Category: Population

Introduction

Background (Demographics and Population Growth Policies)

The fact that the global population rates have been growing exponentially over the past few decades clearly is a reason for concern given the challenges that it implies, with poverty being the key challenge. Indeed, a closer look at the global statistics will show graphically that the increase in the number of people in certain states has been rather drastic. For example, the recent UNO data indicates that the 2100 population growth (PG) rates will reach a stunning mark of 11,000,000,000 people, which is nearly a 1.5-fold increase (146%) compared to the 2016 statistics. The fact that the identified variable is linked directly to social phenomena such as poverty, unemployment, an increase in development of health issues, rise in crime rates, etc., cannot possibly be doubted, according to the existing studies on PG.

The Malthusian theory of population also points to the negative effects that an increase in population growth will inevitably trigger. According to the principal tenets of the theory, the population growth occurs at an exponential rate (particularly, the number of people supposedly doubles with every generation), whereas the amount of food increases in an arithmetic progression. Therefore, the number of people is certain to exceed the amount of the produced food at a certain point of the humankind’s evolution. The implications of the theory are beyond drastic.

Population Growth

The population growth policies, in their turn, can be viewed as the constraining factor for the issues associated with consistent increases in the number of people. According to the official statement of the Bangladeshi authorities, the population growth rates have been reduced significantly after the introduction of the pro-choice opportunities and the promotion of family planning as the foundation for childbirth-related decision-making (see Fig. 1).

At this point, though, one must mention that the actual number of Bangladeshi residents has been on the rise since the 1970s (see Fig. 2). Nevertheless, the opportunities for controlling the childbirth rates that the introduction of family planning options has provided to the residents of the state allows changing the growth rate from exponential to static.

Compared to Egypt, the identified change in the childbirth rate can be viewed as a significant progress. According to the global ranking based on the identified criterion, Egypt lands on the 64^th place based on its population growth rates, whereas Bangladesh takes the 76^th spot, according to the recent statistical analysis. Although the difference might seem minor, the gap in 12 positions makes it quite clear that Bangladeshi authorities made quite a breakthrough in designing the policies that allow for a more rigid control of the issues related to high childbirth rates (i.e., poverty, crime, health concerns, etc.), whereas Egypt could use assistance.

Research Question, Thesis, and Aim of Research

The current study aims to answer the following question: Will the introduction of birth control tools and the concept of Planned Parenthood in Egypt, in general, allow for a drop in some of the most controversial and topical social issues in the state, such as crime rates increase, a rise in the number of maternal deaths, overpopulation, and the following threat of economic collapse caused by high unemployment rates, etc., and how can the cultural constraints (particularly, the Muslim concept of abortions as a sin) be addressed?

By introducing the principles of family planning into the Egyptian communities, one is likely to create premises for a change in people’s mindset, allowing them to focus on the wellbeing of their families as opposed to following the cultural traditions blindly, thus contributing to a gradual reduction in the population growth so that the crime-, economy-, and society-related issues also are reduced.

Method and Approach

The study is aimed, therefore, at considering the effects of the introduction of family planning principles into the environment of Egyptian society and the subsequent drop in population growth rates as the tool for eradicating some of the crucial issues related to the increase in crime rates, maternal deaths, health issues, societal problems, etc. For these purposes, a qualitative case study will be used as the research tool.

Population Growth and Relevant Theories in Demographics and International Relations

Population Growth and Theories of Development

It should be noted that the Malthusian theory has been criticized for the lack of substantial proof. The mathematical aspect of the framework has received an especially harsh criticism. For instance, numerous researchers pointed to the lack of evidence for the formula that Malthus suggested (i.e., the exponential growth of the population and the arithmetic growth of the food supplies). Nevertheless, the prognoses proposed by Malthus are very unsettling. Specifically, the neo-Malthusian theory, which implies a more careful analysis of the critical factors and provides profound empirical evidence for its postulates, warns about the threats that overpopulation conceals, stressing the issue of water scarcity: “The world’s population is growing at a very rapid rate and will lead to a growth in the demand for food. This would constitute the single most important cause of pressure on water resources.” Similarly, the essential tenets of the structural human ecology theory indicate that overpopulation and the lack of vital resources are linked closely. Hence, the current approach to childbirth in certain states needs to be reconsidered, and the principles of family planning have to be promoted on both the state the personal levels.

The rates of population growth, however, hinge on a variety of social, political, economic, and cultural factors. Therefore, the increase in the population varies depending on the state. As a result, the issue of family planning programs’ significance has been raised several times in Egypt, even though local scholars have been maintaining a rather rigid position against the innovation: “Recently, some Egyptian scholars have questioned the continuing need for government support of family planning programs.”

It should be borne in mind that the benefits of the family planning programs’ implementation are only going to become apparent in the long run. Expecting an immediate change and a rapid improvement can hardly be justified, given the time that it takes for the effects to take place.

Furthermore, the fact that the concept of family planning contradicts the principles of Islam needs to be brought up as the crucial impediment to overcome when promoting the change. As stressed above, faith plays a tremendous role in the decision to introduce the concept of family planning into the environment of a particular state or community. Nevertheless, it is imperative to make sure that the principles of family planning should coexist with the religious beliefs of the target population. The latter should not be coerced into making the choices that they consider wrong; however, it is essential to provide them with options.

Scrutinizing closely some of the recent examples of successful family planning policy implementation, one should consider the example of Bangladesh. The state authorities managed to make the religious beliefs of the target audience compatible with the idea of family planning. Naturally, the process of promoting the identified notion in the target environment was also challenging and required a lot of diplomacy and time. Nevertheless, the state currently implements the policy of family planning without any objections from the Bangladeshi residents, which can be viewed as an impressive achievement.

The existing evidence shows that the effects of reinforcing the significance of family planning and allowing the members of the population make sensible choices are beyond impressive. For example, in Bangladesh, maternal mortality rates have dropped by a stupendous 40%.

Liberal Feminism and Birth Control

Freedom for women can be considered one of the foundational tenets of liberal feminism. Freedom of choice when it comes to the issue of childbirth is among the crucial issues that liberal feminism implies. Although Bangladesh is a Muslim state, and, therefore, its core values are based on the idea of patriarchy, women are provided with an opportunity to fight for their independence, freedom, and equal rights because the identified concepts are included in the set of provisions of the Bangladeshi constitution. Therefore, being established on a legal level, the principles of equality can be promoted actively in the identified environment.

In fact, the postulates of liberal feminism imply that the relationships exist in the context of the liberal society, including the domains such as electoral institutions, workplace environments, and the private environment of the household. In other words, the concept of abstract individualism is viewed as the foundation value on which the social justice and the relationships in the society should be based.

The principles that the proponents of liberal feminism herald as the basis for the relationships between the two genders, thus, can be interpreted as the main reasoning behind the promotion of birth control policies. The attempt to envision women primarily as wives and mothers, while admittedly having positive reasons behind it, restricts the options that women have as far as their personal development is concerned and, thus, infringes their rights.

This model not only excluded the health needs of women who were not wives or mothers (particularly younger or older women, lesbian/transgender women, sex workers etc.), but it also ignored the gendered constraints of women trying to access health care, whether related to their reproductive role or not.

Therefore, the idea of childbirth control sits well with the principles of liberal and neo-liberal feminism. The philosophy, thus, can provide a foil for the promotion of a sensible approach to family planning in Egypt similar to the compromise that was made in Bangladesh.

Bangladesh

Background on the Issue of Overpopulation

As stressed above, Bangladesh is one of the countries that has been suffering from overpopulation for quite a while. However, the opportunities to promote feminist ideas in the identified environment, thus, making the state policies more flexible, can become the leeway for improving the current state of affairs in the social, political, and economic landscape of the country.

Analysis of the Governmental Approach

A closer look at the effects of the governmental approach will reveal that the focus on the introduction of the birth control policy into the social context of Bangladesh has affected it positively on a number of levels. As explained above, the increase in the population rates has become less sharp over the past few decades; in fact, shortly after the enhancement of the identified principles in the target environment, the population growth has been showing the tendency to drop, though not quite consistently.

Much to its credit, the government approached the subject matter very wisely. Instead of foisting the new principles onto the local population with no regard for the traditional values of the people living there, the state authorities focused on the reinforcement of the state legal regulations, which detailed that every person is entitled to the same set of rights in the state. Therefore, the introduction of the family planning principles was viewed as a sensible addition to people’s lives and the chance to improve the well-being of the citizens. In other words, the change was promoted at the national level with the government standing firmly behind it. The support from the state authorities has clearly had its effect on the change in the birth rates in the identified environment.

However, as impressive as the effect of the state government supporting the changes Bangladesh is, the activists who are at the helm of the feminist movement deserve even more credit. Changing not only the political but also the social, the political, and the economic backgrounds of the state, women use social media actively to convey the ideas about the necessity to reconsider the policy that prevents providing the Bangladeshi citizens with equal rights.

Apart from the effect that women engaged in media communication have on the changes in the cultural and social context of the state environment, the alterations on the political level need to be addressed. As stressed above, Bangladesh was one of the first and among the few Muslim states that attempted to engage religious authorities into the process of changing the landscape of gender relationships as opposed to giving them too much power or, on the contrary, preventing them from intruding into the identified realm. As a result, prerequisites for cooperation and negotiations instead of an ongoing conflict that the situation could have resulted in were created.

Social Response to Family Planning Programs

Naturally, it would be wrong to claim that the issue of family planning has not faced any objections from some of the orthodox representatives of the Muslim religion. The social response to the alterations to the concept of femininity from the Muslim perspective, which the introduction of the planned parenting principle implied, varied from positive to mixed to negative. It should be noted, though, that the negative responses were surprisingly few. The reasons for the change to occur so smoothly concerns primarily the fact that the Muslim religion, which remains official in Bangladesh, defines the ethical standards in the society, therefore preventing the principles of childbirth control and the acceptance of planned parenthood as the basis for a cultural and economic breakthrough to take place. Understandably enough, there are objections voiced by a more conservative part of the population. Furthermore, one must take into account the threats that the proponents of the movement subject them to because they have to counter not only conservative members of the public but also their extremist behavior that poses a threat to their lives:

In just over a month, two bloggers were hacked to death in public places, following three similar attacks in 2013, in which one blogger died. A decade back, a famous author also fell victim to a similar attack for his critical views on Islam. Indeed, over the last decade, Bangladesh saw a rise in the number of large and small militant Islamist organizations, which carried out explosives attacks on courts and at cultural events around the country.

Nevertheless, one must admit that the instances such as the one referred to above are rather rare occurrences. Although the women in Bangladesh are exposed to a consistent jeopardy, they clearly contribute to the change that it bound to alter both the social and the economic landscapes of the state. The sharp focus on the economic implications of the subject matter combined with the attempt to invite the representatives of the Muslim religion to cooperate can be viewed as a solid step in the right direction.

Egypt

The Demographic Scene and Population Growth

The issues related to population growth appear to go out of proportion in Egypt currently. Having been on the rise for quite a while, the identified indicator of the national well-being points directly to the threat of economic, social, and cultural problems to start brewing. Seeing that the consistent increase in population rates entails an inevitable reduction in the quality of life, it can be assumed that the rise in the unemployment rates is to be expected rather soon in Egypt. The increase in 2.18% that could be witnessed last year makes it quite clear that the state is on the verge of serious demographic issues. Furthermore, the drastic correlation between the births and the deaths of infants (2,585,028 and 564,088 correspondingly) makes it evident that the country is likely to have to deal with the issues related to unemployment, poverty, and crime in the nearest future due to the increase in the population and the lack of opportunities for jobs and education.

Governmental Approaches to the Issue

The case of Egypt, as opposed to the one of Bangladesh, can be viewed as a mismanagement of the birth control policy. Succumbing to the cultural and religious traditions (namely, the Muslim religion), the state authorities have contributed to a massive rise in the population growth rates. The results are quite predictable: a subsequent increase in the population rates caused to the aggravation of a range of social and economic concerns. The population growth rates have been on the increase over the past few decades, and they have still a propensity to rise with every passing year. Seeing that the steep rise in the population rates triggers a vast array of social, economic, and financial issues, as established above, there is not much surprise to the fact that Egypt has not seen much progress for a significant amount of time. The increasingly large number of child deaths, as well as maternal ones, along with other health concerns, points to the fact that the current approach to family planning – or, to be more accurate, the lack thereof – requires reconsideration.

What Has Gone Wrong?

When considering the problems in the current policy of the Egyptian authorities, as far as the issue of family planning is concerned, one must admit that the religious issues clearly remain the primary stumbling block for the representatives of the feminist movement and the proponents of the Planned Parenthood principles to suggest their tools for improving the current situation. The reluctance that the Egyptian Muslim authorities displayed toward the issue of birth control points to the fact that the very discussion of the subject matter is currently impossible in the realm of Egyptian society.

Case Studies Comparison

It could be suggested that the case of Bangladesh should be viewed as the prime example of how the issue under analysis can be addressed. To be more exact, the idea of inviting the proponents of the family planning movement and the representatives of the Muslim religion in Egypt to cooperate and find the solution that could satisfy all parties should be viewed as the possible solution to the current situation.

One must admit, though, that the use of the strategy that has already been chosen by Bangladesh requires that the approach should be customized to the specifics of Egypt. The first difference that needs to be brought up as the essential obstacle in implementing the policy is that the feminist movement in favor of family planning is unlikely to be welcomed and supported by the members of the state government. Unlike Bangladesh, whose constitution states clearly that all people are entitled to the same set of rights and privileges, Egypt is much more rigid on the gender issue.

Therefore, active use of media along with the enhancement of the participants’ security, including both the digital and the physical one, needs to be considered as the priority.

Finally, the issue of the social stigma that contraceptives and birth control are linked closely needs to be addressed. As explained above, the arguments against the use of birth control measures in Egypt stem from prejudices, which are cultural, in general, and religious, in particular. By stressing the significance of cooperation between the states, the members of the feminist movement, and the religious authorities, one is likely to alter the current situation in Egypt, therefore, providing the target population with more opportunities as far as the family planning issues and the economic opportunities are concerned. It would be far too optimistic to assume that the decision to cooperate will have an immediate effect on the status of family planning and the use of contraception as a concept. Quite the opposite: it is expected that most of the representatives of the Egyptian society, as people of the Muslim faith, are bound to confront the idea of providing women with the right to use the concept of family planning and, therefore, prevent regular childbirth.

Conclusion

In order to manage the problems that the reinforcement of the identified policies are certain to cause, one should consider using not only the support of the religious authorities, but also the evidence and arguments from the women who participate in the movement. Particularly, the use of the modern online media and especially the social network platforms that invite the readers to take part in the discussion must be viewed as the essential tools for carrying out the change in the Egyptian society. Afterward, significant changes can be expected.

References

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Bangladesh,” Worldometers, n. d. Web.

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Khalifa, Mona, Julie DaVanzo, and David M. Adamson. “Population Growth in Egypt – A Continuing Policy Challenge.”RAND, last modified 2000. Web.

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Religion and Rights in Bangladesh: Maintaining a Delicate Balance,”Open Democracy, 2015. Web.

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The current world population is over 6 billion, half of the population lives in poverty. The global population trends have been increasing drastically and it is estimated that in fifty years time the world population will be about between 9 billion. The rapid global population increase has put humanity and the world at peril.

Overpopulation will lead to a number of problems in the world. To begin with, the environment will face the problem of air and water pollution. Habitats will be destroyed to create room for human habitation thus less cropland.

Due to dropping water table, levels and rising temperatures there will be a shortage of food. Water scarcity will be a major challenge because of the uneven distribution of safe drinking water. People with no access to clean water will die of water borne diseases. The natural resources will face exhaustion due to the great pressure of the population.

People will have to shift to cities and urban. The rapid urbanization will put a strain on governments’ ability to provide basic services such as sewerage, water, electricity and infrastructure. It is projected that more than half of the world’s total population will be living in urban areas. The scarcity of resources may lead to conflict as people complete for the available ones.

For instance, China the most populous nation in the world will increase in population despite its one child policy. This will lead to enormous demand for resources. The country will face a problem of feeding its population due to reduced food production because of reduced water levels and the rising temperatures due to global warming.

It will be forced to depend on other countries for its food supply especially on the United States which is a world major grain producer accounting for more than half of the world’s total grain production (Brown 1).

Modern medicine, declined mortality rates, improved sanitation are forces responsible for the world’s population explosion. The world’s population is growing the fastest in the developing world. The population is expected to hit 9 billion “nearly all of this growth will take place in developing countries” (“Population Growth” 1). The developing world has a high population due to high fertility levels.

In this part of the world, many females in the reproductive age are potential mothers. For instance in Africa where fertility rates are high the average number of children per woman is five. In addition, the mortality rates have gone down in the developing world.

With reduced deaths, many children survive and grow into adulthood due to improved sanitation and availability of medication. On the contrary, Population growth is declining in the developed world. These countries have gone through the four stages of demographic transitions and the fertility rate is low thus the declining population.

The implications for countries with declining and aging population are shortage of labor due to lack of energetic young men and women in the labor sector. For example, Japan is estimated to have 40 per cent of its population above 65 years by 2050. The shortage of labor leads to low taxes collection. This influences the economic development of the countries as they spend more money on payout the retirees (Bremner et al 4).

Consequently, the governments of these countries will be forced to take measures to drive the fertility rates up to cover up the deficit in population. In other instance, they might have to increase immigration although many people do not favor this solution. The countries are not headed for extinction per se because there is hope for rebound in fertility rates.

Moreover, the immigrants to the developed countries have high fertility rates for example the Hispanic community in the United States and immigrants in the United Kingdom. However, the demographics of such countries are likely to change with the immigrants surpassing the natives in terms of population size.

Fertility rates are lower in developed countries because there is a less young generation in the childbearing age and the opposite is true in developing nations. Furthermore, due to delayed marriages and high cost of living that forces couples to have fewer children and penetration of reproductive health education unlike in the developing countries (“Population Growth” 1).

The population in the US is growing at about 1% per year. The population is heading in the aging direction in about 50 years time because the big number of baby boomers will have aged and the countries that contribute to immigrants such as Mexico will have a higher aging population than the US hence less immigrants for the US and consequently a rapid aging population (Brown 1).

The planet cannot sustain that a 10 billion population due the strain on natural resources. Policies have to be put in place to improve the situation otherwise the population will slow down due to diseases caused by lack of safe drinking water and hunger from shortage of food.

Potential solutions to the problem of overpopulation include. Educate people about family planning methods so that they can have fewer numbers of children. The education must include religious leaders because in some religions such as Islam and catholic women give birth to high number of children. For example, the teachings of catholic are against use of contraceptives expect the natural or rhythm method.

Economic development can be a solution to the problem of rapid population growth because it will deter people from having many children due to the high cost of living as it has happened in Thailand and Mexico.

Finally, failure to reverse population growth rates will create a major problem for humanity simply because the natural resources available will not be able to sustain the explosive population many will face starvation with reduced food production due to global warming. Conflicts are more likely to arise to from the young nations as more developed nations such as Japan become less economically stable with their aging population.

United Arab Emirates

The UAE has a population problem because its population is male dominated. Thus, results into an imbalance between males and females. The fertility rate in this region has been lowered by the society’s acceptance and use of family planning methods and services.

The governments in the region have embraced policies to drive down the high fertility rate. The conservative culture and religion is opening up and more girls are going to school and taking up careers thus a delay in marriage.

Women are making gains in their rights albeit slowly hence the women are becoming empowered. On the other hand, many labor immigrants come to the UAE and this will raise the population in the region as many immigrants come to seek employment in the region that has a high economy and can afford to employ them. However, it is important to note that most of the immigrants are male hence contributes to the sex imbalance in UAE.

Works Cited

Bremner, Jason et al. “World Population Highlights.” Population Bulletin 63 (2008): 1-11.

Brown, Lester. Interview by NOVA. “Voices of concern “. pbs.org. 2004. Web.

Population growth over human history. globalchange.umich.edu. Web.

According to Davis and Mukamal (1078), hypothesis testing is the process of evaluating the strength of the relationship between various sections of the data. It involves the use of instrumental methods to determine the correlations evidenced by the sampled data. When modeling hypothesis tests, there is the preparation of specific hypotheses from the data sampled and the resultant findings are employed to conclude on data relationships. In the hypothesis modeling process, the given situation is in the form of a question, and to investigate it, the question is modified into alternate and null hypotheses. These two statements are comprehensive and jointly exclusive when covering the possible truth of the research question regarding the predictor and the outcome of the population. The null hypothesis clarifies the deficiency involved in the relationship between the forecaster and result whereas the alternate hypothesis illustrates the occurrence of the relationship (Davis and Mukamal 1079).

Moreover, Prins (1384) states that the statistical test determines how the facts displayed by the data can be beyond the question given to support one hypothesis statement as a way of deciding between the presented null and alternate hypothesis by employing the likelihood ratio test. The likelihood ratio test compares two models defined by some constraints statistically. Therefore, the likelihood test functions provide flexibility in testing a variety of questions. For instance, what is the difference statistically between the thresholds in conditions A and B? Nevertheless, these comparisons between models are only legitimate if the assumptions used in both models are accurate (Prins 1384)

However, all tests in statistics have assumptions, and violation of these assumptions makes the statistical test unreliable. For instance, when using a t-test to calculate the hypothesis test about a single population mean, there is a specification of the probability of the variable of the sample. In case there are assumptions violations, the allotment of test statistic t will no longer be t leading to the assessment of p determined to be contradictory. Besides, the assumptions in the parametric test are in theory. In practice, these assumption violations highly affect the model’s outcomes. This means the failure to observe the assumptions, the p-value, formulate the statistical test t invalidly (Quinn and Keuogh 44).

As a result, the following are the assumptions of the t-test and the various ways of inspecting them. Firstly, there is an assumption that samples are taken from a normally distributed population (Mordkoff 1). This assumption exceedingly affects the results of statistics t not unless the distribution, which has occurred, is exceptionally symmetrical. To check the distribution and symmetry of the sample data, there is the use of dotplots regulator, pplots, or boxplots. Moreover, doing variable alteration improves data normality. However, the formal significance normality test is not vastly used because the test is reliant on the size of the sample and may reject the hypothesis in some conditions that t statistical test is still consistent after doing other tests (Quinn and Keuogh 44).

Secondly, there is an assumption applied when deriving the samples from the population with equivalent variances. The level of its variances disturbing the t-test is elevated, and moderates when the extents of samples are the same. To fix the non-normality, the variance must be similar, and investigation occurs to the alike spreads in each boxplot for individual samples. However, it is recommended to accomplish the first-round test when the population variance is equivalent to the f ratio test. This is because the f ratio test is susceptible to non-normality than the cosseted t-test. Additionally, the F test is unsuccessful in sensing the dissimilarity in variance brought up by the coverage of the sample used and this could nullify the t-test or make the t-test results to asymmetrical variances, which will not demonstrate the unpleasant consequence of the failure to examine the assumption on values of t statistic (Quinn and Keuogh 44).

Thirdly, there is an assumption at the phase of creating the data, which involves taking an arbitrary sample from a definite population. If there is no likelihood of sampling haphazardly from the population, then, trying the broad dissimilarity hypothesis for the samples occurs through the randomization test. The used t-test has high sensitive normality assumptions and equal variance when the sizes of the sample are not equal. Consequently, there is a recommendation on the use of samples of equal sizes when designing studies (Quinn and Keuogh 44).

Lastly, there are incidences of outliers in many statistical tests. Outliers are the excessive values from a given sample. Moreover, they are dissimilar to other observational results. Hence, the outliers have tough implications on statistic test in consideration of errors type I and II. Outliers affect both the parametric and non-parametric t-tests even when the tests were undertaken are on their grading (Causineau and Chartier 59). However, ranking reduces the t-test’s vulnerability to outliers (Roberts and Tarassenko 272). Methods that are widely used to identify outliers include Dixons Q test, Grubbs test, Chauvenets criterion, Pierces Criterion, distance, and density-based (Weisstein).

In conclusion, unless for the t-tests that have the capability of making rectifications to violated assumptions, other tests are hard to make necessary corrections. This is because it is always hard to determine the extent of violation of assumption and the value of caused significance changes by the occurrence of the violation (Carpenter 6).

Works Cited

Carpenter, Arthur L. n.d. Simple Test of Hypotheses for Non-statistician: What They are and Why They Can Go Bad. PDF file. Web.

Causineau, Dennis and Sylvain Chartier. “Outliers Detection and Treatment: A Review.” International Journal of Psychology 3.1 (2010): 58-67. Print.

Davis, Roger B. and Kenneth J. Mukamal. “Hypothesis Testing Means: Statistical Primer for Cardiovascular Research”. Circulation. 114 (2006): 1078-1082. Print.

Mordkoff, Toby J. Assumptions of Normality. 2011. PDF file. Web.

Prins, Nicolaas. “Testing Hypothesis Regarding Psychometric functions: Robustness to Violation of Assumptions.” Journal of Vision 10.7 (2010): 1384. Print.

Quinn, Gerry P. and Michael J. Keuogh. Hypothesis Testing. 2001. PDF file. Web.

Roberts, Stephen and Lionel Tarassenko. “A Probabilistic Resource Allocating Network for Novelty Detection.” Neural Computation 6.2 (1995): 270–284. Print.

Weisstein, Eric W. 2013. “Cauchy Distribution” MathWorld. Web.

Abstract

The paper sought to analyze the physiologically-structured population models and their ordinary differential equations reduction. It includes the formulation of a physiologically structured model of the population as a non-autonomous linear integral equation of the population level birth rate (Diekmann et al., 2019). The study seeks to find out when the model that is structured physiologically allows reduction to an ODE that does not involve losing relevant information.

Description of the Problem

There are models which have been developed but have not considered the reduction of a population in a more detailed structured population. The paper seeks to solve the problem of understanding the conditions under which the individual processes against survival, growth, and fission do the developed equations lead to an honest representation of a cell-based model that is size-structured (Diekmann et al., 2019). The particular focus is on cell fission into two parts that are equal. The derivation of an ODE system through the description of the asymptotic time behavior of the population death, reproduction, growth, and depends on the environmental condition, and it is mainly through a single common factor.

The developed problems are as below:

Where:

• X(t) refers to the bacterial biomass and
• S(t) is the substrate concentration in the growth tank at time t

The model was formulated while considering bacterial biomass and not considering issues concerning cells individually. It considered the assumption of results that are substrate in how cells grow and specifically not directly focusing on cell fission. In addition, the issue of ODE-reducibility was considered from different angles and while working under restrictions that are not similar. It analyzed the birthrate at the population level while considering the assumption of a number that is finite specifically for states-at-birth and also at dynamics which are at the individual state. Furthermore, the paper gives new examples, particularly working with division or fission into two parts that are equal. There is also the promotion of equation formation that is the renewal of a structured population that is physiological in nature.

Methods Used

There are several methods used during the study in achieving the desired results. The first method used is the constructive method. It was developed for solving first order partial differential equations without non-local terms. The methods, therefore, solve the equations through generation expansion as applied in the renewal equations theory. The renewal equation took center stage through the use of birth rate history as a state. It is also through dynamical system definition through history updating (Diekmann et al., 2019). This paper, therefore, focuses on renewal equations as the beginning point in modelling populations that are physiologically structured.

Another method is through computation that involves elimination of everything that is non-linear in the combination of the present components. It, in most cases, also involves E-dependent coefficients in the process. This is followed by finding out if the remainder can be written as a linear combination of new functions of x, which is then added to the new functions as additional components (Diekmann et al., 2019). This is a generally acceptable ODE test for reducibility that begins with output functionals collection, which one requires.

New Results

The paper showed the renewal equation leads to the condition for ODE reducibility. There are equations that involve per capita growth, rates of reproduction, and death. The analysis of k=1 and k=2 specifically for fission into two equal parts, the researchers reviewed a paper submitted by Diekmann et al and realized that it was completely under specific restrictions (Diekmann et al., 2019). The paper also found the connection between the example by Diekmann (1983), the equation indicated below;

,and the example of Metz and Diekmann (1986), thus allowing for the reduction in large time behavior to an ODE. It needs proper dedication of resources for the suitable solution to occur; otherwise, the topic is open for anyone interested to pursue further studies. Scholars are also encouraged to share new findings in the field so that more researchers can utilize the literature in their studies.

Possible Extensions

Researchers should focus more on ODE reducibility as it is a field that has not been fully exploited. Its challenging nature makes it difficult for most researchers to study. This move leaves a gap that needs to be filled by new research. This, therefore, encourages researchers to venture into the field and seek out answers for the problems. Such a move contributes to knowledge that can be used in future studies. The ordinary differential equation can also be utilized by other fields’ expertise in solving the challenges they face. The reduction of order method is an area where most researchers have not fully ventured and explored. The ODE is continually evolving to make the field grow; hence any little contribution by any research can have a significant impact in this field.

Reference

Diekmann, O., Gyllenberg, M., & Metz, J. A. (2019). On models of physiologically structured populations and their reduction to ordinary differential equations. Journal of Mathematical Biology, 80(1-2), 189-204. Web.

This article presents a significant research process with essential findings concerning the consistent prevalence of heroin among Puerto Ricans. The qualitative study integrates focus groups as the main procedure for gathering information. This research process reveals that Puerto Ricans suffering from substance abuse depict close similarities in their social backgrounds (Senreich, 2018). Aspects such as unemployment, discrimination, and denial of opportunities in public institutions result in persistently high rates of heroin use. The findings obtained from the focus groups can be helpful in evidence-based practice (EBP) among social workers (Cano, 2020). For instance, the knowledge acquired can help implement practical solutions against the vice among identified victims. In this case, social workers’ knowledge concerning diagnosis and treatment improves, indicating competence among social workers (Senreich, 2018). Most importantly, the article’s integration in EBP is critical for controlling negative social attributes that contribute to the high prevalence of substance abuse among Puerto Ricans.

Several benefits exist from using the qualitative method identified in the article. The researcher’s purpose was to identify the causes of the prevalent use of heroin among a selected group of participants. The researchers save costs and time by implementing effective data gathering approaches (Senreich, 2018). In this case, the article integrated focus groups, including Puerto Ricans engaging in substance abuse and counselors. Participants and research officials engaged informal meetings aimed at identifying inherent reasons supporting substance abuse among minority populations (Cano, 2020). A qualitative method is also beneficial as it is flexible and open-ended. It implies that the procedure used to collect, analyze, and interpret data can present significant findings not expected (Senreich, 2018). In this case, the data gathering process allowed the researchers to integrate attitudes and behavioral attributes contributing to substance abuse among Puerto Ricans.

An objective research study integrates both qualitative and quantitative methods for accurate and relevant findings. In this article, a quantitative approach would have complemented the qualitative method used in identifying high rates of heroin use among Puerto Ricans. For instance, identifying the affected category of the population using the drugs would help develop practical conclusions and recommendations (Senreich, 2018). The article notes that most substance abusers are dropout teenagers and young adults facing unemployment, discrimination, and systemic marginalization. Additionally, qualitative data regarding the total number of individuals engaging in heroin use would help develop social solutions to immoral practices (Cano, 2020). For instance, educational initiatives intended to improve access to academic institutions by minorities would require numerical statistics (Senreich, 2018). Most importantly, the quantitative approach would also complement the qualitative method’s findings by measuring the impact of findings, conclusions, and recommendations.

References

Cano, M. (2020). English use/proficiency, ethnic discrimination, and alcohol use disorder in Hispanic immigrants. Social Psychiatry and Psychiatric Epidemiology, 1-10.

Senreich, E. (2018). The persistent high rates of heroin use among the Puerto Rican population in the United States: A qualitative study. Social Work in Public Health, 33(7-8), 419-438.

This paper aims to discuss the problems, faced by underserved population and its various subgroups in Bayou Region of Louisiana. Although Bayou Region is a real location with its own demographic peculiarities, the analysis is based on a hypothetical scenario which throws light on some problems of the people, living in this place or any other sparsely-populated area of the United States. Yet, these facts, presented in this scenario are not always accurate.

One of the major heath care issues is that a great number of people cannot get instantaneous access to medical services as many of them live in remote villages or reservations while physician medical clinics are located mostly in towns. Consequently, we can presume that very often patients cannot receive appropriate treatment in a timely manner. In this case, we need to speak about the entire community. According to the information, which has been provided, there are more than twenty health-care institutions but they are not organized in a network. Moreover, their number seems to be insufficient.

Furthermore, we need to say that a large portion of the population is only half-employed. Their insurance cannot cover all the expenses, and some of them have practically no financial resources to afford medical services. Another segment of the community which requires close attention is homeless people and their children. Their problems are manifold: first, they are forced to live in insanitary conditions which can give rise to many diseases. Secondly, we can argue that this layer of the society always has to fight against bureaucracy and continuous paper work and it seems that Bayou Region is not an exception from this rule.

Finally, it should be mentioned that the health care professionals of BR experience significance difficulties in screening at-risk groups. Statistical surveys indicate significant rates of depression, post-traumatic stress disorder, schizophrenia and substance abuse and alcohol dependence among the residents of Bayou Region. This phenomenon may be explained by the fact that there is no specific action plan for monitoring the community and promoting healthy lifestyle. Thus, we can conclude that the delivery of health care services in this region is hindered by underdeveloped infrastructure, low income level and bureaucratic mechanisms which are very cumbersome.

On the whole, these problems are familiar to many residents of remote areas. The case of Bayou Region is just another example. Some scholars confirm our initial hypothesis that one of the most issues is screening and prevention of diseases (Kilpatrick & Johnson, 1999, p 427). Medical workers are unable to monitor all segments of the population. Additionally, several studies report that the members of such communities have a lower utilization of mental health services and this eventually increases the number of patients, suffering from psychological disorders (Kilpatrick & Johnson, 1999, p 171).

Under such circumstances, the role of community nurses becomes crucial but according to the recent statistical data the US health care system experiences acute shortage of these medical workers (Isaacs & Knickman, 2004). Many of them are unwilling to work in rural areas because they have practically no opportunities for promotion. Moreover, their wages are not very high. All of these problems prevent patients from receiving appropriate treatment.

It appears that the US health care officials have not developed any specific policies for helping the residents of rural areas. It is necessary to launch outreach programs in order to give assistance and consultations to the underserved segments of population. Unfortunately, this is not always done. Certainly, we need to take into account scheduling process as many members of the staff cannot possibly leave hospitals and it is utterly impermissible to put them off their duties. Again, we have to emphasize the idea that current financing of such outreach programs is inadequate.

In sharp contrast, urban dwellers have to struggle against different difficulties. For instance, in such as New York, healthcare providers complain that hospitals are clearly overcrowded. Practitioners and nurses cannot deal with the increasing inflow of patients (Ginzberg et al, 1997, p 105). In this regard, we need to remember the number of illegal immigrants in large cities. Undoubtedly, they must not be overlooked or ignored by medical workers.

They do everything within their power but it has to be admitted that at this stage they cannot help everyone. Apart from that, a great number of urban dwellers are only partially insured. Homelessness is another stressing issue in urban areas (Ginzberg et al, 1997). In densely populated cities, housing is even less affordable that in such places as Bayou Region. It should be borne in mind that current economic recession has made many people redundant, and they cannot pay for the insurances. In turn, all of these factors impose a heavy burden on health care organizations.

Judging from this discussion, we can come to the conclusion that the problems of underserved population in the United States are relatively the same. First, there are practically no screening procedures, which can identify the disease at early stages or even prevent it. Secondly, bureaucratic structure of many health care organizations frequently causes unnecessary delays. Finally, many hospitals are understaffed, especially if we are speaking about nurses. These are the major reasons why the percentage of underserved population is so high.

Reference List

Ginzberg E., Berliner, H, S, & Ostow M. (1997). Improving health care of the poor: the New York City experience. New York: Transaction Publishers.

Isaacs. S. L & Knickman J.R. (2004). Generalist Medicine and the U.S. Health Care System. San Francisco: Jossey-Bass.

Kilpatrick, A. O, & Johnson, J.A. (1999) Handbook of health administration and policy. New York: CRC Press.

Louisiana House of Representatives. (2007). State and Local Government in Louisiana: An Overview. Web.

Perry, A, M. (2009) Bayou Region. LED (Louisiana Economic Development). Web.

Performing quantitative research is impossible without gathering and analyzing appropriate data. However, the research process may become complicated due to possible confusion of population with a sample, since both notions can be erroneously perceived as a group subjected to research. Therefore, it is necessary to understand an important difference — while population refers to an entire group, the sample encompasses only a specific group selected within the population for research purposes. Consequently, sampling can be defined as a method used to select a required sample from the whole population.

Several sampling methods are empirically distinguishable and can be used depending on the researcher’s needs. In general, all commonly used methods can be separated into two techniques — probability and non-probability sampling (Sharma, 2017). Probability sampling utilizes random selection from the population, which makes it more accurate, whereas non-probability sampling is based on judgment. Furthermore, probability-based methods can be divided into simple random sampling, systematic sampling, stratified sampling, and cluster sampling (Sharma, 2017). All these methods have their advantages and flaws — for instance, simple random, an independent selection of subjects, is easy and unbiased but limited to smaller populations. On the other hand, the method of cluster sampling is feasible for large populations since it uses naturally occurring groups as subjects (Sharma, 2017). However, it is also highly susceptible to selection bias and sampling errors.

The same situation applies to various non-probability sampling schemes. Quota, purposive, self-selection, and snowball sampling offer certain advantages, such as speed, the possibility to make generalizations, or access to hidden populations (Sharma, 2017). However, this utility comes at the increased cost of research bias and sampling errors. Regardless of an appropriate sampling technique, one should understand that sample consists only of a fraction selected from the population. In this regard, sampling must be perceived as a means of extracting the necessary fraction from the whole group.

Reference

Sharma, G. (2017). Pros and cons of different sampling techniques. International Journal of Applied Research, 3(7), 749-752.

Point and Confidence Interval Estimates for a Population Mean

A confidence interval (CI) is used to indicate the reliability of an estimate. There are two types of CIs, that is, interval estimate and point estimate, and both of these can be used to estimate the mean for a population sample. These two forms of confidence intervals are computed from observations and differ from dataset to dataset, but frequently include the parameter of interest, and should be obtained within specified errors of margin if the experiment is repeated. An error of margin is a parameter that is used to indicate the random sampling error in an experimental survey. The larger the value, the less reliable the estimate for the parameter is, in this case the mean.

Interval estimates for a sample mean can be contrasted from point estimates for the population mean, for example, the value could be given as a 95% confidence interval. A point estimate is a single figure that is given as an approximation of the for a population mean of some observation or statistical experiment. In contrast, an interval estimate specifies a range within which the approximated mean of the population lies. Both of these estimates are used to indicate the reliability of a population.

Since the estimate for the population mean is applicable to the whole population, it must have certain characteristics, i.e. it must be consistent within the whole population, it must be unbiased (must be centered on the actual population mean), and it must have a small standard error (the lower the error, the more reliable it is). For example, if we have a population sample (x₁, x₂… x_n), a point estimate for the mean, say θ, would be one that can be used in place of the individual observations. The estimator can be rejected if it cannot be used in place many observations (Groebner, 2007).

An interval estimate gives more information about a population characteristic than a point estimate. Confidence intervals are sometimes referred to as confidence intervals. The general formula for computing the confidence value is:

• Point Estimate ± (Critical Value)(Standard Error)

In order to obtain the confidence interval for the mean when δ is known, we use the following equation:

Consider a 95% confidence interval, in this case,

Determining the Required Sample Size for Estimating a Population Mean

In order to find an estimator for the population mean, the surveyor must use a minimum population size that is based on certain parameters of the population. Before a surveyor picks a sample size, he must know all or part of the following: the standard deviation, δ, of the population, the maximum acceptable difference (margin of error), the mean, and the desired confidence level. In most sample surveys, the surveyor will not have the exact value of δ. Generally, the standard deviation of the population will be approximated from the following: findings of a previous study, the value from a pilot survey, from secondary data, or from the surveyor’s assessment of the dataset. The maximum acceptable difference is the maximum error that the researcher can accept, that is, the maximum deviation of the estimate from the true mean that is permitted. The desired confidence level is the degree of confidence of the surveyor that the estimate of the mean does not deviate from the true mean by a significant value, or by more than the maximum acceptable deviation. Generally, most studies use a 95% confidence level.

The margin of error (e), is added or subtracted from the point estimate to find the value at any point.

Given the margin of error, e, and level of confidence, (1 – α), the required sample size can be obtained using the following formula,

For example, if the standard deviation is 53, the level of confidence is 95% and the margin of error is ±5, then the required sample size, n, is found as:

Hence, the required sample size, which must be a whole number, is estimated as 304.

If δ is unknown, it can be approximated through the sample size method, i.e. use a δ that is thought to be as close as possible to the true δ, or, select a value of δ that is derived from a pilot survey, as outlined above (Groebner, 2007).

Estimating a Population Proportion

An interval estimate for the proportion of a population that follows a normal distribution, p, can be computed by adding the margin of uncertainty to the sample proportion, Ρ. Recalling that a population is normally distributed if the sample size is large, and the standard deviation is given as:

We can approximate this with the sample data:

The upper and lower confidence limits for the population proportion are computed using the following formula:

Where,

z is the standard normal value for the level of confidence desired

is the sample proportion, and n is the sample size.

The maximum error of the population proportion is obtained by adding or subtracting the margin or error from both sides, given by the formula:

Hence, P lies between – e < P< + e

Work Cited

Groebner, David F. Business Statistics: A Decision-Making Approach. New Jersey: Prentice-Hall, Inc., 2007.

When planning to conduct a scientific study, it is important to calculate the sample size. From a practical perspective, it is not economical to study all members of a population. Thus, only a few participants are chosen. Although a smaller number in comparison with the population typifies the sample size, it is utilized to represent the whole populace. Thus, a population can be defined as a complete set of elements or participants from which a few can be included in a study sample. An important aspect to realize is that the findings obtained from a sample can be used to make conclusions about the whole populace.

The formula that is applied in determining sample size (n) does not involve the population size (N). The observation is true in the context of statistics, but it is not true from a business point of view. The formula is as follows:

Where:

• n- Sample size in a study
• – Variance of the population
• Z²– Variance/Error²
• Error²– Square of error

In the above formula, the term population size (N) does not appear. It is apparent that N could have an impact on the sample size by influencing other terms in the formula. The first term that is affected by the population size is the variance, which is used to imply the extent to which data elements of a sample are spread out in a distribution. Relatively small values of variance indicate that data in a sample distribution are close to the anticipated mean. On the other hand, relatively high values of the variance imply that data are spread out, i.e., they are not close to the expected sample mean. Large populations lead to small variances, while small populations result in high variances. Thus, it would be important for business researchers to consider the size of the populace because it would affect the distribution of data elements in relation to the expected mean. For example, populace sizes of 14 million and 20 million households would give different variances, which would impact the sample mean and conclusions about the population. In fact, the populace size of 14 million households would lead to a bigger variance than would be obtained from the populace size of 20 million households.

The square of error (Error²) in the formula would also be impacted by the populace size (N). In many studies, scientists expect some deviations (errors), which can negatively impact the study findings if they are not controlled. It is worth to note that errors can be accepted to some extent, beyond which they make the study results unreliable and invalid. From a practical perspective, the populace size positively correlates with the square of errors. Thus, small populations are associated with relatively large errors, while large populations are associated with relatively small errors that greatly improve the reliability levels of the findings. For example, a population size of 20 million households would have a smaller error than a populace size of 14 million households.

Therefore, it appears that researchers in the field of business should aim at using large populations from which they would select samples. Large populations should give rise to large sample sizes and vice-versa. From a marketing point of view, it is important to use large populations to minimize the chances of making errors that would lead to wrong conclusions about dynamics in the market.

It has being noted that teaching mathematics to liberal arts students is a major problem. This has being caused by poor teaching motivation methods embraced by their teachers. These students have negative attitude towards the subject and often excels poorly in this subject. Therefore there is a great need for the teaching profession to find effective ways they can motivate their students in mathematics to reinforce these students interests in mathematics to improve their mathematics performances. This can well be achieved if some mathematical courses are introduced in liberal arts. This will help greatly these group of students to acquire the fundamental mathematics concepts that of great importance in their day to day lives. This will allow the liberal arts students being well prepared in this subject to develop a positive attitude towards the subject that will yield to excellence performances in mathematics. (LeGere1991).This can be achieved through: (1)Relating mathematics to contemporary significant issues; (2) Employing projects and open- ended problems;

According to Schwartz 1992, the greatest problem area in mathematics education today is effectively teaching liberal arts/non-science majors. These students generally are poorly prepared in mathematics and have negative attitudes about the subject. Every teacher of liberal arts mathematics courses has probably heard students ask questions such as, “Why do I need to learn this mathematics?”, “When will I ever use this material?”, and “What connections are there between this mathematics and my life?” Traditionally, liberal arts/non-science mathematics courses have had a set of unrelated topics, such as set theory, logic, probability and statistics, and number theory. Recently, modern topics, such as management theory, social choice, and elementary computer applications have replaced some of the older topics. Generally, these courses have been taught by the typical lecture-discussion method.

The lack of coherence between topics, the perceived irrelevance of the material, and the scarcity of opportunities for student involvement often result in apathetic classes. Classes are typically large, for budgetary reasons, and this contributes to the problem. Since the students need the course only to fulfill a graduation requirement and not as a prerequisite for any other course, they usually try to obtain a satisfactory grade with a minimum of effort. As a result teachers are often reluctant to teach such courses. Students generally end the course with their negative feelings reinforced, thankful that they will never again be expected to do any mathematics. To remove this problem and stimulate liberal students to perform well in mathematics effective teaching methodologies are needed.(Herbert1980)

Teachers should relate mathematics to prevailing important issues while teaching liberal arts. This is although the non-science students are often opposed to to mathematics, they are generally apprehensive with environmental and other social issues. Hence, their interest can be stimulated by a course that relates fundamental mathematical concepts and exercises to such important current issues as population growth, resource scarcity, international relations, hunger, the arms race, health concerns, such as AIDS and nutrition, and a wide variety of environmental issues, including air and water pollution, acid precipitation, ozone layer depletion, destruction of tropical rain forests and other habitats, potential global warming, and soil erosion and depletion.

Therefore mathematical courses that involve exercises that use real- world data connected to issues that evident on every day life. It can be a very positive move for motivating these in mathematics. The mathematics course should encompass the use of significant data such as data on ecological issues to add to class interest, the course can be related to current newspaper and magazine articles and TV and radio reports and programs. Use of news reports concerning international matter shows the students the significance of their erudition. The course can also be related to events such as Earth Day (an annual event in many countries), and U. N. conferences related to population, hunger, ecology, resources, refugees, and the arms race. The many news reports about these conferences provide an abundance of data that can be converted into mathematical exercises. Films, videotapes, and slide shows can also be used to broaden coverage of issues. (Johnson1983)

There are many sources for mathematical exercises related to current issues. In addition to the graphs, charts, and statistics in daily newspapers and weekly newsmagazines, groups concerned about global threats are excellent sources of material. For example, the Population Reference Bureau produces an annual World Population Data Sheet which contains statistics for the worlds nations and regions for birth and death rates, per capita GNPs (gross national products), population doubling times, infant mortality rates, and other demographic data, that can be used for problems on ratios, percents, averages, and correlations, and for the construction of various graphs and charts. The annual Statistical Abstract of the United States also provides a variety of graphs and charts from which many mathematical exercises can be constructed. Material on particular countries can be obtained from their ministries. (Gross 1993)

Of late there have been increased published materials that offer data, charts, and graphs that can be extremely supportive for generating mathematical paradigm and models in liberal arts teaching. These materials ranges from the annual editions of The State of the World and Vital Signs, both produced by the World Watch Institute, the annual Earth Journal, produced by Buzzword Magazine, and the annual Information Please Environmental Almanac, produced by the World Resources Institute.

By employing such teaching resources in liberal arts teaching will greatly improve the attitude of these students on mathematics that will result to good performances. For instance when teaching the relation ship of high population effects on national resources. Teachers can give the learners some touchable basic mathematical examples such calculating determining how many squire feet of rain forests are destroyed to produce a half pound hamburger. Also the teacher should assists the learners to see the energy consumption of their countries by drawing line graphs using data from statistics from appropriate sites. The use of this real examples in teaching the liberal arts mathematics will help them to create a positive attitude towards the subject since they will anxious to know quantitatively impacts of what they learn in their day to day lives. Thus leading to good mathematics performances for this group of students. (George 1977, 45)

Use of projects and open-ended problems linking mathematics to ecological issues provides many opportunities for open-ended problems and projects. This can entail asking students to gather data from surveys they have organized and carried to carry some calculations, draw graphs. The data can either be that of the population of people in a certain market or the population of people who are benefiting from relief food from a specific place. (Brown 1998, 199).I recommends teachers who teach mathematics to liberal arts to try these teaching strategies and test how the performances of their students will greatly improve.

References

Brown, L. (Ed.): State of the World 1998. – New York: W. W. Norton.

George, R.: The Mathematics of the Energy Crisis. – Westmond, N. J: Intergalactic, 1977.

Gross, F.: The Power of Numbers: A Teacher’s Guide to Mathematics in a Social Science Context. – Cambridge, MA: Educators for Social Responsibility, 1993.

Herbert, L.; Ferleger, L.: Statistics for Social Change. – Boston: South End Press, 1980.

Johnson, M.: Writing in Mathematics Classes: A Valuable Tool for Learning. – In: Mathematics Teacher 76 (1983).

Schwartz, R. H.: Revitalizing Liberal Arts Mathematics. – In: Mathematics and Computer Education Journal (1992).

Bibliography

1. LeGere, A.: Collaboration and Writing in the Mathematics Classroom. – In: Mathematics Teacher (1991).
2. Schoenfeld,A. (Ed.): A Source Book for College Mathematics Teaching. – The Mathematics Association of America, Washington, D. C. , 1990.
3. Schwartz, R. H.: Population, Tree-Diagrams and Infinite Series. – In: UMAP Consortium Newsletter (1992).
4. Schwartz, R. H.: Mathematics and Global Survival. – Needham Heights, MA: Ginn Press, 1998 (4th edition).