Plato, Aristotle, Sophism Essay Example | Topics and Well Written Essays - 500 words

Plato, Aristotle, Sophism - Essay Example Plato’s point is that the prisoners would be highly mistaken in that they would be taking terms in their own language to refer to the shadow that comes before their eyes rather than to the real things that bring the shadows. In this manner, Plato meant that the general terms of human language are not names of the physical objects or things that we can see. They are names of objects or things that we are not able to see, objects or things we can only comprehend with our mind. Plato’s aim in the republic is to explain what is important for us to attain this reflective understanding of real world. It remains the fact that our very ability to think and speak depends upon the forms. The terms of the language that people use, we get their real or true meaning by naming the forms that the things we perceive engage in. In addition, Plato tried to illustrate human nature for education and our want for education. For Aristotle, a firm understanding of what a cause is, and how many types of causes there are, is significant for successful explanation of the world around us. Aristotle identifies four kinds of things that can be applied in answer to a why-question: a. The material cause, which Aristotle stated that in every change, something gets a new determination first before undergoing any form of change, which is in potency to a new determination, then after that thing has been acted upon, it gets a new actualization. b. Formal cause, which states that every individual thing is made up not only matter, but also form. Form is the rule of determination, which accounts for the individual thing being the kind of thing it is. On the other hand, matter is the rule of potentiality, with the ability to become other than it is. c. Efficient cause, which states that, in addition to the causes that are aspects of material things, form, and matter, Aristotle, says that there must be a source for any change or motion. This according to Aristotle might be

Plato’s Allegory of the Cave Essay Example for Free

Plato’s Allegory of the Cave Essay Plato’s Allegory of the cave is a written dialogue between his brother, Glaucon and his mentor, Socrates. Socrates asked Glaucon to imagine a cave inhabited with prisoners since childhood, with legs and hands chained fixedly so that all they could see was the wall. They came to believe that the shadows of the cave were real. Socrates then explained that once the prisoners were freed from the cave, the lights from the outside world would first pain their eyes, and hurt them. Some might even hide back into the cave to avoid it. But those who have opened their eyes starts to widen, sees everything and realizes that the sun is the source of all the light. They recognizes that what they see now is truth, and the shadows that they once thought was true was an illusion. What Plato is trying to say is that the goal of education is merely to open up ones sight, drag everyone out of the cave, so that they can see further and be more open-minded. He explains that it’s not to simply be full of knowledge, but to have the right desires. This can be explained by his little analysis of how the man first was blinded by the light and would want to crawl back into the cave, but he slowly was adapted to the light and realizes that everything that he once thought was real was an illusion. He implies that everyone has some sort of an illusion, but to break free from that illusion would result in improvement. He’s implying that in life, we need to keep breaking illusions to be further educated.

Significance of Feet in Plato’s Symposium Essay -- Philosophy Religion

The Significance of Feet in Plato’s Symposium Plato’s Symposium presents an account of the party given at the house of Agathon, where Socrates and Alcibiades are in attendance. The men at the party take turns eulogizing the god Eros. In Agathon’s eulogy, he describes Eros as a soft and tender being. When Socrates speaks, however, he makes a correction of his host’s account, by saying the soft and tender thing is the beloved, and not the lover, as Agathon would have it. When Alcibiades enters the party toward the end of the dialogue, he complains that Socrates is deceiving Agathon. Alcibiades was once the lover of Socrates, and if he knows anything about his beloved, it is that Socrates is a tough man who can drink without getting drunk and wander the streets of Athens day in and day out without shoes to protect his feet. Though it may seem preposterous that feet matter in a dialogue about love, throughout the Symposium, the condition of the character’s feet helps determine who is the lover and who is the beloved, and furthermore, that those who run away from love in shame are cowardly and those who stand still are noble. Alcibiades could love Socrates for the very reason that he is tough and unwavering. At the beginning of the dialogue, when Socrates is on his way to Agathon’s house he "retreat[s] to a neighbor’s porch and stands there, and when [Agathon] call[s] him, he is unwilling to come in" (236).1 Though Agathon and the other men want Socrates to join them, Aristodemus who is "most in love with Socrates at the time" says: " No, no, leave him alone. That is something of a habit with him. Sometimes he moves off and stands stock still wherever he happens to be" (236). Aristodemus shows respect and admiration for... ...that Socrates would never be so weak as to run off or "wander about in distraction" (282) as Alcibiades did when he loved Socrates. Socrates is therefore remarking on the absurdity of his own statement. Throughout the dialogue, the condition of the character’s feet helps determine who is the lover and who is the beloved, just as the tendency to stand firm or to run away helps determine who is noble and who is base. Socrates is noble and firm, as Alcibiades "takes off in headlong flight." And, Socrates is beloved by Alcibiades and Aristodemus for his steadfastness, but tries to be tender before Agathon to win his love as well. Alcibiades assertion that Socrates is deceitful is correct: and Socrates does try to get all the beauties for himself. Work Cited 1. Plato, Symposium, in The Dialogues of Plato, trans. Seth Benardete (New York: Bantam Books, 1986).

Intertestament Period

Exploring the Bible 3 October, 2011 Inter-Biblical Period: 400 Years The Old Testament focuses on God’s covenant with his people and the stories of men of great faith and also men of prophecy, foretelling Christ coming. The New Testament, however, focuses on the redemption side. The Redemption side is Christ’s story here on Earth and his ministry to other people. Malachi, the last book written in the Old Testament, was written around 424 B. C. Then 418 years after, the book of Matthew, the first book of the New Testament, was written.The Intertestamental Period is the period of history between the two testaments. Since God did not speak to his people directly through prophets, this period is also known as the 400 years of silence. Now, what happened in those four hundred years? The Intertestamental Period was a time where many empires came along, new religious groups formed, and where theological and literature pieces were developed that molded Israel going into the tim e of Christ. Politics in a nation help dictate and run the people, in Israel’s case, they had to deal with many empires coming in.During the Intertestamental Period, they had five key periods of leaders and kingdoms. The Persian Empire arrived around 430 B. C. and they ruled until 332 B. C. The Persian rule was recorded as being mild and tolerant. (Scott, 1783) The Persians were taken over by the Greeks. The Greek period went from 331 B. C. to 167 B. C. (Scott, 1783) Alexander the Great, son of King Philip of Macedonia, was given the control of the Greek army around the age of 20. (Halley, 402) With the control of the army, he was able to sweep over and take over Egypt, Assyria, Babylon, and Persia.Alexander the Great was very compassionate towards the Jews and he spared the city of Jerusalem. He established Greek cities in his conquered places and he did it with the plan to spread Greek culture and language throughout the world including Israel. After Alexander’s deat h, his empire was separated to his four generals. (Halley, 402) Egypt and Palestine was given to Ptolemy and under him the conditions of the Jews were at a peaceful state. In Egypt, Alexandria was the influential center of Judaism. In 167 B. C. was the beginning of the Period of Independence also known as the Maccabean period.Mattathias was a priest and he was upset at Antiochus motivation to destroy the Jews, so he gathered Jews and raised a revolt. Mattathias had five sons, Judas, Jonathan, Simon, John, and Elezar. (Halley, 404) The Maccabean period lasted until 63 B. C. and the Romans followed after. Palestine was conquered by the Romans under Pompey. Idumean was the first to be appointed ruler of Judea. His son, Herod the Great, succeeded him. (Halley, 404) Herod was a shrewd politician who wanted to get good with the Jews. He was very cruel and brutal towards his people.Politics and leaders are key roles in establishing a nation. In this case, these five empires helped develop rules and guidelines that led the people and that eventually grew and prepared Israel for the coming of Christ. During the 400 years of silence, religion was another area that grew in Israel. During this time, the religion was not set up was polytheistic or monotheistic but instead it was groups of people. For example the four main groups of the religions groups or parties are Pharisees, Sadducees, Zealots, and Herodians.These four groups followed what they believed and chose to do whatever they wanted to do. The Pharisees is the group most people probably know because of their role in the Gospels and the fact Paul was one. They took the Law and believed it was their job to take it and to put into new conditions. (Hester, 325-326) The accepted both the Torah and tradition. The Pharisees rejected anyone who didn’t live by the standards they did. On the opposite side were the Sadducees. They were the second main group and they had more power than the Pharisees until 70 A.D. whe n their temple was destroyed. (Hester, 325-326) The Sadducees consisted of the wealthy priests. They were a conservative group and they didn’t try to take the Law and put into a new situation but they instead took and limited themselves to the five books of Moses. After 70 A. D, the Pharisees were the leaders of the Jewish people by giving them a religious life aside from the temple. The other two parties were more political than religious groups. The Zealots were a nationalist group that opposed the Romans.Simon, one of Jesus’ disciples, was a Zealot. The other group, the Herodians, basically supported Herod and his dynasty and supported what he did for the nation. These four groups are the religious groups that developed the main teachings and guidelines people believe until Christ’s ministry began. Even though religion and politics are a big part of a civilization, another key role is the literature and writing of a nation. Literature and writing in Israel or any nation is an influential part because it is what the people follow and speak.In Israel, Hebrew was the language of the Old Testament. This was the main language of religion. Latin was the language of Roman and it was commonly spoken among the people. Greek was another language and it was the language that tied the Roman Empire together. The most common language of Palestine in Jesus’ day was Aramaic. (Halley, 410) Most writings are believed to have been written in Aramaic and translated into Greek. (Halley, 410) The Old Testament was written in Hebrew but was spoken in Greek. Septuagint is a translation of the Old Testament into Greek.The Torah was translated first. It was called the Septuagint because of the 70 translators and Septuagint means 70 in Greek. (Halley, 409) Another type of translation was Targums and they are translations of the Old Testament into Aramaic. They were first oral translations, paraphrases, and interpretations these types of languages and writi ng were influential in the development of the social status during the Intertestamental Period. Even though writing is important the question is what was the theology being taught during this period?Theology is important because it is what the people studied and how they viewed God. The Apocrypha, is writings which the writer assumed the name of a hero long since dead, and rewrote history in terms of prophecy. (Halley, 406) Some of this included, Books of Enoch, Assumption of Moses, Ascension of Isaiah, Book of Jubilees, Psalms of Solomon, and more. These books are the books that were written by people in the period of the 400 years of silence, and they preached about the prophecies that the prophets spoke and also the birth of Christ and the end times.This theology is the closest literature that the people had to connect to God and the prophecies that was spoken. All of these books spoke of Christ coming and his ministry that he will live for and preach. The Intertestamental Period consisted of political powers overtaking each other, religious groups, and theological books and literature that preached Christ coming and that developed the nation of Israel. The political side consisted of Persian, Greek, Egyptian, the Maccabees, and the Romans. The eligious side had the Pharisees, Sadducees, Zealots, and Herodians and they all had different views and theories. And the language part consisted of Latin, Greek, Hebrew, and Aramaic. The theology piece was the Apocrypha, which re-emphasized the prophets foretelling. All these pieces and parts put together made up the Intertestamental Period and it formed Israel and it carried over into the ministry of Jesus. The 400 years of silence at last was broken, and God sent his son Jesus Christ to testify the truth and overcome the world.Work Cited Scott, J. Julius Jr. â€Å"Time Between the Testaments. † ESV Study Bible: English Standard Version. Wheaton, IL: Crossway Bibles. 2008. 1783-1789. Print. Halley, Dr. Henry H. Halley’s Bible Handbook: with New International Version. Grand Rapids, Michigan. Zondervan Publishing House. 2000. 402-412. Print. Hester, H. I. The Heart of Hebrew History: A Study of the Old Testestament. Liberty, Missouri, The Quality Press, Inc. 1962. 313-330. Print.

The Theory of the Canadian Politics

Power and control is what the Canadian government try†s to establish. Government engagement in business is becoming too strong as â€Å"in 1992, 6 of Canada†s 100 largest corporations were operated by, government or government agencies throughout Canada†, (Canadian Democracy pg. 88). The problem with government control in business is that it restricts sole proprietors from investing in certain aspects of business such as Ontario†s liquor industry. In order to restrict government control of business three issues need to be taken into consideration; laws allowing privatization on crown corporations, more government attention establishing social issues that are out of reach to Canadian entrepreneurs for example education and health care. Furthermore incorporated laws need to be developed in order to minimize government control. As Canada moves into the 20th century economic policies are becoming too government controlled and as a result of this Canada is leaning towards a government run bureaucracy. Minimal government control of Canada†s business will contribute significantly to a more prosperous Canadian economy. Which is why it is important for government control to be minimal and allow for privatization of crown corporations. In Ontario liquor sales are down 22% compared to a provinces such as Quebec who have a privatized liquor industry. Since Ontario currently has a greater population than Quebec, obviously liquor sales are higher due to privatization of the industry, which allows more business to be established which makes it more convenient for the consumer to buy because there are many more liquor stores. However the liquor industry is only one of the primary companies that are government controlled. â€Å"Canadians who buy their gas at Petro Canada who watch CDC, TVO or Radio Quebec or who watch other provincial educational T. V networks, who travel Via rail or who pay electricity bills†(Canadian Democracy), are all contributing to government run alliances. To prove privatization has been successful the Quebec liquor industry, as well as the MBA programs at Universities such as Queens and McMaster are both that privatization is a success. Since the government was unable to fund such programs to it†s potential, privatization was an obligation in order to have an efficient program (statistics shown on Queens and McMaster university web pages). Students in the MBA programs at these schools agree that privatization was effective as they now have more advanced resources to work with. Clearly facts have proven that a privatized business/program/industry, is more beneficial not only a Canadian economy but as well as society. Another problem that Canadian†s face is that the government doesn†t focus enough of his attention on social issues concerning Canadians such as; education, health care, environment, welfare, foreign inspections on imports and Income tax. Many of Canada†s collage and university students are moving to the United States because the job opportunities are more abundant and income taxes are much lower. â€Å"Because the bulk of the education is paid for by the government (taxpayers) we are effectively subsidizing US industry by providing skilled labor. Many conservative people argue that this is one reason why students should bear the entire cost of their education. This alternative would close the post-secondary door on many poor people in this country† (www. watleo. ca/~cdsmith/txs/education). An alternative is that the government should turn aside the money that they are currently paying colleges and universities into student loans. Once the student graduates and receives employment this loan could be paid back in the form of income taxes. However if the student decides to work in the US their income tax will no longer compensate for the student loan, as they will have to pay off the loan personally. This method of educational funding would increase the number of students attending post secondary education because finance is a capital reason why some students do not attend collage or university. Another issue facing the Canadian education system is private schools vs. public schools. Canadian statistics prove that students attending private schools are more likely to attend university compared to people enrolled in the public system. The government needs to offer better educational funding in order to provide additional teachers to cut class sizes, as well as more resources. These changes will decrease class size, which will offer more one on one student to teacher attention that will help students prepare for a higher education. Healthcare is another major social issue that the government needs to establish; it is important for Canada to reduce the cost of basic healthcare services. Although Canada†s healthcare administration costs are low compared to the US, the US is associated with higher levels of medical research and development as well as increased amounts of advanced equipment i. e. â€Å"the city of Philadelphia has more MRI machines than all of Canada combined†(www. watleo. uwaterloo. ca/cdsmith. health. html. com). In Canada there are many people who abuse the healthcare system, in order to stop this, government should stop charging a basic fee for medical services and offer a tax refund for those who do not use the medical service provided. This would increase the efficiency of the current universal medical system in Canada and wouldn†t cost the government any money, as the refund would be proportional to the savings, because not as many people would then use the medical system. In addition to social issues in need of government assistance, environmental funding is needed to research issues such as air pollution, water conditions, the ozone and animal habitat which is always an on going environmental issue. On the topic of environment and health, the Toronto Star reported that inspection laws associated with NAFTA need to be stricter as poisonous strawberries imported form Mexico are currently being sold in the Canadian market. The National Post is reporting that 85% of Canadians are upset by the amount of money they pay in tax. StatsCan figures released on April 14th state that Canadians incomes are not keeping pace with economic growth. â€Å"We†re all working harder than ever and we†re not seeing our take home pay rise. It†s got to stop. Governments must take action to cut taxes and increase real take home pay†(Ontario PC Party). The government also needs to flatten income taxes because each province in Canada pays a different income tax rate Quebec than BC being the highest. We could also lower our income taxes if we were to be like the US and impose income tax on inheritance and lottery winnings, which would allow government to reduce personal income tax. A further social government issue that needs to be worked on is welfare. Welfare laws need to be modified because people on welfare have no incentive to look for a job or enroll in training. Welfare fraud is a big issues in some provinces of Canada (fraud and welfare abuse cost Ontario taxpayers an estimated $100 million last year), as well Canada needs to eliminate welfare payments to jail inmates as this is contributing to the millions of dollars wasted each year. It is made clear through the facts and evidence stated why the government needs to focus more upon social issues such as welfare, taxes, education, healthcare and foreign import laws. These government run programs are out of reach to entrepreneurs. Thus Canada needs the government to concentrate and control theses issues, rather than infringing upon Canadian business and economy. Lastly Since research has proven no laws limiting legitimate government control over large companies these laws need to be established and enforced because the government already has to much control over the Canadian economy. Laws need to be created in order to monitor government control and creation of monopolies. Trough extensive research currently there were no laws found that entail government control of business. Government already has a big impact on the pass of laws so this could be why there are no laws limiting their control of business i. e. Ontario liquor, CBC, Via rail, TVO, Petro Canada etc. However the part of business the government must stay in charge of are companies that produce money, the mint, nuclear power plants, army, police forces etc. These are all government run agencies that can not be privatized due to obvious reasons. If anyone could produce money than the Canadian dollar would have no value and the Canadian economy would crash, i. e. the Russian economy. The issue concerning too much government control over business that can be run by entrepreneurs can be resolved by passing â€Å"fair† laws concerning government control over business. This law would limit the amount of control and what the government can control in the Canadian economy. In conclusion as facts have proven above Canadian economic policies are becoming too government controlled, which is resulting in a government run bureaucracy. As government monopolies are becoming to strong it begins to decrease entrepreneurial opportunities for Canadian citizens and is why all business should be privatized. If the government were to allow privatization of some industries it would allow for more job opportunity, profit and a better Canadian economy. As well the government should concentrate more on social issues concerning Canadians such as healthcare, education, taxation etc, these issues could be emphasized on more by privatizing some crown corporations. Lastly law issuing limiting government control is allowed will result in a more prosperous economy. Therefore if all of these issues are handle more efficiently as outlined than Canada†s economy will be booming and it will create more opportunity in the job market for Canadian citizens.

a orange essays

a orange essays Just saying the name Agent Orange gets the attention of every Vietnam veteran, and I dare say most of the Australian and American public, not to mention the Vietnamese. It has been argued about, written about, researched and debated, published in magazines and newspapers, talked about on radio and television. It was the subject of documentaries, legal battles, and in Australia a Royal Commission that lasted some two years and cost 3.8 million dollars. Agent Orange was the code name for a herbicide developed for the military, primarily for use in tropical climates. Although the genesis of the product goes back to the 1940s, serious testing for military applications did not begin until the early 1960s. The purpose of the product was to deny an enemy cover and concealment in dense terrain by defoliating trees and shrubbery where the enemy could hide. The product Agent Orange (a code name for the orange band that was used to mark the drums it was stored in) was principally effective agai nst broad-leaf foliage, such as the dense jungle-like terrain found in Southeast Asia. The product was tested in Vietnam in the early 1960s, and was brought into ever widening use during the height of the war in 1967-68, though its use was diminished and eventually discontinued in 1971. Agent Orange was a 50-50 mix of two chemicals, known conventionally as 2,4,D and 2,4,5,T. the combined product was mixed with kerosene or diesel fuel and dispersed by aircraft, vehicle, and hand spraying. An estimated 19 million gallons of Agent Orange were used in South Vietnam during the war. The earliest health concerns about Agent Orange were about the products contamination with TCDD, or dioxin. TCDD is one of a family of dioxins, some found in nature, and are cousins of dibenzofurans and PCBs. Dioxin is formed by burning chlorine-based chemical compounds with hydrocarbons. The major source of dioxin in the environment (95...

Compare the heights of girls and boys in year 8 and the sixth form Essays

Compare the heights of girls and boys in year 8 and the sixth form Essays Compare the heights of girls and boys in year 8 and the sixth form Essay Compare the heights of girls and boys in year 8 and the sixth form Essay The purpose of this exercise is to compare the heights of girls and boys in year 8 and the sixth form, in an attempt to show the following. That in year 8, girls and boys will have more similarities in height, but girls are more likely to be taller than the boys. In sixth form there will be greater differences between the heights of boys and girls and the boys are more likely to be taller. That there will be a much greater difference in the heights of boys between year 8 and sixth form than between the girls. I will attempt to show this by measuring the heights of boys and girls in year eight and the sixth form. In each case a sample size of fifty will be used in order to produce statistically valid results according to the central limit theorem. This will be done, by obtaining a sample that accurately represents each group. Firstly a list of boys and a list of girls in year eight and the sixth form will be formulated giving each student a number. Then a random number generator will be used to select fifty boys and fifty girls from each year group. I will measure the selected groups independently using the measuring device illustrated below. This device includes two, one meter rulers fixed against the wall, the mark for 0 cm is in line with the floor, and the second ruler is fixed alongside the first so that its 0 cm mark is in line with the firsts 100cm mark. Then two, 30cm rulers with millimetre measurements are fixed, one either side of the second meter ruler, at 140-170cm and 170-200cm, again the 0cm marks for each of these rulers were fixed in line with 140cm and 170cm respectively. This will help to ensure accurate measurements. Students will be asked to remove their shoes, stand up straight, flat on their feet, with their backs against the wall. The setsquare will be held against the wall and the individuals head. The height will be read and recorded. Each height will be measured to millimetres to ensure obtaining a reading that is as accurate as possible. The results were as follows. RESULTS OF THE SIXTH FORM BOYS HEIGHTS GIRLS HEIGHTS 1 175.1 26 181.2 1 153.5 26 155.4 2 178.5 27 186.7 2 169.8 27 162.7 3 187.6 28 180.0 3 156.6 28 152.0 4 175.6 29 185.2 4 157.5 29 160.2 5 178.8 30 184.9 5 174.8 30 155.5 6 180.7 31 182.2 6 175.1 31 153.5 7 172.4 32 160.3 7 160.8 32 160.2 8 175.0 33 179.8 8 168.2 33 170.2 9 180.9 34 167.6 9 167.7 34 158.0 10 167.9 35 171.3 10 159.0 35 162.7 11 170.6 36 178.9 11 157.1 36 168.2 12 175.6 37 170.0 12 171.2 37 165.1 13 177.7 38 161.6 13 164.3 38 149.2 14 184.8 39 169.2 14 165.3 39 158.1 15 176.6 40 166.2 15 163.4 40 172.6 16 185.3 41 173.1 16 148.9 41 173.3 17 187.5 42 183.2 17 153.9 42 168.7 18 186.4 43 167.0 18 157.2 43 162.5 19 180.3 44 171.1 19 159.3 44 157.6 20 165.3 45 174.8 20 169.2 45 158.0 21 176.8 46 181.2 21 153.5 46 164.7 22 175.7 47 179.0 22 169.9 47 152.3 23 179.2 48 165.9 23 166.7 48 160.8 24 190.3 49 183.0 24 147.5 49 161.8 25 190.8 50 176.4 25 157.5 50 156.7 RESULTS OF YEAR EIGHT BOYS HEIGHTS GIRLS HEIGHTS 1 149.4 26 149.9 1 158.7 26 164.1 2 146.0 27 165.2 2 160.6 27 148.9 3 145.0 28 150.2 3 170.2 28 166.1 4 164.8 29 151.2 4 159.6 29 153.5 5 163.7 30 141.9 5 163.6 30 156.0 6 165.2 31 156.2 6 157.4 31 161.9 7 161.1 32 166.2 7 159.0 32 160.2 8 153.7 33 141.9 8 166.8 33 160.0 9 162.2 34 144.8 9 162.8 34 164.2 10 159.7 35 148.6 10 148.2 35 150.8 11 161.6 36 152.7 11 143.0 36 150.1 12 147.1 37 160.3 12 165.1 37 148.9 13 143.4 38 146.8 13 170.2 38 157.3 14 146.2 39 145.9 14 160.9 39 168.2 15 148.2 40 154.6 15 154.3 40 165.8 16 155.8 41 153.2 16 154.8 41 159.0 17 165.0 42 162.1 17 167.3 42 156.7 18 168.2 43 160.0 18 143.5 43 148.9 19 150.6 44 147.2 19 156.6 44 154.9 20 148.8 45 160.9 20 160.6 45 161.3 21 161.9 46 151.4 21 145.0 46 154.9 22 146.8 47 147.1 22 159.6 47 158.7 23 153.0 48 166.8 23 162.5 48 167.9 24 150.0 49 150.0 24 151.3 49 165.8 25 164.4 50 147.3 25 153.2 50 157.0 STEM AND LEAFS THE MEAN AND STANDARD DEVIATION First calculate the mean of each group. This will be done by adding together all of the heights in one group and dividing it by 50 (the sample size). This will be repeated for each of the remaining groups. This calculation is represented by the following formula. Next calculate the standard deviation, which is found by taking one group e.g. Year 8 boys and subtracting the mean from one of the scores. The result of this is squared. This is repeated for every score and the mean of the squared differences is worked out. This is the variance. To find the standard deviation you must square root the variance. This calculation is represented by the following formula. USEFUL BACKGROUND INFO THE CENTRAL LIMIT THEOREM If the sample size is large enough then the distribution of the sample means is approximately normal, irrespective of the distribution of the parent population. The mean of the distribution of the sample means is approximately equal to the parent population. The variance of the distribution of sample means is approximately the variance of the parent population divided by the sample size. These approximations get closer as the sample size gets bigger. These results are known as the Central Limit Theorem. Symbolically if, X ~ (unknown)(?, ?à ¯Ã‚ ¿Ã‚ ½) then ? n ~ (?, ?à ¯Ã‚ ¿Ã‚ ½/n) Provided n is sufficiently large. (n ? 30 is usually a good size). Standard error The standard deviation of the distribution of sample means is called the standard error. 1 s.e = ?à ¯Ã‚ ¿Ã‚ ½/n (variance of the distribution of x = ?à ¯Ã‚ ¿Ã‚ ½/n) In previous calculations I have only worked out the mean x and variance sà ¯Ã‚ ¿Ã‚ ½ of my sample. I cannot calculate confidence intervals for population mean à ¯Ã‚ ¿Ã‚ ½ because I do not know ?à ¯Ã‚ ¿Ã‚ ½. Unfortunately sà ¯Ã‚ ¿Ã‚ ½ is not an unbiased estimator of ?à ¯Ã‚ ¿Ã‚ ½ (i.e. the mean of the distribution of sà ¯Ã‚ ¿Ã‚ ½ is not equal to ?à ¯Ã‚ ¿Ã‚ ½). However Is an unbiased estimator of population variance, and I can use this as an estimate of ?à ¯Ã‚ ¿Ã‚ ½ when calculating standard error in order to produce confidence intervals for à ¯Ã‚ ¿Ã‚ ½. So in order to calculate the standard errors for each of my groups I must first calculate an estimate for ?à ¯Ã‚ ¿Ã‚ ½, using the above formulae. CALCULATIONS FOR THE ESTIMATES ?à ¯Ã‚ ¿Ã‚ ½ AND STANDARD ERRORS I have previously calculated the mean (x) and standard deviation (sà ¯Ã‚ ¿Ã‚ ½) for each of my groups. I will now calculate an estimate for ?à ¯Ã‚ ¿Ã‚ ½ in order to calculate the standard errors and formulate confidence intervals for each of my groups. To estimate ?à ¯Ã‚ ¿Ã‚ ½ I will use the previously stated formula. And then using these estimates for ?à ¯Ã‚ ¿Ã‚ ½ I will calculate the standard errors using the formula. CONFIDENCE INTERVALS If we have one sample mean x then P(à ¯Ã‚ ¿Ã‚ ½ 1s.e x à ¯Ã‚ ¿Ã‚ ½ + 1s.e), but this can be rearranged à ¯Ã‚ ¿Ã‚ ½ 1s.e x x à ¯Ã‚ ¿Ã‚ ½ + 1s.e == à ¯Ã‚ ¿Ã‚ ½ x 1s.e à ¯Ã‚ ¿Ã‚ ½ x + 1s.e So P(x 1s.e à ¯Ã‚ ¿Ã‚ ½ x + 1s.e) = 0.68 I.e. There is a 68% chance that à ¯Ã‚ ¿Ã‚ ½ lies within ? 1s.e of my x. This is known as a confidence interval for the population mean. In this investigation I am going to calculate confidence intervals for each of the individual groups. I am going to use 95% confidence intervals so I will have to calculate a z score for 95%. The z score is the number of standard deviations units away from the mean value. Using numerical methods, accurate tables have been constructed for the area under the normal curve. The table gives the area to the left of (or below) any given z- value. The area under the Standard normal curve that is to the left of z is denoted by ?(z).This is represented by the following diagram. I will obtain a z score for 95% as follows. The area to the left of the z = ?(z) = 0.95 + 0.25 = 0.975 Using the normal distribution table z = 1.96.This will be used in calculating the confidence intervals for each group. CALCULATIONS OF CONFIDENCE INTERVALS. Using my previous calculations for the z score for 95%, the mean of each sample and the standard errors. I will now formulate confidence intervals for each of my groups. YEAR EIGHT BOYS x = 154.1 s.e = 1.19 z score 95% = 1.96 (this will be used for all the groups) P(à ¯Ã‚ ¿Ã‚ ½ 1.96s.e x à ¯Ã‚ ¿Ã‚ ½ + 1.96s.e) = 0.95 P(x 1.96s.e à ¯Ã‚ ¿Ã‚ ½ x + 1.96s.e) = 0.95 P(154.1 1.96 x 1.19 à ¯Ã‚ ¿Ã‚ ½ 154.1 + 1.96 x 1.19) = 0.95 95% C.I for à ¯Ã‚ ¿Ã‚ ½ is (151.8 156.3) . YEAR EIGHT GIRLS x = 158.3 s.e = 0.94 P(à ¯Ã‚ ¿Ã‚ ½ 1.96s.e x à ¯Ã‚ ¿Ã‚ ½ + 1.96s.e) = 0.95 P(x 1.96s.e à ¯Ã‚ ¿Ã‚ ½ x + 1.96s.e) = 0.95 P(153.8 1.96 x 0.94 à ¯Ã‚ ¿Ã‚ ½ 153.8 + 1.96 x 0.94) = 0.95 95% C.I for à ¯Ã‚ ¿Ã‚ ½ is (156.5 160.1) SIXTH FORM BOYS x = 176.7 s.e = 1.08 P(à ¯Ã‚ ¿Ã‚ ½ 1.96s.e x à ¯Ã‚ ¿Ã‚ ½ + 1.96s.e) = 0.95 P(x 1.96s.e à ¯Ã‚ ¿Ã‚ ½ x + 1.96s.e) = 0.95 P(176.7 1.96 x 1.08 à ¯Ã‚ ¿Ã‚ ½ 176.7 + 1.96 x 1.08) = 0.95 95% C.I for à ¯Ã‚ ¿Ã‚ ½ is (174.6 178.8) SIXTH FORM GIRLS x = 161.4 s.e = 0.99 P(à ¯Ã‚ ¿Ã‚ ½ 1.96s.e x à ¯Ã‚ ¿Ã‚ ½ + 1.96s.e) = 0.95 P(x 1.96s.e à ¯Ã‚ ¿Ã‚ ½ x + 1.96s.e) = 0.95 P(161.4 1.96 x 0.99 à ¯Ã‚ ¿Ã‚ ½ 161.4 + 1.96 x 0.99) = 0.95 95% C.I for à ¯Ã‚ ¿Ã‚ ½ is (159.5 163.3) These results are represented by the following graphs. ANALYSIS OF RESULTS The first set of box plots shows the 95% confidence intervals for à ¯Ã‚ ¿Ã‚ ½ for the year eight boys and the girls. They show that the majority of girls and boys in year eight are very similar in height, but girls are more likely to be taller. The second set of box plots shows the 95% confidence intervals for à ¯Ã‚ ¿Ã‚ ½ for the sixth form boys and girls. They show that there is a great difference in the heights of the girls and boys, and it is the majority of boys that are taller. The third set of box plots shows the 95% confidence intervals for à ¯Ã‚ ¿Ã‚ ½ for the boys in year eight and sixth form. They show that the majority of boys are a lot taller in the sixth form. The fourth set of box plots shows the 95% confidence intervals for à ¯Ã‚ ¿Ã‚ ½ for the girls in year 8 and the sixth form the majority of girls in both years are much more similar in height than the boys. The majority of girls in sixth form are more likely to be taller. CONCLUSION AND ANALYSIS My results have shown my hypothesis to be correct. They show that: In year 8 girls and boys have more similarities in height, but girls are more likely to be taller than the boys. In sixth form there are greater differences between the heights of boys and girls and the boys are more likely to be taller. There is a much greater difference in the heights of boys between year 8 and sixth form than between the girls. I believe these results are fairly reliable, as I would say that I obtained my samples of girls and boys in year eight and the sixth form as accurately as I possibly could have given the facilities and amount of time that was available. I also think that using these samples I fairly accurately estimated the confidence intervals for the population mean of each group. I dont think there is much I could have done to improve the method of obtaining the sample or the method I used to measure the heights. The accuracy of my results would improve by using a larger sample size e.g. 100 girls and boys from each year group, according to the central limit theorem. However, this was not possible due to the amount of people available to measure and the amount of time allocated. I could have improved the sample further by taking groups of students from different schools in different areas, this may have given a more accurate representation of the population, as the ranges of heights in different areas for each group may be more varied. However this would have been very difficult to do and would have taken too long, also I dont think it would have shown any great difference in my findings, as the heights of boys and girls in each group throughout the region are most likely to be fairly similar to those I measured. If I had had more time it would have been interesting to find out where exactly the changes in the heights of boys and girls actually occurs. This could have been done by taking a sa mple of fifty girls and fifty boys from each of the years in between year eight and the sixth form, and again calculate confidence intervals to see when the boys go from being the same height or shorter than the girls to being much taller than them.