Hypothesis Test Example of Calculating Probability

An important part of inferential statistics is hypothesis testing. As with learning anything related to mathematics, it is helpful to work through several examples. The following examines an example of a hypothesis test, and calculates the probability of type I and type II errors. We will assume that the simple conditions hold. More specifically we will assume that we have a simple random sample from a population that is either normally distributed or has a large enough sample size that we can apply the central limit theorem. We will also assume that we know the population standard deviation. Statement of the Problem A bag of potato chips is packaged by weight. A total of nine bags are purchased, weighed and the mean weight of these nine bags is 10.5 ounces. Suppose that the standard deviation of the population of all such bags of chips is 0.6 ounces. The stated weight on all packages is 11 ounces. Set a level of significance at 0.01. Question 1 Does the sample support the hypothesis that true population mean is less than 11 ounces? We have a lower tailed test. This is seen by the statement of our null and alternative hypotheses: H0 : ÃŽ ¼11.Ha : ÃŽ ¼ 11. The test statistic is calculated by the formula z (x-bar - ÃŽ ¼0)/(ÏÆ'/√n) (10.5 - 11)/(0.6/√ 9) -0.5/0.2 -2.5. We now need to determine how likely this value of z is due to chance alone. By using a table of z-scores we see that the probability that z is less than or equal to -2.5 is 0.0062. Since this p-value is less than the significance level, we reject the null hypothesis and accept the alternative hypothesis. The mean weight of all bags of chips is less than 11 ounces. Question 2 What is the probability of a type I error? A type I error occurs when we reject a null hypothesis that is true. The probability of such an error is equal to the significance level. In this case, we have a level of significance equal to 0.01, thus this is the probability of a type I error. Question 3 If the population mean is actually 10.75 ounces, what is the probability of a Type II error? We begin by reformulating our decision rule in terms of the sample mean. For a significance level of 0.01, we reject the null hypothesis when z -2.33. By plugging this value into the formula for the test statistics, we reject the null hypothesis when (x-bar – 11)/(0.6/√ 9) -2.33. Equivalently we reject the null hypothesis when 11 – 2.33(0.2) x-bar, or when x-bar is less than 10.534. We fail to reject the null hypothesis for x-bar greater than or equal to 10.534. If the true population mean is 10.75, then the probability that x-bar is greater than or equal to 10.534 is equivalent to the probability that z is greater than or equal to -0.22. This probability, which is the probability of a type II error, is equal to 0.587.

Crime Rates an Econometric Analysis - 4665 Words

Crime Rates: An Econometric Analysis using population, unemployment and growth Table of Contents I. Introduction A.) Background of the Study B.) Problem Statement C.) Objectives D.) Significance of the Study E.) Scope and Limitations II. Review of Related Literature III. Operational Framework A.) Variable List B.) Model Specification C.) A-priori Expectations IV. Methodology A.) Data B.) Preliminary Tests V. Results and Discussions VI. Conclusion and Recommendations VII. Bibliography VIII. Appendices INTRODUCTION A. Background of the Study Ever since the first civilizations, ever since the dawn of government and morals, crime has accompanied mankind in his everyday†¦show more content†¦The researcher would have preferred to add potential influences such as literacy rates and family incomes however such data are collected in periods of 3 years or greater, thus leaving the researcher no choice but to analyze based on annually recorded economic variables. REVIEW OF RELATED LITERATURE Prior to 1968, crime was viewed as a social and moral construct. However this changed when Gary Becker’s Crime and Punishment: An Economic Approach was published in 1968. Becker takes into account the Opportunities and Costs of committing a crime; that a criminal would steal because the benefits are greater than the costs. Wadsworth (2001), on the other hand, states that employment is a key factor that affects crime rates. It is not the fact of unemployment that pushes them into being criminals but the hardships that they experience. The ideals in this study will follow suite to the ideals of the said researchers. The variables of CPI, Income, unemployment and population density all play important roles in determining crime rates. OPERATIONAL FRAMEWORK A. Variable List The variables used in the model are described as follows: Variable Name | Description | Lncrimerate | Crime Rate | d_l_gnipc | Gross National Income Per Capita | d_l_urbanpop | Population in Urbanized Areas | d_l_cpi | Consumer Price Index | d_l_popdens | Population Density | d_l_unemp | Number of Unemployed Persons | B. ModelShow MoreRelatedThe Effect Of Justice Expenditure On The Crime Rate Since 19921392 Words   |  6 PagesExpenditure on the Crime Rate Since 1992 1. Introduction The crime rate is the result of problems in public safety. As one of the most important public good, safety has generates positive externalities that helps urban development. However, due to the fact tighter police protection budgets yield the local law enforcement to maintain or improve the efficiency of policing with fewer hands and resources. Will an increase in justice expenses bring us a safer local environment? 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Part A Aj Davis Department Store - 1479 Words

Course Project Part A September 15, 2013 Applied Managerial Statistics Professor Mayers Brief Introduction The following report presents a detailed statistical analysis of AJ Davis Department Store credit customers. Data was collected from a sample of 50 AJ Davis credit customers on five variables which are Location, Size, Income, Years, and Credit Balance. Out of the five variables, Location,Size, and Income is emphasize more in this analysis. AJ Davis Department Store is very determined to find out more information about their credit customers. So by doing a in-depth analysis of the variables and their relationships through graphical, numerical summary and interpretation should give a detailed summary of their†¦show more content†¦Descriptive Statistics: Years Variable Location Mean StDev Variance Median Range IQR Mode Years Rural 12.46 4.94 24.44 13.00 16.00 7.00 13, 15, 18 Suburban 6.467 2.949 8.695 6.000 9.000 5.000 10 Urban 10.045 3.982 15.855 10.000 17.000 5.000 10 The 2nd pairing of variables I combined together is Location and Years. I demonstrated the variables in a dot plot to illustrate the number of years the customer has been living in that location. The most years was more than 18 years and the location was in an urban area. The highest amount of dots was 10 years. The shape of the distribution is symmetric. The last pairing of variables I combined together is Income and Size and it demonstrated in a scatter plot. The household size of 7 or 8 has the highest income is with over $69,000 and more. The shape of distribution is positive linear relationship. Regression Analysis: Income ($1000) versus Size The regression equation is Income ($1000) = 33.5 + 2.78 Size Predictor Coef SE Coef T P Constant 33.499 3.523 9.51 0.000 Size 2.7824 0.6844 4.07 0.000 S = 12.0983 R-Sq = 25.6% R-Sq(adj) = 24.1% Conclusion As the result shows, the urban location is where most of AJ Davis Department Store comes from with 44%. The urban location also has a higher creditShow MoreRelatedEssay on Aj Davis Department Store Part B892 Words   |  4 PagesAJ Davis Department Store Part B AJ Davis Department Store Introduction The following information will show whether or not the manager’s speculations are correct. He wants to know the following information: Is the average mean greater than $45,000? Does the true population proportion of customers who live in an urban area exceed 45%? Is the average number of years lived in the current home less than 8 years? 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The average (mean) annual income was less than $50,000 †¢ Null Hypothesis is the average annual income is ≠¥ to $50,000. o Ho:  µ ≠¥ 50,000 †¢ Alternate Hypothesis is the average annual income is than $50,000. o Ha:  µ 50,000 †¢ Analysis Plan significance level is: a = 0.05 †¢ n 30 the z test was used to test the hypothesis †¢ Alternative Hypothesis Ha:Read MoreEssay on Math 533 - Course Project Part a948 Words   |  4 PagesAJ DAVIS is a department store chain, which has many credit customers and wants to find out more information about these customers. 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The Vulnerability and Responsibility of the Street Children free essay sample

A street child is a young person, under the age of fifteen, who lives and sleeps in the streets, whose family ties are broken and who cant or wont return home. Street children live in the streets without their families. Each child has to learn how to survive alone, since no adult takes responsibility of them. Often they are very young and completely ignored by their families. These children dont like to be called as street children. They live, or rather struggle to survive; they are usually in the downtown area, near stations and shopping malls, places, which are lit up at night or in garbage dumps, in railway stations or under bridges in most major cities of developing countries around the world. To avoid being caught, street children are the last to go to sleep and the first to get up: thats why theyre often worn out, and can be found sleeping during the day. They group together at night. Amongst children in the street, there are; children who work, children from slums, runaways and delinquents. Other children may spend most of their time in the street, but maintain certain link to their families. Street children are impoverished of a family care and protection. They need care, shelter, and education and because of not having the basic needs, they are forced to come on the streets in their childhood. Most of the street children have to work in their younger age for their families. Their lives stay at a risk in different means both physically and mentally. During their work they are badly treated with and also humiliated. Children may end up on streets for several reasons such as family breakdown, armed conflict, poverty, natural disaster, abuse, and exploitation, dislocation through migration, urbanization, acculturation, and diseases like AIDS is another cause. The population of the street children is growing rapidly around the clock. This is happening due to undeveloped rural areas and the inhabitants over there that doesn’t have any land, property or proper shelter. These reasons have pressurized the families to migrate to a city which leads to overpopulation in a city. Street children especially exist in the major cities of a developing country because of mistreatment and neglect. Unfortunately, even the developed countries don’t have the facilities to undertake them. This has resulted children from these families have to come out on the streets to support themselves and their families. Majority of the children are sent out to the city to earn. Some of them are thrown away from their homes because of abusive family relations and thus places like bus stops, railway stations and market places become their homes and also their work places. Lack of education, food, medication and other factors are critical issues in this problem. Street children are engaged in some kind of economic activity ranging from begging to vend. The street children are vulnerable during work and as well as in their daily life routine. Street children are often found in bazaars, commercial areas, hotels and parks. Thousands of children mostly work in different areas of a city as a table boy, helpers for steel work, street hawkers, and hotel workers, beggars at streets, truck boy and working with transport industry. They also try to earn a living through collecting garbage, breaking bricks or pushing rickshaws. Some of them are just involved in petty crimes. Most of the children are insulted in their work and they go home at the end of the day. They contribute their earnings to their families. Due to the economic fragility of the families, they lead a permanent life on the streets. While growing up, these children usually experience severe mortification, social antipathy and considerable danger. As a result, they often grow hostile towards the society. This hatred and distrust draws the naive street children to involve in criminal activities. Thus, they play an effective role in tainting the society. There are children belonging to very poor families who are deprived and unprivileged and sometime marginalized. Among them there are street children without parents, home or any type of shelter. There are slum children living in the street side or by the side of rail line. These children begin each of the days with the tension of collecting foods. In which age, they should go to school; they have to go in search of livelihood. They have to work hard till the night. Even sometime they have to be involved in different types of risky jobs, which are threats to their lives. Street children have many problems than any other normal child can have. In the course of their life they have to go through hardships, pain and sadness. It is very natural to see a child in the streets starving, crying for new clothes and many more but nobody reaches a hand towards them because they are distinguished from the society. Many of the street children who have run away from home have done so because they were beaten. Tragically, their homelessness can lead to further abuse through exploitative child labour. Many children of the streets become emotionally scarred. They refuse to speak for months as they get scarred of the unknown people, any new and unknown encounter traumatizes them. These children often feel guilty and blame themselves for their mistreatment. Such damage to their minds can take years to recover even in the most loving environment and it is impossible on the streets to even heal the person because of society’s bad behavior towards them. Helping street children requires offering them choices within their environment and assisting them to make their own decisions, which will in time lead away from the street. The mistaken belief that rescuing street children involves removing them from the street as quickly as possible often does more harm than good, resulting in failed rehabilitation and a return to the street. Female children are often discriminated against in the society. Their health, education, prosperity and freedom are all impacted. The society is very narrow-minded towards the girls. Child marriage is another way in which the girls are disadvantaged. In addition, to limiting educational possibilities and stunting personal development early marriages carry health risk. In many parts of Bangladesh, child marriage is still continued even after the government had put an end to it. India’s female population is less than male. Poor health is another major problem for street children. Half of all street children in many countries are malnourished but the proportion of the street children is higher. Street dwellers in the city are extremely vulnerable in terms of their health needs, hygiene and utilization of healthcare services and this marginalized group is neglected by public. These children are not only underweight, but their growth is also underdeveloped; for example a twelve year old child can be mistaken for an eight year old. They work and live among garbage and open sewers. They are not only exposed to diseases but they are also unlikely to be vaccinated or receive medical treatment. The children often don’t get the fresh food, clean water to eat or drink. They grow up in unhygienic environment, so they are infected with many illnesses. They have an uncertain supply of food, education, medical treatment and are at a high risk of addiction and sickness. Most street children have not been vaccinated at all. They usually cannot afford, and do not trust, doctors or medicines. If they receive any treatment at all it will often be harmful, as with kids whose parents place scalding metal on their bellies as a remedy for persistent stomach pain. They suffer from exhaustion, injury, exposure to dangerous chemicals, plus muscle and bone afflictions. Poverty is the primary cause of the street children crisis. Poverty dumps a crowd of problems onto a child. Not only do these problems cause immediate suffering, they also conspire to keep the child poor throughout his/her life. In order to survive, a child has to sacrifice education and training; without skills the child will, as an adult, remain at the bottom of the economic heap. Though the problems of the street children will not decrease but it is our responsibility to help them with their tribulations. Every person should help a child to overcome their poverty, scared ness of the society and many more. If we help the children, they will open up to our society and they will be happier. People also should cooperate with the children to make their lives better and also to make a country peaceful about these reasons. We should firstly volunteer with many non-governmental organizations. We should do encouragement for the rights of abused children to make awareness among the general public. We should try to change the mentality approach about these children because these neglected children need to be loved and care like every other child. There should be a project based service delivery package for this population which can make awareness about rights and responsibilities. There should be a participative approach with children otherwise we cannot bring a productive change. Participative approach of skilled and motivated human resource is an essential part of success of project. We should help out these children by having a shelter base school where we can try our level best to change the society’s mentally approach about these deprived children. So come and join hands with us for this purpose for the prosperity of country and these children. Because they have not reached the age of majority, street children have no representation in the governing process. They have no vote themselves nor by proxy through their parents, from whom they likely are alienated. Nor do street children have any economic leverage. Governments, consequently, may pay little attention to them. We should try to convince the government to take responsibility towards them. Non-Governmental Organizations (NGOs) may also be blamed for encouraging children to live in the streets by making street life more bearable or attractive through the services they provide. We should make residential rehabilitation programs which can provide an environment isolated from the streets where activities are focused on assisting children to recover from drug addiction. We should alleviate the worst aspects of street life for children by providing services to them in the streets. We should provide them with a loving and care atmosphere which will make the children to gain confidence and self-esteem. We should create a help centre where the children can have fun and it can be a safe place for them. We should at least try to run a free school by the help of the government so that the children who work can get a basic education. This will help in improving the economic state of a country. We should also provide nutrition, medical treatment and health centre, repatriation to their families, temporary and permanent shelter. We should spend some moments with the street children so that their hatred towards the society can lessen for example ask their names, what they like to do or their favorite games. We could let them feel like family if we eat together with them. They will enjoy their meal more happily. We should also share some of our memories with them to create a smile in their face. By doing these things we can improve the lives of the street children, their attitude towards the society and help them in many ways. We all can enjoy being citizens of a country. Our and their lives can change if we are a little conscious about the problems. Everyone can live peacefully in the future and together. Everyone should help out each other when it is needed so that our lives can be much better