Pitfalls Of Relativism Essays (2294 words) - Relativism,

Pitfalls Of Relativism Pitfalls of Relativism The year was 1943. Hundreds of Jewish people were being marched into the gas chambers in accordance with Adolf Hitler's orders. In the two years that followed, millions of Jews were killed. Only a fraction survived the painful ordeals at the Nazi German prison camps. However, all of the chaos ended as World War II came to a close: the American and British soldiers had won and Hitler's Third Reich was no more. A certain ethical position would state that the anti-semantic Nazi German culture was neither right nor wrong in its actions. In fact, it is this view of the cultural relativist that assume all actions considered right in a culture to be good for that culture alone. Moreover, the relativist claims that these actions can not be judged according to their ethical correctness because there is no absolute standard by which they could be compared. In the above case, this position would not allow for the American and British soldiers to interfere with the Nazis; the relativist would claim that the Allies were wrong in fighting the Germans due to a cultural disagreement. In truth, it is the relativist position, which has both negative logical and practical consequences, and negligible benefits. The first logical consequence of relativism is that the believer must contradict himself in order to uphold his belief. The view states that all ethics are relative while putting forth the idea that no absolute standard of rightness exists. If this is the case, then what is cultural relativism relative to? From a purely logical point of view, this idea is absurd, for in assuming that something is relative one must first have some absolute by which it is judged. Let the reader consider this example to reinforce the point. A young woman is five feet tall, and her older friend is six feet tall. The younger female considers herself short because she looks at her friend and sees that her friend is taller than she is. It would be illogical to say that the first woman is short if she were the only female in existence; if this were the case then there would not be anyone for her to be relative to in height. However, this logical fallacy is what the relativist assumes by stating that there is no standard of rightness for relativity. Quite simply, the cultural relativist is stating that he is relative to an absolute, which he considers non-existent. Another logical error that the relativist makes lies in his Cultural Differences Argument (Rachels p.454). The premise of this argument is that different cultures have different moral codes (Rachels p.454). The conclusion that the relativist derives is that there is no objective 'truth' in morality, [and therefore] right and wrong are only matters of opinion [that] vary from culture to culture (Rachels p.454). The main logical problem with this argument is that the stated conclusion does not necessarily need to be the case if the premise is given. The premise states what different people believe to be true, and the conclusion jumps to the assumption that this belief must necessarily be the case. Let the reader consider this instance, which closely follows the form of the above given argument. Assume that there is a society that believes that sunning as much as possible in the nude can only benefit a person. Due to scientific study, it has been experimentally shown that overexposure to the sun's ultraviolet rays can cause skin cancer. Being in the American culture, people know this to be true and therefore would disagree with sunning too often. According to the relativist, since the two cultures disagree concerning the practice of sunning there is no objective truth about it. However, this is a faulty conclusion because empirical evidence shows that the first culture would be wrong in its beliefs. In truth, one cannot derive a substantive conclusion about a subject (morally) from the mere fact that people disagree about it (Rachel p.454). Having discussed the logical consequences of relativism, it is necessary to expound upon the effects of its practice. The first of these repercussions is that culture determines what is functionally right and wrong. This means that the individual has

Every AP Statistics Practice Test Available Free and Official

Every AP Statistics Practice Test Available Free and Official SAT / ACT Prep Online Guides and Tips Are you taking the AP Statistics exam soon and want to make sure you’re prepared?One of the best ways to measure your progress and figure out which areas you need to focus on is to take practice exams.There are a lot of AP Statisticspractice exams available; however, some are higher-quality than others. Taking a poorly written practice exam can cause you to study the wrong things and give you an inaccurate picture of what the real AP exam will be like. In this guide, I’ll go over every AP Statistics practice test available, explain if and how you should use each one, and end with a schedule you can follow to help you incorporate practice tests into your study plans. Official AP Statistics Practice Exams Official practice exams are those that have been created by the College Board (the organization that develops and administers all the actual AP exams). They are always the top resources to use because you can be sure that they accurately reflect the format and content of the real AP exam. There are three types of official practice resources: Complete Practice Tests The College Board has released two complete examswhich are linked below. 2012 AP Statistics Released Exam 1997 AP Statistics Released Exam Both links include the complete exam, an answer key, and scoring information. Both of these are very useful study resources, even the 1997 exam since the AP Stats exam hasn't changed much since then. This is the current examis three hours long with two sections. Students can use a graphing calculator for the entire exam. Multiple-Choice Section: 40 questions 90 minutes Worth 50% of total score Free-Response Section: 6 questions (5 free response and one investigative task) 90 minutes Worth 50% of total score The only major difference between the current format and the format of the 1997 exam is that the 1997 exam had 35 multiple-choice questions instead of 40. The content the exam tests has remained consistent, so, despite its age, this test is still a great resource to use and will give you a good idea of what your AP exam will be like. The 2012 exam has the same format as the current exam. Multiple-Choice Questions The College Board often reuses multiple-choice questions, so there are not many released official multiple-choice questions available for AP Stats. Besides the multiple-choice questions from the released exam, the only official multiple-choice questions you can use in your studying are in the AP Statistics Course Description. Beginning on page 19, there are 18 multiple-choice questions, along with an answer key. Free-Response Questions Compared to multiple choice, there are many more official free-response questions you can use to study and, since they are recent, they’ll give you a very accurate idea of what to expect on the real exam. The College Board has released free-response questions from 1998-2017which means you have dozens of official free-response questions to use for your studying.All the free-response questions include answer keys and sample responses. Unofficial AP Statistics Practice Tests and Quizzes Even though they weren’t created by the College Board, many unofficial practice AP Statistics exams are still high-quality and can be a great study resource. For each resource listed below, I explain what it includes and how you should use it. Barron’s Barron’s has a free, high-quality, and complete practice exam that you can take either timed or untimed. Multiple-choice questions are automatically graded after you complete the exam, and there are guidelines for self-scoring your free-response sections. This practice test is similar to the real AP test in both content and format, so you should definitely use it as you study. The next section of this guide has guidelineson how to use this resource and others. Shmoop Shmoop is the only resource on this list that requires you to pay to access any of its resources. Paying its monthly fee of about $25 gets you access to a diagnostic exam, four full-length practice tests, and additional practice questions. With a paid subscription, you also get access to Shmoop’s resources for the SAT, ACT, and other AP exams. Stat Trek This is a complete, 40-question, multiple-choice test. You can take the test timed or untimed, and you can choose to see the answer to each question immediately after you answer it or wait until the end of the exam to see what the correct answers were. Some of these questions are a bit easier than those found on the real AP exam, but this is still a solid resource. McGraw-Hill McGraw-Hill has a 25-question multiple-choice quiz (although the questions are randomly selected from a larger pool, so if you take the quiz multiple times you may get more than 25 questions out of it). The quiz is automatically graded and has brief answer explanations. You can only take the quiz in untimed mode. This is one of the higher-quality short quizzes available with questions similar in content to those you’ll see on the real AP exam. Albert.io Albert.io organizes its practice questions into the four Big Ideas of AP Statistics, and the Big Ideas are further broken down into more specific topics, each with relevant short quizzes which can be useful if you’re studying and want to easily find questions on certain subjects. The questions are ranked as easy, moderate, or difficult, they aren’t timed, and you see the correct answer (plus a detailed explanation) after you answer each question. You will have to sign up for a free account, which includes a limited number of credits you can use to answerquestions. If you want to access more questions beyond your initial allotment, you'll have to buycredits or earn them by referring friends. Varsity Tutors The Varsity Tutors resources include four diagnostic tests and 139 short practice quizzes, organized by topic. The four diagnostic tests each contain 40 multiple-choice questions and, like the Stat Trek test, they are similar to, but a bit easier than, the real AP exam. You’re timed while taking the exams and, as a bonus, after you complete the exam, the questions are organized into different categories so you can see which categories you did best in and which categories you should focus your studying on. For this site, I’d recommend mostly using the diagnostic tests since most of the individual quizzes are so short (only 1-3 questions) that it can be frustrating to continually start and finish separate quizzes. Free Test Online Free Test Online has a 32-question multiple-choice quiz. This is shorter than the multiple-choice section of the real AP exam, but this is a good resource to use if you want a shorter study session. The quiz is not timed and is automatically graded after you complete it. Kansas State University Quiz and Answer Key This is a 25-question multiple-choice quiz from Kansas State University’s Department of Mathematics. The questions are good quality, although you do have to grade the quiz yourself (the correct letter is in bold in the answer key). This another good option if you want a to answer some practice questions but don’t want to take a full exam. Dan Shuster This site has 24 quizzes (12 multiple choice and 12 free response). They were created by an AP Statistics teacher and follow his curriculum schedule. Each multiple-choice quiz has 10 questions, and short answer explanations are given after you complete each quiz. Each free-response quiz has three questions as well as answer explanations. The free-response questions especially are shorter and easier than you’ll find on the real AP exam, but you can still use this resource if you want to do some quick, targeted studying. How to Use These AP Statistics Practice Tests Knowing how to use each of these resources will make your studying more effective, as well as prepare you for what the real AP Statistics exam will be like. Read the guide below to learn how and when you should use these practice tests and quizzes. First Semester Right now you’re still learning a lot of key information, so during your first semester of AP Stats you should focus on quizzes and free-response questions on topics you’ve already covered. Begin using these materials about halfway through the semester. Multiple-Choice Practice For multiple-choice practice, take unofficial quizzes that let you choose which subjects you want to be tested on. This lets you review content you’ve already learned and avoid questions on material you haven’t covered yet. The best resources for this are Albert.io, Varsity Tutors, and Dan Shuster because their quizzes are clearly organized by specific subject. Free-Response Practice For free-response questions, use the official released free-response questions from the Official Practice Exams section. There are a lot of questions available, so look through them to find questions you can answer based on what you’ve already learned. It’s best if you answer a group of them (up to six) together at a time to get the most realistic preparation for the actual AP exam. It also helps to time yourself when answering these questions, particularly as it gets later in the semester. Try to spend about 12 minutes each on the first five questions and 30 minutes on the investigative task (which will be the last question in the section). Second Semester Second semester is when you can begin taking complete practice exams and continue reviewing content you’ve already learned. Follow these five steps: Step 1: Complete Your First Complete Practice Exam About a month or two into this semester, after you’ve covered a majority of the content you need to know for the AP exam, take your first complete practice exam. For this first practice test, I recommend using the 1997 official practice exam. You should take this test timed and in one sitting, then correct it when you’re finished. If you haven’t already, this is a good time to set a score goal for yourself. Aim for at least a 3 since this is the lowest passing score for the exam. However, if you scored a 3 or higher on this first practice exam, it’s a good idea to set your goal score even higher, to a 4 or 5. Getting a higher score on the AP Stats exam looks more impressive to colleges, and it can sometimes get you more college credit. Step 2: Analyze Your Score Results After you’ve figured out your score, look over each problem you answered incorrectly and try to figure out why you got the question wrong.As you’re doing this, look for patterns in your results. Are you finding that you got a lot of questions on experimental design wrong? Did you do well on multiple choice but struggled with free response? Figuring out which problems you got wrong and why is the best way to stop repeating your mistakes and make improvements for future exams. Even if it seems tedious, don’t be tempted to skip this step! Step 3: Focus on Your Weak Areas By now, you should have a good idea of the areas or techniques you need to work on to raise your score.If there are specific content areas you need to work on, review them by going over your notes, reading a review book, and answering multiple-choice and free-response questions that focus specifically on those topics. If you’re struggling with your test-taking techniques, for example, running out of time on the exam or misreading questions, the best way to combat these issues is to answer a lot of practice questions under realistic testing conditions. Step 4: Take Another Practice Exam After you’ve spent time improving your weak areas, it’s time to see the results of your hard work.Take and score another complete practice exam, timed and finished in one sitting. This is a good time to use the 2012 official released exam or the Barron's exam. Step 5: Review Your Results to Determine Your Future Study Plan Now you’re able to see how much you’ve improved, and in which areas, since you took your first complete practice exam.If you’ve made improvements and have reached or are close to your target score, you may only need to do some light studying from now until the AP exam. However, if you haven’t improved a lot, or you’re still far from your score goal, you’ll need to analyze the way you’ve been reviewing and think of ways to improve. The most common reason for not improving is not actively studying, and only passively leafing through your notes or reviewing missed questions. Active studying takes longer and requires more effort, but it’s the best way to see significant improvements. As you’re studying, make sure you really understand exactly where you made a mistake for every practice question you answer incorrectly. Also, when you’re reviewing your notes, stop every few minutes and mentally go over what you just learned to make sure you’re retaining the information. You can repeat these steps as many times as you need to in order to make improvements and reach your target score. If you need more complete practice tests, you can create your own by combining a set of official free-response questions with 40 unofficial multiple-choice questions. Stat Trek and Varsity Tutors are probably the best resources to use for the multiple-choice questions since each of their exams already have 40 questions combined for you. Conclusion: Where to Find AP Statistics Practice Exams If you want to score well on the AP Statistics exam, you’ll almost certainly need to take some practice tests. Official resources are the best to use, but there are also lots of high-quality unofficial quizzes and tests that you should be using. During your first semester, focus on answering individual free-response and multiple-choice questions on topics you’ve already covered in class. For your second semester, follow these steps: Take and score your first complete practice exam Analyze your score results Focus your studying on weak areas Take and score another complete practice exam Review your results to determine your future study plan What's Next? Wonderingwhich other math classes you should take? Math is often the trickiest subject to choose classes for, but out guide will help you figure out exactly which math classes to take for each year of high school. How many AP classes should you take?Get your answer based on your interests and your college goals. Want some tips on how to study for your AP exams?Check outour five-step plan on how to study for AP exams. Want to improve your SAT score by 160 points or your ACT score by 4 points?We've written a guide for each test about the top 5 strategies you must be using to have a shot at improving your score. Download it for free now:

Critically Analyse the Role and Value of 'The Community' in Global Essay

Critically Analyse the Role and Value of 'The Community' in Global Justice Theory - Essay Example A modern example of how the world has rallied for social good is used to put the input of the global community in the topic under discussion into perspective. An example of â€Å"the invisible children†, an organization that produced the â€Å"Kony 2012† film that recently went viral for social good is used in this case. Introduction Social justice, is defined as the fair and appropriate implementation of laws in line with the natural law to all people regardless of their ethnicity, gender, wealth status, race, religious beliefs, political affiliations and so on with equality and without discrimination. Social justice begins with the acquisition of civil rights, defined as the privileges associated with citizenship of a particular country. These include the right to freedom, proper governance, justice and fairness in the implementation of the laws of the land together with human and natural rights like the privilege to hold public office subject to an individual’ s conduct (Kuper, 2000)1. From the definition, social justice begins at the local community level to the level of a country before going global. Global social justice cannot therefore be achieved if individual countries have not created room for its actualization. A deeper meaning of global social justice To have an in-depth understanding of global social justice, the following four areas must be properly explained; equal citizenship, entitlement to a social minimum, equality of opportunities and fair distribution of resources. With a proper appreciation of these issues, the social justice in a global context will be clearly realised. According to Simon Maxwell (2008)2, in his publication to the Overseas Development Institute (ODI), the above four areas have the following meaning: Equal citizenship. This is not just being a resident of a particular nation and earning a living within the confines of the country’s borders, but has a much wider requisite of freedom, equality and solidarity expressed by citizens of a country and by an extension the world. The voice of a citizen must be heard and they should be in a position to hold public institutions accountable to be considered full beneficiaries of social justice. Guarantee of social minimum. This has the implication of investments in social protection to ensure that all the civil liberties so achieved are not ceded but instead expanded to cover areas that are yet to experience social justice. It therefore requires vigilance on the part of citizens to ensure that all achievements with regards social justice are properly safeguarded from malicious interest groups seeking to steal any gains from a country’s citizens. Equality of opportunities This deals with the chance to reap the benefits of economic, social and cultural gains. Members of a country or society must have equal access to gains opportunities to education, health and fair administration of justice with the option of holding anyone attem pting to deny these opportunities to account. A society that avails equal chances to its members is therefore considered to have provided social justice to its constituents. Fair distribution The social justice agenda if facing problems thanks to the issue of distribution. This is one topic rarely discussed in most circles because it touches on the elite. America is considered on the nations

Hinduism and Judaism Essay Example | Topics and Well Written Essays - 1000 words

Hinduism and Judaism - Essay Example There are interesting similarities and differences in their foundational teachings that show some of the elements that unite people, and some of the ways in which cultural context informs religious development. Hinduism traces its roots back to approximately 2500 B.C. It was not a religion that began with one particular historical event, but was rather a gradual development of beliefs by peoples in the Indus Valley (Zaehner, pg. 15). Its sacred literature has two categories: sruti and smriti. The sruti were heard, or divinely revealed, and include the Vedas (the most ancient Hindu scriptures), the Upanishads, the Brahmanas, and the Aranyakas. The Vedas contain the creation account, regulations for sacrifices, and prayers. According to Hindu tradition, these texts were secretly taught by a prophet to a disciple (David S. Noss, 55). The smriti are texts that were remembered or passed down orally. The difference is that these were written by humans rather than by the gods. The smriti consist of the epics, the Sutras and the Puranas. The epics are long poems about events in the lives of heroic warriors. The Sutras relate to such ideas as dharma, yoga, and Vedanta. The most significant of these was the Laws of Manu, or the Manusmriti, which concerned proper law and conduct for Hindus. The Puranas are mythological writings, containing the stories of the gods and goddesses (Knott, pg. 24-25). The Indus Valley civilization cohered around two cities, Mohenjo-Daro and Harappa. Between 2500 and 2000 B.C., the nomadic Indo-Aryans began to migrate into this area, just as the Indus Valley peoples began to disappear. The Vedas were the scriptures of the Indo-Aryans and are most commonly acknowledged as the basis for Hinduism, and they are also said to be Hinduism's supreme authority (Morgan, pg. 32). The Vedic conception of rita, or cosmic order, later served as the basis for the ideas of dharma and karma. The gods served as guardians of this idea of rita and had to be propitiated regularly by sacrifice. (Morgan, pg. 33). And so with the idea of sacrifice came a collection of regulations and technicalities for the sacrifice process (Kinsley, pg. 92). During this time, the priesthood came to assume a good amount of power in society. Such new doctrines as the four stages of life, the idea of transmigration, and the origins of the caste system (Morgan, pg. 48). While in its foundat ional stages, Hinduism had claimed that the soul could die, either on Earth or even in heaven, but this change had the soul being reborn in an endless cycle, seeking release, or moksha from this unending existence. Despite the fact that Judaism started far from Hinduism, there are many striking similarities to complement the differences between the two faiths. According to such sources as the Tanakh and the Talmud, the Jewish faith is based on a covenant between God and Abraham, established approximately in 2000 B.C., and renewed between God and Moses around 1200 B.C. Unlike Hinduism, Judaism is monotheistic (Huns Kung, pg. 88). Like Hinduism, Judaism relies on its texts and traditions to provide its central authority. Like the Vedas, the Torah underwent a brief period of

Sarbanes-Oxley Act Essay Example | Topics and Well Written Essays - 500 words

Sarbanes-Oxley Act - Essay Example Examples include WorldCom and Enron companies. According to section 404(a) of the Act, it is a requirement that managements of companies assess their Internal Controls effectiveness and report on the same over financial reporting. Also, Subsection (b) of the same section, calls on independent auditors to attest to the assessments done by the managements. This is regarding Internal Controls effectiveness. The opinion of the study is that it is not too much a regulation. As stipulated by the Securities and Exchange Commissions report, the enactment of section 404 of the Sarbox Act have proven too costly. The outlays are incredibly high to companies, which has led to some attempts at their reduction while upholding effectiveness like the reforms of year 2007. (sec.gov, 2009) However, the cost of implementation is far much less than the 2001-2002 business scandals cost. These are with the inclusion of Global Crossing, Tyco, WorldCom and Enron companies, which shook the confidence of investors a great deal. (Hallberg, 2008 p390) A case at hand to support the argument that, it is not too much regulation is that of the collapsed Enron Company. This is where, in October 2001, Enron company had made public their third quarter earnings report where they purported to have realized an after tax Earnings of USD 1.01 billion. On the same date of reporting, Enron had cut down equity of shareholders by USD 1.2 billion, which it claimed to be the rectification of accounting errors. In November, 2001 the company filed with the SEC Form 8-K, considering the current events. This form 8-K stipulated that Enron would like to restate their financial statements from year 1997 through 2001 June. These statements led to a shocking fall of income of USD 569 million. In the same year, 2001, Enron Company filed for bankruptcy which triggered investigations. (De Vay, 2006 p3) As said earlier, as stipulated by the Securities and Exchange

Identifying Problems When Obtaining Population Parameters

Identifying Problems When Obtaining Population Parameters We estimate population parameters, such as the mean, based on the sample statistics. It is difficult to get a precise value or point estimation of these figures. A more practical and informative approach is to find a range of values in which we expect the population parameters will fall. Such a range of values is called a confidence interval. 1. CONFIDENCE INTERVAL Definition The confidence interval is a range of values constructed from sample data so that the population parameter is likely to occur within that range at a specified probability. The specified probability is called the level of confidence. The shape of the probability distribution of the sample mean allows us to specify an interval of specific probability that the population mean,  µ, will fall into. 1.1 Large Sample Or Standard Deviation Is Known Case 1: The standard deviation à Ã†â€™ is known; or It is a large sample (i.e. at least 30 observations). The Central Limit Theorem states that the sampling distribution of the sample means is approximately normal. We can use the tables in the Appendix to find the appropriate Z value. Key Points The standard normal distribution allows us to draw the following conclusions: 68% of the sample means will be within 1 standard deviations of the population mean,  µ. 95% of the sample means will be within 1.96 standard deviations of the population mean,  µ. 99% of the sample means will lie within 2.58 standard deviations of the population mean. These intervals are called the confidence interval. The standard deviation above (i.e. the standard error) is referring to the standard deviation of the sampling distribution of the sample mean. Locating 0.475 in the body of the table, read the corresponding row and column values, the value is 1.96. Thus, the probability of finding a Z value between 0 and 1.96 is 0.475. Likewise, the probability of being in the interval between -1.96 and 0 is also 0.475. When we combine these two, the probability of being in the interval of -1.96 to 1.96 is therefore 0.95. 1.1.1 How do you compute a 95% confidence interval? Assume our research involves the annual starting salary of business graduates in a local university. The sample mean is $39,000, while the standard deviation of the sample mean is $250. Assume our sample contains more than 30 observations. The 95% confidence interval is between $38,510 and $39,490. Found by $39,000 +/- 1.96($250) In most situations, the population standard deviation is not available, so we estimate it as follows: (Standard Error) Conclusions: 95% confidence interval 99% confidence interval Confidence interval for the population mean (n > 30) Z depends on confidence level Example 1 The Hong Kong Tourist Association wishes to have information on the mean annual income of tour guides. A random sample of 150 tour guides reveals a sample mean of $45,420. The standard deviation of this sample is $2,050. The association would like answers to the following questions: (a) What is the population mean? The best estimate of the unknown population value is the corresponding sample statistic. The sample mean of $45,420 is a point estimate of the unknown population mean. (b) What is a reasonable range of values for population mean? The Association decides to use the 95% level of confidence. To determine the corresponding confidence interval, we use the formula: The endpoints would be $45,169 and $45,671 and they are called confidence limits. We could expect about 95% of these confidence intervals contain the population mean. About 5% of the intervals would not contain the population mean annual income, i.e. the  µ. Figure 2 Probability distribution of population mean 1.2 Small Sample Or Standard Deviation Is Unknown Case 2: The sample is small (i.e. less than 30 observations) or, the population standard deviation is not known. The correct statistical procedure is to replace the standard normal distribution with the t distribution. The t distribution is a continuous distribution with many similarities to the standard normal distribution. 1.2.1 Standard normal distribution versus t distribution Figure 3 Z distribution versus t distribution The t distribution is flatter and more spread out than the standard normal distribution. The standard deviation of the t distribution is larger than the normal distribution. Confidence interval for a sample with unknown population mean, à Ã†â€™. The confidence interval is Assume the sample is from a normal population. Estimate the population standard deviation (à Ã†â€™) with the sample standard deviation (s). Use t distribution rather than the Z distribution. Example 2 A shoe maker wants to investigate the useful life of his products. A sample of 10 pairs of shoes that had been walked for 50,000 km showed a sample mean of 0.32 inch of sole remaining with a standard deviation of 0.09 cm. Constructing a 95% confidence interval for the population mean, would it be reasonable for the manufacturer to conclude that after 50,000 km the population mean amount of sole remaining is 0.3 cm? Assume the population distribution is normal. The sample standard deviation is 0.09 cm. There are only 10 observations and hence, we use t distribution Estimation: = 0.32, s = 0.09, and n = 10. Step 1: Locate t by moving across the row for the level of confidence required (i.e. 95%). Step 2: The column on the left margin is identified as df. This refers to the number of degrees of freedom. The number of degree of freedom is the number of observations in the sample minus the number of samples, written n-1.(i.e. 10-1=9). Step 3: Confidence Interval = The endpoints of the confidence interval are 0.256 and 0.384. Step 4: Interpretation the manufacturer can be reasonably sure (95% confident) that the mean remaining tread depth is between 0.256 and 0.384 cm. Because 0.3 is in this interval, it is possible that the mean of the population is 0.3. 2. CHOOSING AN APPROPRIATE SAMPLE SIZE The necessary sample size depends on three factors: Level of confidence wanted: To increase level of confidence, increase n. Margin of error the researcher will tolerate: To reduce allowable error, increase n. Variability in the population being studied: For a more widely dispersed sample, increase n. We can express the interaction among these three factors and the sample size in the following formula: Sample size for estimating the population mean, Note: n: Sample size Z: Standard normal value S: Estimate of population standard deviation E: Maximum allowable error Example 3 An accounting student wants to know the mean amount that independent directors of small companies earn per month as remuneration for being a director. The error in estimating the mean is to be less than $100 with a 95% level of confidence. The student found a report by the government that estimated the standard deviation to be $1000. What is the required sample size? Maximum allowable error, E, is $100. Value of Z for a 95% level of confidence is 1.96, and the estimate of the standard deviation is $1000. Substitute into , we get n = [ (1.96) (1000) ] 2 = 19.62 = 384.16 100 The sample of 385 is required to meet the requirements. If the students want to increase the level of confidence, e.g. 99%, this requires a larger sample. Z = 2.58, so n = [ (2.58) (1000) ] 2 = 25.82 = 665.64 100 Sample = 666 3. WHAT IS A HYPOTHESIS? Definitions Hypothesis is a statement about a population parameter developed for the purpose of testing. Hypothesis testing is a procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement. In statistical analysis, we always make a claim about the population parameters, i.e. a hypothesis. We collect data and then use the data to test the assertion. 4.1 Five-Step Procedure For Testing A Hypothesis Figure 4 How to test a hypothesis 4.1.1 Step 1: State null hypothesis (H0) and alternative hypothesis (H1) The first step is to state the hypothesis being tested. It is called the null hypothesis. We either reject or fail to reject the null hypothesis. Failing to reject the null hypothesis does not prove that H0 is true. The null hypothesis is a statement that is not rejected unless our sample data provide convincing evidence that it is false. The alternative hypothesis is a statement that is accepted if the sample data provide sufficient evidence that the null hypothesis is false. Example 4 A journal has disclosed that the mean age of commercial helicopters is 15 years. A statistical test of this statement would first need to determine the null and the alternate hypotheses. The null hypothesis represents the current or reported condition. It is written H0:  µ = 15. The alternate hypothesis is that the statement is not true, i.e. H1:  µ à ¢Ã¢â‚¬ °Ã‚   15. 4.1.2 Step 2: Select a level of significance The level of significance is the probability of rejecting the null hypothesis when it is true. A decision is made to use the 5% level, 1% level, 10% level or any other level between 0 and 1. We must decide on the level of significance before formulating a decision rule and collecting sample data. Type I error: Rejecting the null hypothesis, H0, when it is true. Type II error: Accepting the null hypothesis when it is false. Example 5 Suppose AA Watch Ltd has informed bracelet suppliers to bid for contract on the supply of a large amount of bracelets. Suppliers with the lowest bid will be awarded a sizable contract. Suppose the contract specifies that the watch producers quality-assurance department will take samples of the shipment. H0: The shipment of bracelet contains 6% or less substandard bracelets. H1: More than 6% of the boards are defective. A sample of 50 bracelets received August 2 from BB Metals Ltd revealed that four bracelets, or 8%, were substandard. The shipment was rejected because it exceeded the maximum of 6% substandard bracelets. If the shipment was actually substandard, the decision to return the bracelets to the supplier was correct. However, suppose the four substandard bracelets selected in the sample of 50 were the only substandard bracelets in the shipment of 4,000 bracelets. Then only 1/10 of 1% were defective (4/4000 = 0.001). In that case, less than 6% of the entire shipment was substandard and rejecting the shipment was an error. We may have rejected the null hypothesis that the shipment was not substandard when we should have accepted the null hypothesis. By rejecting a true null hypothesis, we committed a Type I error. AA Watch Ltd would commit a Type II error if, unknown to the company an incoming shipment of bracelet from BB Metals Ltd contained 15% substandard bracelets, yet the shipment was accepted. How could this happen? Suppose two out of the 50 bracelets in the sample (4%) tested were substandard, and 48 out of the 50 were good bracelets. As the sample contained less than 6% substandard bracelets, the shipment was accepted but it could be purely by chance that the 48 good bracelets selected in the sample were the only acceptable ones in the entire shipment. In conclusion: Null Hypothesis Accepts H0 Rejects H0 H0 is true Correct decision Type I error H0 is false Type II error Correct decision 4.1.3 Step 3: Select the test statistics There are many test statistics. In this chapter, we use both Z and t as the test statistic. Definition A test statistic is a value, determined from sample information, used to determine whether to reject the null hypothesis. In hypothesis testing for the mean ( µ) when à Ã†â€™ is known or the sample size is large, the test statistic Z is computed by: The Z value is based on the sampling distribution of , which follows the normal distribution when the sample is reasonably large with a mean () equal to  µ, and a standard deviation , which is equal to . We can thus determine whether the difference between and  µ is statistically significant by finding the number of standard deviations is from  µ, using the formula above. 4.1.4 Step 4: Formulate the decision rule Definition A decision rule is a statement of the specific conditions under which the null hypothesis is rejected and the conditions under which it is not rejected. The region or area of rejection defines the location of all those values that are so large or so small that the probability of their occurrence under a true null hypothesis is rather remote. The area where the null hypothesis is not rejected is to the left of 1.65. The area of rejection is to the right of 1.65. A one-tailed test is being applied. The 0.05 level of significance was chosen. The sampling distribution of the statistic Z is normally distributed. The value 1.65 separates the regions where the null hypothesis is rejected and where it is not rejected. The value 1.65 is the critical value. The critical value is the dividing point between the region where the null hypothesis is rejected and the region where it is not rejected. Figure 5 Area of rejection for the null hypothesis 4.1.5 Step 5: Make a decision The final step in hypothesis testing is computing the test statistic, comparing it to the critical value, and making a decision to reject or not to reject the null hypothesis. Based on the information, Z is computed to be 2.34, the null hypothesis is rejected at the 0.05 level of significance. The decision to reject H0 was made because 2.34 lies in the region of rejection, i.e. beyond 1.65. We would reject the null hypothesis, reasoning that it is highly improbable that a computed Z value this large is due to sampling variation. Had the computed value been 1.65 or less, say 0.71, the null hypothesis would not be rejected. It would be reasoned that such a small computed value could be attributed to chance. Example 6 A large car leasing company wants to buy tires that average about 60,000 km of wear under normal usage. The company will, therefore, reject a shipment of tires if tests reveal that the life of the tires is significantly below 60,000 km on the average. The company would be glad to accept a shipment if the mean life is greater than 60,000 km. However, it is more concerned that it will have sample evidence to conclude that the tires will average less than 60,000 km of useful life. Thus, the test is set up to satisfy the concern of the car leasers that the mean life of the tires is less than 60,000 km. The null and alternate hypotheses in this case are written H0:  µ à ¢Ã¢â‚¬ °Ã‚ ¥ 60,000 and H1:  µ In this problem, the rejection region is pointing to the left, and is therefore in the left tail. Summary: If H1 states a direction, we use a one-tailed test. If no direction is specified in the alternate hypothesis, we use a two-tailed test. Figure 6 One-tailed test 5. TESTING FOR POPULATION MEAN WITH KNOWN POPULATION STANDARD DEVIATION 5.1 Two-tailed Test ABC Watch Ltd manufactures luxury watches at several plants in Europe. The weekly output of the Model A33 watch at the Swiss Plant is normally distributed, with a mean of 200 and a standard deviation of 16. Recently, because of market expansion, mechanisation has been introduced and employees laid off. The CEO would like to investigate whether there has been a change in the weekly production of the Model A33 watch. To put it another way, is the mean output at Swiss Plant different from 200 at the 0.01 significant levels? 5.1.1 Step 1: State null hypothesis and alternate hypothesis The null hypothesis is The population mean is 200. H0:  µ = 200. The alternate hypothesis is The mean is different from 200. H1:  µ à ¢Ã¢â‚¬ °Ã‚   200. 5.1.2 Step 2: Select the level of significance The 0.01 level of significance is used. This is ÃŽÂ ±, the probability of committing a Type I error, and it is the probability of rejecting a true null hypothesis. 5.1.3 Step 3: Select the test statistic The test statistic for the mean of a large sample is Z. Figure 7 Normalise the standard deviation 5.1.4 Step 4: Formulate the decision rule The decision rule is formulated by finding the critical values of Z from Appendix D. Since this is a two-tailed test, half of 0.01, or 0.005, is placed in each tail. The area where H0 is not rejected, i.e. area between the two tails, is 0.99. Appendix D is based on half of the area under the curve, or 0.5. Then 0.5 0.005 is 0.495, so 0.495 is the area between 0 and the critical value. The value nearest to 0.495 is 0.4951. Then read the critical value in the row and column corresponding to 0.4951. It is 2.58. Decision rule: Reject H0 if the computed Z value is not between -2.58 and +2.58. Do not reject H0 if Z falls between -2.58 and +2.58. Figure 8 Two-tailed test 5.1.5 Make a decision and interpret the result Compute Z and apply the decision rule to decide whether to reject H0. The mean number of watches produced weekly for last year is 203.5. The standard deviation of the population is 16 watches. Because 1.55 does not fall in the rejection region, H0 is not rejected. We conclude that the population mean is not different from 200. So we would report to the CEO that the sample evidence does not show that the production rate at the Swiss plant has changed from 200 per week. The difference of 3.5 units between the historical weekly production rate and the mean number of watches produced weekly for last year can reasonably be attributed to sampling error. Figure 9 Rejection regions for the two-tailed test So did we prove that production rate is still 200 per week? No! Failing to disprove the hypothesis that the population mean is 200 is not the same thing as proving it to be true. 5.2 P-value In Hypothesis Testing Definition P-value is the probability of observing a sample value as extreme as, or more extreme than, the value observed, given that the null hypothesis is true. How confident are we in rejecting the null hypothesis? This approach reports the probability of getting a value of the test statistic at least as extreme as the value actually obtained. This process compares the probability called the P-value, with the significant level. If the P-value If the P-value > significant level, H0 is not rejected. A very small P-value, such as 0.0001, indicates that there is little likelihood the H0 is true. If a P-value of 0.2033 means that H0 is not rejected, there is little likelihood that it is false. Figure 10 P-value P-value Interpretation Less than 0.1 Some evidence that H0 is not true Less than 0.05 Strong evidence that H0 is not true Less than 0.01 Very strong evidence that H0 is not true Less than 0.001 Extremely strong evidence that H0 is not true The probability of finding a Z value of 1.55 or more is 0.0606, found by 0.5 0.4394. The probability of obtaining an greater than 203.5 if  µ = 200 is 0.0606. To compute the P-value, we need to be concerned with the region less than -1.55 as well as the values greater than 1.55. The two-tailed P-value is 0.1212, found by 2(0.0606). The P-value of 0.1212 is greater than the significance level of 0.01, so H0 is not rejected. Chapter Review The Central Limit Theorem states that the sampling distribution of the sample means is approximately normal. The standard error refers to the standard deviation of the sampling distribution of the sample mean. We use t distribution when the sample is less than 30 observations and the population standard deviation is not known. The necessary sample size depends on 1) level of confidence wanted ; 2) margin of error the researcher will tolerate; 3)variability in the population.   By rejecting a true null hypothesis, we committed a Type I error. We would reject the null hypothesis when it is highly improbable that a computed Z value this large is due to sampling variation. What You Need To Know Confidence interval: A range of values constructed from sample data so that the population parameter is likely to occur within that range at a specified probability. Hypothesis: A statement about a population parameter developed for the purpose of testing. Hypothesis testing: A procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement. Critical value: The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected. P-value: The probability of observing a sample value as extreme as, or more extreme than, the value observed, given that the null hypothesis is true. Work Them Out 1. The average number of days in outdoors assignments per year for salespeople employed by an electronic wholesaler needs to be estimated with a 0.90 degree of confidence. In a small sample, the mean was 150 days and the standard deviation was 14 days. If the population mean is estimated within two days, how many salespeople should be interviewed? A 134 B 152 C 111 D 120 2. A random sample of 85 staff of managerial grade revealed that a person spent an average of 6.5 years on the job before being promoted. The standard deviation of the sample was 1.7 years. Using the 0.95 degree of confidence, what is the confidence interval for the population mean? A 6.19 and 6.99 B 6.15 and 7.15 C 6.14 and 6.86 D 6.19 and 7.19 3. The mean weight of lorries travelling on a particular highway is not known. A state highway authority needs an estimate of the mean. A random sample of 49 lorries was selected and finds the mean is 15.8 tons, with a standard deviation of 3.8 tons. What is the 95 per cent interval for the population mean? A 14.7 and 16.9 B 14.2 and 16.6 C 14.0 and 18.0 D 16.1 and 18.1 4. A bank wants to estimate the mean balances owed by platinum Visa card holders. The population standard deviation is estimated to be $300. If a 98% confidence interval is used and an interval of $75 is desired, how many platinum cardholders should be taken into sample? A 84 B 82 C 62 D 87 5. A sample of 20 is selected from the population. To determine the appropriate critical t-value, what number of degrees of freedom should be used? A 20 B 19 C 23 D 27 6. If the null hypothesis that two means are equal is true, where will 97% of the computed z-values lie between? A  ± 2.58 B  ± 2.38 C  ± 2.17 D  ± 1.68 7. Suppose we are testing the difference between two proportions at the 0.05 level of significance. If the computed z is -1.57, what is our decision? A Reject the null hypothesis B Do not reject the null hypothesis C Review the sample D Own judgment 8. The net weights of a sample of bottles filled by a machine manufactured by Dame, and the net weights of a sample filled by a similar machine manufactured by Putne Inc, are (in grams): Dame: 5, 8, 7, 6, 9 and 7 Putne: 8, 10, 7, 11, 9, 12, 14 and 9 Testing the claim at the 0.05 level that the mean weight of the bottles filled by the Putne machine is greater than the mean weight of the bottles filled by the Dame machine, what is the critical value? A 2.215 B 2.175 C 1.782 D 1.682 9. Which of the following conditions must be met to conduct a test for the difference in two sample means? A Data must be of interval scale B Normal distribution for the two populations C Same variances in the two populations D All the above are correct 10. Take two independent samples from two populations in order to determine if a statistical difference on the mean exists. The number for the first sample and the number in the second sample are 15 and 12 respectively. What is the degree of freedom associated with the critical value? A 24 B 25 C 26 D 27 SHORT QUESTIONS A consumer group would like to estimate the mean monthly water charge for a single family house in June within $5 using a 99% level of confidence. Similar research has found that the standard deviation is estimated to be $25.00. What would be the sample size? The manager of the Kingsway Mall wants to estimate the mean amount spent per shopping visit by customers. A sample of 20 customers reveals the following amounts spent. $48 $42 $46 $51 $23 $41 $54 $37 $52 $48 $50 $46 $61 $61 $49 $61 $51 $52 $58 $43 What is the best estimate of the population mean? Determine a 99 per cent confidence interval. Interpret the result. Would it be reasonable to conclude that the population mean is $50? What about $60? ESSAY QUESTION 1. ABC Film Ltd knows that a certain favourite movie ran an average of 84 days, and the corresponding standard deviation was 10 days. The manager of New Westminster district was interested in comparing the movies popularity in his region with that in all of Canadas other theatres. He randomly selected 70 theatres in his region and found that they showed the movie for an average of 82 days. (a) State appropriate hypotheses for testing whether there was a significant difference in the length of the pictures run between theatres in the New Westminster district and all of Canadas other theatres. (b) Test these hypotheses at a 1% significance level.

Tobacco: The Cost-effectiveness of Current Smoke-free Policies Essay

1. Introduction Tobacco use constitutes a global epidemic that results in 5 million deaths each year (World Health Organization, 2008). If current trends in tobacco use continue, the number of tobacco-related deaths is expected to rise to 8 million deaths annually by 2030 – with 80 percent of these deaths taking place in low and middle-income countries (LMICs) (Mathers & Loncar, 2006). Currently, about 10 percent of the world’s smokers live in India (World Health Organization, 2008). The 2009-2010 Global Adult Tobacco Survey, a nationally representative household survey, found that 34.6% of adults over the age of 15 in India currently use tobacco (International Institute for Population Sciences (IIPS), 2010). The prevalence of tobacco smoking in Gujarat, India, including those using smokeless as well as smoked tobacco is estimated to be 19.8% among males and 1.5% among females(International Institute for Population Sciences (IIPS), 2010). Most smokers in India consume bidis, small cigarettes containing, on average, 25 percent less tobacco than the average cigarette (Jha et al., 2008). Despite the smaller amount of tobacco in bidis, they can produce more nicotine, carbon monoxide, and tar than the average manufactured cigarette because of the way smokers puff on them (Mackay J et al., 2006). One recent nationally representative case-control study found t hat about 70% of smoking-related deaths in India take place during productive years of life between 30-69 years of age (Jha et al., 2008). In addition, the study projected that smoking will kill one million people each year starting in 2010 (Jha et al., 2008). Since 2005, the World Health Organization (WHO)'s Framework Convention on Tobacco Control (FCTC) offers a ... ... or sub-national setting. In the past several years, low and middle-income countries have seen an increased number of smoke-free policies (World Health Organization, 2009). However, some of these policies do not meet the FCTC’s recommendations or are poorly enforced at the sub-national level (World Health Organization, 2009). Therefore, it is important to examine the cost-effectiveness of current smoke-free policies to provide decision makers with the evidence needed to strengthen existing policies to meet FCTC requirements. Additionally, given the exceptions in India’s current smoke-free legislation and the high levels of exposure to secondhand smoke found in recent data, there is a particular need for transparent cost-effectiveness analysis of smoke-free legislation in India(International Institute for Population Sciences (IIPS), 2010; Trostle et al., 1999).