Example of Normal Approximation of a Binomial Distribution

Example of Normal Approximation of a Binomial Distribution The binomial distribution involves a discrete random variable. Probabilities in a binomial setting can be calculated in a straightforward way by using the formula for a binomial coefficient. While in theory, this is an easy calculation, in practice it can become quite tedious or even computationally impossible to calculate binomial probabilities. These issues can be sidestepped by instead using a normal distribution to approximate a binomial distribution.  We will see how to do this by going through the steps of a calculation. Steps to Using the Normal Approximation First, we must determine if it is appropriate to use the normal approximation. Not every binomial distribution is the same. Some exhibit enough skewness that we cannot use a normal approximation. To check to see if the normal approximation should be used, we need to look at the value of p, which is the probability of success, and n, which is the number of observations of our binomial variable. In order to use the normal approximation, we consider both np and n( 1 - p ). If both of these numbers are greater than or equal to 10, then we are justified in using the normal approximation. This is a general rule of thumb, and typically the larger the values of np and n( 1 - p ), the better is the approximation. Comparison Between Binomial and Normal We will compare an exact binomial probability with that obtained by a normal approximation. We consider the tossing of 20 coins and want to know the probability that five coins or less were heads. If X is the number of heads, then we want to find the value: P(X 0) P(X 1) P(X 2) P(X 3) P(X 4) P(X 5). The use of the binomial formula for each of these six probabilities shows us that the probability is 2.0695%. We will now see how close our normal approximation will be to this value. Checking the conditions, we see that both np and np(1 - p) are equal to 10. This shows that we can use the normal approximation in this case. We will utilize a normal distribution with mean of np 20(0.5) 10 and a standard deviation of (20(0.5)(0.5))0.5 2.236. To determine the probability that X is less than or equal to 5 we need to find the z-score for 5 in the normal distribution that we are using. Thus z (5 – 10)/2.236 -2.236. By consulting a table of z-scores we see that the probability that z is less than or equal to -2.236 is 1.267%. This differs from the actual probability but is within 0.8%. Continuity Correction Factor To improve our estimate, it is appropriate to introduce a continuity correction factor. This is used because a normal distribution is continuous whereas the binomial distribution is discrete. For a binomial random variable, a probability histogram for X 5 will include a bar that goes from 4.5 to 5.5 and is centered at 5. This means that for the above example, the probability that X is less than or equal to 5 for a binomial variable should be estimated by the probability that X is less than or equal to 5.5 for a continuous normal variable. Thus z (5.5 – 10)/2.236 -2.013. The probability that z

Investing in Back Bay-Beacon Hill Essay Example | Topics and Well Written Essays - 1000 words

Investing in Back Bay-Beacon Hill - Essay Example As the report declares the main aspects of the real estate industry in Beacon Hill are as follows. First, properties in the area are valued at a price that is either double or triple of its value ten years ago. This presents an opportunity for profits for investors and developers in the long run. Second, properties in the area are either rented as apartments ranging from $1,600 for one-bedroom apartments up to $2,100 for two-bedroom apartments. Third, speculators forecasted the â€Å"back slope† in the northern part of the Beacon Hill area to increase value in the future. Fourth, due to state legislature, which established the Beacon Hill area as an historic district, constructions of new buildings have been prohibited. As a result, the values of existing buildings were enhanced, while land values were limited. This paper stresses that given the aforementioned characteristics of the real estate market in Beacon Hill, Alexander concluded that buying a property in the â€Å"back slope† would be most profitable because of the relatively lower values of properties there. Furthermore, he realized that it would be best to purchase a property, which would require an extensive amount of renovation to have better control on the property’s income potential. The property that interested Alexander was a 4-unit apartment in the â€Å"back slope.† The property is located in the middle of the block and is set 100 feet from the road.

Management Skills Case Study Example | Topics and Well Written Essays - 750 words

Management Skills - Case Study Example Since decisions and plans made by the vacuum process approach normally do not consider the consequences of counter-decisions that are likely to be taken by the affected parties, this conventional approach could be possibly shortsighted. This paper will describe how the coevolutionary gaming theory facilitates the group decision making. Often a decision made on the basis of the current and historical data may bring worse outcomes if reactionary moves are not taken into account appropriately. It can be illustrated using the example of a new market entrant who plans to implement a low price strategy to dominate over the existing market players. Based on the data available, this low price strategy may seem a simple approach because it is just to trim down prices and to charge lower than what the competitors charge for the same product. However, this plan conveniently assumes that the existing market players who share the entire market will simply ignore the newcomer. What the available data do not tell is that the market leaders are in a better position to engage in a price war than the new market entrant because they will probably have a stronger financial base. Taking advantages of the economies of scale, market leaders can aggressively compete with the newcomer to the extent where they set prices below the breakeve n point of the new market entrant. At this point, the newcomer would struggle to survive because the firm’s operating expenses outweigh its sales revenues. According to Perc and Szolnoki (2010), the coevolutionary gaming approach is capable of addressing the limitation of the conventional decision making practice because when making decision and plans based on coevolutionary gaming approach, possible counter-decisions and plans of affected parties are considered, and the impacts of those reactions are well scrutinized for the purpose of making changes to the decisions and plans being developed before they are

Different Ethnic Groups Essay Example for Free

Different Ethnic Groups Essay After going through and being asked to react to the following statement, â€Å"Students who dine solely with members of their own ethnic group and participate in ethnic student organizations and activities contribute to a decline of ethnic relations on campus,† I have come to realized that as any other statements there’s always two sides and a lot more to consider before coming to a consent as a whole. In my opinion this is very important and should always view both sides of the story Many types of these ethnic group’s actions could make it easy and lead to a decline of ethnic relations on campus. For instance, let’s say if ethnic students did everything together as a whole and only participated in ethnic student activities, then it would clearly segregate the campus. Unfortunately no a days, this world for the most part is over the whole â€Å"whites and blacks racial dividedness† and no one should be seeking to head back on that path. Now, campus cafeterias could show a sign of being like that again with several different ethnic groups scattered around. If each ethnic group had its individual organization, I fell that the members could get extremely complacent and start to sense a feeling of superiority. This eventually could lead to conflict with other ethnic groups throughout the campus. It is very also important to look at how other students will view these ethnic groups. Other students may not like these groups, which once again could cause conflict. Another scenario, students may feel threatened by these groups, therefore making them not want to attend class or socialize around campus. There could also be a chance that these ethnic groups could try to bring down a disliked professor or even another group. Universities un-affiliated with a religion may deal with groups with strong religious beliefs and could try to implement their religion into the university. Even schools that are represented by a religion are in danger of an ethnic group with other strong religious beliefs. With all of these problems with ethnic groups, could quickly multiply as they also act as networking groups to get more students who share the same background and beliefs to attend the school. Even graduates could carry this on to their future employers, although experts claim that no network group has ever set out to bring a company down, companies are the next things for college students and bring the risk of having their employees become far too separate from each other. The whole idea of diversity in a company is to make the work place and as uniformly supportive to all cultural backgrounds. By allowing these groups to form, the companies are moving further away from supposedly a fair and diverse population of workers. The segregation of ethnic groups from the rest of the student body largely contributes to a decline of ethnic relations on campus. If I were to see the other side of this and disagree, I would believe that if students of the same ethnic background only dined and participated in ethnic student activities, then it would not contribute to a decline of ethnic relations on campus. Instead, it would have students who tend to stay surrounded by people of the same ethnic background could feel more free, comfortable and accepted; thus bringing improvements leading to higher grades in the classroom and a more humble attitude towards life. Also being placed within a group will most likely encourage students to go out socialize, take part in campus activities, and enjoy campus life. This will lead to them interacting with people from different backgrounds. These ethnic groups can also play a vital role in campus activity by hosting fundraisers, parties, or sporting events. Another benefit to being part of an ethnic group is that many voices are better than one. So by having your voice heard out there will help put an end to any discrimination that was happening and will strengthen the campus’ ethnic relations. Future students looking to attend the school in the years to come may visit the campus and see a group of people who share the same ethnicity or interest as themselves and allure them to come to that university. These groups working as a form of networking could be a massive tool in attracting a diverse student body while at the same time strengthening ethnic relationships. Having been asked to support the statement or disagree, I would have to say that I agree with the statement. In order to be diverse, the student body as a whole it needs to be integrated at all times. It is one thing to live and be part of a group with people who share the same background; however, they should also be with people of other ethnicities and backgrounds. One group only interacting with themselves and not acknowledging other groups forms poor ethnic relations habits. I also feel that most ethnic groups will express a religion that will stir up controversy with other groups for one reason or another. By using the groups as a way to persuade new students to attend the school, the groups will quickly grow and possibly cause even more conflict to the university. Ethnic groups will bring nothing more than poor ethnic relations habits to campus and future graduates workplaces causing only a decline in ethnic relations.

Biography of Wyatt Earp :: essays research papers

Wyatt Earp was born in Monmouth, Illinois Monday, March 19, 1848 and died Friday January 13, 1929 in Los Angeles, California of the flu. Wyatt Earp is mostly famous for his gun fight at the OK Corral in Tombstone, Arizona 1881. But, there are more interesting facts about his life and you will hear them today. In 1876 Wyatt Earp was hired as an assistant Marshall in Dodge City, Kansas, one of the worst places in America at the time. In Dodge City daylight muggings and nighttime shootings were an everyday thing. When Wyatt became a lawman there he was able to stop the crimes without killing anybody, he put them in jail. One of his techniques was called ?buffaloing? where he would knock a criminal on the head with his favorite weapon, the Colt 45. In 1878 Wyatt met Doc Holliday. They became friends when Doc shot a man who was aiming at Wyatt?s back. Doc was also involved in the battle at the OK Corral. Doc suffered from Tuberculosis and went from 200 pounds to 122 pounds in the year after the gunfight. He died in 1883. The gunfight at the ok Corral started and ended in thirty violent seconds, but, it must have felt like thirty minutes to the fighters. Eight people were ready to fight. On one side were Wyatt, his brothers and Doc Holliday. On the other side were the Clantons and their gang of Cowboys (horse thieves and cattle rustlers of the time). They all had no intention of being disarmed. Frank McLaury, a Cowboy, had the best aim of the group. He was shot by Wyatt and was the first to die. Everybody started fighting. In the end, almost all the cowboys died, Ike Clanton begged for his life and was thrown in jail. Morgan Earp was shot in the shoulder. Virgil Earp was shot in the leg. Doc was shot in the hip. Only Wyatt came out unharmed.

Government Spending on Healthcare

Government Spending on Healthcare Talia Oliver 10/22/2012 HCS/440 Donna Lupinacci, MSN The article I read was written by Margaret Cuomo, M. D. and it focused on health care costs and how the government is doing unnecessary spending in health care. According to the article, the government has spent about $750 billion dollars on medical care that was not needed. Some of the areas where the author believes that the money has been wasted have been in unnecessary services, excessive administrative costs, inflated prices, prevention failures and fraud.The issue is that this spending is not actually contributing to the improvement of patient health. The author the author had stated that â€Å"Flaws in the current system of cancer treatment contribute to unnecessary spending† (Cuomo, 2012). There is so much money being spent on the blood tests, diagnostic scans and other medical procedures that deal with cancer and it is costing about $200 billion every year. With their being so much unnecessary spending on healthcare, Cuomo discussed how a group called the IOM committee was able to come up with ways in order to cut the spending and be able to continue to provide quality health service.Government Regulation on Media in AmericaThe article discusses that the IOM believes that â€Å"Eliminating wasteful spending for just one year ($750 billion) would be equal to more than 10 years of Medicare cuts† (Cuomo, 2012). This wasteful spending according to the article can’t continue to happen and it is important that we understand where the money is being spent and how the excessive spending can be changed. My opinion of the article is that there is too much money being spent by the government on healthcare that is not contributing to ensuring the improvement of patient health.When you look at the unnecessary medical care, there are services that are not needed but the money is being wasted when it could be spent somewhere else. Cancer is a serious issue and I believe it is important for patients to get assistance to help treat cancer. However, there are times that some patients are a bit paranoid and believe they have cancer and request tests to be sure. This is not necessary if the patient shows no signs of having cancer and the money can be saved. I believe that the article is very valid with saying that a lot of the spending is going to unnecessary things.The reason why I believe this is because there are those patients who are able to afford care and can’t get covered for care due to the fact that there is not enough funding available. The article is also valid in identifying ways that the unnecessary costs can be cut without disrupting the quality of care given to the patients. The ability to save money by cutting the unnecessary spending can help the money go to services like Medicare. I believe that the spending discussed in this article is way too much on unnecessary services that are not helping to improve patientâ€⠄¢s health.There is no reason that $750 billion is being wasted and it could go to services that can help those who can’t afford healthcare and those on Medicare. I believe that the government could focus that money on prevention methods that would help to avoid patients requiring more care. Prevention efforts are important and yet the government doesn’t invest enough money in it. The article had stated that recently â€Å"The House of Representatives voted 236 to 183 to repeal the Affordable health Care Act’s Prevention and Public Health Trust Fund† (Cuomo, 2012).Instead of trying to prevent the spending on unnecessary services and issues, repealing this act just ensures that there will be more costs to come due to the fact that prevention efforts are not being taken seriously. In conclusion, this article talked about the fact that a lot of the government spending on healthcare is being wasted on unnecessary services. The government is spending money on healthcare that can be cut in order to spend money where it is needed in healthcare.If the government focused more on being able to cut costs and provide quality care, $750 billion can go to Medicare services or even to state agencies that are trying to help uninsured patients get proper care. The government spending on healthcare should be spent on ensuring the improvement of patient’s health. References Cuomo, M. (2012, September 25). Margaret I. Cuomo, M. D. : The Truth About Health Care Costs. Breaking News and Opinion on The Huffington Post. Retrieved October 23, 2012, from http://www. huffingtonpost. com/margaret-i-cuomo-md/health-care-costs_b_1901742. html

Living With Roommates