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Submitted By gangan

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Words 303

Pages 2

Consider a population of size N with the following elements: .

Now suppose we perform a random sampling of size n of the population. The total number of unique samples that can be drawn is, where

To simplify the notation, we’ll letbe equal to.

Now sincerepresents the total number of unique samples that can be drawn, we know that there will bedifferent sample means, .

Therefore the mean of the sample mean will simply be

We don’t know the exact shape of eachbut we know that it will contain n random elements from the population

Here’s the ABSOLUTELY CRUCIAL PART of the proof. We know for a fact that the a’s from the population will appear on and off in each, the important thing is to ask how many times will each a appear? This result requires a proof even beyond my own capability, and I’m simply going to quote the result from my own observation:

“If the population size is N and the sample size is n, each element in the population will appeartimes in the sampling distribution.”

But we know that

Again we will adopt a simpler notation temporarily. Let

Reminder: represents the number of times each element in the population will appear in the sampling distribution.

We know that

where

Here’s the clincher. Since each element in the population will appeartimes, I can perform the following tour de force

Which simplifies to

Before I restore the original expressions forand, I need to perform a sleight of hand maneuver for

Convince yourself that the above is consistent with the factorial notation. The reason for me doing this will become very apparent soon.

With that done,now becomes

You should realise that many terms will cancel off, leaving us with just

But this is just the expression for, the population mean.…...

...LAW 2 MEAN? The line connecting the Sun to a planet sweeps equal areas in equal times. PROOF OF LAW 2 CONTINUED… Insert (4) into (14) (17) (17) is simplified when t=0 (18) PROOF OF LAW 2 CONTINUED… C is always constant so (17)=(18) (19) Differential area equation for polar coordinates Differential area equation is half of the constant function. (20) PROOF OF LAW 2 CONTINUED… LAW 2 is PROVED!!!! KEPLER’S LAWS OF PLANETARY MOTION 1. Planets move around the Sun in ellipses, with the Sun at one focus. 2. The line connecting the Sun to a planet sweeps equal areas in equal times. 3. The square of the orbital period of a planet is proportional to the cube of the semimajor axis of the ellipse. TERMINOLOGY OF LAW 3 Semi-Major Axis a Semi-Minor Axis b Major Axis Minor Axis a=semi-major axis b=semi-minor axis T=period WHAT DOES LAW 3 MEAN? The square of the orbital period of a planet is proportional to the cube of the semimajor axis of the ellipse. PROOF OF LAW 3 There is a concept from the proof of Law 1 that is needed…Eccentricity Two formulas (31) and (32) (32) (31) PROOF OF LAW 3 CONTINUED… Kepler’s law says (33) Formula 1 for area Formula 2 uses (20) PROOF OF LAW 3 CONTINUED… Combining Formula 1 and 2 and solving for T (34) rmax is equal to (31) when θ=π PROOF OF LAW 3 CONTINUED… a is found using the maximum position vector (35) PROOF OF......

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...Best Answer Answer Quite simply, criminal law deals with violations of statutory crime and its punishment, civil law deals with non-criminal things such as contracts/business, and things such as divorce, etc. You can be charged under both criminal and civil proceedings though. A perfect example is OJ Simpson being aquitted of criminal charges but being found guilty in civil proceedings. Probably the biggest difference between the two is what is called "BURDEN OF PROOF" In criminal proceedings the burden is "BEYOND ALL REASONABLE DOUBT" for someone to be convicted. This means that the evidence points substantially towards guilt and there is little to no question of guilt. For a jury in this case you would need everyone in agreement or the case is a mistrial. Unaminity is the key. In civil proceedings all that is required is "preponderance of evidence" which means that if it is found that it was more likely than not that an incident occured then you can be convicted. This kind of proceeding is really a "who has the best evidence" type of case. Only a majority of jury members decision is needed in this case. ANSWER 2 1. Criminal Law cases must prove the person's guilt beyond reasonable doubt, whilst civil cases must be proved on the balance of probability 2. The main difference however is who the law is aimed at. Theoretically criminal law is a considered to be harming all members of society. Consequently, the victim of the criminal case is all of......

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...The Proof of Miracles Miracle is such a strong word in the world, but what classifies a miracle? Do miracles even exist? According to the Merriam-Webster dictionary, a miracle is defined as an extraordinary event manifesting divine intervention in human affairs. Through my personal experience, I can say that miracles do exist. I went from having fun, to going blind in one eye, to having sight in both eyes better than 20/20 within less than two months. My story starts off with me and my three brothers playing paintball, a sport with two teams where the object of the game is to shoot the other team to win. We played about three games of paintball and the last game had just ended. We all took off our masks and my oldest brother was out of bullets, so we thought. He started to shoot the ground and all that came out of his gun was CO2 (a chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom, which makes a gas at standard temperature and pressure). After he shot the ground, my oldest brother started shooting at my feet and slowly moved the gun up my body until he got to my face. Then out of nowhere, two paintballs came out of the gun and hit me in the mouth and the eye. Everything changed from joking around to sheer panic. Everyone dropped their gear, and my oldest brother picked me up in his arms, and rushed me into the house. The paintballs that hit me were filled with red paint, so it was hard to tell what was blood and what was paint. My......

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...Title: Proof of God’s Existence By: Vicki Williams PHI208 Ethics and Moral Reasoning Instructor: Barbara Palomino de Velasco June 10, 2013 The Bible diagnoses humanity's deepest need and then prescribes what it claims is the only antidote: God saves sinners. Is there an intellectual basis for believing in God? Or is being religious just a matter of faith? Christianity invites scrutiny into its claims and its reasons for having faith and these deserve examination... Some people feel that acceptance of God is entirely a matter of faith. But the Scriptures actually claim that it is the truth which is the basis for this faith (Rom. 10:17). Truth is only truth if it is objective truth, that is, it is true for everyone regardless of time or circumstances. Thus, God is either true (and there can be objective proofs to support this), or He is not true and only subjective 'truth' can be offered for 'proof'. Some ancient Philosophers have argued that the fact mankind can imagine there being a God is a proof that there must be one. While this argument does have some merit, it mainly has limitations. Taken to its natural application this means that if anything can be imagined it must exist. Based on the nightmares I had as a three and four year old boy- I really hope this theory isn't true or I'm not going to sleep well tonight! On the other hand, 'beliefs' and 'truth' are sometimes not entirely related. Tertullian once bragged that the main reason he so readily......

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...LOGICAL INFERENCE & PROOFs Debdeep Mukhopadhyay Dept of CSE, IIT Madras Defn • A theorem is a mathematical assertion which can be shown to be true. A proof is an argument which establishes the truth of a theorem. Nature & Importance of Proofs • In mathematics, a proof is: – a correct (well-reasoned, logically valid) and complete (clear, detailed) argument that rigorously & undeniably establishes the truth of a mathematical statement. • Why must the argument be correct & complete? – Correctness prevents us from fooling ourselves. – Completeness allows anyone to verify the result. • In this course (& throughout mathematics), a very high standard for correctness and completeness of proofs is demanded!! Overview • Methods of mathematical argument (i.e., proof methods) can be formalized in terms of rules of logical inference. • Mathematical proofs can themselves be represented formally as discrete structures. • We will review both correct & fallacious inference rules, & several proof methods. Applications of Proofs • An exercise in clear communication of logical arguments in any area of study. • The fundamental activity of mathematics is the discovery and elucidation, through proofs, of interesting new theorems. • Theorem-proving has applications in program verification, computer security, automated reasoning systems, etc. • Proving a theorem allows us to rely upon on its correctness even in the most critical scenarios. Proof Terminology •......

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...Simple Random Sample A simple random sample X 1 , , X n , taken from a population represented by a random variable X with mean and standard deviation , has the following characteristics. Each X i , i 1,, n , is a random variable that has the same distribution as X, and thus the same mean and standard deviation . The X i ’s are independent random variables implying the following identity: V X 1 X n V X 1 V X n 2 2 n 2 The sample mean, X where X X1 X n , is a continuous random variable n with mean X and standard deviation X of the mean. Sampling Distribution of the Mean , known as the standard error n Result 1: If X is Normal, then X is Normal and Z of the sample n. X n , regardless of the size Result 2: If n is large enough, n 30 , then X is approximately normal and X Z , regardless of the distribution of the population random variable X. n Sampling Distribution of the Proportion For a binomial random variable X with parameters n (sample size) and (probability of X success or population proportion), the sample proportion is given by p . n The sample proportion p is a continuous random variable with mean p and standard deviation p 1 n . Result 3: If n is large enough, n 5 and n1 5 , then p is approximately p normal and Z . 1 n Why? We want to use the sample mean X to......

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...provided with an understanding of the parts of the envelope, common types of envelopes, the materials that make up an envelope and how they are printed. We will then give you some helpful information on working with an envelope manufacturer and conclude with a basic postal guide. So, welcome to the school of the envelope, a journey through the life of that silent messenger you use every business day. We hope you enjoy your journey. 2 I. A Brief History of the Envelope The envelope is an essential form of communication that has been used throughout history. Biblical references to messages being sent in closed containers or under seal can be found in various books of the Bible. The modern envelope was created in 17 th century Europe as a means to ensure that communications between merchants and nobles were kept confidential. Postage at that time was so costly that only the wealthiest could afford to use an envelope. An Act of the British Parliament of 1840 created the model for most modern postal systems. That act also created penny postage and the first prepaid “postal wrapper” which was an envelope. These early Mulready Envelopes were used as messages themselves, but within months, citizens were sending a message in the envelope and using the prepaid wrapper as an envelope. Envelopes in America were first produced in the mid 1840s, mostly by hand. An envelope blank was cut out using a template and a shoemaker’s knife. The cut blank was then gummed, except for the seal......

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...precedents * If b is the result of a and c is equal to a then b is the result of c as well * Proof of acts * How do we prove things * Presumptions in law are set in hundreds of years of legal precedents * The party that has the burden of overcoming the presumption of that law has the burden of proof * At common law over the course of 300 years, three burdens of proof evolve. * The Default Position: * When in doubt, this is the burden of proof * Preponderance of the evidence * More likely than not * Whoever has the burden of proof, has the obligation to present anything over 50% * Ex: * Two people suing each other in a car accident * Person files compliant, other files counter, both have the burden * Have to prove the other ran the red light * Joint whiteness says that the traffic signal was working so someone did run the light * Person A has 17 witnesses that says B ran the light * B has 3 witnesses * From a quantitative basis, there is no question here * A has proven that he has the greater weight * B has someone that doesn’t know either party * Evidence is the default burden of proof * If a jury decides the evidence is even, whoever has the burden of proof loses. * Common law begins to realize that in some situations for a variety of reasons, there is a......

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...What is a proof? Yes- there is a big leap when we prove something from everyday axioms. But do they not assume the title of axioms once proved? If they were axioms, what is the need to prove it? Does it not reflect the inability of a person to actually see the ‘obvious’ when he is searching for proofs? Let me be clear that I enjoy a well written and elegant proof and am immensely discontent when I am unable to prove a seemingly obvious statement. But is not the fact that my intuition suggests that the statement has to be true reason enough? You may argue that a proof leaves nothing to chance and gives you a guarantee that the statement has to be true. But when you dissect a proof right down to its base axiom, there you will have to rely on an assumption- that our intuition deems valid. I am not here to argue on the basis of the correctness in our underlying assumption of the base axiom. Proofs are merely a way to deduce results from a given premise. And the premise here is the truth of the axiom. I am uncomfortable about the fact that people are willing to accept the intuition behind the base axiom but not the intuition behind the results that follow. True, in most cases it is easier to be aware of the former- and the latter may be hard to see as obvious. A case in point would be Fermat’s last theorem. Andrew Wiles did come up with an absolutely marvellous proof of the theorem- something that puzzled the greatest minds for three and a half centuries. But in those three......

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...an appropriate letter to the applicant. The applicant attends an interview when required and a rejection or acceptance decision is sent by the tutor to the admissions department who write an appropriate letter to the applicant. Applicants applying for HE courses apply via UCAS who send the application to the college on behalf of the student. All applicants who have been offered a place must enrol on the course before they are accepted as students in the college. Applicants turning up for enrolment have to fill in an enrolment form which must be signed by the course tutor. The enrolment form is then taken to registry who process the student. The applicant’s details are entered onto the registry system. The student must provide some proof of how payment is to be made for the course. Registry can take a deposit for applicants paying privately. Once the applicants financial situation is ascertained a photograph of the applicant is taken and a college ID card is issued. Students details are also sent to the library to synchronise the library database so student can start borrowing books. Student information is also sent to the finance department who collect monies from either the local council or the students. Once students are enrolled at the college they are sent a letter to inform them when their induction will take place. They are also sent a timetable....

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...subjective or emotional value. Here the conclusion is derived as there is a clear link between emotional reactivity of investors and trading performance. 3. Do you think that the results of Sam’s study are generalizable to other organizations? Why (not)? This study is lack in generalizability. For the purpose of this study Sam has chosen 50 traders form a training programme. But the question is whether the 50 samples represent the real situation. Therefore, if this research is to be generalized to some other organization or industry then the samples should contain the representatives those of industries. 4. Sam explains to Akrisious that even though he has tried to meet all the hallmarks of scientific investigation, he has not included all the variables that influence the financial performance of traders. Using vocabulary Akrisious has never heard of (such as omitted variable bias and biased and inconsistent parameter estimates) Sam argues that his study may lack rigor and that therefore his study is probably not 100% scientific. a. What is rigor? Means carefulness, scrupulousness and the degree of exactitude in research investigation. A good theoretical base and around methodological design would add rigor to a purposive study. b. Do you think that despite the apparent lack of rigor, Sam’s efforts have still been worthwhile? Yes, Rigor is an important characteristics for a successful research. But it not that the only factor which determine the......

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...Figure the means and standard deviations for the each group: governors and CEOs. b. Explain, to a person who has never had a course in statistics, what you have done. c. Note the ways in which the means and standard deviations differ, and speculate on the possible meaning of these differences, presuming they are representative of U.S. governors and large corporations’ CEOs in general. 7. Radel and colleagues (2011) conducted a study of how feeling overly controlled makes you desire—even unconsciously—more freedom. In their study, 52 Canadian undergraduates played a video game in a laboratory and were randomly assigned to one1 of the following: • An automony deprivation condition, in which they were told to follow instructions precisely, constantly given instructions over a loudspeaker, and carefully observed on everything they did. • A neutral condition, which was much more laid back. After this activity, they were asked to do a lexical decision task (a standard approach for measuring unconscious responses), in which they were shown a series of words and nonwords in random order and had to press C if it was a real word or N if not. Half of the real words were related to autonomy (such as freedom and choice), and half were neutral (such as whisper and hammer). The key focus of the study was on how long it took people to press the button (response latency) for each kind of real word, averaged over the many words of each type. The table below shows the mean and......

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...penalties associated with various quality parameters and impurity levels of the coal. In many cases, predicting and controlling the quality of the coal accurately can mean the difference between economic success and failure for a coalmine. The heating value is a quality of particular interest. It is the basis of payment in most coal supply contracts. In the United States, the heating value of coal is measured in British thermal units per pound (BTU/lb), by using a national standard test procedure (ASTM D-2015). ASTM standards also indicate that repeated measurements should give values within 100 BTU/lb. Normally, the only way to determine heating value is to send a carefully prepared sample off to a laboratory and wait for the results. This can lead to disputes, if the customer and the supplier don’t agree on which lab to use or if the results from different laboratories give drastically different results. In an effort to determine the extent of any intra- or inter-laboratory variation in heating value measurements, one coal company prepared 50 samples of coal from a common area (so that results were expected to be very close). They shipped the samples in two batches to five reputable laboratories. Each batch consisted of five samples sent to each lab; the two batches were sent at different times. The samples were drawn randomly, packed in water (to minimize oxidation effects) and sent in sealed plastic containers. The labs were not aware that an experiment was being run,......

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...Law and Health Care- Proof of Negligence Vickie Young Dr. L. Forbes Strayer University January 22, 2012 Identify and explain the four elements of proof necessary for a plaintiff to prove a negligence case. Negligence is defined as “conduct that falls below the standards of behavior established by law for the protection of others against unreasonable risk of harm. A person has acted negligently if he or she has departed from the conduct of a reasonably prudent person acting under similar circumstances” (Legal Dictionary,n.a.). This conduct is also referred to as “duty of care” (Showalter, 2008, pg.47, para.1). Though negligence is one of the most common types of lawsuits relative to the medical industry, it requires significant proof. The four elements that must be present to prove negligence are as follows: duty of care, breach of said duty, injury, and causation (Showalter, 2008, pg.47, para.1). Duty of care has been defined previously as the reasonable and expected behavior of one in a particular situation. It is somewhat of a given standard that is simply understood. If there is a breach of this expected behavior and there is injury to person or property, and the breach is shown to have caused the injury, then the person or persons, who did not perform or behave as expected, may be liable for the injuries it caused. The challenge for proving negligence in a medical arena is complex since the jury may not have the professional experience to......

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...Were the groups in this study independent or dependent? Provide a rationale for your answer. 2. t = −3.15 describes the difference between women and men for what variable in this study? Is this value significant? Provide a rationale for your answer. 3. Is t = −1.99 significant? Provide a rationale for your answer. Discuss the meaning of this result in this study. 4. Examine the t ratios in Table VI. Which t ratio indicates the largest difference between the males and females post MI in this study? Is this t ratio significant? Provide a rationale for your answer. 5. Consider t = −2.50 and t = −2.54. Which t ratio has the smaller p value? Provide a rationale for your answer. What does this result mean? 6. What is a Type I error? Is there a risk of a Type I error in this study? Provide a rationale for your answer. 7. Should a Bonferroni procedure be conducted in this study? Provide a rationale for your answer. 8. If researchers conducted 9 t-tests on their study data. What alpha level should be used to determine significant differences between the two groups in the study? Provide your calculations. 9. The authors reported multiple df values in Table VI. Why were different df values reported for this study? 10. What does the t value for the Physical Component Score tell you about men and women post MI? If this result was consistent with previous research, how might you use this knowledge in your practice?...

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