Effect of Diluted Thyroxine on Highland Amphibians

Effect of Diluted Thyroxine on Highland Amphibians 1.1. Replication of an experiment on extremely diluted thyroxine and highland amphibians (Study 1) A key issue in science is the reproducibility of experiements. The purpose of the study was to investigate whether the initial experiment using diluted thyroxine and highland amphibians was reproducible. One particular experiment was reported as being reproducible by the initial researchers as well as independent researchers. This experiment tested fouty-eight hourly applications of Thyroxine 30X against Water 30X on the development of highland amphibians. Treatment commenced from the two-legged stage on. Parameters measured were the number of frogs that reached the 4-legged stage and the number that reached the tail-reduction stage. The initial study published results in 1990, the study was replicated and the results published in 2000 and after reanalysing the results, published again in 2010. All 3 studies reported that metamorphosis occurred more slowly in those treated with Thyroxine 30X compared to Water 30X. The author replicated the study again, reanalysed and combined the results of the initial team and the independent researchers with his own results. The methods as set out by the initial study were followed as closely as possible. 8 basins were randomly divided between each group, giving each group 4 basins, each with 20 frogs, totalling 80 frogs per group. A handling error caused cross-contamination of two basins which were subsequently excluded from the study. Therefore 60 frogs across 3 basins per group were considered. The Thyroxine 30X frogs showed a clear trend of delayed metamorphosis into the 4-legged stage as well as for the tail-reduction stage. However, due to the small sample size, the results were not statistically significant. The author’s conclusion was that the results of his own replication experiment, though not statistically significant, were in line with those of the initial study, as well the other independent researchers. 1.2. The effect of homoeopathically prepared thyroxine on highland frogs: influence of electromagnetic fields (Study 2) One of the principles of homoeopathy, the law of similars, can be demonstrated by hyper-stimulating frogs by immersing them in a thyroxine solution (10à ¢Ã‚ Ã‚ »Ãƒ ¢Ã‚ Ã‚ ¸ parts by weight, unsuccussed) and then inducing the reverse reaction by subjecting them to a homoeopathically prepared solution (10à ¢Ã‚ Ã‚ »Ã‚ ¹Ã‚ ³) of thyroxine. The study used the amphibian model to investigate the effects of various electromagnetic fields on homoeopathically prepared thyroxine solution. A microwave, mobile phone, x-ray luggage inspection device and a red light barcode scanner were used. A replication of the basic experiment conducted previously was used as a control group. All experiments were performed blind. The starting point was defined as the point at which the 2-legged stage begins. The experiment continued until the 4-legged stage. 149 basins were used, each containing 20 frogs. From this, 21 groups were formed according to treatment. 6 different experiments were performed. In each experiment one group (100-200 frogs) were treated with the control substance and the other group (100-200 frogs) were treated with standard test solution. This part of the study serves as a pilot study to the influence of environmental factors on homoeopathic preparations. With each experiment, the control group was compared to the standard test solution as well as to test solutions exposed to the various electromagnetic fields. In total, 860 frogs were treated with homoeopathically prepared thyroxine and 860 frogs were treated with standard control solution. In addition, 1160 frogs were treated with the various solutions exposed to electromagnetic fields. The findings suggest that homoeopathically prepared thyroxine 30D has an inhibitory effect on the metamorphosis of frogs while this effect is blocked when exposed to the microwave oven or the mobile phone. This is in agreement with the assumptions of manufacturers of homoeopathic preparations. Noted/Noticeable Flaws Study 1: Methods as set out by the original researcher were followed as closely as possible to replicate the study, however, there was a slight deviation in that the amphibians were stored at dimmed daylight at a temperature of eight degrees Celsius for a period of three days to prevent the tadpoles from developing beyond the designated starting stage before the experiment scheduled to start. This may have delayed the onset of the two-legged stage and in turn may have affected the sensitivity of the larvae to the thyroxine 30X. Even thought they have produced interesting findings, the highland amphibian Rana temporaria is not easily available for further research. This particular species is not available from breeders, permits need to be obtained and then collected from the field. Proffesional expertise and experience is required for collection, transportation and handling. The numbers in each study are too small to be statistically significant until they are pooled together. Even though the study was replicated as closely as possible, individual factors may have influenced the findings and so pooling the results may not be as accurate as expected. Study 2: Using water solutions exposed to electromagnetic fields could serve as an additional control group. The natural environment of the amphibian should try and be mimicked as far as possible. It may be hypothesised that the increase in temperature in a laboratory environment compared to that of the natural biotope may be the stimulus that make the highland amphibian sensitive to the homoeopathically prepared thyroxine. In both studies, the results are statistically significant only when the results are pooled together and an adequate number of animals are included. Special efforts should be made to increase the size of homoeopathic studies in general to make the results more reliable and credible. More Valid and Credible Findings Ideally the natural habitat of the amphibians should be mimicked as far as possible to avoid influencing developmental stages. Future experiments should be conducted on a species that is more readily available. An expert on amphibians should be employed to supervise the study and offer advice on the handling and management of the animals. Given that the study was reproducible on numerous occasions, I think the findings are valid and credible. Thyroxine 30X, a dilution beyond avogardo’s number, produced a clear trend by slowing down metamorphosis. Contribution Towards Evidence Based Homoeopathy These studies do contribute towards evidence based homoeopathy. They are scientific, quantitative designs and are confirmatory of the law of similars. Thyroxine is stimulatory in crude form when exposed to an amphibian. Without thyroxine, amphibians would not undergo metamorphosis from tadpole to juvenile frog. Thyroxine in homoeopathic preparation had an inhibitory effect and was confirmed on numerous occasions by reproducing the same experiments by independent researchers.

Discussion of the Dispossession of Lolita’s sexuality in Vladimir Nabokov’s Lolita

In the novel Lolita, the concept of sexuality is greatly emphasized by the author. It is apparent in the character of Lolita and Humber that one is in control over the other. Humbert possesses greater power over Lolita to make all her needs possible.On the other hand Lolita uses sex as her way of gaining things that she desires. There is a big complex relationship between the tow since Lolita is in dispossession of her sexuality and Humbert is in possession of it.The sexual relationship of the two main characters can be considered unnatural because Lolita is only on her 12 years. But since Lolita is in dispossession of her sexuality she agreed to the consummation of their relationship although giving the consent to sexual relations with an old man is really no consent.She can by no means what so ever be held accountable for, or be said to legitimately consent to, the sexual relationship between the two.   The question of power and control between the two main characters in Vladimir Nabokov’s Lolita is crucially linked to the fundamental weakness and vulnerability of both the erotically obsessed Humbert Humbert and the childish object of his obsession, Lolita herself.The first instance of Humbert’s attempting to bribe and pacify Lolita with material offerings occurs at the point in the narrative where she first learns of her mother’s death.   This is also the point, following their first full-fledged sexual encounter, where Lolita initiates the practice of sardonic reference to the immorality and illegality of their relationship.   She has actually suffered internal injury from their intercourse and does not yet seem to have a manipulative purpose for verbally goading her stepfather/lover.â€Å"You chump,† she said, sweetly smiling at me.   â€Å"You revolting creature.   I was a daisy-fresh girl, and look what you’ve done to me.   I ought to call the police and tell them you raped me.   Oh, you dirty, dirty old man.† (Nabokov 141)Humbert then informs her of Charlotte Haze’s death.   Lolita’s grief and indignation are announced by the fact that in their next lodging they take separate rooms.   Humbert has already cited a list of purchases he has made in hopes of mollifying her—everything from a box of candy to a travel clock to a new wardrobe of summer dresses.   But it is her complete childish dependency and vulnerability that bring her to him.â€Å"At the motel we had separate rooms, but in the middle of the night she came sobbing into mine, and we made it up very gently.   You see, she had absolutely nowhere else to go† (Nabokov 142)Lolita â€Å"works† hard in order to get her allowance. It is atypical for her to do a work like that at her age because it means that she needs to have sex with him to get everything that she needs and wants.   Humber will ask Lolita to perform sexual favors for him so that he can give him allowance in retur n.This is one of the visible proofs that he is in control over her sexuality. The control is not only based on a psychological level but also on a physical level. Everything that she needs, from a simple candy or cartoon magazine and a visit to a movie house will not be easily granted if she will not perform any sexual encounter with Humbert. Although there are several times that she gains fulfillment by getting her desires, she is still in no control of the situation or her sexuality.â€Å"Her weekly allowance, paid to her under condition she fulfill her basic obligations, was twenty-one cents at the start of the Beardsley era-and went up to one dollar five before its end.This was more than generous arrangement seeing she constantly received from me all kinds of small presents and had for the asking any sweetmeat or movie under the moon-although, of course, I might fondly demand an additional kiss, or even a whole collection of assorted caresses, when I knew she coveted very badly some item of juvenile amusement† (Nabokov 183-184).Lolita becomes, in effect—and as a result of forces brought relentlessly to bear on her essentially vital and resilient nature, she is a live-in prostitute of the most venal nature. â€Å"O Reader!   Laugh not, as you imagine me, on the very wrack of joy noisily emitting dimes and quarters, and great big silver dollars like some sonorous, jingly and wholly demented machine vomiting riches; and in the margin of that leaping epilepsy she would firmly clutch a handful of coins in her little fist† (Nabokov 184).That understated â€Å"little fist† line is the moral center of this passage.   Lolita is still a child.   Whatever her learned capacity for degenerate bargaining, what she loses—and what she knows she is losing—by her required performance of sexually â€Å"paradisal philters† is the carefree childhood to which she is entitled by the developmental norms of her society.Even in school. Lolita experiences great trouble connecting with the boys in the way her teachers feel she should. She displays behavior that is really differently from the other girls. This also strengthens the fact of the argument that Humbert is in possession of her sexuality, and she cannot do anything about it.   â€Å"Dolly Haze, she said is a lovely child but, the onset of sexual maturing seems to give her trouble† (Nabokov 193).  Humbert deprives Lolita of every possibility of salutary contact with males her own age.   In fact he deprives her of performing any and all activities that will drive her away from his desires for addictive gratification.   Part of this is his fear that Lolita will tell other people the kind of life thay have whenever they are together.Participation in school theatricals becomes a particular source of tension for Humbert.   This issue is resolved, in typical fashion, by an unusually exciting favor.   During a visit to her school, Humber t comes to a classroom where Dolly is quietly studying in the company of another girl. â€Å"I sat beside Dolly†¦and unbuttoned my overcoat and for sixty-five cents plus the permission to participate in the school play, had Dolly put her inky, chalky, red-knuckled hand under the desk† (Nabokov 198).

The Relationship Between Life Expectancy at Birth and Gdp Per Capita

The relationship between Life Expectancy at birth and GDP per capita (PPP) Candidate: Teacher: Candidate number: Date of submission: Word Count: 2907 Section 1: Introduction In a given country, Life Expectancy at birth is the expected number of years of life from birth. Gross domestic product per capita is defined as the market value of all final goods and services produced within a country in one year, divided by the size of the population of that country. The main objective of the present project is to establish the existence of a statistical relation between Life Expectancy (y) at birth and GDP per capita (x).First, we will present in Section 2 the data, from an official governmental source, containing Life Expectancy at birth and GDP per capita of 48 countries in the year 2003. We will put this data in a table ordered alphabetically and at the end of the section we will perform some basic statistical analysis of these data. These statistics will include the mean, median, modal cl ass and standard deviation, for both Life Expectancy and GDP per capita. In Section 3 we will find the regression line which best fits our data and the corresponding correlation coefficient r.It is natural to ask if there is a non-linear model, which better describes the statistical relation between GDP per capita and Life Expectancy. This question will be studied in Section 4, where we will see if a logarithmic relation of type y=A ln(x+C) + B, is a better model. In Section 5 we will perform a chi square test to get evidence of the existence of a statistical relation between the variables x and y. In the last section of the project, other than summarizing the obtained results, we will present several possible directions to further investigation. Section 2: Data collectionThe following table shows the GDP per capita (PPP) (in US Dollars), denoted xi, and the mean Life Expectancy at birth (in years), denote yi, in 48 countries in the year 2003. The data has been collected through an online website (2). According to this website it represents official world records. Country| GDP – 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. Burundi| 600| 43,02| 10. Canada| 29800| 79,83| 1. Central African Republic| 1100| 41,71| 12. Chile| 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. Dominican 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. Georgia| 2500| 64,76| 29. Germany| 27600| 78,42| 30. Ghana| 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. Indonesia| 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 Kingdom| 27700| 78,16| 48. United States| 37800| 77,14| Table1: GDP per capita and Life Expectancy at birth in 48 countries in 2003 (source: reference [2]) Statistical analysis: First we compute some basic statistics of the data collected in the above table.Basic statistics for the GDP per capita: Mean: x=i=148xi48 = 12900 In order to compute the median, we need to order the GDP values: 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 value of the two central values (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 have the following table of frequencies: Class| 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| Table 2: Frequencies of GDP per capita with classes of USD 1000 With this choice of classes, the modal class is 2000-2999 (with a frequency of 5). If instead we consider classes of USD 5000 (0-4999, 5000-9999, †¦) the modal class is the first: 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, 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 middle value of the two central values:Median= 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 above that there are three 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 have been found 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. 3 7| 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| 235 175004| 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| 38002088| 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 parameter of the GDP and Life Expectancy: Sxy =148 i=148(xi-x)(yi-y)= 73011. 6 Section 3: Linear regression We start our investigation by studying the line best fit of the data in Table 1. This will allow us to see whether there is a relation of linear dependence between GDP and Life Expectancy. The regression line for the variables x and y is given by the following formula: y-y  ? =SxySx2(x-x ) By using the values found above we get: y= 62. 51 + 0. 5926*10-3 x The Pearson's correlati on coefficient is: r = 0. 6380 The following graph shows the data on Table 1 together with the line of best fit computed Figure 1: Linear regression. The value of the correlation coefficient r ~ 0. , is evidence of a moderate positive linear correlation between the variables x and y. On the other hand it is apparent from the graph above that the relation between the variables is not exactly 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], â€Å"the main reason for this non-linear relationship [between GDP per capita and Life Expectancy] is because people consume both needs and wants.People consume needs in order to survive. Once a person’s needs are satisfied, they could then spend the rest of their money on non-necessities. If everyone’s needs are satisfied, then any increase in GDP per capita would barely affect Life Expectancy. â€Å" There are various other reasons that one can think of, to explain the non-linear relationship between GDP per capita and Life Expectancy. For example 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 more 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 Inequality Index, that describe the distribution of wealth among the population. Moreover, the wealth of the population is not the only 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 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. Let’s try to guess what could be a reasonable 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 part 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 close to 1 when x is close to 0 (if one has a little amount 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 function xn= f(x) sa tisfying the above requirements: xn= log (cx + 1)/c, (2) where c is some positive parameter. This function is chosen mainly for two reasons. On one hand it satisfies the requirements 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 software 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 depende nce between x and y.Equation (3) can be rewritten in the following equivalent 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 curve 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 curve of type y= Aln(x) + B (i. e. equation of type four 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 simply add C to all the x values and redo the same plot as for C= 0 with the new independent variable x1= x + C. We omit showing the graphs containing the curve of best fit for all the possible 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 decide 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, after 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 remove 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 square test (? 2? test) We conclude our investigation by making a chi square 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 correlated. H1: GDP and Life Expectancy are correlated * Observed frequency: The observed frequencies are obtained directly from Ta ble 2: | Below y? | Above y? | Total|Below 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 calculate the chi square variable: ?2? = ( f0-fe)2/fe = 8. 85 In order to decide whether we accept or not the alternative hypothesis H1, we need to find the number of degrees of freedom (df) and to fix a level of confidence .The number of degrees of freedom is: df= (number of rows – 1) (number of columns –1) = 1 The corresponding critical 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 confidence = 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 dependence 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 used in the project. A second, probably more interesting, possible improvement 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 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 important direction of improvement of the present project is related to the somewhat arbitrary 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 dictated 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. ;lt;http://www. gapminder. org;gt;. 2. â€Å"GDP – per Capita (PPP) vs. Infant Mortality Rate. Index Mundi – Country Facts. W eb. 4Jan. 2012. <http://www. indexmundi. com/g/correlation. aspx? v1=67>. 3. â€Å"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. â€Å"Table: Chi-Square Probabilities. † Faculty & Staff Webpages. Web. 4 Jan. 2012. <http://people. richland. edu/james/lecture/m170/tbl-chi. html>.

The Pharmaceutical Industry and Technological Advancement

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