. This will allow us to confirm the existence of a relation between the variables x and y. For this purpose we formulate the following null and alternative hypotheses. H0 GDP and Life Expectancy are not cor cogitate to. H1 GDP and Life Expectancy are correlated * Observed frequency The observed frequencies are obtained directly from Table 2 Below y? higher up y? totBelow x 14 1 15 Above x 16 17 33 Total 30 18 48 Table 6 Observed frequencies for the chi square test * Expected 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 now prefigure the chi square variable ?2? = ( f0-fe)2/fe = 8. 85 In order to decide whether we encounter or not the alternative guessing H1, we need to find the number of ground levels of freedom (df) and to fix a take aim of confidence .The number of degrees of freedom is df= (number of rows 1) (number of columns 1) = 1 The corresponding life-sustaining values of chi square, depending on the choice of level of confidence , are given in the following table (see reference 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 greater than any of the above critical values, we conclude that even with a level of c onfidence = 0. 005 we can accept the alternative hypothesis H1 GDP and Life Expectancy are related.The above test shows that there is some relation between the two variables x (GDP per capita) and y (Life Expectancy at birth). Our goal is to further investigate 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 definitely some statistical relation between the two variables (with a confidence level = 0. 005). The study of linear regression shows that there is a moderate positive linear correlation between the two variables, with a correlation coefficient r 0. . This linear model can be greatly improved replacing the linear dependance with a different type of relation. In particular 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 anomalous 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 improvement Of course one possible improvement 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 list countries, data related to different years (other than 2003), and one could even think of studying data referring to local regions within a single country.All this can be found in literature but we decided to restrict to the data presented in this project because we considered it enough as an application of the mathematical and statistical tools employ in the project. A second, probably more interesting, possible improvement of the project wou ld be to consider other economic factors that can affect the Life Expectancy at birth of a country. Indeed the GDP per capita is vindicatory a measure of the average wealth of a country and it does not take in account the distribution of the wealth.There are however several economic indices that measure the dispersion 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 inequality Index reference (measuring the dispersion of wealth in a country).This would have 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 grave direction of improvement of the present project is related to the somewhat compulsive choice of the logarithmic model used to describe the relation between GDP and Life Expectancy. Our choice of the function y= Aln(x+C) +B, was mainly inflict by the statistic package at our disposal in the excel software used 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 mixed 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. Infant 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. n z/ weeklyfeatures/WF6. hypertext markup language>. 4. Table Chi-Square Probabilities. Faculty & Staff Webpages. Web. 4 Jan. 2012. <http//people. richland. edu/james/ babble out/m170/tbl-chi. html>.