11.5: Global Production
11.5.1 Hegemony and Economic Ascendency
At times industrialization has propelled countries to great economic heights. Britain, The United States and Japan all rode an industrial wave to international prominence. In those countries and others, a (largely mythical) golden age centers around a time when low-skilled workers could earn a sufficient wage to secure economic security. This is more-or-less what the “American Dream” was. Deindustrialization has changed the economic trajectories of these countries and the people living in these countries. However, it must be noted that post-industrial countries that have not seen rapid increases in poverty. Wages have been largely stagnant for decades, but they have not generally gone down. The largest difference has to do with relative prosperity for industrial and post-industrial countries. Countries such as Japan, the UK or the US are no longer far wealthier than their neighbors. In the same way that flooding a market with a particular product reduces the value of that product, flooding the world with industrial capacity lowersthe relative value of that activity. Developing countries function as appendages to the larger economies in the world. The poor serve the needs of the wealthy. Unindustrialized countries buy goods from developed countries, or they license or copy technology and make the products themselves.
11.5.2 Space and Production
In the context of a globalized market, a factory built in one market may not built in another. This is not to say that producing goods is a zero-sum game, but there are limits to the amount of any good that can be sold. It’s a valid question to ask why transnational corporations (TNCs) have bought into China at rates far greater than in Cuba, Russia, or other Communist or formerly Communist (to varying degrees) countries? There is only so much spare capacity for production in the world. If one giant country (China) is taking all the extra capacity, then there will be none left for others. FDI is simply easier in China, since there is more bang for the buck. This is largely a function of population. The population of China is roughly two times the population of Sub Saharan Africa. And China has a single political/economic running class, as opposed to 55 different sets of often fractious political classes. If the industrialization of Africa happens at all, it will occur after China and its immediate neighbors who have been drawn into its larger economic functioning have largely finished their own industrializing. An example of this proximate effect is seen in the shift of some industries from China to Vietnam and Indonesia.
China’s industrialization had to do with promoting itself as a huge cheap labor pool, and as a gigantic market for goods. It successfully leveraged both of these characteristics to attract foreign investment, and to gain foreign technology from the companies that have invested in producing goods there. Industrialization overall seems to have slowed. The speed at which China industrialized has not been matched by other countries following China. One current idea is that the world is in a race between industrial expansion and rapid over capacity of production. In other words, the reason that industrialization isn’t expanding as rapidly as before is that we are already making enough goods to satisfy demand. Remember that goods require demand. Unsold goods don’t produce any income. If the factories in the world are already producing enough, or even too much, then new factories are much less likely to be built. Technological advances and the massive industrialization of China might have ended the expansion of industry.
It also appears that the highest levels of manufacturing income are well in the past. According to economist Dani Rodrik, the highest per capita incomes from manufacturing occurred between 1965 and 1975, and has fallen dramatically since then. This is even considering inflation. Many countries industrializing now only see modest improvements in income. This is related to supply and demand. When there are fewer factories, they make comparatively more money. When factories are everywhere, they are competing with everyone.
11.5.3 Trade
Even more than expansion of industrial production, the world has seen an expansion of trade. Global trade has produced an intricate web of exchanges as products are now designed in one country, parts are produced in 10 others, assembled in yet another country, and then marketed to the world. Consider something as complex as an automobile. The parts of a car can be sourced from any of dozens of countries, but they all have to be brought to one spot for assembly. Such coordination would have been impossible in the past.
Individuals can buy directly from another country on the internet, but most international trade is business-to-business. TNCs are able to conduct an internal form of international trade in goods that can be moved and produced in a way that is most advantageous for the company. Tax breaks, easy credit and banking privacy laws exist to siphon investment from one place to another.
Because of global trade, improvements in communication and transportation have enabled some companies to enact just in time delivery, in which the parts need for a product only arrive right before they are needed. The advantage of this is that a company has less money trapped in components in a storage facility, and it becomes easier to adjust production. Once again, such coordination at a global scale was not possible even in the relatively recent past.
11.5.4 Deindustrialization
Historically industrialized countries were the wealthy countries of the world. Industrialization, however, is now two centuries old. In the last decades of the twentieth century, deindustrialization began in earnest in the United Kingdom, the United States, and many other places. Factories left and the old jobs left with them. Classical economics holds that such jobs had become less valuable, and that moving them offshore was a good deal for everyone. Offshored goods were cheaper for consumers, and the lost jobs were replaced by better jobs. The problem with this idea is that it separates the condition of being a consumer from the condition of being a worker. Most people in any economy are workers. They can only consume as long as they have an income, and that is tied to their ability to work. Many workers whose jobs went elsewhere found that their new jobs paid less than their old jobs.
11.5.5 What Happens After Deindustrialization?
The simple answer to the above question is this. The service economy happens! As manufacturing provides fewer jobs, service industries tend to create new jobs. This is a very delicate balance, however. If you are a 50-year old coal minerwhose job has been eliminated by automation, it is very difficult for you to simply change jobs and enter the ‘service sector’. This transition is very damaging to those without the right skills, training, education, or geographic location. Many parts of the American Midwest, for example, have become known as the “rust belt” as industrial facilities closed, decayed, and literally rusted to the horror of those residents who once had good jobs there. The city of Detroit, for example, lost nearly half of its urban population from 1970-2010. Meanwhile the state of Illinois loses one resident every 15 minutes as job growth has weakened in the post-industrial age. However, job growth and productivity in the service economy have strengthened and provide more job opportunities today than the industrial era ever did in the U.S.
There are 3 sectors to every economy:
- Primary (agriculture, fishing, and mining)
- Secondary (manufacturing and construction)
- Tertiary (service related jobs)
The vast majority of economic growth in the post-industrialized world comes in the tertiary sector. This doesn’t mean that all tertiary jobs pay well. Just ask any fast-food worker if the service sector is making them rich! However, service sector jobs are very dynamic and offer tangible opportunities to millions of people around the world to earn a living providing services to somebody else. We can further break down the service sector into 1) public (the post office, public utilities, working for the government) 2) business (businesses providing services to other businesses) 3) consumer (anything that provides a service to a private consumer e.g. hotels, restaurants, barber shops, mechanics, financial services). Traditionally, service sector jobs worked very much like manufacturing jobs in that employees worked regular hours, earned benefits from the employer, gained raises through increased performance, and went to work somewhere outside of their home. Many service jobs in the 21st century, however, have been categorized as the gig economy , in which workers serve as contractors (rather than employees), have no regular work schedules, don’t earn benefits, and often work in isolation from other workers rather than as a part of a team. Examples of ‘jobs’ in the gig economy include private tutor, Uber/Lyft driver, AirBNB host, blogger, and YouTuber. Work, in this economy, is not necessarily bound by particular places and spaces in the way that it did in manufacturing. Imagine a steel worker calling in to tell his/her boss that they’re just going to work from home today! Even public schools have adapted to this model in the following manner. As schools cancel class due to weather, the new norm is to hold class online, whereby students do independent work submitted to the teacher even though nobody is at school. As such, some workers are freed up from the traditional constraints of time and place and can choose to live anywhere as long as they maintain access to a computer and the Internet. Services like fiverr. com facilitate a marketplace for freelance writers compose essays for others or for graphic artists to sell their design ideas directly to customers without every meeting one another.
The global marketplace continues to be defined as a place where the traditional relationships between employer and employee are changing dramatically. A word of caution is necessary here, however. As many choose to celebrate the freedom that accompanies flexible work schedules, there is also a darker side in that the traditional ‘contract’ and social cohesive element between workers and owners is very much at risk. One defining factor of the 20th century was the development of civil society that fought for and won a host of protective measures for workers, who otherwise could face abusive work conditions. Child labor laws, minimum wage, environmental safety measures, overtime pay, and guards against discrimination were all based upon an employer-employee relationship that seems increasingly threatened by the gig economy. Uber drivers can work themselves to exhaustion since they are not employees. AirBNB hosts can skirt environmental safety precautions since they do not face the same safety inspections required at hotels. These are just a few examples, but they are very worth consideration. Regardless of the positives and negatives, the new service economy is having a transformative effect upon all facets of society. Although, the authors of this textbook are all geography professors with PhD’s from a variety of universities, perhaps the next version of this textbook will simply draw upon the gig economy to seek the lowest cost authors who are willing to write about all things geographical. Will you be able to tell the difference? (We hope so!!!)


15.6: Eukaryotic Transcription - Initiation of Transcription in Eukaryotes
- Describe how transcription is initiated and proceeds along the DNA strand
Steps in Eukaryotic Transcription
Eukaryotic transcription is carried out in the nucleus of the cell by one of three RNA polymerases, depending on the RNA being transcribed, and proceeds in three sequential stages:
- Initiation
- Elongation
- Termination.
Initiation of Transcription in Eukaryotes
Unlike the prokaryotic RNA polymerase that can bind to a DNA template on its own, eukaryotes require several other proteins, called transcription factors, to first bind to the promoter region and then help recruit the appropriate polymerase. The completed assembly of transcription factors and RNA polymerase bind to the promoter, forming a transcription pre-initiation complex (PIC).
The most-extensively studied core promoter element in eukaryotes is a short DNA sequence known as a TATA box, found 25-30 base pairs upstream from the start site of transcription. Only about 10-15% of mammalian genes contain TATA boxes, while the rest contain other core promoter elements, but the mechanisms by which transcription is initiated at promoters with TATA boxes is well characterized.
The TATA box, as a core promoter element, is the binding site for a transcription factor known as TATA-binding protein (TBP), which is itself a subunit of another transcription factor: Transcription Factor II D (TFIID). After TFIID binds to the TATA box via the TBP, five more transcription factors and RNA polymerase combine around the TATA box in a series of stages to form a pre-initiation complex. One transcription factor, Transcription Factor II H (TFIIH), is involved in separating opposing strands of double-stranded DNA to provide the RNA Polymerase access to a single-stranded DNA template. However, only a low, or basal, rate of transcription is driven by the pre-initiation complex alone. Other proteins known as activators and repressors, along with any associated coactivators or corepressors, are responsible for modulating transcription rate. Activator proteins increase the transcription rate, and repressor proteins decrease the transcription rate.
The Three Eukaryotic RNA Polymerases (RNAPs)
The features of eukaryotic mRNA synthesis are markedly more complex those of prokaryotes. Instead of a single polymerase comprising five subunits, the eukaryotes have three polymerases that are each made up of 10 subunits or more. Each eukaryotic polymerase also requires a distinct set of transcription factors to bring it to the DNA template.
RNA polymerase I is located in the nucleolus, a specialized nuclear substructure in which ribosomal RNA (rRNA) is transcribed, processed, and assembled into ribosomes. The rRNA molecules are considered structural RNAs because they have a cellular role but are not translated into protein. The rRNAs are components of the ribosome and are essential to the process of translation. RNA polymerase I synthesizes all of the rRNAs except for the 5S rRNA molecule.
RNA polymerase II is located in the nucleus and synthesizes all protein-coding nuclear pre-mRNAs. Eukaryotic pre-mRNAs undergo extensive processing after transcription, but before translation. RNA polymerase II is responsible for transcribing the overwhelming majority of eukaryotic genes, including all of the protein-encoding genes which ultimately are translated into proteins and genes for several types of regulatory RNAs, including microRNAs (miRNAs) and long-coding RNAs (lncRNAs).
RNA polymerase III is also located in the nucleus. This polymerase transcribes a variety of structural RNAs that includes the 5S pre-rRNA, transfer pre-RNAs (pre-tRNAs), and small nuclear pre-RNAs. The tRNAs have a critical role in translation: they serve as the adaptor molecules between the mRNA template and the growing polypeptide chain. Small nuclear RNAs have a variety of functions, including “splicing” pre-mRNAs and regulating transcription factors. Not all miRNAs are transcribed by RNA Polymerase II, RNA Polymerase III transcribes some of them.
Modeling transcription : This interactive models the process of DNA transcription in a eukaryotic cell.
Key Points
- Eukaryotic transcription is carried out in the nucleus of the cell and proceeds in three sequential stages: initiation, elongation, and termination.
- Eukaryotes require transcription factors to first bind to the promoter region and then help recruit the appropriate polymerase.
- RNA Polymerase II is the polymerase responsible for transcribing mRNA.
Key Terms
- repressor : any protein that binds to DNA and thus regulates the expression of genes by decreasing the rate of transcription
- activator : any chemical or agent which regulates one or more genes by increasing the rate of transcription
- polymerase : any of various enzymes that catalyze the formation of polymers of DNA or RNA using an existing strand of DNA or RNA as a template


21.2.2: Plotting Points and Intervals on the Number Line
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Learning Outcomes
- Plot a point on the number line
- Plot an interval on the number line
The number line is of fundamental importance and is used repeatedly in statistics. It is a tool to visualize all of the possible outcomes of a study and to organize the results of the study. Often a diagram is placed above the number line to provide us with a picture of the results. By the end of this section, you will be able to plot points and intervals on a number line and use these plots to understand the possible outcomes and actual outcomes of studies.
Drawing Points on a Number Line
A number line is just a horizontal line that is used to display all the possible outcomes. It is similar to a ruler in that it helps us describe and compare numbers. Similar to a ruler that can be marked with many different scales such as inches or centimeters, we get to choose the scale of the number line and where the center is.
Example \(\PageIndex{1}\)
The standard normal distribution is plotted above a number line. The most important values are the integers between -3 and 3. The number 0 is both the mean (average) and median (center).
- Plot the number line that best displays this information.
- Plot the value -1.45 on this number line.
Solution
- We sketch a line, mark 0 as the center, and label the numbers -3, -2, -1, 0, 1, 2, 3 from left to right.
- To plot the point -1.45, we first have to understand that this number is between -1 and -2. It is close to half way between -1 and -2. We put a circle on the number line that is close to halfway between these values as shown below.
Example \(\PageIndex{2}\)
When working with box plots, we need to first set up a number line that labels what is called the five point summary: Minimum, First Quartile, Median, Third Quartile, and Maximum. Suppose the five point summary for height in inches for a basketball team is: 72,74,78,83,89. Plot these points on a number line
Solution
When plotting points on a number line, we first have to decide what range of the line we want to show in order to best display the points that appear. Technically all numbers are on every number line, but that does not mean we show all numbers. In this example, the numbers are all between 70 and 90, so we certainly don't need to display the number 0. A good idea is to let 70 be on the far left and 90 be on the far right and then plot the points between them. We also have to decide on the spacing of the tick marks. Since the range from 70 to 90 is 20, this may be too many numbers to display. Instead we might want to count by 5's. Below is the number line that shows the numbers 70 to 90 and counts by 5's. The five point summary is plotted on this line.
Exercise
A histogram will be drawn to display the annual income that experienced registered nurses make. The boundaries of the bars of the histogram are: $81,000, $108,000, $135,000, $162,000, and $189,000. Plot these points on a number line.
Plotting an Interval on a Number Line
Often in statistics, instead of just having to plot a few points on a number line, we need to instead plot a whole interval on the number line. This is especially useful when we want to exhibit a range of values between two numbers, to the left of a number or to the right of a number.
Example \(\PageIndex{3}\)
A 95% confidence interval for the proportion of Americans who work on weekends is found to be 0.24 to 0.32, with the center at 0.28. Use a number line to display this information.
Solution
We just draw a number line, include the three key numbers: 0.24, 0.32, and 0.28 and highlight the part of the interval between 0.23 and 0.31.
Example \(\PageIndex{4}\): rejection region
In Hypothesis testing, we sketch something called the rejection region which is an interval that goes off to infinity or to negative infinity. Suppose that the mean number of hours to work on the week's homework is 4.2. The rejection region for the hypothesis test is all numbers larger than 7.3 hours. Plot the mean and sketch the rejection region on a number line.
Solution
We plot the point 4.2 on the number line and shade everything to the right of 7.3 on the number line.


Why learn about experiences and emotions related to death and dying?
“Everything has to die,” he told her during a telephone conversation.
“I want you to know how much I have enjoyed being with you, having you as my friend, and confidant and what a good father you have been to me. Thank you so much.” she told him.
“You are entirely welcome.” he replied.
He had known for years that smoking will eventually kill him. But he never expected that lung cancer would take his life so quickly or be so painful. A diagnosis in late summer was followed with radiation and chemotherapy during which time there were moments of hope interspersed with discussions about where his wife might want to live after his death and whether or not he would have a blood count adequate to let him precede with his next treatment. Hope and despair exist side by side. After a few months, depression and quiet sadness preoccupied him although he was always willing to relieve others by reporting that he ‘felt a little better’ if they asked. He returned home in January after one of his many hospital stays and soon grew worse. Back in the hospital, he was told of possible treatment options to delay his death. He asked his family members what they wanted him to do and then announced that he wanted to go home. He was ready to die. He returned home. Sitting in his favorite chair and being fed his favorite food gave way to lying in the hospital bed in his room and rejecting all food. Eyes closed and no longer talking, he surprised everyone by joining in and singing “Happy birthday” to his wife, son, and daughter-in-law who all had birthdays close together. A pearl necklace he had purchased 2 months earlier in case he died before his wife’s birthday was retrieved and she told him how proud she would be as she wore it. He kissed her once and then again as she said goodbye. He died a few days later.
[1]
A dying process that allows an individual to make choices about treatment, to say goodbyes and to take care of final arrangements is what many people hope for. Such a death might be considered a “good death.” But of course, many deaths do not occur in this way. Not all deaths include such a dialogue with family members or being able to die in familiar surroundings; people may die suddenly and alone, or people may leave home and never return. Children sometimes precede parents in death; wives precede husbands, and the homeless are bereaved by strangers.
In this module, we will look at death and dying, grief and bereavement, palliative care, and hospice to better understand these last stages of life.
Modification, adaptation, and original content.
Authored by
: Sarah Carter for Lumen Learning.
Provided by
: Lumen Learning.
License
:
CC BY: Attribution


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2: Matrix Methods for Mechanical Systems
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2.1: A Uniaxial Truss
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We now investigate the mechanical prospection of tissue, an application extending techniques developed in the electrical analysis of a nerve cell. In this application, one applies traction to the edges of a square sample of planar tissue and seeks to identify, from measurement of the resulting deformation, regions of increased 'hardness' or 'stiffness.' For a sketch of the associated apparatus, visit the Biaxial Test site.
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2.2: A Small Planar Truss
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We return once again to the biaxial testing problem, introduced in the uniaxial truss module. It turns out that singular matrices are typical in the biaxial testing problem.
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2.3: The General Planar Truss
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2.4: Chapter 2 Exercises
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3.4.3: Effective Nuclear Charge
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Skills to Develop
- Explain the difference between nuclear charge and effective nuclear charge
The reason electrons are attached to atoms is the Coulomb's law attraction between the positively charged nucleus and the negatively charged electrons. Without the nuclear charge holding on to the electrons, they would have no reason to stay in orbitals near nuclei. So it makes sense that energy of the orbitals and their size depend on the nuclear charge. For instance, equivalent orbitals get lower in energy and smaller (more density closer to the nucleus) when nuclear charge increases. (If you want to see where nuclear charge is in the orbital equations, follow this link and click on an orbital; then scroll down to see the equation. Z eff is the effective nuclear charge.)
There are 2 reasons an electron might not spend that much time actually right next to the nucleus, even though there is an attraction. One is angular momentum . Remember from physics that angular momentum is basically the momentum in the "around the center" direction multiplied by r, the distance from the center. Thus, if the electron is "in" the nucleus, r = 0, and angular momentum is 0. Remember also the second quantum number is the angular momentum quantum number ℓ, which corresponds to s, p, d and f orbitals. For s orbitals, ℓ = 0, and angular momentum is zero, so the electron can hang out at the nucleus. For the other shapes (p,d,f) angular momentum is not zero, and there is a node at the nucleus. So only s orbitals can hang out right at the nucleus.
The other reason an electron might not be able to get that close to the nucleus is because of the other electrons. Electrons repel each other, and because of the exclusion principle there can only be 2 electrons in each orbital. So think about lithium, element 3. Its first 2 electrons are in 1s. The third electron is in 2s. Because the principal quantum number is bigger, the 2s electron is usually farther from the nucleus than the 1s electrons. That means that often the 1s electrons are between the 2s electron and the nucleus. This means that the total Coulomb attraction felt by the 2s electron is smaller than you would calculate using just nuclear charge and distance, because there is also the electron-electron repulsion. This is the basis of the effective nuclear charge . Because of the electron-electron repulsions, the outer electrons in a many-electron atom feel less attraction toward the nucleus, so they feel like the nuclear charge is smaller than it actually is. So effective nuclear charge is always smaller than actual nuclear charge.
Effective nuclear charge depends on the type of electron. Electrons in s orbitals, even 4s or 5s, still spend some time right at the nucleus, and when they are there, they feel the full nuclear charge, so on average the s electrons feel a nuclear charge closer to the actual nuclear charge. Electrons in d or f orbitals really don't get very close to the nucleus, so they really get blocked, or shielded by inner electrons. They feel a smaller effective nuclear charge than s electrons. And electrons in p orbitals are in between.
Usually if you do any calculation of orbitals for many-electron atoms, you will use effective nuclear charge instead of actual nuclear charge. The size and energy of the orbitals will depend on effective nuclear charge, not on actual nuclear charge. Size and energy of orbitals determines some very important chemical properties, including the size of the element (as an atom, ion, or in a molecule) and how easily it loses or gains electrons. Electrons take up most of the space in an atom, so orbital size tells you size. Losing electrons requires the energy by which they are bound, which is the roughly same as the orbital energy. Adding electrons only works if the orbital where they will go is lower energy than where they came from.


10.6: Selecting Copyright for Page (Meta-Tags)
Each page should be marked with a copyright. This can be selected via the "Licensing" meta-tag in the collapsible "Page setting" panel on each page (see below).
Selecting the appropriate license requires more discussion about the nature of the license.
Creative Commons Licensing
The Creative Commons licenses are the most ubiquitous licensing in OER and grant "baseline rights", such as the right to distribute the copyrighted work worldwide for non-commercial purposes, and without modification. The details of each of these licenses depend on the version, and comprises a selection out of four conditions shown below.
Other licensing is possible including the GNU General Public License (GNU GPL) and the GNU Free Documentation License (GNU FPL). These are less common licensing in OER.
CC Licensing
Creative commons licenses are built in the following way.
First use CC - to indicate that the material is covered by a Creative Commons copyright, then use one or more of the following to indicate further restrictions
CC- 0 means that the material is public domain and there are no conditions
BY means that the author much be acknowledged, if this is the only condition then the copyright would be CC-BY
SA means that if the work is shared the conditions must be the same as the original. One could have CC-SA. CC-BY-SA is a common license
NC meant that only non-commercial use of the work is permitted.
ND means thar no changes can be made in the material.
Cannot ARBITRARILY Mix and Match Content from Different Licensing
Rights in an adaptation can be expressed by a CC license that is compatible with the status or licensing of the original work or works on which the adaptation is based.
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Avoid adding any content into the Libretexts that has a ND license on it. These do not play well with the community remixing aspect of the LibreTexts project. There are a few texts with this annoying license on the platform, but not many.


11.6: Comparison of the Chi-Square Tests
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You have seen the \(\chi^{2}\) test statistic used in three different circumstances. The following bulleted list is a summary that will help you decide which \(\chi^{2}\) test is the appropriate one to use.
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Goodness-of-Fit:
Use the goodness-of-fit test to decide whether a population with an unknown distribution "fits" a known distribution. In this case there will be a single qualitative survey question or a single outcome of an experiment from a single population. Goodness-of-Fit is typically used to see if the population is uniform (all outcomes occur with equal frequency), the population is normal, or the population is the same as another population with a known distribution. The null and alternative hypotheses are:
- \(H_{0}\): The population fits the given distribution.
- \(H_{a}\): The population does not fit the given distribution.
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Independence:
Use the test for independence to decide whether two variables (factors) are independent or dependent. In this case there will be two qualitative survey questions or experiments and a contingency table will be constructed. The goal is to see if the two variables are unrelated (independent) or related (dependent). The null and alternative hypotheses are:
- \(H_{0}\): The two variables (factors) are independent.
- \(H_{a}\): The two variables (factors) are dependent.
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Homogeneity:
Use the test for homogeneity to decide if two populations with unknown distributions have the same distribution as each other. In this case there will be a single qualitative survey question or experiment given to two different populations. The null and alternative hypotheses are:
- \(H_{0}\): The two populations follow the same distribution.
- \(H_{a}\): The two populations have different distributions.
Review
The goodness-of-fit test is typically used to determine if data fits a particular distribution. The test of independence makes use of a contingency table to determine the independence of two factors. The test for homogeneity determines whether two populations come from the same distribution, even if this distribution is unknown.
Exercise \(\PageIndex{1}\)
Which test do you use to decide whether an observed distribution is the same as an expected distribution?
- Answer
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a goodness-of-fit test
Exercise \(\PageIndex{3}\)
Which test would you use to decide whether two factors have a relationship?
- Answer
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a test for independence
Exercise \(\PageIndex{4}\)
Which test would you use to decide if two populations have the same distribution?
Exercise \(\PageIndex{5}\)
How are tests of independence similar to tests for homogeneity?
- Answer
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Answers will vary. Sample answer: Tests of independence and tests for homogeneity both calculate the test statistic the same way \(\sum_{i \cdot j} \frac{(O-E)^{2}}{E}\). In addition, all values must be greater than or equal to five.
Exercise \(\PageIndex{6}\)
How are tests of independence different from tests for homogeneity?
Bringing It Together
Exercise \(\PageIndex{7}\)
- Explain why a goodness-of-fit test and a test of independence are generally right-tailed tests.
- If you did a left-tailed test, what would you be testing?
- Answer a
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The test statistic is always positive and if the expected and observed values are not close together, the test statistic is large and the null hypothesis will be rejected.
- Answer b
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Testing to see if the data fits the distribution “too well” or is too perfect.


7.12: Daily Life in Western Europe and the Byzantine Empires
In both Western Europe and Byzantium, the vast majority of the population was made up of farmers. In Western Europe, some of these were what we call dependent farmers , living on the lands of aristocrats and giving over much of their surplus to their landlords. But in many villages, the majority of farmers might live on their own land and even enjoy a form of self-government. Although some slavery existed—especially in zones of conflict like the Mediterranean—compared to the days when vast estates had been worked by unfree labor (see Chapter Six), the workers on the estates of the Frankish aristocracy or those free and independent farmers enjoyed greater freedom than had their Roman counterparts. But their life was precarious. Crop yields were low, at ratios of around 3:1—meaning only giving back about three times as much as was planted—and the average Carolingian farmer frequently did not get adequate calories.
So too did most of the population of the Byzantine Empire live in small villages, living at a subsistence level, and selling what rare surplus they had. Byzantium, like its Western European counterpart, was fundamentally rural.
The nobles of Western Europe were generally part of a warrior aristocracy. These aristocrats often outfitted and equipped themselves based on the wealth of their lands. Their values were those of service to their king and loyalty and bravery in battle. Nobles would often not live on their lands but follow the royal court, which would itself travel from place to place rather than having a fixed location. Battle may have been frequent, but until Charlemagne, the scale of battle was often small, with armies numbering a few hundred at most.
Along with its warrior aristocracy, gender roles in the Frankish kingdom—like those of the Roman Empire that came before it—reflected a patriarchal society. The Christian religion generally taught that wives were to submit to their husbands, and the men who wrote much of the religious texts often thought of women in terms of weakness and temptations to sexual sin. “You,” an early Christian writer had exclaimed of women, “are the devil’s gateway…you are the first deserter of the divine law…you destroyed so easily God’s image, man…” 4 The warlike values of the aristocracy meant that aristocratic women were relegated to a supporting role, to the management of the household. Both Roman and Germanic law placed women in subordination to their fathers and then, when married, to their husbands.
That said, women did enjoy certain rights. Although legally inferior to men in Roman Law (practiced in the Byzantine Empire and often among those peoples who were subjects of the Germanic aristocracies), a wife maintained the right to any property she brought into a marriage. Women often played a strong economic role in peasant life, and, as with their aristocratic counterparts, peasant women often managed the household even if men performed tasks such as plowing and the like. And the Church gave women a fair degree of autonomy in certain circumstances. We often read of women choosing to become nuns, to take vows of celibacy, against the desires of their families for them to marry. These women, if they framed their choices in terms of Christian devotion, could often count on institutional support in their life choices. Although monasticism was usually limited to noblewomen, women who became nuns often had access to an education. Certain noblewomen who became abbesses could even become powerful political actors in their own right, as did Gertrude of Nivelles (c. 621 – 659), abbess of the monastery of Nivelles in what is today Belgium.
Endnotes
4 Tertullian, On the Apparel of Women, 1:1


6.11: The Circumstellar Habitable Zone
The area around a star within which a planet or planets with sufficient mass and atmospheric pressure can support liquid water at the planet’s surface is called the circumstellar habitable zone ; also called the Goldilocks Zone . First theorized in 1953, numerous exoplanets have been discovered within this Zone. Not all biologists and astronomers agree that a planet must exist in the circumstellar habitable zone to support life or even intelligent life; many point to extremophiles found on Earth; organisms which flourish in extreme Earth conditions that would be harmful to most life.
It appears that one in five stars have an Earth-sized planet in a circumstellar habitable zone. There could be as many as 40 billion circumstellar habitable zone planets in the Milky Way Galaxy.


10.6: The pH of Weak Acid Solutions
COOH ] is 0.50 M and both [CH 3 COO – ] and [H 3 O + ] are zero. A small amount of CH 3 COOH will ionize; let’s call this x , making the change for [CH 3 COOH] “- x ”, increasing both [CH 3 COO – ] and [H 3 O + ] by the amount “+ x ”. Finally, the equilibrium concentration of [CH 3 COOH] will be (0.50 M – x) and both [CH 3 COO – ] and [H 3 O + ] will be x . The completed table is shown below.
For a solution of a strong acid, calculating the [H 3 O + ] concentration is simple; because the acid is 100% dissociated, the concentration of hydronium ions is equal to the molar concentration of the strong acid (this is, of course, only true for a monoprotic acid such as HCl or HNO 3 ; for H 2 SO 4 , [H 3 O + ] = 2 × [H 2 SO 4 ], etc.). For a weak acid, however, the hydronium ion concentration will be much, much less than the molar concentration of the acid and [H 3 O + ] must be calculated using the value of K a . We can approach this using an ICE table, like we did for previous equilibrium problems. If we prepared a solution of acetic acid that was exactly 0.50 M, then initially [CH 3| [CH 3 COOH] | [CH 3 COO – ] | [H 3 O] + | |
|---|---|---|---|
| === Initial === | 0.50 M | 0 | 0 |
|
Change
|
- x | + x | + x |
|
Equilibrium
|
0.50 M - x | x | x |
The expression for K a for acetic acid is given in equation in section 10.5 . Substituting for our equilibrium values:
\[K_{a}=1.8\times 10^{-5}=\frac{[H_{3}O^{+}][CH_{3}COO^{-}]}{[CH_{3}COOH]}=\frac{x^{2}}{0.50-x} \nonumber \]
\[x^{2}+9.0\times 10^{-6}x-1.8\times 10^{-5}=0 \nonumber \]
The above equation is a quadratic equation and we could solve it using the standard quadratic formula. This is not necessary, however, because acetic acid is a weak acid and by definition, very little of the dissociated form will exist in solution, making the quantity x very, very small. If x is much, much less than 0.50 M (our initial concentration of acetic acid), then (0.50 M – x) 0.50 M and the equation simplifies to:
\[K_{a}=1.8\times 10^{-5}=\frac{[H_{3}O^{+}][CH_{3}COO^{-}]}{[CH_{3}COOH]}=\frac{x^{2}}{0.50} \nonumber \]
\[x=[H_{3}O^{+}]=\sqrt{(1.8\times 10^{-5})\times 0.50}=3.0\times 10^{-3}M \nonumber \]
We can test our assumption by substituting for x ; (0.50 – 0.0030 ) = 0.497, which rounds to 0.50 to two significant figures. Because the concentration of hydronium ion is very small for a weak acid, for most typical solutions, the concentration of hydronium ion can be estimated simply as:
\[[H_{3}O^{+}]=\sqrt{(K_{a}\times C_{0}} \nonumber \]
where C 0 is the initial molar concentration of the weak acid.
- Nitrous acid (HNO 2 ) is a weak acid with a K a of 4.3 × 10 -4 . Estimate the hydronium ion concentration and the p H for a 0.50 M solution of nitrous acid in distilled water.
- Acetic acid is a weak acid with K a = 1.8 × 10 -5 . For a solution of acetic acid in water, the [H 3 O + ] is found to be 4.2 × 10 -3 M. What is the concentration of unionized acetic acid in this solution?
\[\ce{CH3COOH(aq) + H2O(l) <=> CH3COO^{–}(aq) + H3O^{+}(aq)} \nonumber\]
- A solution is prepared in which acetic acid is 0.700 M and its conjugate base, acetate anion is 0.600 M. As shown above, the K a of acetic acid is 1.8 x 10 -5 ; what will the p H of this solution be?
- What concentration of the weak acid, acetic acid ( K a = 1.8 × 10 -5 ) must you have in pure water in order for the final pH to be 2.38?


5.1: Introduction on Probability
Students will be able to:
- Understand and use probability terminology correctly.
- Calculate the probability of a given event for equally likely outcomes.
- Determine the compliment of a given event and find its probability.
Toss a thumb tack one time. Do you think the tack will land with the point up or the point down?
We cannot predict which way the tack will land before we toss it. Sometimes it will land with the point up and other times it will land with the point down. Tossing a tack is a random experiment since we cannot predict what the outcome will be. We do know that there are only two possible outcomes for each trial of the experiment: lands points up or lands point down. If we repeat the experiment of tossing the tack many times we might be able to guess how likely it is that the tack will land point up.
- A random experiment is an activity or an observation whose outcome cannot be predicted ahead of time.
- The result of an experiment is called an outcome .
- A trial is one repetition of a random experiment.
- The sample space is the set of all possible outcomes for a random experiment.
- An event is a subset of the sample space. An event can be thought of as a collection of outcomes that are grouped together.
- A simple event is an event that consists of only one outcome.
- The probability of an event is the measure of how likely it will happen. It can be expressed as a number between 0 and 1, or as a percentage between 0% and 100% The probability of an impossible event is 0 (or 0%), and the probability of a certain event is 1 (or 100%).
Do you think chances of the tack landing point up and the tack landing point down are the same? This is an example where your intuition may be wrong. Having only two possible outcomes does not mean each outcome has a 50/50 chance of happening. In fact we are going to see that the probability of the tack landing point up is about 66%.
To begin to answer this question, toss a tack 10 times. For each toss record whether the tack lands point up or point down.
|
Trial |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|---|---|---|---|---|---|---|---|---|---|---|
|
Up/Down |
Up |
Down |
Up |
Up |
Down |
Up |
Up |
Down |
Up |
Down |
The random experiment here is tossing the tack one time. The possible outcomes for the experiment are that the tack lands point up or the tack lands point down so the sample space is S = {point up, point down}. We are interested in the event E that the tack lands point up, E = {point up}.
Based on our data we would say that the tack landed point up six out of 10 times. The fraction, , is called the relative frequency. Since we would guess that probability of the tack landing point up is about 60%.
Let’s repeat the experiment by tossing the tack ten more times.
|
Trial |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|---|---|---|---|---|---|---|---|---|---|---|
|
Up/Down |
Up |
Up |
Up |
Down |
Down |
Up |
Up |
Down |
Up |
Up |
This time the tack landed point up seven out of 10 times or 70% of the time. If we tossed the tack another 10 times we might get a different result again. The probability of the tack landing point up refers to what happens when we toss the tack many, many times. Let’s toss the tack 150 times and count the number of times it lands point up. Along the way we will look at the proportion of landing point up.
|
Trials |
Number Up |
Total Number of Up |
Total Number of Trials |
Proportion |
|---|---|---|---|---|
|
0-10 |
6 |
6 |
10 |
6/10=0.60 |
|
11-20 |
7 |
13 |
20 |
13/20=0.65 |
|
21-30 |
8 |
21 |
30 |
21/30=0.70 |
|
31-40 |
6 |
27 |
40 |
27/40=0.68 |
|
41-50 |
8 |
35 |
50 |
35/50=0.70 |
|
51-60 |
8 |
43 |
60 |
43/60=0.72 |
|
61-70 |
4 |
47 |
70 |
47/70=0.67 |
|
71-80 |
7 |
54 |
80 |
54/80=0.68 |
|
81-90 |
5 |
59 |
90 |
59/90=0.66 |
|
91-100 |
6 |
65 |
100 |
65/100=0.65 |
|
101-110 |
10 |
75 |
110 |
75/110=0.68 |
|
111-120 |
5 |
80 |
120 |
80/120=0.67 |
|
121-130 |
8 |
88 |
130 |
88/130=0.68 |
|
131-140 |
4 |
92 |
140 |
92/140=0.66 |
|
141-150 |
7 |
99 |
150 |
99/150=0.66 |
When we have a small number of trials the proportion varies quite a bit. As we start to have more trials the proportion still varies but not by as much. It appears that the proportion is around 0.66 or 66%. We would have to do about 100,000 trials to get a better approximation of the actual probability of the tack landing point up.
The tack landed point up 99 out of 150 trials. The probability P of event E is written as:
\[P(E)=\frac{\# \text { of trials with point up }}{\text { total number of trials }}=\frac{99}{150} \approx 0.66\]
We would say the probability that the tack lands point up is about 66%.
Equally Likely Outcomes
In some experiments all the outcomes have the same chance of happening.  If we draw a single card from a well shuffled deck of cards, each card has the same chance of being selected. We call outcomes like these equally likely. Drawing names from a hat or drawing straws are other examples of equally likely outcomes. The tack tossing example did not have equally likely outcomes since the probability of the tack landing point up is different than the probability of the tack landing point down.
An experiment has equally likely outcomes if every outcome has the same probability of occurring.
For equally likely outcomes, the probability of event A , P(A) , is:
\[P(A)=\dfrac{ \text {number of ways for A to occur}}{\text { total number of outcomes }}.\]
Round Off Rule : Give probabilities as a fraction or as a decimal number rounded to three decimal places.
A standard deck of 52 playing cards consists of four suits (hearts, spades, diamonds and clubs). Spades and clubs are black while hearts and diamonds are red. Each suit contains 13 cards, each of a different rank : An Ace (which in many games functions as both a low card and a high card), cards numbered 2 through 10, a Jack, a Queen, and a King.
The image below gives an example of a complete deck of 52 cards.
Draw a single card from a well shuffled deck of 52 cards. Each card has the same chance of being drawn so we have equally likely outcomes. Find following probabilities of the given events.
- P (card is red)
P (card is red) = \(\dfrac{ \text {number of red cards}}{\text { total number of cards }} = \dfrac{26}{52} = \dfrac{1}{2}\)
The probability that the card is red is \(\dfrac{1}{2}\).
- P (card is a heart)
P (card is a heart) = \(\dfrac{ \text {number of hearts}}{\text { total number of cards }} = \dfrac{13}{52} = \dfrac{1}{4}\)
The probability that the card is a heart is \(\dfrac{1}{4}\).
- P (card is a red 5)
P (card is a red 5) = \(\dfrac{ \text {number of red fives}}{\text { total number of cards }} = \dfrac{2}{52} = \dfrac{1}{26}\)
The probability that the card is a red five is \(\dfrac{1}{26}\).
 The sample space is S = {1, 2, 3, 4, 5, 6}. Find the following probabilities of the given events.

- P (roll an odd number)
The event roll an odd number is E = {1, 3, 5}.
P (roll an odd number) = \(\dfrac{ \text {number of ways to roll an odd number}}{\text { total number of ways to roll a die }} = \dfrac{3}{6} = \dfrac{1}{2}\)
The probability of rolling an odd number is \(\dfrac{1}{2}\).
- P (roll a number less than five)
The event roll a number less than five is F = {1, 2, 3, 4}.

A small bookcase contains five math, three English and seven science books. A book is chosen at random. What is the probability that a math book is chosen?
Solution
Since the book is chosen at random each book has the same chance of being chosen and we have equally likely outcomes.
\[P(\text { math book })=\frac{\text { number of ways to choose a math book }}{\text { total number of books }}=\frac{5}{15}=\frac{1}{3}\]
The probability a math book was chosen is \(\dfrac{1}{3}\).
Three Ways of Finding Probabilities
There are three ways to find probabilities. In the tack tossing example we calculated the probability of the tack landing point up by doing an experiment and recording the outcomes. This was an example of an empirical probability. The probability of getting a red jack in a card game or rolling a five with a fair die can be calculated from mathematical formulas. These are examples of theoretical probabilities. The third type of probability is a subjective probability. Saying that there is an 80% chance that you will go to the beach this weekend is a subjective probability. It is based on experience or guessing.
- A theoretical probability is based on a mathematical model where all outcomes are equally likely to occur.
- An empirical probability is based on an experiment or observation and is the relative frequency of the event occurring.
- A subjective probability is an estimate (a guess) based on experience or intuition.
Law of Large Numbers
When studying probabilities, many times the law of large numbers will apply. If you want to observe what the probability is of getting tails up when flipping a coin, you could do an experiment. Suppose you flip a coin 20 times and the coin comes up tails nine times. Then, using an empirical probability, the probability of getting tails is 9/20 = 45%. However, we know that the theoretical probability for getting tails should be 1/2 = 50%. Why is this different? It is because there is error inherent to sampling methods. However, if you flip the coin 100 times or 1000 times, and use the information to calculate an empirical probability for getting tails up, then the probabilities you will observe will become closer to the theoretical probability of 50%. This is the law of large numbers.
The law of large numbers means that with larger numbers of trials of an experiment the observed empirical probability of an event will approach the calculated theoretical probability of the same event.
Complements
If there is a 75% chance of rain today, what are the chances it will not rain? We know that there are only two possibilities. It will either rain or it will not rain. Because the sum of the probabilities for all the outcomes in the sample space must be 100% or 1.00, we know that
P (will rain) + P (will not rain) = 100%.
Rearranging this we see that
P (will not rain) = 100% - P (will rain) = 100% - 75% = 25%.
The events E = {will rain} and F = {will not rain} are called complements.
The complement of event E , denoted by \(\overline{E}\), is the set of outcomes in the sample space that are not in the event E . The probability of \(\overline{E}\) is given by \(P(\overline{E}) = 1- P(E)\).
Draw a single card from a well shuffled deck of 52 cards.
- Look at the suit of the card. Here the sample space S = {spades, clubs, hearts, diamonds}. If event E = {spades} the complement \(\overline{E}\) = {clubs, hearts, diamonds}.
- Look at the value of the cards. Here the sample space is S = {A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K}. If the event E = {the number is less than 7} = {A, 2, 3, 4, 5, 6} the complement \(\overline{E}\) = {7, 8, 9, 10, J, Q, K}.
A train arrives on time at a particular station 85% of the time. Does this mean that the train is late 15% of the time? The answer is no. The complement of E = {on time} is not \(\overline{E}\) = {late}. There is a third possibility. The train could be early. The sample space is S = {on time, early, late} so the complement of E = {on time} is \(\overline{E}\) = {early or late}. Based on the given information we cannot find P (late) but we can find P (early or late) = 15%.
Impossible Events and Certain Events
Recall that \(P(A)=\dfrac{ \text {number of ways for A to occur}}{\text { total number of outcomes }}\). What does it mean if we say the probability of the event is zero? \(P(A)=\dfrac{ \text {number of ways for A to occur}}{\text { total number of outcomes }} = 0\). The only way for a fraction to equal zero is when the numerator is zero. This means there is no way for event A to occur. A probability of zero means that the event is impossible.
What does it mean if we say the probability of an event is one? \(P(A)=\dfrac{ \text {number of ways for A to occur}}{\text { total number of outcomes }} = 1\). The only way for a fraction to equal one is if the numerator and denominator are the same. The number of ways for A to occur is the same as the number of outcomes. There are no outcomes where A does not occur. A probability of 1 means that the event always happens.
- \(P(A) = 0\) means that A is impossible
- \(P(A) = 1\) means that A is certain