Confidence Interval for the Difference of Two Population Proportions

Confidence intervals are one part of inferential statistics.  The basic idea behind this topic is to estimate the value of an unknown population  parameter by using a statistical sample.  We can not only estimate the value of a parameter, but we can also adapt our methods to estimate the difference between two related parameters.  For example we may want to find the difference in the percentage of the male U.S. voting population who supports a particular piece of legislation compared to the female voting population. We will see how to do this type of calculation by constructing a confidence interval for the difference of two population proportions.  In the process we will examine some of the theory behind this calculation.  We will see some similarities in how we construct a confidence interval for a single population proportion as well as a confidence interval for the difference of two population means. Generalities Before looking at the specific formula that we will use, lets consider the overall framework that this type of confidence interval fits into.  The form of the type of confidence interval that we will look at is given by the following formula: Estimate /- Margin of Error Many confidence intervals are of this type. There are two numbers that we need to calculate.  The first of these values is the estimate for the parameter.  The second value is the margin of error.  This margin of error accounts for the fact that we do have an estimate.  The confidence interval provides us with a range of possible values for our unknown parameter. Conditions We should make sure that all of the conditions are satisfied before doing any calculation. To find a confidence interval for the difference of two population proportions, we need to make sure that the following hold: We have two simple random samples from large populations.  Here large means that the population is at least 20 times larger than the size of the sample. The sample sizes will be denoted by n1 and n2.Our individuals have been chosen independently of one another.There are at least ten successes and ten failures in each of our samples. If the last item in the list is not satisfied, then there may be a way around this.  We can modify the plus-four confidence interval construction and obtain robust results.  As we go forward we assume that all of the above conditions have been met. Samples and Population Proportions Now we are ready to construct our confidence interval.  We start with the estimate for the difference between our population proportions. Both of these population proportions are estimated by a sample proportion.  These sample proportions are statistics that are found by dividing the number of successes in each sample, and then dividing by the respective sample size. The first population proportion is denoted by p1.  If the number of successes in our sample from this population is k1, then we have a sample proportion of k1 / n1. We denote this statistic by  pÌ‚1.  We read this symbol as p1-hat because it looks like the symbol p1 with a hat on top. In a similar way we can calculate a sample proportion from our second population.  The parameter from this population is p2.  If the number of successes in our sample from this population is k2, and our sample proportion is  pÌ‚2 k2 / n2. These two statistics become the first part of our confidence interval. The estimate of p1 is pÌ‚1.  The estimate of p2 is pÌ‚2.  So the estimate for the difference p1 - p2 is pÌ‚1 - pÌ‚2. Sampling Distribution of the Difference of Sample Proportions Next we need to obtain the formula for the margin of error.  To do this we will first consider the   sampling distribution of  pÌ‚1  . This is a binomial distribution with probability of success p1 and  n1 trials. The mean of this distribution is the proportion p1.  The standard deviation of this type of random variable has variance of p1  (1 - p1  )/n1. The sampling distribution of pÌ‚2 is similar to that of pÌ‚1  .  Simply change all of the indices from 1 to 2 and we have a binomial distribution with mean of p2 and variance of p2 (1 - p2 )/n2. We now need a few results from mathematical statistics in order to determine the sampling distribution of pÌ‚1 - pÌ‚2.  The mean of this distribution is p1 - p2.  Due to the fact that the variances add together, we see that the variance of the sampling distribution is p1  (1 - p1  )/n1 p2 (1 - p2 )/n2.  The standard deviation of the distribution is the square root of this formula. There are a couple of adjustments that we need to make.  The first is that the formula for the standard deviation of pÌ‚1 - pÌ‚2 uses the unknown parameters of p1 and p2.  Of course if we really knew these values, then it would not be an interesting statistical problem at all.  We would not need to estimate the difference between p1 and  p2..  Instead we could simply calculate the exact difference. This problem can be fixed by calculating a standard error rather than a standard deviation.  All that we need to do is to replace the population proportions by sample proportions.  Standard errors are calculated from upon statistics instead of parameters. A standard error is useful because it effectively estimates a  standard deviation.  What this means for us is that we no longer need to know the value of the parameters p1 and p2.  .Since these sample proportions are known, the standard error is given by the square root of the following expression: pÌ‚1 (1 -  pÌ‚1 )/n1   pÌ‚2 (1 -  pÌ‚2 )/n2. The second item that we need to address is the particular form of our sampling distribution.  It turns out that we can use a normal distribution to approximate the sampling distribution of  pÌ‚1  - pÌ‚2.  The reason for this is somewhat technical, but is outlined in the next paragraph.   Both  pÌ‚1 and  pÌ‚2   have a sampling distribution that is binomial.  Each of these binomial distributions may be approximated quite well by a normal distribution.  Thus pÌ‚1  - pÌ‚2 is a random variable.  It is formed as a linear combination of two random variables.  Each of these are approximated by a normal distribution.  Therefore the sampling distribution of pÌ‚1  - pÌ‚2 is also normally distributed. Confidence Interval Formula We now have everything we need to assemble our confidence interval.  The estimate is (pÌ‚1 - pÌ‚2) and the margin of error is z* [ pÌ‚1 (1 -  pÌ‚1 )/n1   pÌ‚2 (1 -  pÌ‚2 )/n2.]0.5.  The value that we enter for z* is dictated by the level of confidence C.  Ã‚  Commonly used values for z* are 1.645 for 90% confidence and 1.96 for 95% confidence.  These values for  z* denote the portion of the standard normal distribution where exactly  C percent of the distribution is between -z* and z*.   The following formula gives us a confidence interval for the difference of two population proportions: (pÌ‚1 - pÌ‚2) /- z* [ pÌ‚1 (1 -  pÌ‚1 )/n1   pÌ‚2 (1 -  pÌ‚2 )/n2.]0.5

Chapter Review 1-3 Principles of Supply Chain Management

Chapter Review 1-3 Principles of Supply Chain Management, 3ed Wisner, Leong, Tan 2012 Chapter Review 1-3 Chapter One: Introduction To Supply Chain Management A Supply Chain is the steps necessary for a manufacturer to procure materials, build a product, and transport the product to consumers. The consumers buy the products based on a combination of cost, quality, availability, maintainability and reputation factors. They hope these products will live up to their needs and expectations. An example of a supply chain that I was involved in while serving in the Air Force would be when I worked in the Supply Squadron. One of our customers, the jet engine maintenance shop, would need a part to repair and F101 engine for use on a†¦show more content†¦This is also likely to lower raw material inventory and total assets. E-Procurement brought many benefits to companies. It has proven to be a time saver and cost effective. It is more accurate as the information is only entered once instead of twice. Before the users had to enter the information and the buyer had to reenter the inf ormation creating another error point in the process. E-procurement is more flexible as it can be used without the restrictions of location or time of day. Status of orders can be looked at without having to check paper trails. There are different reasons why a firm may use a single supplier instead of favoring multiple suppliers. Using a single supplier gives a firm the chance to build a stronger relationship with the supplier. Costs would be lower due to larger purchases keeping the cost per unit down and transportation would be cheaper as the firm can take advantage of full truckloads. Single sourcing would also make sense if the firm’s requirements are too small. It would not be worthwhile to split the order among multiple suppliers. There is a disadvantage with sole sourcing as well. If the purchase was for a proprietary product or process and the supplier holds the patents to them, the firm has no choice but to buy from them. Multiple sourcing is advantageous fo r many reasons. 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Retrofitting and Rehabilitation of Civil Engineering Structures free essay sample

We obviously knew why should it be a non destructive test, because we aren’t allowed to destruct the entire structure in the name of testing. Roadware 10 Minute Concrete Menderâ„ ¢ Roadware needle tip crack injection costs less to install and is more effective than pressure applied epoxy injection methods.   according to top concrete repair contractors. We can get in, permanently repair the cracks, and get out in the same time it use to take just to install the old injection ports. Evaluation and Repair of Fire-Damaged Buildings †¢ Typically, repair materials are similar to the original construction materials. Timber structures may be repaired with new timbers or composites of steel and timber members, and steel structures are normally repaired with steel. †¢ Concrete and masonry structural elements are frequently repaired with fiber reinforced polymers (FRP) or externally bonded steel members using epoxy adhesive. Concrete structures are occasionally repaired with shotcrete as well. †¢ Selecting the appropriate repair material is a critical step in the repair process. We will write a custom essay sample on Retrofitting and Rehabilitation of Civil Engineering Structures or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page The repair material must be compatible with the base material, project needs, technical resources available, financial constraints, and multiple other project specific criteria. Two case studies are presented to illustrate various types of structural damage and repairs. †¢ The first summarizes the assessment and repair for arson damage to a reinforced concrete stadium structure. †¢ The second addresses the evaluation and repair of damage to a steel framed office building. Slab separation and joist spalling Case Study 1: Stadium Fire An arson fire occurred on an elevated level of a reinforced concrete stadium structure. After the fire department approved the area for entry, initial observations were made. The fire severely damaged the concrete in an area approximately 25 feet by 25 feet above the fire; the concrete joists were deeply spalled, the slab separated from the joists, and severe cracking was present The fire was intense to the extent that the seating area directly above the fire suffered heat damage Case Study 2: Occupied Steel High Rise Fire Fire occurred in an occupied space of a steel framed high rise. The steel framing was protected by a sprayed fire resistive material. The exposed metal deck supports a composite concrete floor system.. Heat from the fire caused some buckling of the metal decks near the fire Buckled metal deck. Note separation of concrete from metal deck through exploratory opening The evaluation effort included steel hardness readings of the structural steel framing members. In addition, concrete cores were extracted from the floor slab where the fire occurred and in the deck directly above the fire for compression testing. No testing was considered necessary in the adjacent room with deformed plastic due to the low temperature indication. Test results confirmed that steel hardness was in the expected range to indicate minimal heat damage to the steel frame. Concrete cores indicated that compressive strengths exceeded design requirements The assessment concluded that the steel frame did not require repairs. However, to prevent displacements of the separated concrete slab above the metal deck, structural repairs were required . The repairs included adding new beams to support the metal deck and the addition of grout to fill the voids between the concrete and metal deck FINALLY †¢ All structures should be evaluated in a systematic manner to determine the extent, if any, of required repairs. A variety of testing methods and tools are available to evaluate the effects on both the materials and structural elements. †¢ Evaluations, combined with an engineering analysis, allow effective and economical repair details to be developed and installed as needed.

A Gmat Essay for Your Next Class - How to Write an Excellent Gmat Essay For This Topic

A Gmat Essay for Your Next Class - How to Write an Excellent Gmat Essay For This TopicGmat essay writing involves all aspects of the topic. It is all about 'content' not just about teaching the content. I will explain what I mean in the following article.The content of your Gmat essay is the meat and potatoes of it. This is where you put your essay and your topic into the larger context of the subject. If you do not prepare well, then you have a large chance of losing the readers interest. You may also find it hard to understand and comprehend everything that you have to write down, especially if you are brand new to this field.Your topics for your Gmat essay should be the most important. But do not make it as the last topic. This would mean that you have come up with a truly interesting topic, but the reader is now reluctant to read on to the meat and potatoes of your essay.The next step would be to build up the importance of your topic from the very first sentence. The next paragra ph must mention the need for the topic, how it relates to your topic and how this topic can help solve the problems your students might be facing.When you get ready to write the next paragraph, start with the conclusion of your topic. This is one of the most difficult parts of the entire essay, especially if you have come up with a really good topic. Now is the time to run through your thesis statement and convince the reader about why they should read further. Go through each and every paragraph of your dissertation outline and pick out the important parts.Write them down and do not let yourself become forgetful. With each sentence, try to make it more coherent and informative, so that you can have a strong argument for your readers to identify with.When you finish with your Gmat essay, make sure that you leave it in such a way that the reader would come back to it time and again. In fact, you would probably have a second or third draft to add some extra information and one more de tail.

The Pitfall of Argumentative Essay Topics Essay Topics Against Smoking

The Pitfall of Argumentative Essay Topics Essay Topics Against Smoking The Tried and True Method for Argumentative Essay Topics Essay Topics Against Smoking in Step by Step Detail Searching endlessly and visiting dozens of sites which only promise a wide variety of fresh and great ideas, but, actually, have the exact same topics. Our qualified academic experts will aid you in finding a great persuasive essay topic according to all of your requirements and compose a custom-tailored model answer. You can select a topic that apparently appears unique and fascinating, but when it has to do with finding relevant details on it there is none. The issue, nevertheless, is that not all businesses can be relied upon in order to deliver quality essays on time so you need to be cautious in selecting one. An essay generator is an alternative for those students that have problems to seek out something to speak about on their essays, they don't have the creativity neither the patient to select a topic type instead of even a subject place. To choose which subject you're likely to discuss, it's essential to see the complete collection of good persuasive speech topics from the specific area of study. Still, figuring out the ideal topic for your essay isn't your only concern for a student. A student ought to keep in contact with the hottest trends and know which persuasive essay topics are related to sound convincing in regards to defending personal opinions. Use transition words to earn your text coherent and simple to read. To make your life simpler and give you a topic that has all of the characteristics to qualify as a fantastic topic, we developed the idea to make an essay topics generator. For example, let's say you decide on the very first topic from our list. The Hidden Gem of Argumentative Essay Topics Essay Topics Against Smoking So, the best method to compose a fantastic persuasive essay is to discover a theme you're familiarized with and wish to share your experience with the reader. To get started writing with fluency about something it's essential that you feel attracted to that subject area, it is simpler to discuss something that you like than something about you don't feel interested in. Deciding upon a persuasive essay topic may take a great deal of time without ending up with the terrific solution. In the example of deficiency of appropriate proof, your words won't be believed. The upcoming few paragraphs will compose most of your essay. Always bear in mind an ideal persuasive essay needs to be persuasive. Qualities of an excellent persuasive essay topic The topic ought to be specific. Possessing no notion of the persuasive essay topics, you just do not understand what to write about. ARandom topic generatoris made to help you locate the most fascinating and appropriate topic for your essay. There are several intriguing topics that could be become a persuasive essay if you take the opportunity to consider about doing it. The Characteristics of Argumentative Essay Topics Essay Topics Against Smoking Be attentive to the size it must be and begin writing the narrative essay outline. As any guide on the best way to compose a persuasive essay will inform you, your essay has to be organized in paragraphs with a logical progression from 1 paragraph to the next. In general, you can observe that writing a persuasive essay isn't a brain surgery. Just adhere to the guidelines stated above, and you'll be well on your way to writing a great persuasive essay. Argumentative Essay Topics Ess ay Topics Against Smoking - Dead or Alive? The problem is that cigarette smoking is a mass phenomenon around the world, and it has raised the level of epidemics as a result of range of smokers. The issue of smoking has an important part in modern society. It is a huge phenomenon all around the world, so you will find plenty of studies from authoritative sources. It should be banned for everyone. Things You Won't Like About Argumentative Essay Topics Essay Topics Against Smoking and Things You Will On the opposite side, acquiring a list of good persuasive essay topics is insufficient. Persuasive essays are a really good approach to encourage the reader to check at a particular topic in a different light. Argumentative essay topics are so important since they are debatableand it's critical to at all times be critically considering the world around us. If you're not so knowledgeable about the subject of the assigned essay it's going to be even harder to produce valuable thought s and thought provoking concepts. 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Topic Ideas For College Essays - How to Select Topic Ideas For Your Essay

Topic Ideas For College Essays - How to Select Topic Ideas For Your EssayIf you want to write an argumentative essay that will stand out from the crowd, then there are some topic ideas for college English courses that you need to keep in mind. By having a focus on some specific points of view, you will be better able to formulate your argument.You should always start by writing about a point of view that has a strong opinion on the subject matter at hand. For example, if you are writing about immigration, your point of view could be 'immigration is good for the country'. This is a general argumentative statement that holds the views of all members of the debate. The premise is the basis of the argument, and the foundation is the structure of the essay.Your next idea can also be a general one. You could begin your essay with an introduction, for example, by stating, 'Immigration is an idea worth exploring'. Your introduction should encourage readers to read on and give them a sense of what you will be talking about in your argument. Your point should also be related to the topic that you have already decided upon. This way, they will understand what they are getting into.Your third idea should involve a special form of argument, so you need to ensure that you choose it carefully. For example, if you are going to talk about how you would react if you had a certain disease, your topic should relate to a public policy debate about the appropriate treatment for a particular disease. Usually this would involve something very sensitive, so you need to make sure that you avoid any sensitive content.Your next topic should include an element of surprise. You could give a hypothetical answer to a question or try to throw your reader off guard. In order to do this successfully, you must know what questions you will be addressing. Try to learn as much as you can about the topic before you begin. By doing this, you will be better able to come up with interesting and believab le answers.To make sure that you don't fall into the trap of giving out answers that have only a weak and weakly held argument, you need to be clear about what you intend to say. Find a topic for which you have a strong background, and that you can really back up with evidence. Then you need to work hard to build your argument. Then you can begin your essay with a strong argument and a great foundation.Topic ideas for college essays can include anything from religion to the environment. The same general format applies, but the topics themselves can vary. You just need to be clear about what you want to talk about, and decide whether or not you can really build a strong argument about the topic.It can be challenging to build a strong argument. Many students get bogged down by facts or over think their arguments. You need to remember that the topics you select should have a strong foundation to support them, and you should then be able to build a convincing argument that people will r eally appreciate.

Chemistry Extended Essay Topics Ib - the Conspiracy

Chemistry Extended Essay Topics Ib - the Conspiracy Nevertheless, in chemistry extended essays it's perfectly reasonable to formulate the research question for a statement or as a hypothesis in place of a genuine question. Except for selecting a topic, researching, and pinning down the major question, it's crucial to choose the structure as a protracted essay must be long. Be that as it could, the topic for your essay is a critical thing that has to be chosen carefully and with higher precision. Chemistry papers can be challenging, not on account of the topic, but due to deciding on the topic. You should remember an elongated essay format demands an article of 4000 words. You may believe that it is only the cover of your essay, but it's an important component, and you need to treat it correctly. This kind of essay isn't the easiest one to cope with. You won't be able to crank out a 4,000-word essay in a week and receive an A. All essays have to be supervised by means of a school supervisor. Nowadays it's very difficult to locate a trustworthy essay writing service. A protracted essay is one which you won't finish overnight. You will construct your whole extended essay around it. The Do's and Don'ts of Chemistry Extended Essay Topics Ib Relevant chemical formulas (like structural formulas), balanced equations (such as state symbols) and mechanisms ought to be included. Another concept is to write about is how environment impacts the body with assorted diseases. Additionally, by definition, some topics aren't proper for an elongated essay in physics, which is an experimental science with a certain strategy and techniques. In such situations, the element of private analysis and evaluation is very important. 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