8.14A: Habitats and Energy Metabolism of Crenarchaeota
- Outline the various types of energy metabolism used by Crenarchaeota
The Crenarchaeota are Archaea that have been classified as either a phylum of the Archaea kingdom, or in a kingdom of its own. Archaea exist in a broad range of habitats, and as a major part of global ecosystems, they may contribute up to 20% of earth’s biomass.
The first-discovered archaeans were extremophiles. Indeed, some archaea survive high temperatures, often above 100 °C (212 °F), as found in geysers, black smokers, and oil wells. Other common habitats include very cold habitats and highly saline, acidic, or alkaline water. However, archaea also include mesophiles that grow in mild conditions, in marshland, sewage, the oceans, and soils.
Extremophile archaea are members of four main physiological groups. These are the:
- halophiles
- thermophiles
- alkaliphiles
- acidophiles
These groups are not comprehensive or phylum-specific, nor are they mutually exclusive, since some archaea belong to several groups. Nonetheless, they are a useful starting point for classification Halophiles live in extremely saline environments such as salt lakes. Thermophiles grow best at temperatures above 45 °C (113 °F), in places such as hot springs; hyperthermophilic archaea grow optimally at temperatures greater than 80 °C (176 °F). Other archaea exist in very acidic or alkaline conditions.
Recently, several studies have shown that archae exist not only in mesophilic and thermophilic environments but are also present, sometimes in high numbers, at low temperatures as well, as found in cold oceanic environments.
Chemical reactions and energy sources
Archaea exhibit a great variety of chemical reactions in their metabolism and use many sources of energy. These reactions are classified into nutritional groups, depending on energy and carbon sources.
Some archaea obtain energy from inorganic compounds such as sulfur or ammonia (they are lithotrophs). These include nitrifiers, methanogens and anaerobic methane oxidisers. In these reactions one compound passes electrons to another (in a redox reaction), releasing energy to fuel the cell’s activities. One compound acts as an electron donor and one as an electron acceptor. The energy released generates adenosine triphosphate ( ATP ) through chemiosmosis, in the same basic process that happens in the mitochondrion of eukaryotic cells.
Other groups of archaea use sunlight as a source of energy (phototrophs). However, oxygen–generating photosynthesis does not occur in any of these organisms. Many basic metabolic pathways are shared between all forms of life; for example, archaea use a modified form of glycolysis (the Entner–Doudoroff pathway) and either a complete or partial citric acid cycle. These similarities to other organisms probably reflect both early origins in the history of life and their high level of efficiency.
Some Euryarchaeota are methanogens living in anaerobic environments such as swamps. This form of metabolism evolved early, and it is even possible that the first free-living organism was a methanogen. A common reaction involves the use of carbon dioxide as an electron acceptor to oxidize hydrogen. Methanogenesis involves a range of coenzymes that are unique to these archaea, such as coenzyme M and methanofuran. These reactions are common in gut-dwelling archaea. Acetic acid is also broken down into methane and carbon dioxide directly, by acetotrophic archaea. These acetotrophs are archaea in the order Methanosarcinales, and are a major part of the communities of microorganisms that produce biogas.
Other archaea use CO2 in the atmosphere as a source of carbon, in a process called carbon fixation (they are autotrophs). This process involves either a highly modified form of the Calvin cycle or a recently discovered metabolic pathway called the 3-hydroxypropionate/4-hydroxybutyrate cycle. The Crenarchaeota also use the reverse Krebs cycle while the Euryarchaeota also use the reductive acetyl-CoA pathway. Carbon–fixation is powered by inorganic energy sources. No known archaea carry out photosynthesis.
Archaeal energy sources are extremely diverse, and range from the oxidation of ammonia by the Nitrosopumilales to the oxidation of hydrogen sulfide or elemental sulfur by species of Sulfolobus, using either oxygen or metal ions as electron acceptors.
Phototrophic archaea use light to produce chemical energy in the form of ATP. In the Halobacteria, light-activated ion pumps like bacteriorhodopsin and halorhodopsin generate ion gradients by pumping ions out of the cell across the plasma membrane. The energy stored in these electrochemical gradients is then converted into ATP by ATP synthase (photophosphorylation).
Some marine Crenarchaeota are capable of nitrification , suggesting these organisms may affect the oceanic nitrogen cycle, although these oceanic Crenarchaeota may also use other sources of energy. Vast numbers of archaea are also found in the sediments that cover the sea floor, with these organisms making up the majority of living cells at depths over 1 meter below the ocean bottom.
Key Points
- The first-discovered archaeans were extremophiles.
- Extremophile archaea are members of four main physiological groups: halophiles, thermophiles, alkaliphiles, and acidophiles.
- Some archaea obtain energy from inorganic compounds such as sulfur or ammonia (they are lithotrophs).
- Other groups of archaea use sunlight as a source of energy (phototrophs) or CO2 in the atmosphere as a source of carbon (autotrophs).
Key Terms
- extremophile : An organism that lives under extreme conditions of temperature, salinity, and so on. They are commercially important as a source of enzymes that operate under similar conditions.
- phototroph : An organism that carries out photon capture to acquire energy. They use the energy from light to carry out various cellular metabolic processes.
- autotroph : Any organism that can synthesize its food from inorganic substances, using heat or light as a source of energy.


1.3A: Homeostatic Control
Homeostasis is maintained by the body’s responses to adverse stimuli, ensuring maintenance of an optimal physiological environment.
- Model the feedback process of homeostasis
Key Points
- Homeostatic control mechanisms have at least three interdependent components: a receptor, integrating center, and effector.
- The receptor senses environmental stimuli, sending the information to the integrating center.
- The integrating center, generally a region of the brain called the hypothalamus, signals an effector (e.g. muscles or an organ ) to respond to the stimuli.
- Positive feedback enhances or accelerates output created by an activated stimulus. Platelet aggregation and accumulation in response to injury is an example of positive feedback.
- Negative feedback brings a system back to its level of normal functioning. Adjustments of blood pressure, metabolism, and body temperature are all negative feedback.
Key Terms
- homeostasis : The ability of a system or living organism to adjust its internal environment to maintain a stable equilibrium, such as the ability of warm-blooded animals to maintain a constant body temperature.
- negative feedback : A feedback loop in which the output of a system reduces the activity that causes that output.
- positive feedback : A feedback loop in which the output of a system is increased by the mechanism’s own influence on the system that creates that output.
EXAMPLES
When an individual doesn’t have enough to eat, the body adjusts by slowing down metabolism so that he or she expends fewer calories. This adaptation conserves the limited energy available from the inadequate diet.
Concept of Homeostasis
Homeostasis regulates an organism ‘s internal environment and maintains a stable, constant condition of properties like temperature and pH. Homeostasis can be influenced by either internal or external conditions and is maintained by many different mechanisms. All homeostatic control mechanisms have at least three interdependent components for the variable being regulated:
- A sensor or receptor detects changes in the internal or external environment. An example is peripheral chemoreceptors, which detect changes in blood pH.
- The integrating center or control center receives information from the sensors and initiates the response to maintain homeostasis. The most important example is the hypothalamus, a region of the brain that controls everything from body temperature to heart rate, blood pressure, satiety (fullness), and circadian rhythms (sleep and wake cycles).
- An effector is any organ or tissue that receives information from the integrating center and acts to bring about the changes needed to maintain homeostasis. One example is the kidney, which retains water if blood pressure is too low.
The sensors, integrating center, and effectors are the basic components of every homeostatic response. Positive and negative feedback are more complicated mechanisms that enable these three basic components to maintain homeostasis for more complex physiological processes.
Positive Feedback
Positive feedback is a mechanism in which an output is enhanced in order to maintain homeostasis. Positive feedback mechanisms are designed to accelerate or enhance the output created by a stimulus that has already been activated. Positive feedback mechanisms are designed to push levels out of normal ranges. To achieve this, a series of events initiates a cascading process that builds to increase the effect of the stimulus. This process can be beneficial but is rarely used because it may become uncontrollable. A positive feedback example is blood platelet accumulation and aggregation, which in turn causes blood clotting in response to an injury of the blood vessels.
Negative Feedback
Negative feedback mechanisms reduce output or activity to return an organ or system to its normal range of functioning. Regulation of blood pressure is an example of negative feedback. Blood vessels have sensors called baroreceptors that detect if blood pressure is too high or too low and send a signal to the hypothalamus. The hypothalamus then sends a message to the heart, blood vessels, and kidneys, which act as effectors in blood pressure regulation. If blood pressure is too high, the heart rate decreases as the blood vessels increase in diameter ( vasodilation ), while the kidneys retain less water. These changes would cause the blood pressure to return to its normal range. The process reverses when blood pressure decreases, causing blood vessels to constrict and the kidney to increase water retention.
Negative Feedback Loop : The hypothalamus secretes corticotropin-releasing hormone (CRH), which directs the anterior pituitary gland to secrete adrenocorticotropic hormone (ACTH). In turn, ACTH directs the adrenal cortex to secrete glucocorticoids, such as cortisol. Glucocorticoids not only perform their respective functions throughout the body but also prevent further stimulating secretions of both the hypothalamus and the pituitary gland
Temperature control is another negative feedback mechanism. Nerve cells relay information about body temperature to the hypothalamus. The hypothalamus then signals several effectors to return the body temperature to 37 degrees Celsius (the set point). The effectors may signal the sweat glands to cool the skin and stimulate vasodilation so the body can give off more heat.
If body temperature is below the set point, muscles shiver to generate heat and the constriction of the blood vessels helps the body retain heat. This example is very complex because the hypothalamus can change the body’s temperature set point, such as raising it during a fever to help fight an infection. Both internal and external events can induce negative feedback mechanisms.
Homeostatic Control : This image illustrates the feedback mechanisms of homeostatic controls.


2.39: Basso Continuo
This page discusses a musical practice found in almost every Baroque piece: the use of basso continuo. With the end of the Baroque period, continuo fell out of fashion and was rarely heard in the music of the Classical era and beyond. The exception to this was in secco recitative in Classical opera, which continued to make use of sparse, improvised harmony on the harpsichord, though not of the harpsichord-cello pairing of Baroque continuo. This means that the presence of a basso continuo line in a piece of music is a strong indication that the piece is from the Baroque period.
The text in this reading is from a larger article about figured bass . Figured bass is a system of numbers and symbols written beneath the continuo line that indicated the harmonies that were to be improvised by the instrument playing the chords, usually the harpsichord. While figured bass is an important part of the practice of basso continuo, and you should be familiar with that definition, it’s not necessary in this class to dig any deeper into the music theory behind figured bass. Your main concern should be to understand the role of continuo in Baroque music and the instruments that most often performed it, so the information on this page will suffice, unless you wish to study the topic in greater depth for your own information.
Basso Continuo
Basso continuo parts, almost universal in the Baroque era (1600–1750), provided the harmonic structure of the music. The phrase is often shortened to continuo , and the instrumentalists playing the continuo part, if more than one, are called the continuo group . The titles of many Baroque works make mention of the continuo section, such as J. S. Bach’s Concerto for Two Violins, Strings and Continuo in D Minor.
The makeup of the continuo group is often left to the discretion of the performers, and practice varied enormously within the Baroque period. At least one instrument capable of playing chords must be included, such as a harpsichord, organ, lute, theorbo, guitar, regal, or harp. In addition, any number of instruments which play in the bass register may be included, such as cello, double bass, bass viol, or bassoon. The most common combination, at least in modern performances, is harpsichord and cello for instrumental works and secular vocal works, such as operas, and organ for sacred music. Typically performers match the instrument families used in the full ensemble: including bassoon when the work includes oboes or other winds, but restricting it to cello and/or double bass if only strings are involved. Harps, lutes, and other handheld instruments are more typical of early 17th-century music. Sometimes instruments are specified by the composer: in L’Orfeo (1607) Monteverdi calls for an exceptionally varied instrumentation, with multiple harpsichords and lutes with a bass violin in the pastoral scenes followed by lamenting to the accompaniment of organo di legno and chitarrone , while Charon stands watch to the sound of a regal.
The keyboard (or other chording instrument) player realizes a continuo part by playing, in addition to the indicated bass notes, upper notes to complete chords, either determined ahead of time or improvised in performance. The figured bass notation, described below, is a guide, but performers are also expected to use their musical judgment and the other instruments or voices as a guide, and experienced players often incorporate motives found in the other instrumental parts. Modern editions of such music usually supply a realized keyboard part, fully written out for a player, in place of improvisation. With the rise in historically informed performance, however, the number of performers who improvise their parts, as Baroque players would have done, has increased.
Basso continuo, though an essential structural and identifying element of the Baroque period, continued to be used in many works, especially sacred choral works, of the classical period (up to around 1800). An example is C. P. E. Bach’s Concerto in D minor for flute, strings and basso continuo. Examples of its use in the 19th century are rarer, but they do exist: masses by Anton Bruckner, Beethoven, and Franz Schubert, for example, have a basso continuo part for an organist to play.
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Basso continuo. Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Figured_bass#Basso_continuo . License : CC BY-SA: Attribution-ShareAlike


8.2: One-Dimensional Bifurcations
A bifurcation occurs in a nonlinear differential equation when a small change in a parameter results in a qualitative change in the long-time solution. Examples of bifurcations are when fixed points are created or destroyed, or change their stability.
We now consider four classic bifurcations of one-dimensional nonlinear differential equations: saddle-node bifurcation, transcritical bifurcation, supercritical pitchfork bifurcation, and subcritical pitchfork bifurcation. The corresponding differential equation will be written as \[\overset{.}{x}=f_r(x),\nonumber\] where the subscript \(r\) represents a parameter that results in a bifurcation when varied across zero. The simplest differential equations that exhibit these bifurcations are called the normal forms , and correspond to a local analysis (i.e., Taylor series expansion) of more general differential equations around the fixed point, together with a possible rescaling of \(x\).
Saddle-Node Bifurcation
The saddle-node bifurcation results in fixed points being created or destroyed. The normal form for a saddle-node bifurcation is given by \[\overset{.}{x}=r+x^2.\nonumber\]
The fixed points are \(x_* = ±\sqrt{-r}\). Clearly, two real fixed points exist when \(r < 0\) and no real fixed points exist when \(r > 0\). The stability of the fixed points when \(r < 0\) are determined by the derivative of \(f(x) = r + x^2\), given by \(f' (x) = 2x\). Therefore, the negative fixed point is stable and the positive fixed point is unstable.
Graphically, we can illustrate this bifurcation in two ways. First, in Fig. \(\PageIndex{1}\)(a), we plot \(\overset{.}{x}\) versus \(x\) for the three parameter values corresponding to \(r < 0\), \(r = 0\) and \(r > 0\). The values at which \(\overset{.}{x} = 0\) correspond to the fixed points, and arrows are drawn indicating how the solution \(x(t)\) evolves (to the right if \(\overset{.}{x}> 0\) and to the left if \(\overset{.}{x}< 0\)). The stable fixed point is indicated by a filled circle and the unstable fixed point by an open circle. Note that when \(r = 0\), solutions converge to the origin from the left, but diverge from the origin on the right. Second, in Fig. \(\PageIndex{1}\)(b), we plot a bifurcation diagram illustrating the fixed point \(x_*\) versus the bifurcation parameter \(r\). The stable fixed point is denoted by a solid line and the unstable fixed point by a dashed line. Note that the two fixed points collide and annihilate at \(r = 0\), and there are no fixed points for \(r > 0\).
Transcritical Bifurcation
A transcritical bifurcation occurs when there is an exchange of stabilities between two fixed points. The normal form for a transcritical bifurcation is given by \[\overset{.}{x}=rx-x^2.\nonumber\]
The fixed points are \(x_* = 0\) and \(x_* = r\). The derivative of the right-hand-side is \(f' (x) = r − 2x\), so that \(f' (0) = r\) and \(f ' (r) = −r\). Therefore, for \(r < 0\), \(x_* = 0\) is stable and \(x_* = r\) is unstable, while for \(r > 0\), \(x_* = r\) is stable and \(x_* = 0\) is unstable. The two fixed points thus exchange stability as \(r\) passes through zero. The transcritical bifurcation is illustrated in Fig. \(\PageIndex{2}\).
Supercritical Pitchfork Bifurcation
The pitchfork bifurcations occur in physical models where fixed points appear and disappear in pairs due to some intrinsic symmetry of the problem. Pitchfork bifurcations can come in one of two types. In the supercritical bifurcation, a pair of stable fixed points are created at the bifurcation (or critical) point and exist after (super) the bifurcation. In the subcritical bifurcation, a pair of unstable fixed points are created at the bifurcation point and exist before (sub) the bifurcation.
The normal form for the supercritical pitchfork bifurcation is given by \[\overset{.}{x}=rx-x^3.\nonumber\]
Note that the linear term results in exponential growth when \(r > 0\) and the nonlinear term stabilizes this growth. The fixed points are \(x_* = 0\) and \(x_* = ±\sqrt{r}\), the latter fixed points existing only when \(r > 0\). The derivative of \(f\) is \(f' (x) = r − 3x^2\) so that \(f' (0) = r\) and \(f' (±\sqrt{r}) = −2r\). Therefore, the fixed point \(x_* = 0\) is stable for \(r < 0\) and unstable for \(r > 0\) while the fixed points \(x = ±\sqrt{r}\) exist and are stable for \(r > 0\). Notice that the fixed point \(x_* = 0\) becomes unstable as \(r\) crosses zero and two new stable fixed points \(x_* = ±\sqrt{r}\) are born. The supercritical pitchfork bifurcation is illustrated in Fig. \(\PageIndex{3}\).
Supercritical Pitchfork Bifurcation
In the subcritical case, the cubic term is destabilizing. The normal form (to order \(x^3\)) is \[\overset{.}{x}=rx+x^3.\nonumber\]
The fixed points are \(x_* = 0\) and \(x_* = ±\sqrt{-r}\), the latter fixed points existing only when \(r ≤ 0\). The derivative of the right-hand-side is \(f' (x) = r +3x^2\) so that \(f' (0) = r\) and \(f' (±\sqrt{-r}) = −2r\). Therefore, the fixed point \(x_* = 0\) is stable for \(r < 0\) and unstable for \(r > 0\) while the fixed points \(x = ±\sqrt{-r}\) exist and are unstable for \(r < 0\). There are no stable fixed points when \(r > 0\).
The absence of stable fixed points for \(r > 0\) indicates that the neglect of terms of higher-order in \(x\) than \(x^3\) in the normal form may be unwarranted. Keeping to the intrinsic symmetry of the equations (only odd powers of \(x\)) we can add a stabilizing nonlinear term proportional to \(x^5\). The extended normal form (to order \(x^5\)) is \[\overset{.}{x}=rx+x^3-x^5,\nonumber\] and is somewhat more difficult to analyze. The fixed points are solutions of \[x(r+x^2-x^4)=0.\nonumber\]
The fixed point \(x_* = 0\) exists for all \(r\), and four additional fixed points can be found from the solutions of the quadratic equation in \(x^2\):
\[x_*=\pm\sqrt{\frac{1}{2}(1\pm\sqrt{1+4r})}.\nonumber\]
These fixed points exist only if \(x_*\) is real. Clearly, for the inner square-root to be real, \(r ≥ −1/4\). Also observe that \(1 −\sqrt{1+4r}\) becomes negative for \(r > 0\). We thus have three intervals in \(r\) to consider, and these regions and their fixed points are \[\begin{array}{rlll} r<-\frac{1}{4}: & x_*=0 &\text{(one fixed point);} \\ -\frac{1}{4}<r<0: & x_*=0, &x_*=\pm\sqrt{\frac{1}{2}(1\pm\sqrt{1+4r})} &\text{(five fixed points);} \\ r>0: &x_*=0, &x_*=\pm\sqrt{\frac{1}{2}(1+\sqrt{1+4r})} &\text{(three fixed points).}\end{array}\nonumber\]
Stability is determined from \(f' (x) = r + 3x^2 − 5x^4\). Now, \(f' (0) = r\) so \(x_* = 0\) is stable for \(r < 0\) and unstable for \(r > 0\). The calculation for the other four roots can be simplified by noting that \(x_*\) satisfies \(r + x_*^2 − x_*^4 = 0\), or \(x_*^4 = r + x_*^2\). Therefore, \[\begin{aligned}f'(x_*)&=r+3x_*^2-5x_*^4 \\ &=r+3x_*^2-5(r+x_*^2) \\ &=-4r-2x_*^2 \\ &=-2(2r+x_*^2).\end{aligned}\]
With \(x_*^2=\frac{1}{2}(1\pm\sqrt{1+4r})\), we have \[\begin{aligned} f'(x_*)&=-2\left(2r+\frac{1}{2}(1\pm\sqrt{1+4r})\right) \\ &=-\left( (1+4r)\pm\sqrt{1+4r}\right) \\ &=-\sqrt{1+4r}\left(\sqrt{1+4r}\pm 1\right).\end{aligned}\]
Clearly, the plus root is always stable since \(f' (x_*) < 0\). The minus root exists only for \(−\frac{1}{4}< r < 0\) and is unstable since \(f' (x_*) > 0\). We summarize the stability of the various fixed points:
\[\begin{array}{rll} r<-\frac{1}{4}:&x_*=0&\text{(stable);} \\ -\frac{1}{4}<r<0:&x_*=0,&\text{(stable)} \\ &x_*=\pm\sqrt{\frac{1}{2}(1+\sqrt{1+4r})}&\text{(stable);} \\ &x_*=\pm\sqrt{\frac{1}{2}(1-\sqrt{1+4r})}&\text{(unstable);} \\ r>0:&x_*=0&\text{(unstable)} \\ &x_*=\pm\sqrt{\frac{1}{2}(1+\sqrt{1+4r})}&\text{(stable).}\end{array}\nonumber\]
The bifurcation diagram is shown in Fig. \(\PageIndex{4}\), and consists of a subcritical pitchfork bifurcation at \(r = 0\) and two saddle-node bifurcations at \(r = −1/4\). We can imagine what happens to \(x\) as \(r\) increases from negative values, supposing there is some small noise in the system so that \(x = x(t)\) will diverge from unstable fixed points. For \(r < −1/4\), the equilibrium value of \(x\) is \(x_* = 0\). As \(r\) increases into the range \(−1/4 < r < 0\), \(x\) will remain at \(x_* = 0\). However, a catastrophe occurs as soon as \(r > 0\). The \(x_* = 0\) fixed point becomes unstable and the solution will jump up (or down) to the only remaining stable fixed point. Such behavior is called a jump bifurcation. A similar catastrophe can happen as \(r\) decreases from positive values. In this case, the jump occurs as soon as \(r < −1/4\).
Since the stable equilibrium value of \(x\) depends on whether we are increasing or decreasing \(r\), we say that the system exhibits hysteresis . The existence of a subcritical pitchfork bifurcation can be very dangerous in engineering applications since a small change in a problem’s parameters can result in a large change in the equilibrium state. Physically, this can correspond to a collapse of a structure, or the failure of a component.


11: Radian Measure
-
- 11.2: Arc Length
- So suppose that we have a circle of radius r and we place a central angle with radian measure 1 on top of another central angle with radian measure 1, as in Figure 4.2.1(a). Clearly, the combined central angle of the two angles has radian measure 1+1 = 2, and the combined arc length is r + r = 2r.
-
- 11.4: Circular Motion- Linear and Angular Speed
- So suppose that an object moves along a circle of radius r, traveling a distance s over a period of time t, as in Figure 4.4.1. Then it makes sense to define the (average) linear speed ν of the object as: \(v=\frac{s}{t}\). Let θ be the angle swept out by the object in that period of time. Then we define the (average) angular speed ω of the object as: \(ω = \frac{θ}{ t}\).
Thumbnail: Angle \(θ\) and intercepted arc \(\overparen{AB}\) on circle of circumference \(C = 2πr\).
Contributors
-
Michael Corral (Schoolcraft College). The content of this page is distributed under the terms of the GNU Free Documentation License, Version 1.2 .


6.2: National and Global Workplace Trends
The Gig Economy
Since the global financial crisis in 2008, people have been looking at new ways of working that will not leave them so exposed to losing their income. Self-employment is not a new concept — tradespeople and creatives are used to finding work from various sources — but the emergence of centralized digital platforms selling services or products has led to the explosive growth of the gig economy . Many workers are turning away from a “9 to 5” job in favor of independent work where there is no long-term relationship between buyer and seller, hence the use of the term “gig.” This type of work also can be known as a “side hustle” (Lee, 2018).
To understand the scale and motivations of the growing independent workforce, the McKinsey Global Institute conducted a 2016 survey involving 8,000 workers across the United States, the United Kingdom, Germany, Sweden, France and Spain. The survey found that independent work was becoming more mainstream with digital platforms creating large marketplaces that connect buyers and sellers who can communicate in real time. The 2016 data indicated 20 to 30 percent of the workforce across the US and EU-15 countries were involved in independent work (Lee, 2018).
Some examples of companies offering independent or gig work include Uber and Lyft for car-sharing and delivery services, Airbnb and VRBO for short-term or vacation rentals, and Etsy for selling artistic creations. For people looking for local services in the home or office, TaskRabbit and Airtasker make light work of everyday tasks. Food delivery companies that hire independent contractors include DoorDash, UberEats and GrubHub.
For workers who can deliver via the internet, digital platforms have opened up global markets, creating a surge in home workers offering services to startups or businesses needing a more flexible workforce or specialized skills for one-off projects. On platforms like Upwork, Fiverr, and People Per Hour, freelancers provide creative services that include website design, content creation, coding, and consultancy. Even legal services are being transformed through platforms such as Lawyers on Demand, Lawpath and Legal Zoom, making the legal process of setting up a business easier and cheaper (Lee, 2018).
Why are more people choosing to join the gig economy?
Flexible work schedules and independent work offers flexibility to students funding higher education, to parents caring for young children, and to Millennials and Gen Y looking to swap a corporate career for a portfolio career. Freelancing also offers a top-up income or a “side hustle” for people already in part-time or full-time jobs (Lee, 2018).
Greater Control
Freelancers control their work and pricing. They may also choose who they work with to match their skills, experience, interests, and values. This choice gives greater rewards, not limited to or even necessarily financially but perhaps also in terms of job satisfaction.
Freelancers have greater choice over where they work. This flexibility helps provide an improved work-life balance, especially relevant in the United States where paid time off or vacation leave can be limited, and the corporate culture can involve long hours in the office to show commitment.
Home-based working environments could provide freelancers with protection from office workplaces that can be toxic, especially in large corporations with a vertical hierarchy and one-way communication rather than an open, collaborative culture. Promotion can be influenced by favoritism in the form of the halo effect or by prejudice based on gender, sexuality, race, or disability.
In traditional workplaces, people can feel removed from the end user, which impacts job satisfaction. Freelancers often deal directly with clients and on team projects, they collaborate with colleagues on an equal footing. Knowing that outcomes relate directly to their performance, independent workers can be more motivated to produce high-quality work (Lee, 2018).
What are the challenges of independent working and drawbacks of centralized platforms (apps)?
Gig workers can face issues with income security, employment rights, credit availability and safety. Income security can be affected by a digital platform’s payment structure. With a low-cost, low-skill entry, the pool of workers competing for gigs on these platforms has swelled, causing a race to the bottom on pricing with workers often accepting below minimum wage rates. Earnings can be unpredictable with gaps between gigs causing difficulty in meeting living costs. Self-employed workers also may be denied access to financial products, such as mortgages, personal loans, and personal pensions. Centralized digital platforms are vulnerable to data breaches and cyber-attacks that could make order history, client communications, and outstanding work and payments unavailable. Social isolation is a concern because freelance working can be lonely with associated risks for mental health. Communication and relationships between buyers and sellers are often short-term, and a sense of workplace community could be lacking. In direct service gig work, personal safety could be a concern when gig workers have customers in their cars or homes and are delivering products to private entities.
Much can change in the next decade, but the independent workforce or gig economy is growing. Governments, businesses, and individuals would be wise to prepare for the possibility of a ‘Yellow World’ where “technology creates and supports the open, honest, collaborative community”. Unlike centralized platforms that create a culture of competitive individualism, blockchain-based platforms promote a culture of collaboration and co-operation for everyone’s benefit (Lee, 2018).
Remote and Hybrid Work - Work From Anywhere
Although many technology-based organizations have allowed employees to do some work from home in the last twenty years rather than working only on-site, the COVID-19 global pandemic rapidly increased the incidence of work being conducted from anywhere with an internet connection in many knowledge-based professions. In 2020, many state and local governments issued “stay at home orders” and required most businesses to shut down their in-person operations. Employees suddenly were asked to work from home using communication technologies such as Zoom and Google Meet. Data, company information and work tasks were accessed online through virtual private networks (VPNs) or similar technology. Students and teachers at all educational levels attended and participated in classes using communication apps and online learning management systems such as Google Classroom, Canvas, Blackboard and Brightspace (D2L). In fact, many parents of school-aged children were asked to ensure their children were participating in classes and even were expected to help teach their children leading to the expectation that adults would be present in the home which necessitated working from home if possible.
As the pandemic continued with alternating surges and reductions in cases, organizations began to develop more permanent “work from anywhere” or hybrid policies and procedures. As of the writing of this OER, some Fortune 500 companies allowing for completely remote work include Adobe, Fujitsu, Nationwide, Pratt & Whitney and Cisco (Crossover, 2022). IBM, Amazon and Twitter are allowing teams and/or individual workers to decide where they work, and Alphabet, the parent company of Google, is allowing flexible scheduling including working remotely, in the office or a combination (Crossover, 2022). Many employees in careers where technology allowed work to be completed outside the office or off-site realized important benefits such as greater productivity and flexibility and reductions in commuting time. In fact, a phenomenon known as the “great resignation” occurred when forced changes to work environments caused employees to reconsider their relationships with their employers and workspaces. Many employees realized they wanted more flexibility and time at home for personal pursuits outside of work after realizing how much time they had been spending commuting and being in the office.
An important caveat is that service-oriented careers such as the hospitality industry (restaurants and hotels), barbers/cosmetologists and skilled trades (plumbing, HVAC, electrical, home building/remodeling) were not able to transition, in most cases, to online work. Therefore, many workers in these industries suffered in the first year of the pandemic during shut-downs when their industries were at a standstill and simply were not permitted to operate. Although activity levels in these industries have rebounded and actually increased in several industries as of 2022, employees left these fields when their work was unavailable, also part of the “great resignation,” which has led to staffing shortages.
The hybrid or “work from anywhere” movement continues to be in a state of flux. In late 2021 and early 2022, many employers, including Tesla and Apple, who allowed work from home in 2020 called their employees back to the office at least for parts of the work week. Although many knowledge workers indicated they felt more productive working from home and not having to commute, employers who required their employees to return to the office seemed to value the management control a traditional office-based setting offers, and these employers cited the creativity and collaboration that could occur more readily in face-to-face, in-person work environments. As a potential employee, you will need to determine whether your interests and skills lie in working independently or in a hybrid situation from an off-site location such as your home or in a traditional workplace location requiring in-person interactions with colleagues and supervisors.
Social Media Careers
Individuals use social media to share their stories with the world, and organizations use social media to advertise their products and services. Social media applications (apps) include LinkedIn, Facebook (Meta), Twitter, Instagram, TikTok, Pinterest and many others. The widespread use of social media all over the world has given rise to careers that didn’t exist only 10-20 years ago. Influencers use social media to speak for a brand, business or organization and explain how they use the products or services in their personal lives. Influencers often are compensated by the businesses whose products they use. Bloggers and online writers use social media to share research, personal insights and knowledge with a widespread audience. A popular example is travel writing/blogging where people earn money by traveling and writing about their experiences as they are compensated by the businesses they visit. Bloggers also can earn money by allowing advertisers to post on their blogs. Some occupation titles in social media listed on O*NET and LinkedIn include search marketing strategist, social media specialist/manager and brand marketing coordinator.
All of the careers listed above can facilitate “side hustles” as well. Social media’s pervasiveness and easy accessibility allows anyone with an interest and some technical skills to earn money on a part-time basis by posting stories, videos or blogs and monetizing their posts with advertising. Closely related to social media apps are platforms allowing crafting and selling such as Etsy, Handmade at Amazon, eBay and ArtFire. Creative people, crafters and entrepreneurs use these platforms to sell their own made products or resell others’ products. Whereas in the past, someone might have had a creative hobby and sold their products at local craft fairs or community events, now these selling platforms allow anyone to create an online store for their products. If you are a creative person, you may wish to investigate online opportunities to share your story content, research, reports and products as a way to earn money.
Artificial Intelligence and Automation
Artificial intelligence refers to computers being able to accomplish tasks that in the past could have been done only by humans. Examples include speech, vision and even cognitive tasks such as learning and remembering. Use of large language models such as ChatGPT is becoming widespread in workplace communication and publications. Automation is similar in that job tasks previously done by humans are completed by computers or machines.  The rise of artificial intelligence and automation has created a reduction in manual labor jobs in many industries. When considering potential careers, students are advised to research how artificial intelligence and/or automation has impacted the field and whether increased computer or machine usage will change how humans are employed in that field. Increased use of computers or machines in a certain field may not be negative, however, in that professionals are needed to build, program and operate those machines. Careers related to artificial intelligence and automation include data scientists, computer and information research scientists, mechatronics engineers and technicians, robotics engineers and technicians, software engineers and developers and validation/quality engineers.
 Those strategies might include switching how they power their facilities from fossil fuels such as coal or natural gas to renewable energy sources such as solar, wind and geothermal sources. Other changes might be procedural to save energy such as fewer in-person meetings or less national/international travel. Sustainability is important to many workers and consumers, and organizations are looking for ways to appeal to their potential customers and employees. Examples of careers in renewable energy include environmental scientist, renewable energy analyst, energy engineer and auditor, heating/ventilation/air conditioning (HVAC) mechanic, solar thermal installer and technician.
Educational Attainment and Skill Maintenance
The days of getting on-the-job training in one career and continuing in that career for the next 30 years have vanished. Although on-the-job training still can be found, there is a need for employees to continue increasing their knowledge, skills and abilities to keep up with all-encompassing reliance upon technology in every field. Similarly, in many careers, there is a need to earn continuing education units (CEUs) or similar credentials to maintain qualification for existing positions. In the last 30 years or so, the United States has seen a continual increase in educational levels required for entry-level positions.
For example, as recently as the 1990s, registered nurses could earn a nursing diploma in a hospital-based nursing program then go into practice after passing a national or state exam. Associate degrees in nursing (ADN) became more available after World War II and remain as the most common educational path for registered nurses. Many community hospitals, the United States military and the Veterans’ Administration require higher-level education, however, and bachelors degrees in nursing (BSN) are becoming more common as the entry-level credential for registered nurses (Institute of Medicine, 2011). Nurses also have the option to continue their education with a master’s of science degree in nursing (MSN), and now there are even doctoral-level nursing programs to earn a Doctorate of Nursing Practice (DNP).
Microcredentials are a relatively new method for employees to prove their educational attainment and to maintain skills. These short-term, skills-based training programs, also known as “stackable credentials,” allow workers to earn industry-recognized certifications or qualifications in related fields either as stand-alone credentials or “stack” the microcredentials into a longer educational program such as a bachelor’s degree (Marcus, 2020). Examples of microcredential certifications include project management, data analytics, cyber and network security and blockchain technology.


3.21: Linear Relationships (2 of 4)
Learning Objectives
- Use a correlation coefficient to describe the direction and strength of a linear relationship. Recognize its limitations as a measure of the relationship between two quantitative variables.
The Correlation Coefficient ( r )
The numerical measure that assesses the strength of a linear relationship is called the correlation coefficient and is denoted by r . In this section, we
- define r .
- discuss the calculation of r .
- explain how to interpret the value of r .
- talk about some of the properties of r .
- Correlation coefficient (r)
-
(Definition)
The correlation coefficient ( r ) is a numeric measure that measures the strength and direction of a linear relationship between two quantitative variables.
Calculation: r is calculated using the following formula:
where n is the sample size; x is a data value for the explanatory variable; is the mean of the x-values; is the standard deviation of the x-values; similarly, for the terms involving y. To calculate r, the term is calculated for each individual. These terms are added together, then the sum is divided by (n–1).
However, the calculation of r is not the focus of this course. We use a statistics package to calculate the correlation coefficient for us, and the emphasis of this course is on the interpretation of r ’s value.
Interpretation
Once we obtain the value of r , its interpretation with respect to the strength of linear relationships is quite simple, as this walkthrough illustrates:
Use the simulation below to investigate how the value of relates to the direction and strength of the relationship between the two variables in the scatterplot.
In the simulation, use the slider bar at the top of the simulation to change the value of the correlation coefficient ( r ) between −1 and 1. Observe the effect on the scatterplot. Click on the “Switch Sign” button to jump between positive and negative relationships of the same strength.
Click here to open this simulation in its own window.
Contributors and Attributions
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution


29.0: Chapter Overview
-
- Last updated
- Save as PDF
Learning Objectives
- Define pericyclic reactions as concerted processes involving cyclic movement of electrons.
- Explain the Woodward-Hoffmann Rules for predicting outcomes based on molecular orbital symmetry.
- Outline subclasses like cycloadditions, electrocyclic reactions, and sigmatropic rearrangements.
- Describe how pericyclic reactions proceed through cyclic transition states and molecular orbitals.
- Provide key examples like Diels-Alder, electrocyclic ring closures, and Claisen rearrangements.
- Discuss how pericyclic reactions are used in synthesis of complex molecules and materials.
Pericyclic reactions represent a fascinating class of organic reactions characterized by the concerted movement of electrons around a cyclic array of atoms. These reactions involve a cyclic transition state where bonding changes occur with the involvement of π electrons. The term "pericyclic" stems from the Greek roots "peri," meaning around, and "cyclo," referring to cycle or ring, encapsulating the cyclic nature of these reactions.
Key features of pericyclic reactions include their stereospecificity, concertedness, and often high regio- and stereochemical control. They are governed by orbital symmetry rules, primarily the Woodward-Hoffmann rules, which provide a theoretical framework for predicting the outcome of these reactions based on the symmetry properties of the molecular orbitals involved.
Pericyclic reactions encompass several important subclasses, including:
-
Cycloadditions : In cycloaddition reactions, two or more unsaturated molecules combine to form a cyclic product. Diels-Alder reaction is a classic example, where a conjugated diene reacts with a dienophile to form a cyclohexene ring.
-
Electrocyclic Reactions : These reactions involve the reorganization of π electrons in a conjugated system to form a cyclic compound. Examples include the ring-opening and ring-closing reactions of cyclobutene and cyclohexadiene systems.
-
Sigmatropic Rearrangements : Sigmatropic rearrangements involve the migration of a σ-bond (a single bond) along a conjugated system. The Claisen rearrangement and the Cope rearrangement are prominent examples of sigmatropic shifts.
-
[2+2] Cycloadditions : These reactions involve the formation of a four-membered ring from two unsaturated reactants.
Pericyclic reactions find wide applications in organic synthesis due to their efficiency, selectivity, and the ability to form multiple bonds stereospecifically in a single step. Moreover, their predictable nature based on orbital symmetry principles makes them valuable tools for designing complex molecular architectures. Understanding and mastering pericyclic reactions are essential for synthetic chemists aiming to construct intricate organic molecules efficiently and with precision.


11.2: Two Population Means with Unknown Standard Deviations
-
- Last updated
- Save as PDF
- The two independent samples are simple random samples from two distinct populations.
-
For the two distinct populations:
- if the sample sizes are small, the distributions must be normal.
- if the sample sizes are large, knowing the distributions are not important.
The test comparing two independent population means with unknown and possibly unequal population standard deviations is called the Aspin-Welch \(t\)-test. The degrees of freedom formula was developed by Aspin-Welch.
The comparison of two population means is very common. A difference between the two samples depends on both the means and the standard deviations. Very different means can occur by chance if there is great variation among the individual samples. In order to account for the variation, we take the difference of the sample means, \(\bar{X}_{1} - \bar{X}_{2}\), and divide by the standard error in order to standardize the difference. The result is a t-score test statistic.
Because we do not know the population standard deviations, we estimate them using the two sample standard deviations from our independent samples. For the hypothesis test, we calculate the estimated standard deviation, or standard error , of the difference in sample means , \(\bar{X}_{1} - \bar{X}_{2}\).
The standard error is:
\[\sqrt{\dfrac{(s_{1})^{2}}{n_{1}} + \dfrac{(s_{2})^{2}}{n_{2}}}\]
The test statistic ( t -score) is calculated as follows:
\[\dfrac{(\bar{x_{1}}-\bar{x_{2}}) - (\mu_{1} - \mu_{2})}{\sqrt{\dfrac{(s_{1})^{2}}{n_{1}} + \dfrac{(s_{2})^{2}}{n_{2}}}}\]
where:
- \(s_{1}\) and \(s_{2}\), the sample standard deviations, are estimates of \(\sigma_{1}\) and \(\sigma_{1}\), respectively.
- \(\sigma_{1}\) and \(\sigma_{2}\) are the unknown population standard deviations.
- \(\bar{x}_{1}\) and \(\bar{x}_{2}\) are the sample means. \(\mu_{1}\) and \(\mu_{2}\) are the population means.
The number of degrees of freedom (\(df\)) requires a somewhat complicated calculation. However, a computer or calculator calculates it easily. The \(df\) are not always a whole number. The test statistic calculated previously is approximated by the Student's t -distribution with \(df\) as follows:
Degrees of freedom
\[df = \dfrac{\left(\dfrac{(s_{1})^{2}}{n_{1}} + \dfrac{(s_{2})^{2}}{n_{2}}\right)^{2}}{\left(\dfrac{1}{n_{1}-1}\right)\left(\dfrac{(s_{1})^{2}}{n_{1}}\right)^{2} + \left(\dfrac{1}{n_{2}-1}\right)\left(\dfrac{(s_{2})^{2}}{n_{2}}\right)^{2}}\]
When both sample sizes \(n_{1}\) and \(n_{2}\) are five or larger, the Student's t approximation is very good. Notice that the sample variances \((s_{1})^{2}\) and \((s_{2})^{2}\) are not pooled. (If the question comes up, do not pool the variances.) It is not necessary to compute the degrees of freedom by hand. A calculator or computer easily computes it.
Two Independent Samples with Statistics Calculator
Enter in the statistics, the tail type and the confidence level and hit Calculate and the test statistic, t, the p-value, p, the confidence interval's lower bound, LB, and the upper bound, UB will be shown. Be sure to enter the confidence level as a decimal, e.g., 95% has a CL of 0.95.
| Sample Size | Sample Mean | Sample Standard Deviation | |
| First Sample | |||
| Second Sample |
|
|
| t: | p: | LB: | UB: |
Example \(\PageIndex{1}\): Independent groups
The average amount of time boys and girls aged seven to 11 spend playing sports each day is believed to be the same. A study is done and data are collected, resulting in the data in Table \(\PageIndex{1}\). Each populations has a normal distribution.
| Sample Size | Average Number of Hours Playing Sports Per Day | Sample Standard Deviation | |
|---|---|---|---|
| Girls | 9 | 2 | 0.866 |
| Boys | 16 | 3.2 | 1.00 |
Is there a difference in the mean amount of time boys and girls aged seven to 11 play sports each day? Test at the 5% level of significance.
Answer
The population standard deviations are not known. Let g be the subscript for girls and b be the subscript for boys. Then, \(\mu_{g}\) is the population mean for girls and \(\mu_{b}\) is the population mean for boys. This is a test of two independent groups, two population means.
Random variable: \(\bar{X}_{g} - \bar{X}_{b} =\) difference in the sample mean amount of time girls and boys play sports each day.
- \(H_{0}: \mu_{g} = \mu_{b}\ \) or \(\mu_{g} - \mu_{b} = 0\)
- \(H_{a}: \mu_{g} \neq \mu_{b} \) or \( \mu_{g} - \mu_{b} \neq 0\)
The words "the same" tell you \(H_{0}\) has an "=". Since there are no other words to indicate \(H_{a}\), assume it says "is different." This is a two-tailed test.
Distribution for the test: Use \(t_{df}\) where \(df\) is calculated using the \(df\) formula for independent groups, two population means. Using a calculator, \(df\) is approximately 18.8462. Do not pool the variances.
Calculate the test statistic and the p -value using a Student's t -distribution: \(t = -3.1424 \), \(p\text{-value} = 0.0054\)
Graph:
Figure \(\PageIndex{1}\): Normal distribution curve representing the difference in the average amount of time girls and boys play sports all day\[s_{g} = 0.866\]
\[s_{b} = 1\]
So,
\[\bar{x}_{g} - \bar{x}_{b} = 2 - 3.2 = -1.2\]
Half the \(p\text{-value}\) is below –1.2 and half is above 1.2.
Make a decision: Since \(\alpha > p\text{-value}\), reject \(H_{0}\). This means you reject \(\mu_{g} = \mu_{b}\). The means are different.
Go to the Two Independent Samples With Statistics calculator.
Fill in the table:
First Sample Sample size = 9, First Sample Sample Mean = 2, First Sample Standard Deviation = 0.866
Second Sample Sample size = 16, Second Sample Sample Mean = 3.2, Second Sample Standard Deviation = 1.00
Check "\(\neq\)" and click Calculate.
The \(p\text{-value}\) is \(p = 0.0054\).
Conclusion: At the 5% level of significance, the sample data show there is sufficient evidence to conclude that the population mean number of hours that girls and boys aged seven to 11 play sports per day is different (population mean number of hours boys aged seven to 11 play sports per day is greater than the population mean number of hours played by girls OR the population mean number of hours girls aged seven to 11 play sports per day is greater than the population mean number of hours played by boys).
Exercise 10.2.1
Two samples are shown in Table \(\PageIndex{2}\) . Both have normal distributions. The means for the two populations are thought to be the same. Is there a difference in the means? Test at the 5% level of significance.
| Sample Size | Sample Mean | Sample Standard Deviation | |
|---|---|---|---|
| Population A | 25 | 5 | 1 |
| Population B | 16 | 4.7 | 1.2 |
Answer
The \(p\text{-value}\) is \(0.4125\), which is much higher than 0.05, so we decline to reject the null hypothesis. There is not sufficient evidence to conclude that the means of the two populations are not the same.
Note on Sample Sizes:
When the sum of the sample sizes is larger than \(30 (n_{1} + n_{2} > 30)\) you can use the normal distribution to approximate the Student's \(t\).
Example \(\PageIndex{2}\)
A study is done by a community group in two neighboring colleges to determine which one graduates students with more math classes. College A samples 11 graduates. Their average is four math classes with a standard deviation of 1.5 math classes. College B samples nine graduates. Their average is 3.5 math classes with a standard deviation of one math class. The community group believes that a student who graduates from college A has taken more math classes, on the average. Both populations have a normal distribution. Test at a 1% significance level. Answer the following questions.
- Is this a test of two means or two proportions?
- Are the populations standard deviations known or unknown?
- Which distribution do you use to perform the test?
- What is the random variable?
- What are the null and alternate hypotheses? Write the null and alternate hypotheses in words and in symbols.
- Is this test right-, left-, or two-tailed?
- What is the \(p\text{-value}\)?
- Do you reject or not reject the null hypothesis?
Solutions
- two means
- unknown
- Student's t
- \(\bar{X}_{A} - \bar{X}_{B}\)
- \(H_{0}: \mu_{A} = \mu_{B}\) and \(H_{a}: \mu_{A} > \mu_{B}\)
-
Figure 10.2.2.
Therefore, it is a right-tailed test.
-
0.1928
-
Do not reject.
-
At the 1% level of significance, from the sample data, there is not sufficient evidence to conclude that a student who graduates from college A has taken more math classes, on the average, than a student who graduates from college B.
Exercise \(\PageIndex{2}\)
A study is done to determine if Company A retains its workers longer than Company B. Company A samples 15 workers, and their average time with the company is five years with a standard deviation of 1.2. Company B samples 20 workers, and their average time with the company is 4.5 years with a standard deviation of 0.8. The populations are normally distributed.
- Are the population standard deviations known?
- Conduct an appropriate hypothesis test. At the 5% significance level, what is your conclusion?
Answer
- They are unknown.
- The \(p\text{-value} = 0.0878\). At the 5% level of significance, there is insufficient evidence to conclude that the workers of Company A stay longer with the company.
Two Independent Samples with data Calculator
Type in the values from the two data sets separated by commas, for example, 2,4,5,8,11,2. Then enter the tail type and the confidence level and hit Calculate and the test statistic, t, the p-value, p, the confidence interval's lower bound, LB, and the upper bound, UB will be shown. Be sure to enter the confidence level as a decimal, e.g., 95% has a CL of 0.95.
Data1:
Data2:
|
CL: |
| t: | p | LB | UB |
Example \(\PageIndex{3}\)
A professor at a large community college wanted to determine whether there is a difference in the means of final exam scores between students who took his statistics course online and the students who took his face-to-face statistics class. He believed that the mean of the final exam scores for the online class would be lower than that of the face-to-face class. Was the professor correct? The randomly selected 30 final exam scores from each group are listed in Table \(\PageIndex{3}\) and Table \(\PageIndex{4}\).
| 67.6 | 41.2 | 85.3 | 55.9 | 82.4 | 91.2 | 73.5 | 94.1 | 64.7 | 64.7 |
| 70.6 | 38.2 | 61.8 | 88.2 | 70.6 | 58.8 | 91.2 | 73.5 | 82.4 | 35.5 |
| 94.1 | 88.2 | 64.7 | 55.9 | 88.2 | 97.1 | 85.3 | 61.8 | 79.4 | 79.4 |
| 77.9 | 95.3 | 81.2 | 74.1 | 98.8 | 88.2 | 85.9 | 92.9 | 87.1 | 88.2 |
| 69.4 | 57.6 | 69.4 | 67.1 | 97.6 | 85.9 | 88.2 | 91.8 | 78.8 | 71.8 |
| 98.8 | 61.2 | 92.9 | 90.6 | 97.6 | 100 | 95.3 | 83.5 | 92.9 | 89.4 |
Is the mean of the Final Exam scores of the online class lower than the mean of the Final Exam scores of the face-to-face class? Test at a 5% significance level. Answer the following questions:
- Is this a test of two means or two proportions?
- Are the population standard deviations known or unknown?
- Which distribution do you use to perform the test?
- What is the random variable?
- What are the null and alternative hypotheses? Write the null and alternative hypotheses in words and in symbols.
- Is this test right, left, or two tailed?
- What is the \(p\text{-value}\)?
- Do you reject or not reject the null hypothesis?
- At the ___ level of significance, from the sample data, there ______ (is/is not) sufficient evidence to conclude that ______.
(See the conclusion in Example, and write yours in a similar fashion)
Be careful not to mix up the information for Group 1 and Group 2!
Answer
- two means
- unknown
- Student's \(t\)
- \(\bar{X}_{1} - \bar{X}_{2}\)
-
- \(H_{0}: \mu_{1} = \mu_{2}\) Null hypothesis: the means of the final exam scores are equal for the online and face-to-face statistics classes.
- \(H_{a}: \mu_{1} < \mu_{2}\) Alternative hypothesis: the mean of the final exam scores of the online class is less than the mean of the final exam scores of the face-to-face class.
- left-tailed
-
\(p\text{-value} = 0.0011\)
Figure 10.2.3.
- Reject the null hypothesis
-
The professor was correct. The evidence shows that the mean of the final exam scores for the online class is lower than that of the face-to-face class.
At the 5% level of significance, from the sample data, there is (is/is not) sufficient evidence to conclude that the mean of the final exam scores for the online class is less than the mean of final exam scores of the face-to-face class.
First put the data for each group into two lists (such as L1 and L2). Press STAT. Arrow over to TESTS and press 4:2SampTTest. Make sure Data is highlighted and press ENTER. Arrow down and enter L1 for the first list and L2 for the second list. Arrow down to \(\mu_{1}\): and arrow to \(\neq \mu_{1}\) (does not equal). Press ENTER. Arrow down to Pooled: No. Press ENTER. Arrow down to Calculate and press ENTER.
Confidence Intervals for the Difference Between Two Population Means (Independent Samples)
Hypothesis tests are used to provide evidence that one population mean is larger, smaller, or different from another population mean. Recall that if the interest is to estimate how much larger a population mean is from a given value, then a confidence interval is used. In same way that confidence intervals for a population mean is constructed, a confidence interval for the difference between two population means can also be constructed if the samples are independent of each other. The center of the confidence interval will be the difference between the sample means and the margin of error will be the product of the corresponding value of t and the standard error. Putting this together gives the following formula.
Confidence Interval for the Difference Formula
\(\left(\bar{x_2}-\bar{x_1}-t_{\frac{\alpha}{2}}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}},\bar{x_2}-\bar{x_1}+t_{\frac{\alpha}{2}}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\right)\)
Although one can always theoretically use the formula, in practical applications technology is used. The computer calculators built into this section compute the confidence interval without too much difficulty.
Example \(\PageIndex{4}\)
It is well known that college graduates make more money on average than students who do not graduate from college, but a college was interested in how much more money on average their graduates made compared to those who did not complete their degree. Notice that a confidence interval rather than a hypothesis test will be helpful in providing information about this. Suppose a college tracked a total of 300 students who entered the college. 242 of these students successfully completed their degree and ten years after they entered college their mean salary was $72,148 and their standard deviation was $22,263. The 58 students who did not successfully complete their degree had a mean salary of $47,972 ten years after they entered the college and their standard deviation was $16,845.
Answer the following questions:
- Is this a confidence interval for the difference of two means or two proportions?
- Are the population standard deviations known or unknown?
- Which distribution do you use to perform the test?
- What is the random variable?
- What is the lower bound for the 95% confidence interval?
- What is the upper bound for the 95% confidence interval?
- State and interpret the 95% confidence interval.
- Interpret the lower bound for the 95% confidence interval.
- Interpret the upper bound for the 95% confidence interval.
- Explain what it means to be 95% confident in the context of the study.
Answer
- two means
- unknown
- Student's \(t\)
- \(\bar{X}_{1} - \bar{X}_{2}\)
- 18,955
- 29,397
- The 95% confidence interval is [18955, 29397]. With 95% confidence, the population mean salary ten years after entering college is between $18,955 and 29,397 higher for those who successfully complete their degree compared to those who do not complete their degree.
- With 95% confidence, we can state that ten years after entering college, those who complete their degree, on average, make at least $18,955 more money per year than those who do not complete their degree.
- With 95% confidence, we can state that ten years after entering college, those who complete their degree, on average, have a salary that is at most $29,397 higher than those who do not complete their degree.
- If many samples of 242 students who completed their degree and 58 students who did not complete their degree were looked at, then a different confidence interval would result from each of these samples. 95% of these confidence intervals will contain the true population mean difference in salaries and 5% of these confidence intervals will fail to contain the true population mean difference in salaries.
Exercise \(\PageIndex{5}\)
How much more exercise do Californians (CA) get each week compared to Alabamians (AL)? The table below shows the results of a study on exercise. Assume that the distributions of exercise time are normal for both California and Alabama. Come up with and interpret the 90% confidence interval for the difference.
| CA | 162 | 134 | 97 | 168 | 88 | 72 | 297 | 141 | 130 |
| AL | 37 | 111 | 28 | 75 | 92 | 187 | 12 | 62 |
Answer
The 90% confidence interval is [15.74, 119.71]. With 90% confidence it can be concluded that the population mean amount of exercise that Californians get each week is between 15.74 and 119.71 minutes longer than the population mean amount of exercise that Alabamians get each week.
Cohen's Standards for Small, Medium, and Large Effect Sizes
Note: Cohen's Standards are covered in relatively few elementary statistic classes, so this topic may be optional
Cohen's \(d\) is a measure of effect size based on the differences between two means. Cohen’s \(d\), named for United States statistician Jacob Cohen, measures the relative strength of the differences between the means of two populations based on sample data. The calculated value of effect size is then compared to Cohen’s standards of small, medium, and large effect sizes.
| Size of effect | \(d\) |
|---|---|
| Small | 0.2 |
| medium | 0.5 |
| Large | 0.8 |
Cohen's \(d\) is the measure of the difference between two means divided by the pooled standard deviation: \(d = \dfrac{\bar{x}_{2}-\bar{x}_{2}}{s_{\text{pooled}}}\) where \(s_{pooled} = \sqrt{\dfrac{(n_{1}-1)s^{2}_{1} + (n_{2}-1)s^{2}_{2}}{n_{1}+n_{2}-2}}\)
Example \(\PageIndex{6}\)
Calculate Cohen’s d for Example. Is the size of the effect small, medium, or large? Explain what the size of the effect means for this problem.
Answer
\(\mu_{1} = 4 s_{1} = 1.5 n_{1} = 11\)
\(\mu_{2} = 3.5 s_{2} = 1 n_{2} = 9\)
\(d = 0.384\)
The effect is small because 0.384 is between Cohen’s value of 0.2 for small effect size and 0.5 for medium effect size. The size of the differences of the means for the two colleges is small indicating that there is not a significant difference between them.
Example \(\PageIndex{7}\)
Calculate Cohen’s \(d\) for Example. Is the size of the effect small, medium or large? Explain what the size of the effect means for this problem.
Answer
\(d = 0.834\); Large, because 0.834 is greater than Cohen’s 0.8 for a large effect size. The size of the differences between the means of the Final Exam scores of online students and students in a face-to-face class is large indicating a significant difference.
Example 10.2.8
Weighted alpha is a measure of risk-adjusted performance of stocks over a period of a year. A high positive weighted alpha signifies a stock whose price has risen while a small positive weighted alpha indicates an unchanged stock price during the time period. Weighted alpha is used to identify companies with strong upward or downward trends. The weighted alpha for the top 30 stocks of banks in the northeast and in the west as identified by Nasdaq on May 24, 2013 are listed in Table and Table, respectively.
| 94.2 | 75.2 | 69.6 | 52.0 | 48.0 | 41.9 | 36.4 | 33.4 | 31.5 | 27.6 |
| 77.3 | 71.9 | 67.5 | 50.6 | 46.2 | 38.4 | 35.2 | 33.0 | 28.7 | 26.5 |
| 76.3 | 71.7 | 56.3 | 48.7 | 43.2 | 37.6 | 33.7 | 31.8 | 28.5 | 26.0 |
| 126.0 | 70.6 | 65.2 | 51.4 | 45.5 | 37.0 | 33.0 | 29.6 | 23.7 | 22.6 |
| 116.1 | 70.6 | 58.2 | 51.2 | 43.2 | 36.0 | 31.4 | 28.7 | 23.5 | 21.6 |
| 78.2 | 68.2 | 55.6 | 50.3 | 39.0 | 34.1 | 31.0 | 25.3 | 23.4 | 21.5 |
Is there a difference in the weighted alpha of the top 30 stocks of banks in the northeast and in the west? Test at a 5% significance level. Answer the following questions:
- Is this a test of two means or two proportions?
- Are the population standard deviations known or unknown?
- Which distribution do you use to perform the test?
- What is the random variable?
- What are the null and alternative hypotheses? Write the null and alternative hypotheses in words and in symbols.
- Is this test right, left, or two tailed?
- What is the \(p\text{-value}\)?
- Do you reject or not reject the null hypothesis?
- At the ___ level of significance, from the sample data, there ______ (is/is not) sufficient evidence to conclude that ______.
- Calculate Cohen’s d and interpret it.
Answer
- two means
- unknown
- Student’s-t
- \(\bar{X}_{1} - \bar{X}_{2}\)
-
- \(H_{0}: \mu_{1} = \mu_{2}\) Null hypothesis: the means of the weighted alphas are equal.
- \(H_{a}: \mu_{1} \neq \mu_{2}\) Alternative hypothesis : the means of the weighted alphas are not equal.
- two-tailed
- \(p\text{-value} = 0.8787\)
- Do not reject the null hypothesis
-
This indicates that the trends in stocks are about the same in the top 30 banks in each region.
Figure 10.2.4.
5% level of significance, from the sample data, there is not sufficient evidence to conclude that the mean weighted alphas for the banks in the northeast and the west are different - \(d = 0.040\), Very small, because 0.040 is less than Cohen’s value of 0.2 for small effect size. The size of the difference of the means of the weighted alphas for the two regions of banks is small indicating that there is not a significant difference between their trends in stocks.
References
- Data from Graduating Engineer + Computer Careers. Available online at http://www.graduatingengineer.com
- Data from Microsoft Bookshelf .
- Data from the United States Senate website, available online at www.Senate.gov (accessed June 17, 2013).
- “List of current United States Senators by Age.” Wikipedia. Available online at http://en.Wikipedia.org/wiki/List_of...enators_by_age (accessed June 17, 2013).
- “Sectoring by Industry Groups.” Nasdaq. Available online at http://www.nasdaq.com/markets/barcha...&base=industry (accessed June 17, 2013).
- “Strip Clubs: Where Prostitution and Trafficking Happen.” Prostitution Research and Education, 2013. Available online at www.prostitutionresearch.com/ProsViolPosttrauStress.html (accessed June 17, 2013).
- “World Series History.” Baseball-Almanac, 2013. Available online at http://www.baseball-almanac.com/ws/wsmenu.shtml (accessed June 17, 2013).
Chapter Review
Two population means from independent samples where the population standard deviations are not known
- Random Variable: \(\bar{X}_{1} - \bar{X}_{2} =\) the difference of the sampling means
- Distribution: Student's t -distribution with degrees of freedom (variances not pooled)
Formula Review
Standard error: \[SE = \sqrt{\dfrac{(s_{1}^{2})}{n_{1}} + \dfrac{(s_{2}^{2})}{n_{2}}}\]
Test statistic ( t -score): \[t = \dfrac{(\bar{x}_{1}-\bar{x}_{2}) - (\mu_{1}-\mu_{2})}{\sqrt{\dfrac{(s_{1})^{2}}{n_{1}} + \dfrac{(s_{2})^{2}}{n_{2}}}}\]
Degrees of freedom:
\[df = \dfrac{\left(\dfrac{(s_{1})^{2}}{n_{1}} + \dfrac{(s_{2})^{2}}{n_{2}}\right)^{2}}{\left(\dfrac{1}{n_{1} - 1}\right)\left(\dfrac{(s_{1})^{2}}{n_{1}}\right)^{2} + \left(\dfrac{1}{n_{2} - 1}\right)\left(\dfrac{(s_{2})^{2}}{n_{2}}\right)^{2}}\]
where:
- \(s_{1}\) and \(s_{2}\) are the sample standard deviations, and n 1 and n 2 are the sample sizes.
- \(x_{1}\) and \(x_{2}\) are the sample means.
Confidence Interval:
\(\left(\bar{x_2}-\bar{x_1}-t_{\frac{\alpha}{2}}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}},\bar{x_2}-\bar{x_1}+t_{\frac{\alpha}{2}}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\right)\)
Cohen’s \(d\) is the measure of effect size:
\[d = \dfrac{\bar{x}_{1} - \bar{x}_{2}}{s_{\text{pooled}}}\]
where
\[s_{\text{pooled}} = \sqrt{\dfrac{(n_{1} - 1)s^{2}_{1} + (n_{2} - 1)s^{2}_{2}}{n_{1} + n_{2} - 2}}\]
Glossary
- Degrees of Freedom (\(df\))
- the number of objects in a sample that are free to vary.
- Standard Deviation
- A number that is equal to the square root of the variance and measures how far data values are from their mean; notation: \(s\) for sample standard deviation and \(\sigma\) for population standard deviation.
- Variable (Random Variable)
-
a characteristic of interest in a population being studied. Common notation for variables are upper-case Latin letters \(X, Y, Z,\)... Common notation for a specific value from the domain (set of all possible values of a variable) are lower-case Latin letters \(x, y, z,\).... For example, if \(X\) is the number of children in a family, then \(x\) represents a specific integer 0, 1, 2, 3, .... Variables in statistics differ from variables in intermediate algebra in two following ways.
- The domain of the random variable (RV) is not necessarily a numerical set; the domain may be expressed in words; for example, if \(X =\) hair color, then the domain is {black, blond, gray, green, orange}.
- We can tell what specific value x of the random variable \(X\) takes only after performing the experiment.
Contributors and Attributions
-
Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.


12.3B: Mendel’s Law of Dominance
- Explain the concept of dominance versus recessiveness
Alleles Can Be Dominant or Recessive
Most familiar animals and some plants have paired chromosomes and are described as diploid. They have two versions of each chromosome: one contributed by the female parent in her ovum and one by the male parent in his sperm. These are joined at fertilization. The ovum and sperm cells (the gametes) have only one copy of each chromosome and are described as haploid.
Mendel’s law of dominance states that in a heterozygote, one trait will conceal the presence of another trait for the same characteristic. Rather than both alleles contributing to a phenotype, the dominant allele will be expressed exclusively. The recessive allele will remain “latent,” but will be transmitted to offspring by the same manner in which the dominant allele is transmitted. The recessive trait will only be expressed by offspring that have two copies of this allele; these offspring will breed true when self-crossed.
By definition, the terms dominant and recessive refer to the genotypic interaction of alleles in producing the phenotype of the heterozygote. The key concept is genetic: which of the two alleles present in the heterozygote is expressed, such that the organism is phenotypically identical to one of the two homozygotes. It is sometimes convenient to talk about the trait corresponding to the dominant allele as the dominant trait and the trait corresponding to the hidden allele as the recessive trait. However, this can easily lead to confusion in understanding the concept as phenotypic. For example, to say that “green peas” dominate “yellow peas” confuses inherited genotypes and expressed phenotypes. This will subsequently confuse discussion of the molecular basis of the phenotypic difference. Dominance is not inherent. One allele can be dominant to a second allele, recessive to a third allele, and codominant to a fourth. If a genetic trait is recessive, a person needs to inherit two copies of the gene for the trait to be expressed. Thus, both parents have to be carriers of a recessive trait in order for a child to express that trait.
Since Mendel’s experiments with pea plants, other researchers have found that the law of dominance does not always hold true. Instead, several different patterns of inheritance have been found to exist.
Key Points
- Dominant alleles are expressed exclusively in a heterozygote, while recessive traits are expressed only if the organism is homozygous for the recessive allele.
- A single allele may be dominant over one allele, but recessive to another.
- Not all traits are controlled by simple dominance as a form of inheritance; more complex forms of inheritance have been found to exist.
Key Terms
- dominant : a relationship between alleles of a gene, in which one allele masks the expression (phenotype) of another allele at the same locus
- recessive : able to be covered up by a dominant trait