Hypothesis Test for Comparing Two Proportions

In this article we will go through the steps necessary to perform a hypothesis test, or test of significance, for the difference of two population proportions.  This allows us to compare two unknown proportions and infer if they are not equal to each other or if one is greater than another. Hypothesis Test Overview and Background Before we go into the specifics of our hypothesis test, we will look at the framework of hypothesis tests.  In a test of significance we attempt to show that a statement concerning the value of a population  parameter (or sometimes the nature of the population itself) is likely to be true.   We amass evidence for this statement by conducting a statistical sample.  We calculate a statistic from this sample.  The value of this statistic is what we use to determine the truth of the original statement.  This process contains uncertainty, however we are able to quantify this uncertainty The overall process for a hypothesis test is given by the list below: Make sure that the conditions that are necessary for our test are satisfied.Clearly state the null and alternative hypotheses. The alternative hypothesis may involve a one-sided or a two-sided test.  We should also determine the level of significance, which will be denoted by the Greek letter alpha.Calculate the test statistic. The type of statistic that we use depends upon the particular test that we are conducting.  The calculation relies upon our statistical sample.  Calculate the p-value. The test statistic can be translated into a p-value.  A p-value is the probability of chance alone producing the value of our test statistic under the assumption that the null hypothesis is true. The overall rule is that the smaller the p-value, the greater the evidence against the null hypothesis.Draw a conclusion. Finally we use the value of alpha that was already selected as a threshold value.  The decision rule is that If the p-value is less than or equal to alpha, then we reject t he null hypothesis. Otherwise we fail to reject the null hypothesis. Now that we have seen the framework for a hypothesis test, we will see the specifics for a hypothesis test for the difference of two population proportions.   The Conditions A hypothesis test for the difference of two population proportions requires that the following conditions are met:   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.The individuals in our samples have been chosen independently of one another.  The populations themselves must also be independent.There are at least 10 successes and 10 failures in both of our samples. As long as these conditions have been satisfied, we can continue with our hypothesis test. The Null and Alternative Hypotheses Now we need to consider the hypotheses for our test of significance.  The null hypothesis is our statement of no effect.  In this particular type of hypothesis test our null hypothesis is that there is no difference between the two population proportions.  We can write this as H0: p1 p2. The alternative hypothesis is one of three possibilities, depending upon the specifics of what we are testing for:   Ha:  p1 is greater than p2. This is a one-tailed or one-sided test.Ha: p1 is less than p2. This is also one-sided test.Ha: p1 is not equal to p2. This is a two-tailed or two-sided test. As always, in order to be cautious, we should use the two-sided alternative hypothesis if we do not have a direction in mind before we obtain our sample.  The reason for doing this is that it is harder to reject the null hypothesis with a two-sided test. The three hypotheses can be rewritten by stating how p1 - p2 is related to the value zero.  To be more specific, the null hypothesis would become H0:p1 - p2 0.  The potential alternative hypotheses would be written as: Ha:  p1 - p2   0 is equivalent to the statement p1 is greater than p2.Ha:  p1 - p2  Ã‚  0 is equivalent to the statement p1 is less than p2.Ha:  p1 - p2  Ã‚  Ã¢â€°   0 is equivalent to  the statement p1 is not equal to p2. This equivalent formulation actually shows us a little bit more of what is happening behind the scenes.  What we are doing in this hypothesis test is turning the two parameters p1 and p2  into the single parameter p1 - p2.  We then test this new parameter against the value zero.   The Test Statistic The formula for the test statistic is given in the image above.  Ã‚  An explanation of each of the terms follows: The sample from the first population has size n1.  The number of successes from this sample (which is not directly seen in the formula above) is k1. The sample from the second population has size n2.  The number of successes from this sample is k2.The sample proportions are  p1-hat k1 / n1  and p2-hat   k2 / n2 .We then combine or pool the successes from both of these samples and obtain:  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚   p-hat ( k1 k2) / ( n1 n2). As always, be careful with order of operations when calculating.  Everything underneath the radical must be calculated before taking  the square root. The P-Value The next step is to calculate the p-value that corresponds to our test statistic. We use a standard normal distribution for our statistic and consult a table of values or use statistical software.   The details of our p-value calculation depend upon the alternative hypothesis we are using: For Ha: p1 - p2   0, we calculate the proportion of the normal distribution that is greater than Z.For Ha: p1 - p2  Ã‚  0, we calculate the proportion of the normal distribution that is less than Z.For Ha: p1 - p2  Ã‚  Ã¢â€°   0, we calculate the proportion of the normal distribution that is greater than |Z|, the absolute value of Z.  After this, to account for the fact that we have a two-tailed test, we double the proportion.   Decision Rule Now we make a decision on whether to reject the null hypothesis (and thereby accept the alternative), or to fail to reject the null hypothesis.  We make this decision by comparing our p-value to the level of significance alpha. If the p-value is less than or equal to alpha, then we reject the null hypothesis. This means that we have a statistically significant result and that we are going to accept the alternative hypothesis.If the p-value is greater than alpha, then we fail to reject the null hypothesis. This does not prove that the null hypothesis is true. Instead it means that we did not obtain convincing enough evidence to reject the null hypothesis.   Special Note The confidence interval for the difference of two population proportions does not pool the successes, whereas the hypothesis test does.  The reason for this is that our null hypothesis assumes that p1 - p2 0.  The confidence interval does not assume this.  Some statisticians do not pool the successes for this hypothesis test, and instead use a slightly modified version of the above test statistic.

Same Sex Marriages Should Be Legal - 1713 Words

Same-sex marriages have a same gender identity or biological as gay and lesbian couples. Not a long time ago, people did not understand about gays, homosexualities, and these married situations. Therefore, they have many false ideas about gays and think homosexuality is against God and nature. After many years, gay people fought for same-sex rights and had many improvements in understanding these relations. Besides, more people had open minds to homosexuality and accepted gay marriages. According to â€Å"Supreme Court Strike down Bans on Same-Sex marriage,† Richard Wolf and Brad Heath (usatoday.com, June 26,2015) wrote that The US Supreme Court allowed same-sex marriages legally in fifty states. In other words, gays and lesbians have freedom rights to love and marry like any normal person. Millions of Americans were very happy because gay couples became legally during their waiting and struggling a long time, and this was also a historic milestone. Although some people still dispute against this problem, I believe that the law and we should have equal rights for the gay couples become legally because they have the human rights and have the right to pursue their happiness, adopt children, and people should not discriminate against them. The US Supreme Court expanded the freedom right for gay couples to follow their happiness because they have right to love anybody from their heart. People should not discriminate same-sex couples because they also have the living right as normalShow MoreRelatedSame Sex Marriage Should Be Legal1288 Words   |  6 Pages Marriage is not precisely the same as it used to be interpreted. For example, women used to be their husband’s property. Sometimes the women were forced to marry whoever their parents wanted them to marry and most of the time they couldn’t leave the marriage. Nowadays women have more freedom. They can vote, they can run their own business, and they can marry whichever man they want to. 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Appendix a †Meditation Worksheet Free Essays

Appendix A – Meditation Worksheet Veronica P. Sally-Garner PSY/211 ESSENTIALS OF PSYCHOLOGY Thursday, October 20, 2011 Instructor: Terry Browning Appendix A Meditation Worksheet Directions: Locate two resources on the Internet that explain meditation techniques. Copy and paste the web address into the top of the matrix. We will write a custom essay sample on Appendix a – Meditation Worksheet or any similar topic only for you Order Now After reviewing the website, provide a brief summary for each source. Below your summary, list two interesting facts you learned from each site. Try the techniques you located in your Internet search. Provide a brief description of what happened in your experience. Be sure to answer the two questions below the matrix also. Web Address (URL):| http://bestmeditationtechniques. org/the-basics-of-meditation| http://www. mum. edu/meditation. html? gclid=CKOsutyS-KsCFQ1U7Aod9HlUKw| Summary of resource:| Things happen so quickly in this fast-paced world. Decisions are made without thought. This could cause stress to build and affect the way people deal with life. Everyone needs to stop and take a few minutes to calm down. So if life is stressing you out, it’s time to slow down. Meditation can help relieve stress and worries. The practice of meditation is a way to be still for a while and let our minds become focused until we become calm enough to deal with the daily pressures of life. | Mediation can mean different things to different people. Mediation can be defined in three main types: focused attention- concentration, open monitoring-contemplation, and automatic self-transcending – the Transcendental Mediation Program. Transcendental Meditation also known as TM. TM is the most widely practiced, most researched, and most effective method of self-development mediation technique. Two interestingFacts: | 1. Meditation can improve Concentration. Two commonly used exercises are: Mantra – a phrase or a sound that is recited repeatedlyBreathing – achieved by inhaling through the nose, letting the diaphragm (not the chest) expand and exhaling through the mouth 2. Breathing is an integral part of meditation. Proper breathing is required in many practices of meditation and is therefore an im portant tool to be possessed by people who wish to mediate. The practice of proper breathing while meditating slows down a person’s heart rate which leads to a relaxed mood. While this may sound easy, it can be quite challenging when done for the first time. | 1. Transcendental Meditation is the most popular technique. It is practiced by people of all ages, cultures, and religions. More than five million people worldwide practice this simple, natural technique. 2. People who practice Transcendental Meditation find themselves more alert, calmer, more energetic, dynamic, yet somehow wiser, as if you have a broader perspective on their daily activities and experiences. | What happened after you tried each technique? | After trying each technique one should feel relaxed and a sense of heighten awareness. Meditation helps the mind become less distracted. | After trying Transcendental Meditation one should have more confidence and higher self esteem. | 1. Do you think meditation leads to heightened sensation and perception? Explain why or why not. ————————————————- Mediation can lead to heightened sensation and perception. Each is different but ————————————————- complements each other. Sensation is the process by which we sense through touch, smell, ————————————————- sight, taste and smell. Perception is our way of interpreting what these sensations mean and ————————————————- how to make sense of it. 2. Have you considered integrating meditation into your own lifestyle? If so, list one strategy for fitting meditation into your schedule. If not, explain why. ————————————————- I have integrated meditation into my personal lifestyle with physical exercise. Physical ————————————————- exercise promotes overall fitness, helps manage emotional stress and relieves tension. How to cite Appendix a – Meditation Worksheet, Papers

Media Advertising The Army Advertising Campaign Essay Example For Students

Media Advertising The Army Advertising Campaign Essay Targets You! Exploratory Essays Research PapersThe U. S. Army Advertising Campaign Targets You! The United States Army recruitment campaign slogan of An Army of One, is reminiscent of the legendary Musketeers code. These men protected the king and lived by the phrase One for All and All for One. To the Musketeers this represented the power they found in unity as well as in individuality, combined. This relationship, though somewhat subtle, is simply a play on the recent felt need of many to serve and protect their country and freedom. The U.S. Army uses television, print, and now the World Wide Web with an interactive internet website, at www. goarmy.com, for recruiting males and females from right out of high school, to the age of 35, although this particularly is trying to reach the large, new internet using group of young people today. The new Go Army ad campaign from the United States Army, successfully overcomes its slight lack of logical reasoning, by appealing to its audiences interests by using the implicit high reputation that the U.S. Army entails and by employing the surge of patriotic and sense of duty; as well as visually captivating graphics that draw on desires to be adventurous and exotic. The site has visually stimulating graphics of men and women in their various uniforms all set in backgrounds of diverse locations. From the starting page there is a man dressed in winter gear in the foreground, with snow-capped alpine mountain peaks behind him. While the page is loading a flash quote reads, Ive trained in desert, urban, jungle, and alpine. .. Next to this picturesque scene is the army logo, with the phrase written down the side An Army of One. This is characteristic of the sites use of attractive visuals to appeal to its audience. There are various pop-up options available at the bottom of the screen, including; enlistment bonus, learn about the online game, partnership for youth success, and hot shots. Another sidebar also includes ice soldier-pushing the limits, arena football, and career professionals. Each element is meant to appeal to a different aspect of their intended audience. The U. S. army deliberately de-emphasized the logical reasoning of their purpose, to ask their audience to join the army. This was an intelligent decision in and of itself because there really arent many good logical reasons to join the army. They dont make any statements on the website like join the army and become strong and healthy, because obviously there are much safer ways to become strong and healthy. Or, join the army because its the best way to make money, because like before, there are safer ways to make money. They specifically dont make mention of any logos because for one, there just arent any really good reasons, and two, because it isnt necessary due to the number and strength of the messages ethos and pathos. The U.S. army uses the fact that it is so well known and highly regarded worldwide, to support its appeal. The U.S Army has been protecting the country since its creation and everyone knows that. From the very opening page of the website there is the Army emblem right there on the side. This denotes credibility. The opening page also shows someone in full dress uniform. When you see someone in an Army uniform, this alone draws your respect for the person. The website also emphasizes other points through its pop-up options, such as the one that talks about the Army Marksmen team that is to compete in a worldwide competition, the heading reads, Armys top marksmen among worlds best. This strong emphasis on ethos is a wise decision because it is very well manifest. In another pop-up option they use an association with a high-class society to support their credibility. This is seen in the option that reads, Partnership for youth success. Jumpstart your career. Critical accounting theory Essay Its asks you to enter the world of the ice soldiers, once again to make that strong appeal to the desire to be in exotic and exciting situations. This is successful because they show such diverse locations very vividly in their graphics and use such highly descriptive terms as vertical ice and roaring winds that appeals to the flood of extreme sports lovers. The ad does not leave out the casual observer of sports however. The option of Arena Football attracts the attention of sports lovers. The visuals in this part of the ad are especially appealing to fans of sports because they are so full of action. The picture shows an actual play in motion of someone running from the opposing team with the ball, and it is all in close-up. The diction here is also very well chosen because they use phrases like force to be reckoned with. The message in this part of the ad is that there is probably no sport more closely associated with the Army than football. This is another good choice of association. Football has become a very popular sport in the U.S., even surpassing at times the viewers of baseball. As a member of this intended audience, being born in Texas, and therefore having football in my blood, I feel this is strongly appeals to me. Not only as seeing them in a game of this different version of the sport, but also in the things they offer at the games. They offer things like prizes, interactive games, and free jerseys. This is a very successful mode of appealing to sports enthusiasts. The U.S. army also makes an attempt to appeal to a larger audience by offering an online video game. This would attract the attention of many youth who are now into this recent fad. The pop-up screen strengthens this appeal by using such terms as innovative and highly realistic. This Official Army Game, therefore reaches its intended audience of the computer savvy generation that has begun to emerge. The purpose is to get these young people to play this game, so that their interest in signing up and doing the real thing might grow. In my case however, because I only play these games because Im too lazy to do the real thing, this method might not be so effective. But the army even has that base covered by stating that just by playing their new game you can Gain experience as a Soldier in the U.S. Army, without ever leaving your desk. The scene it portrays appears to be a desert campaign, so there may also be an implied pull here at recent events in the Middle East. Therefore there may be more than just a fun appeal here, but also a type of appeal to political beliefs and interests. Overall, the U.S. Army most successfully uses the emotion of patriotism as well as the Armys reputation to successfully appeal to its intended audience. They made their message even more appealing, by presenting the message over a wide medium, which is the World Wide Web. By expanding their audience to sports lovers, marksmen, and the online game players of today, they will gain that much more coverage and attract that many more recruits. These reasons would be enough for me to join if I didnt have any other reasons not to do so. Personally, I feel I owe my country so much for the freedoms it provides me, and the army for protecting those freedoms. I feel a strong sense of duty to serve my country, and I almost enlisted the day the towers fell, but an even deeper emotion runs through me, and that is toward my mother. Because she is truly the one who provides me with all that I have, and if she had let me go that day, I would have. Needless to say, she did not, and so I am here. My mother says I can serve my country better this way.