At the end of this chapter, you should be able to:
Explain why health and well-being in early childhood is so important.
Describe qualities and benefits of high-quality early care and education programs.
Outline what the book will be addressing in regards to safety, health, and nutrition.
Discuss what licensing is and its role in keeping children safe and healthy.
Compare and contrast Title 22 and Title 5 licensing requirements.
A culture of wellness exists when staff and child health and safety are valued, supported, and promoted through health & wellness programs, policies, and environment.
1
Introduction
Evidence shows that experiences in childhood are extremely important for a child’s healthy development and lifelong learning. How a child develops during this time affects future cognitive, social, emotional, language, and physical development, which in turn influences school readiness and later success in life.
Research on a number of adult health and medical conditions points to pre-disease pathways that have their beginnings in early and middle childhood.
During early childhood, the human brain grows to 90 percent of its adult size by age 3. Early childhood represents the period when young children reach developmental milestones that include:
Emotional regulation and attachment
Language development
Cognitive development
Physical development (motor skills)
All of these milestones can be significantly delayed when young children experience inadequate caregiving, environmental stressors, and other negative risk factors. These stressors and factors can affect the brain and may seriously compromise a child’s physical, social-emotional, and cognitive growth and development.
More than any other developmental periods, childhood sets the stage for:
School success
Health literacy
Self-discipline
The ability to make good decisions about risky situations
Eating habits
Conflict negotiation and healthy relationships with family and friends
3
Understanding Childhood Health Concerns
Although young children are typically healthy, it is during this time that they are at risk for conditions such as:
Developmental and behavioral disorders
Child maltreatment
Asthma and other chronic conditions
Obesity
Dental caries (cavities)
Unintentional injuries
While typically nonfatal, these conditions affect children, their education, their relationships with others, and the health and well-being of the adolescents and adults they will become.
5
Emerging Issues in Childhood Health
The keys to understanding childhood health are recognizing the important roles these periods play in adult health and well-being and focusing on conditions and illnesses that can seriously limit children’s abilities to learn, grow, play, and become healthy adults.
Prevention efforts in early and middle childhood can have lasting benefits. Emerging issues in early and middle childhood include implementing and evaluating multidisciplinary public health interventions that address social determinants of health by:
Fostering knowledgeable and nurturing families, parents, and caregivers
Creating supportive and safe environments in home, schools, and communities
Increasing access to high-quality health care
6
Early Childhood Development and Education
Early childhood, particularly the first 5 years of life, impacts long–term social, cognitive, emotional, and physical development.
Healthy development in early childhood helps prepare children for the educational experiences of kindergarten and beyond.
Early childhood development and education opportunities are affected by various environmental and social factors, including:
Early life stress
Socioeconomic status
Relationships with parents and caregivers
Access to early education programs
Early life stress and adverse events can have a lasting impact on the mental and physical health of children.
Specifically, early life stress can contribute to developmental delays and poor health outcomes in the future. Stressors such as physical abuse, family instability, unsafe neighborhoods, and poverty can cause children to have inadequate coping skills, difficulty regulating emotions, and reduced social functioning compared to other children their age.
Additionally, exposure to environmental hazards, such as lead in the home, can negatively affect a child's health and cause cognitive developmental delays. Research shows that lead exposure disproportionately affects children from minority and low–income households and can adversely affect their readiness for school.
The socioeconomic status of young children's families and communities also significantly affects their educational outcomes. Specifically, poverty has been shown to negatively influence the academic achievement of young children. Research shows that, in their later years, children from disadvantaged backgrounds are more likely to need special education, repeat grades, and drop out of high school. Children from communities with higher socioeconomic status and more resources experience safer and more supportive environments and better early education programs.
The Effects of Poverty on Education
“Despite being one of the most developed countries in the world, the United States has one of the highest rates of childhood poverty globally.”
7
“Poverty has a particularly adverse effect on the academic outcomes of children, especially during early childhood.”
8
Research has shown that children from families in poverty enter school with a readiness gap. Contributions to this gap include:
Poor physical development and health (due to poor nutrition and lack of access to medical care)
Challenges with concentration, memory, attentiveness, curiosity, and motivation
9
due to the chronic stress of living in poverty
Greater risk for behavioral and emotional problems
Exposure to environmental hazards (such as lead paint) and violence in their communities.
Two additional things that are important to note:
This gap disproportionately affects Black and Latinx children.
Families experiencing poverty have challenges finding affordable, high-quality early care and education programs and are often in districts with under-resourced schools.
10
Early childhood programs are a critical outlet for fostering the mental and physical development of young children. According to the Center on the Development Child at Harvard University’s
A Science-Based Framework for Early Childhood Policy,
“ The principal elements that have consistently produced positive impacts include:
highly skilled teachers;
small class sizes and high adult-to-child ratios;
age-appropriate curricula and stimulating materials in a safe physical setting;
a language-rich environment;
warm, responsive interactions between staff and children; and
high and consistent levels of child participation.”
11
The National Association for the Education for Young Children says that high quality programs:
Create caring communities of learners in which children develop relationships with each and the teachers and each child and family are included.
Teach to support children’s development and learning by being intentional with the environmental design, materials, and activities and by providing positive guidance for children’s behavior.
Have developmentally appropriate curriculum that helps children learn and grow that sets challenging, yet achievable goals for children, balances adult instruction and group activities with play and child-chosen experiences, provides enough time for deep engagement, and is based on the children’s interests, abilities, and knowledge.
Regularly assesses children’s development and learning to inform their environmental design, curriculum, and interactions with children and their families.
Is founded on partnerships with families in which families are respected and valued, share their goals and concerns, are encouraged to participate, and with clear communication (in a family’s home language whenever possible).
12
And you can check out the checklist
15 Must-Haves for All Child Care Programs
published by The Administration for Children and Families Office of Child Care in Appendix A.
Early childhood development and education programs can also help reduce educational gaps. For example, Head Start is a federally funded early childhood program that provides comprehensive services for children from low–income families. Head Start aims to improve health outcomes, increase learning and social skills, and close the gap in readiness to learn for children from low–income families and at-risk children. Enrolling children in full-day kindergarten after the completion of preschool has also been shown to improve academic achievement.
Furthermore, extended early childhood programs for children up to 3rd grade, also referred to as booster programs, can provide comprehensive educational, health, and social services to complement standard early childhood and kindergarten programs. These programs help sustain and bolster early developmental and academic gains. Characteristic of such programs include:
Low student-teacher ratio
Teacher training
Creation of school-family partnerships
School meals
Provision of transportation to and from school
Health care services and referrals
Home visitation
Supportive social services
Quality education in elementary school is necessary to reinforce early childhood interventions and prevent their positive effects from fading over time. Research also shows that school quality has an impact on both the short– and long–term educational attainment of children, as well as on their health. For example, children who enroll in low–quality schools with limited health resources, safety concerns, and low teacher support are more likely to have poorer physical and mental health.
The developmental and educational opportunities that children have access to in their early years have a lasting impact on their health as adults. The Carolina Abecedarian Project found that the children in the study who participated in a high–quality and comprehensive early childhood education program, including health care and nutritional components, were in better health than those who did not. The study found that, at age 21, the people who participated in the comprehensive early education program exhibited fewer risky health behaviors—for example, they were less likely to binge drink alcohol, smoke cigarettes, and use illegal drugs. This group also self–reported better health and had a lower number of deaths.
Furthermore, by their mid–30s the children who participated in the comprehensive early childhood development and education program had a lower risk for heart disease and associated risk factors, including obesity, high blood pressure, elevated blood sugar, and high cholesterol. These studies show that quality early childhood development and education programs can play a key role in reducing risky health behaviors and preventing or delaying the onset of chronic disease in adulthood. We will look at what high quality programming looks like at the end of the chapter.
Early childhood development and education are key determinants of future health and well–being. Addressing the disparities in access to early childhood development and education opportunities can greatly bolster young children’s future health outcomes.
Additional research is needed to increase the evidence base for what can successfully impact the effects of childhood development and education on health outcomes and disparities. This additional evidence will facilitate public health efforts to address early childhood development and education as social determinants of health.
14
Importance of Considering Holistic Development
Early childhood education (ECE) is not a singular concept, but rather a holistic concept that focuses on several aspects of a child's development, including their cognitive, social, emotional, moral, spiritual, and physical well-being (American Academy of Pediatrics 2012; Chisholm 2004; Tinajero & Loizillon 2012). According to Shonkoff,
et al.
(2011), debates about early childhood education often focus on education and the enhancement of children's cognitive skills and later academic progress. However, these authors stress the importance of viewing children in a holistic manner as part of a bigger system. In this manner, early childhood education becomes a vehicle for enhancing the physical, cognitive, mental, social and economic well-being of individuals and the societies in which they live. Early care and education programs should enhance multiple interrelated dimensions of a child's life; including health and safety, nutrition, emotional well-being, and social competence.
15
Focus on Wellness
Rather than waiting for health issues to arise, families and early childhood education programs should focus on supporting children’s wellness. “Wellness describes the entirety of one’s physical, emotional, and social health; this includes all aspects of functioning in the world (physiological, intellectual, social, and spiritual), as well as subjective feelings of well-being. A child who is doing well frequently experiences joy, delight, and wonder, is secure and safe in his/her family and community, and is continually expanding and deepening his/her engagement with the world around him/her.”
16
Wellness is an active process. It requires awareness and directed, thoughtful attention to the choices we make. Early care and education programs can play a critical role in helping children, families and staff commit to and implement healthy lifestyle choices that promote both physical and mental well-being. The two, in fact, are closely linked. Our feelings, thoughts, and behaviors directly impact our physical health. Similarly, our physical health status has a direct impact on our feelings, thoughts, and behaviors.
18
We must also support children’s mental well-being and help them navigate everyday stress and adversity as well as trauma and significant sources of stress. The American Psychological Association shares that “[b]uilding resilience — the ability to adapt well to adversity, trauma, tragedy, threats or even significant sources of stress — can help our children manage stress and feelings of anxiety and uncertainty.”
19
It is important that children are in an environment that keeps them physically and emotionally safe and healthy and provides sound nutrition. As an early educator, providing these requires attention, planning, and intention.
Introduction to Safety, Health, and Nutrition
This book is divided into three sections. These include:
Safety
Health
Nutrition
Safety
Children are curious and eager to learn. They depend on their caregivers to keep them safe by making sure that nothing within a child’s reach can harm them. Injuries are a serious health risk to young children. But most injuries are predictable and preventable.
20
But it fails to protect children by being enclosed with a fence and gate.
21
ECE programs can prevent risks and unnecessary harm to children by committing to a culture of safety. A culture of safety prioritizes safety at all levels. It encourages programs to learn from past problems and prevent them in the future.
22
Programs should not assume that nothing will ever go wrong. In fact, they should plan that something is going to go wrong. And their goal is to make it as hard as possible for things to go wrong.
23
Chapters 2.1 through 2.7 will focus on ways programs can protect children’s safety.
Chapter 2.1 discusses active supervision and creating a culture of safety
Chapter 2.5 is an example of an accident reporting form
Chapter 2.6 covers preparing for and managing emergencies
Chapter 2.7 discusses child maltreatment (abuse and neglect)
Health
“Health is more than merely the absence of disease—it is an evolving human resource that helps children and adults adapt to the challenges of everyday life, resist infections, cope with adversity, feel a sense of personal well-being, and interact with their surroundings in ways that promote successful development.”
24
As mentioned at the beginning of the chapter, research is showing that many adult health issues, such as high blood pressure, heart disease, and diabetes, are linked to what happens during early childhood (and even prenatally!). We also know that during early childhood there are biological systems that are more sensitive to environmental factors (such as child maltreatment, malnutrition, and recurring issues to infectious disease).
26
It is vital for children and their families to have support for children’s physical, oral, and mental health. This happens through promoting health and protection from illness.
Chapters 3.1 through 3.10 focus on children’s health.
Chapter 3.1 examines promoting health
Chapter 3.2 talks about developmental and health screening
Chapter 3.3 is a list of developmental milestones
Chapter 3.4 looks at preventing illness
Chapter 3.5 addresses supportive health care
Chapter 3.6 is an example of a form used when children are excluded from care
Chapter 3.7 is an example of an exposure notice
Chapter 3.8 describes special health care needs during childhood
Chapter 3.9 addresses physical activity for young children
Chapter 3.10 discusses children’s mental health
Nutrition
Healthy eating and being active are essential to a child's well-being. Children who are under- or over-nourished are at risk for chronic health problems.
27
Early childhood is an important time for developing healthy habits for life. Children’s bodies grow and develop in ways that affect the way they think, eat, and behave.
28
A healthy diet not only affects growth, but also immunity, intellectual capabilities, and emotional well-being. Families and educators must ensure that children receive an adequate amount of needed nutrients to provide a strong foundation for the rest of their lives.
29
Chapters 4.1 through 4.7 focus on nutrition for children.
Chapter 4.1 delves into basic nutrition for children
Chapter 4.2 examines protecting nutrition and wellness
Chapter 4.3 discusses providing good nutrition
Chapter 4.4 addresses menu planning
Chapter 4.5 discusses food safety
Chapter 4.6 is an example of a food allergy management checklist
Chapter 4.7 is an example of a self-assessment of the food your center serves
Introduction to Licensing for Child Care Centers in California
Licensing for child care programs in the state of California is overseen by the Community Care Licensing Division of the California Department of Social Services, whose primary mission is to ensure the health and safety of all children enrolled in licensed child care facilities in California.
 By definition, both types of child care facilities provide nonmedical, age-appropriate care and supervision of infant to school-age children for less than 24 consecutive hours at a time.
Child care centers
provide care in a large-group setting, usually operate in an institutional or commercial building, and must meet a number of requirements regarding their physical facilities, and the services they provide. For the most part, when this book refers to licensing requirements, it will be referencing licensing for child care centers.
Family child care homes
provide care in small-group settings in the licensee’s private house or apartment. They include two types of facilities: small family child care homes that care for no more than 8 children at any one time, and large family child care homes that care for no more than 14 children at any one time.
The key to licensing’s effort to protect the health and safety of children in child care is a thorough and fair licensing and inspection program that ensures all facilities are in compliance. Community Care Licensing is mandated by law to ensure that licensed facilities meet established health and safety standards and that the licensee has access to all relevant information about the department.
Once a facility meets all the requirements to merit a license, licensing’s principal way of helping facilities maintain compliance is through periodic unannounced site visits by licensing program analysts (LPAs). In addition to conducting these inspections, our LPAs work with licensees to address any needed changes, and conduct follow-up visits to confirm that problems have been eliminated. It is also the role of the LPA to investigate any complaints about a facility and consult with licensees to ensure they are aware of their rights and responsibilities.
Licensing also provides ready access to the child care advocate program. The advocates serve as a link between licensing and the community of licensees, provide information about licensed child care to families and the public, act as a liaison with businesses, child care organizations, and resource and referral agencies, and coordinate complaints and concerns on behalf of children in care. As a result, licensing is both an enforcement agency and a resource on health and safety requirements.
31
The complete licensing regulations, often referred to as Title 22, can be found on the Community Care Licensing Division of the California Department of Social Services Child Care Regulations
Online California Code of Regulations, Title 22, Division 12 only
webpage (see Resources for Further Exploration). Highlights of the regulations can be found in Appendix B.
Licensing Requirements for Early Childhood Programs that are State-Subsidized
There are more stringent requirements for programs whose families get tuition assistance from the state of California, commonly referred to as Title 5. The most notable difference in Title 5 regulations is stricter adult-to-child ratios and staff qualifications. See Table 1.1 for a comparison of ratios.
32
Table 1.1 – Title 22 and Title 5 Adult to Child Ratio Comparison
33
,
34
,
35
Age Group
Title 22
Title 5
Infants
1 adult to 4 children
1 adult to 3 children
Preschoolers
1 adult to 12 children
1 adult to 8 children
School-Aged
1 adult to 14 children
1 adult to 14 children
Accreditation from the National Association for the Education of Young Children
While not legal regulations, there are even further requirements in place for programs that choose to pursue voluntary accreditation by the National Association for the Education of Young Children. One of the differences you will notice between licensing and accreditation is some levels of staff qualifications. See Table 1.2 for a comparison of staff qualifications.
Must be at least 18 years old and be supervised at all times by a teacher
Title 5
Aide: same as Title 22
Assistant: Must have 6 ECE units
NAEYC Accreditation
Child Development Associate (CDA) or 12 units in early childhood education, child development, elementary education, and/or early childhood special education
Teacher
Title 22
Must have 12 ECE units and 6 months of experience
Title 5
Associate teacher: 12 ECE units
Teacher: 24 ECE units and 16 GE units
Master Teacher: Teacher requirements
and
6 units in specialization area
and
2 units in adult supervision
NAEYC Accreditation
A minimum of higher education degree in early childhood education, child development, elementary education, or early childhood special education or 60 units with 30 units in early childhood education, child development, elementary education, and/ or early childhood special education. (preferred accredited higher education institutions)
Supervisor/ Director/ Administrator
Title 22
Must have 12 core ECE units and 3 administration units
Title 5
Site Supervisor: AA or 60 units with 24 ECE units
and
5 administration units
and
2 adult supervision units
Director: BA
and
24 ECE units
and
6 administration units
and
2 adult supervision units
NAEYC Accreditation
A minimum of a baccalaureate-level higher education degree in early childhood education, child development, elementary education, or early childhood special education; or 120 college units with 36 units in early childhood education, child development, elementary education, and/or early childhood special education.
While licensing regulations ensure that children stay safe and healthy, high quality care goes even further to provide the best start for young children. One reason that programs choose to become NAEYC accredited is to document the quality of care and education they provide children and families they serve. There are other processes and assessments that programs may use to ensure high quality, as well.
Pause to Reflect
As you progress through this book and course, what connections can you make about how being knowledgeable about health, safety, and nutrition will support early childhood educators in both following licensing and other applicable regulations and ensuring they provide high quality care for young children and their families?
Summary
Early childhood is a critical time in development. Many outcomes, both positive and negative, have their beginnings in these years. It is vital that children’s health and safety be protected. High-quality early care and education programs can play a valuable role in improving outcomes for children.


10.2: Rotational Variables
- Describe the physical meaning of rotational variables as applied to fixed-axis rotation
- Explain how angular velocity is related to tangential speed
- Calculate the instantaneous angular velocity given the angular position function
- Find the angular velocity and angular acceleration in a rotating system
- Calculate the average angular acceleration when the angular velocity is changing
- Calculate the instantaneous angular acceleration given the angular velocity function
So far in this text, we have mainly studied translational motion, including the variables that describe it: displacement, velocity, and acceleration. Now we expand our description of motion to rotation—specifically, rotational motion about a fixed axis. We will find that rotational motion is described by a set of related variables similar to those we used in translational motion.
Angular Velocity
Uniform circular motion (discussed previously in Motion in Two and Three Dimensions ) is motion in a circle at constant speed. Although this is the simplest case of rotational motion, it is very useful for many situations, and we use it here to introduce rotational variables.
In Figure \(\PageIndex{1}\), we show a particle moving in a circle. The coordinate system is fixed and serves as a frame of reference to define the particle’s position. Its position vector from the origin of the circle to the particle sweeps out the angle \(\theta\), which increases in the counterclockwise direction as the particle moves along its circular path. The angle \(\theta\) is called the angular position of the particle. As the particle moves in its circular path, it also traces an arc length s.
The angle is related to the radius of the circle and the arc length by
\[\theta = \frac{s}{r} \ldotp \label{10.1}\]
The angle \(\theta\), the angular position of the particle along its path, has units of radians (rad). There are \(2\pi\) radians in 360°. Note that the radian measure is a ratio of length measurements, and therefore is a dimensionless quantity. As the particle moves along its circular path, its angular position changes and it undergoes angular displacements \(\Delta \theta\).
We can assign vectors to the quantities in Equation \ref{10.1}. The angle \(\vec{\theta}\) is a vector out of the page in Figure \(\PageIndex{1}\). The angular position vector \(\vec{r}\) and the arc length \(\vec{s}\) both lie in the plane of the page. These three vectors are related to each other by
\[\vec{s} = \vec{\theta} \times \vec{r} \ldotp \label{10.2}\]
That is, the arc length is the cross product of the angle vector and the position vector, as shown in Figure \(\PageIndex{2}\).
The magnitude of the angular velocity, denoted by \(\omega\), is the time rate of change of the angle \(\theta\) as the particle moves in its circular path. The instantaneous angular velocity is defined as the limit in which \(\Delta\)t → 0 in the average angular velocity \(\bar{\omega} = \frac{\Delta \theta}{\Delta t}\):
\[\omega = \lim_{\Delta t \rightarrow 0} \frac{\Delta \theta}{\Delta t} = \frac{d \theta}{dt}, \label{10.3}\]
where \(\theta\) is the angle of rotation (Figure \(\PageIndex{2}\)). The units of angular velocity are radians per second (rad/s). Angular velocity can also be referred to as the rotation rate in radians per second. In many situations, we are given the rotation rate in revolutions/s or cycles/s. To find the angular velocity, we must multiply revolutions/s by 2\(\pi\), since there are 2\(\pi\) radians in one complete revolution. Since the direction of a positive angle in a circle is counterclockwise, we take counterclockwise rotations as being positive and clockwise rotations as negative.
We can see how angular velocity is related to the tangential speed of the particle by differentiating Equation \ref{10.1} with respect to time. We rewrite Equation \ref{10.1} as
\[s = r \theta \ldotp\]
Taking the derivative with respect to time and noting that the radius r is a constant, we have
\[\frac{ds}{dt} = \frac{d}{dt} (r \theta) = \theta \frac{dr}{dt} + r \frac{d \theta}{dt} = r \frac{d \theta}{dt}\]
where \(\theta \frac{dr}{dt}\) = 0. Here, \(\frac{ds}{dt}\) is just the tangential speed v t of the particle in Figure \(\PageIndex{1}\). Thus, by using Equation \ref{10.3}, we arrive at
\[v_{t} = r \omega \ldotp \label{10.4}\]
That is, the tangential speed of the particle is its angular velocity times the radius of the circle. From Equation \ref{10.4}, we see that the tangential speed of the particle increases with its distance from the axis of rotation for a constant angular velocity. This effect is shown in Figure \(\PageIndex{3}\). Two particles are placed at different radii on a rotating disk with a constant angular velocity. As the disk rotates, the tangential speed increases linearly with the radius from the axis of rotation. In Figure \(\PageIndex{3}\), we see that v 1 = r 1 \(\omega_{1}\) and v 2 = r 2 \(\omega_{2}\). But the disk has a constant angular velocity, so \(\omega_{1} = \omega_{2}\). This means \(\frac{v_{1}}{r_{1}} = \frac{v_{2}}{r_{2}}\) or v 2 = \(\left(\dfrac{r_{2}}{r_{1}}\right)\)v 1 . Thus, since r 2 > r 1 , v 2 > v 1 .
Up until now, we have discussed the magnitude of the angular velocity \(\omega = \frac{d \theta}{dt}\), which is a scalar quantity—the change in angular position with respect to time. The vector \(\vec{\omega}\) is the vector associated with the angular velocity and points along the axis of rotation. This is useful because when a rigid body is rotating, we want to know both the axis of rotation and the direction that the body is rotating about the axis, clockwise or counterclockwise. The angular velocity \(\vec{\omega}\) gives us this information. The angular velocity \(\vec{\omega}\) has a direction determined by what is called the right-hand rule. The right-hand rule is such that if the fingers of your right hand wrap counterclockwise from the x-axis (the direction in which \(\theta\) increases) toward the y-axis, your thumb points in the direction of the positive z-axis (Figure \(\PageIndex{4}\)). An angular velocity \(\vec{\omega}\) that points along the positive z-axis therefore corresponds to a counterclockwise rotation, whereas an angular velocity \(\vec{\omega}\) that points along the negative z-axis corresponds to a clockwise rotation.
We can verify the right-hand-rule using the vector expression for the arc length \(\vec{s} = \vec{\theta} \times \vec{r}\), Equation \ref{10.2}. If we differentiate this equation with respect to time, we find
\[\frac{d \vec{s}}{dt} = \frac{d}{dt}(\vec{\theta} \times \vec{r}) = \left(\dfrac{d \theta}{dt} \times \vec{r}\right) + \left(\vec{\theta} \times \dfrac{d \vec{r}}{dt}\right) = \frac{d \theta}{dt} \times \vec{r} \ldotp\]
Since \(\vec{r}\) is constant, the term \(\vec{\theta} \times \frac{d \vec{r}}{dt}\) = 0. Since \(\vec{v} = \frac{d \vec{s}}{dt}\) is the tangential velocity and \(\omega = \frac{d \vec{\theta}}{dt}\) is the angular velocity, we have
\[\vec{v} = \vec{\omega} \times \vec{r} \ldotp \label{10.5}\]
That is, the tangential velocity is the cross product of the angular velocity and the position vector, as shown in Figure \(\PageIndex{5}\). From part (a) of this figure, we see that with the angular velocity in the positive z-direction, the rotation in the xy-plane is counterclockwise. In part (b), the angular velocity is in the negative z-direction, giving a clockwise rotation in the xy-plane.
A flywheel rotates such that it sweeps out an angle at the rate of \(\theta\) = \(\omega\)t = (45.0 rad/s)t radians. The wheel rotates counterclockwise when viewed in the plane of the page. (a) What is the angular velocity of the flywheel? (b) What direction is the angular velocity? (c) How many radians does the flywheel rotate through in 30 s? (d) What is the tangential speed of a point on the flywheel 10 cm from the axis of rotation?
Strategy
The functional form of the angular position of the flywheel is given in the problem as \(\theta\)(t) = \(\omega\)t, so by taking the derivative with respect to time, we can find the angular velocity. We use the right-hand rule to find the angular velocity. To find the angular displacement of the flywheel during 30 s, we seek the angular displacement \(\Delta \theta\), where the change in angular position is between 0 and 30 s. To find the tangential speed of a point at a distance from the axis of rotation, we multiply its distance times the angular velocity of the flywheel.
Solution
- \(\omega\) = \(\frac{d \theta}{dt}\) = 45 rad/s. We see that the angular velocity is a constant.
- By the right-hand rule, we curl the fingers in the direction of rotation, which is counterclockwise in the plane of the page, and the thumb points in the direction of the angular velocity, which is out of the page.
- \(\Delta \theta\) = \(\theta\)(30 s) − \(\theta\)(0 s) = 45.0(30.0 s) − 45.0(0 s) = 1350.0 rad.
- v t = r\(\omega\) = (0.1 m)(45.0 rad/s) = 4.5 m/s.
Significance
In 30 s, the flywheel has rotated through quite a number of revolutions, about 215 if we divide the angular displacement by 2\(\pi\). A massive flywheel can be used to store energy in this way, if the losses due to friction are minimal. Recent research has considered superconducting bearings on which the flywheel rests, with zero energy loss due to friction.
Angular Acceleration
We have just discussed angular velocity for uniform circular motion, but not all motion is uniform. Envision an ice skater spinning with his arms outstretched—when he pulls his arms inward, his angular velocity increases. Or think about a computer’s hard disk slowing to a halt as the angular velocity decreases. We will explore these situations later, but we can already see a need to define an angular acceleration for describing situations where \(\omega\) changes. The faster the change in \(\omega\), the greater the angular acceleration. We define the instantaneous angular acceleration \(\alpha\) as the derivative of angular velocity with respect to time:
\[\alpha = \lim_{\Delta t \rightarrow 0} \frac{\Delta \omega}{\Delta t} = \frac{d \omega}{dt} = \frac{d^{2} \theta}{dt^{2}}, \label{10.6}\]
where we have taken the limit of the average angular acceleration, \(\bar{\alpha} = \frac{\Delta \omega}{\Delta t}\) as \(\Delta t → 0\). The units of angular acceleration are (rad/s)/s, or rad/s 2 .
In the same way as we defined the vector associated with angular velocity \(\vec{\omega}\), we can define \(\vec{\alpha}\), the vector associated with angular acceleration (Figure \(\PageIndex{6}\)). If the angular velocity is along the positive z-axis, as in Figure \(\PageIndex{4}\), and \(\frac{d \omega}{dt}\) is positive, then the angular acceleration \(\vec{\alpha}\) is positive and points along the +z- axis. Similarly, if the angular velocity \(\vec{\omega}\) is along the positive z-axis and \(\frac{d \omega}{dt}\) is negative, then the angular acceleration is negative and points along the +z-axis.
We can express the tangential acceleration vector as a cross product of the angular acceleration and the position vector. This expression can be found by taking the time derivative of \(\vec{v} = \vec{\omega} \times \vec{r}\) and is left as an exercise:
\[\vec{a} = \vec{\alpha} \times \vec{r} \ldotp \label{10.7}\]
The vector relationships for the angular acceleration and tangential acceleration are shown in Figure \(\PageIndex{7}\).
We can relate the tangential acceleration of a point on a rotating body at a distance from the axis of rotation in the same way that we related the tangential speed to the angular velocity. If we differentiate Equation \ref{10.4} with respect to time, noting that the radius r is constant, we obtain
\[a_{t} = r \alpha \ldotp \label{10.8}\]
Thus, the tangential acceleration a t is the radius times the angular acceleration. Equations \ref{10.4} and \ref{10.8} are important for the discussion of rolling motion (see Angular Momentum ).
Let’s apply these ideas to the analysis of a few simple fixed-axis rotation scenarios. Before doing so, we present a problem-solving strategy that can be applied to rotational kinematics: the description of rotational motion.
- Examine the situation to determine that rotational kinematics (rotational motion) is involved.
- Identify exactly what needs to be determined in the problem (identify the unknowns). A sketch of the situation is useful.
- Make a complete list of what is given or can be inferred from the problem as stated (identify the knowns).
- Solve the appropriate equation or equations for the quantity to be determined (the unknown). It can be useful to think in terms of a translational analog, because by now you are familiar with the equations of translational motion.
- Substitute the known values along with their units into the appropriate equation and obtain numerical solutions complete with units. Be sure to use units of radians for angles.
- Check your answer to see if it is reasonable: Does your answer make sense?
Now let’s apply this problem-solving strategy to a few specific examples.
A bicycle mechanic mounts a bicycle on the repair stand and starts the rear wheel spinning from rest to a final angular velocity of 250 rpm in 5.00 s. (a) Calculate the average angular acceleration in rad/s 2 . (b) If she now hits the brakes, causing an angular acceleration of −87.3 rad/s 2 , how long does it take the wheel to stop?
Strategy
The average angular acceleration can be found directly from its definition \(\bar{\alpha} = \frac{\Delta \omega}{\Delta t}\) because the final angular velocity and time are given. We see that \(\Delta \omega\) = \(\omega_{final}\) − \(\omega_{initial}\) = 250 rev/min and \(\Delta\)t is 5.00 s. For part (b), we know the angular acceleration and the initial angular velocity. We can find the stopping time by using the definition of average angular acceleration and solving for \(\Delta\)t, yielding
\[\Delta t = \frac{\Delta \omega}{\alpha} \ldotp\]
Solution
- Entering known information into the definition of angular acceleration, we get $$\bar{\alpha} = \frac{\Delta \omega}{\Delta t} = \frac{250\; rpm}{5.00\; s} \ldotp$$Because \(\Delta \omega\) is in revolutions per minute (rpm) and we want the standard units of rad/s 2 for angular acceleration, we need to convert from rpm to rad/s: $$\Delta \omega = 250 \frac{rev}{min}\; \cdotp \frac{2 \pi\; rad}{rev}\; \cdotp \frac{1\; min}{60\; s} = 26.2\; rad/s \ldotp$$Entering this quantity into the expression for \(\alpha\), we get $$bar{\alpha} = \frac{\Delta \omega}{\Delta t} = \frac{26.2\; rpm}{5.00\; s} = 5.24\; rad/s^{2} \ldotp$$
- Here the angular velocity decreases from 26.2 rad/s (250 rpm) to zero, so that \(\Delta \omega\) is −26.2 rad/s, and \(\alpha\) is given to be –87.3 rad/s 2 . Thus $$\Delta t = \frac{-26.2\; rad/s}{-87.3\; rad/s^{2}} = 0.300\; s \ldotp$$
Significance
Note that the angular acceleration as the mechanic spins the wheel is small and positive; it takes 5 s to produce an appreciable angular velocity. When she hits the brake, the angular acceleration is large and negative. The angular velocity quickly goes to zero.
The fan blades on a turbofan jet engine (shown below) accelerate from rest up to a rotation rate of 40.0 rev/s in 20 s. The increase in angular velocity of the fan is constant in time. (The GE90-110B1 turbofan engine mounted on a Boeing 777, as shown, is currently the largest turbofan engine in the world, capable of thrusts of 330–510 kN.) (a) What is the average angular acceleration? (b) What is the instantaneous angular acceleration at any time during the first 20 s?
A wind turbine (Figure \(\PageIndex{9}\)) in a wind farm is being shut down for maintenance. It takes 30 s for the turbine to go from its operating angular velocity to a complete stop in which the angular velocity function is \(\omega\)(t) = \(\Big[\frac{(ts^{−1} −30.0)^{2}}{100.0} \Big]\)rad/s. If the turbine is rotating counterclockwise looking into the page, (a) what are the directions of the angular velocity and acceleration vectors? (b) What is the average angular acceleration? (c) What is the instantaneous angular acceleration at t = 0.0, 15.0, 30.0 s?
Strategy
- We are given the rotational sense of the turbine, which is counterclockwise in the plane of the page. Using the right hand rule (Figure 10.5), we can establish the directions of the angular velocity and acceleration vectors.
- We calculate the initial and final angular velocities to get the average angular acceleration. We establish the sign of the angular acceleration from the results in (a).
- We are given the functional form of the angular velocity, so we can find the functional form of the angular acceleration function by taking its derivative with respect to time.
Solution
- Since the turbine is rotating counterclockwise, angular velocity \(\vec{\omega}\) points out of the page. But since the angular velocity is decreasing, the angular acceleration \(\vec{\alpha}\) points into the page, in the opposite sense to the angular velocity.
- The initial angular velocity of the turbine, setting t = 0, is \(\omega\) = 9.0 rad/s. The final angular velocity is zero, so the average angular acceleration is $$\bar{\alpha} \frac{\Delta \omega}{\Delta t} = \frac{\omega - \omega_{0}}{t - t_{0}} = \frac{0 - 9.0\; rad/s}{30.0 - 0\; s} = -0.3\; rad/s^{2} \ldotp$$
- Taking the derivative of the angular velocity with respect to time gives \(\alpha = \frac{d \omega}{dt} = \frac{(t − 30.0)}{50.0}\) rad/s 2 $$\alpha (0.0; s) = -0.6\; rad/s^{2}, \alpha (15.0\; s) = -0.3\; rad/s^{2}, and\; \alpha (30.0\; s) = 0\; rad/s \ldotp$$
Significance
We found from the calculations in (a) and (b) that the angular acceleration α and the average angular acceleration \(\bar{\alpha}\) are negative. The turbine has an angular acceleration in the opposite sense to its angular velocity.
We now have a basic vocabulary for discussing fixed-axis rotational kinematics and relationships between rotational variables. We discuss more definitions and connections in the next section.


23.5: Exercises
Compute each of the following Galois groups. Which of these field extensions are normal field extensions? If the extension is not normal, find a normal extension of \({\mathbb Q}\) in which the extension field is contained.
- \(\displaystyle G({\mathbb Q}(\sqrt{30}\, ) / {\mathbb Q})\)
- \(\displaystyle G({\mathbb Q}(\sqrt[4]{5}\, ) / {\mathbb Q})\)
- \(\displaystyle G( {\mathbb Q}(\sqrt{2}, \sqrt{3}, \sqrt{5}\, )/ {\mathbb Q} )\)
- \(\displaystyle G({\mathbb Q}(\sqrt{2}, \sqrt[3]{2}, i) / {\mathbb Q})\)
- \(\displaystyle G({\mathbb Q}(\sqrt{6}, i) / {\mathbb Q})\)
Determine the separability of each of the following polynomials.
- \(x^3 + 2 x^2 - x - 2\) over \({\mathbb Q}\)
- \(x^4 + 2 x^2 + 1\) over \({\mathbb Q}\)
- \(x^4 + x^2 + 1\) over \({\mathbb Z}_3\)
- \(x^3 +x^2 + 1\) over \({\mathbb Z}_2\)
Give the order and describe a generator of the Galois group of \(\gf(729)\) over \(\gf(9)\text{.}\)
Determine the Galois groups of each of the following polynomials in \({\mathbb Q}[x]\text{;}\) hence, determine the solvability by radicals of each of the polynomials.
- \(\displaystyle x^5 - 12 x^2 + 2\)
- \(\displaystyle x^5 - 4 x^4 + 2 x + 2\)
- \(\displaystyle x^3 - 5\)
- \(\displaystyle x^4 - x^2 - 6\)
- \(\displaystyle x^5 + 1\)
- \(\displaystyle (x^2 - 2)(x^2 + 2)\)
- \(\displaystyle x^8 - 1\)
- \(\displaystyle x^8 + 1\)
- \(\displaystyle x^4 - 3 x^2 -10\)
Find a primitive element in the splitting field of each of the following polynomials in \({\mathbb Q}[x]\text{.}\)
- \(\displaystyle x^4 - 1\)
- \(\displaystyle x^4 - 8 x^2 + 15\)
- \(\displaystyle x^4 - 2 x^2 - 15\)
- \(\displaystyle x^3 - 2\)
Prove that the Galois group of an irreducible quadratic polynomial is isomorphic to \({\mathbb Z}_2\text{.}\)
Prove that the Galois group of an irreducible cubic polynomial is isomorphic to \(S_3\) or \({\mathbb Z}_3\text{.}\)
Let \(F \subset K \subset E\) be fields. If \(E\) is a normal extension of \(F\text{,}\) show that \(E\) must also be a normal extension of \(K\text{.}\)
Let \(G\) be the Galois group of a polynomial of degree \(n\text{.}\) Prove that \(|G|\) divides \(n!\text{.}\)
Let \(F \subset E\text{.}\) If \(f(x)\) is solvable over \(F\text{,}\) show that \(f(x)\) is also solvable over \(E\text{.}\)
Construct a polynomial \(f(x)\) in \({\mathbb Q}[x]\) of degree \(7\) that is not solvable by radicals.
Let \(p\) be prime. Prove that there exists a polynomial \(f(x) \in{\mathbb Q}[x]\) of degree \(p\) with Galois group isomorphic to \(S_p\text{.}\) Conclude that for each prime \(p\) with \(p \geq 5\) there exists a polynomial of degree \(p\) that is not solvable by radicals.
Let \(p\) be a prime and \({\mathbb Z}_p(t)\) be the field of rational functions over \({\mathbb Z}_p\text{.}\) Prove that \(f(x) = x^p - t\) is an irreducible polynomial in \({\mathbb Z}_p(t)[x]\text{.}\) Show that \(f(x)\) is not separable.
Let \(E\) be an extension field of \(F\text{.}\) Suppose that \(K\) and \(L\) are two intermediate fields. If there exists an element \(\sigma \in G(E/F)\) such that \(\sigma(K) = L\text{,}\) then \(K\) and \(L\) are said to be conjugate fields . Prove that \(K\) and \(L\) are conjugate if and only if \(G(E/K)\) and \(G(E/L)\) are conjugate subgroups of \(G(E/F)\text{.}\)
Let \(\sigma \in \aut( {\mathbb R} )\text{.}\) If \(a\) is a positive real number, show that \(\sigma( a) > 0\text{.}\)
Let \(K\) be the splitting field of \(x^3 + x^2 + 1 \in {\mathbb Z}_2[x]\text{.}\) Prove or disprove that \(K\) is an extension by radicals.
Let \(F\) be a field such that \(\chr(F) \neq 2\text{.}\) Prove that the splitting field of \(f(x) = a x^2 + b x + c\) is \(F( \sqrt{\alpha}\, )\text{,}\) where \(\alpha = b^2 - 4ac\text{.}\)
Prove or disprove: Two different subgroups of a Galois group will have different fixed fields.
Let \(K\) be the splitting field of a polynomial over \(F\text{.}\) If \(E\) is a field extension of \(F\) contained in \(K\) and \([E:F] = 2\text{,}\) then \(E\) is the splitting field of some polynomial in \(F[x]\text{.}\)
We know that the cyclotomic polynomial
\[ \Phi_p(x) = \frac{x^p - 1}{x - 1} = x^{p - 1} + x^{p - 2} + \cdots + x + 1 \nonumber \]
is irreducible over \({\mathbb Q}\) for every prime \(p\text{.}\) Let \(\omega\) be a zero of \(\Phi_p(x)\text{,}\) and consider the field \({\mathbb Q}(\omega)\text{.}\)
- Show that \(\omega, \omega^2, \ldots, \omega^{p-1}\) are distinct zeros of \(\Phi_p(x)\text{,}\) and conclude that they are all the zeros of \(\Phi_p(x)\text{.}\)
- Show that \(G( {\mathbb Q}( \omega ) / {\mathbb Q} )\) is abelian of order \(p - 1\text{.}\)
- Show that the fixed field of \(G( {\mathbb Q}( \omega ) / {\mathbb Q} )\) is \({\mathbb Q}\text{.}\)
Let \(F\) be a finite field or a field of characteristic zero. Let \(E\) be a finite normal extension of \(F\) with Galois group \(G(E/F)\text{.}\) Prove that \(F \subset K \subset L \subset E\) if and only if \(\{ \identity \} \subset G(E/L) \subset G(E/K) \subset G(E/F)\text{.}\)
Let \(F\) be a field of characteristic zero and let \(f(x) \in F[x]\) be a separable polynomial of degree \(n\text{.}\) If \(E\) is the splitting field of \(f(x)\text{,}\) let \(\alpha_1, \ldots, \alpha_n\) be the roots of \(f(x)\) in \(E\text{.}\) Let \(\Delta = \prod_{i \lt j} (\alpha_i - \alpha_j)\text{.}\) We define the discriminant of \(f(x)\) to be \(\Delta^2\text{.}\)
- If \(f(x) = x^2 + b x + c\text{,}\) show that \(\Delta^2 = b^2 - 4c\text{.}\)
- If \(f(x) = x^3 + p x + q\text{,}\) show that \(\Delta^2 = - 4p^3 - 27q^2\text{.}\)
- Prove that \(\Delta^2\) is in \(F\text{.}\)
- If \(\sigma \in G(E/F)\) is a transposition of two roots of \(f(x)\text{,}\) show that \(\sigma( \Delta ) = -\Delta\text{.}\)
- If \(\sigma \in G(E/F)\) is an even permutation of the roots of \(f(x)\text{,}\) show that \(\sigma( \Delta ) = \Delta\text{.}\)
- Prove that \(G(E/F)\) is isomorphic to a subgroup of \(A_n\) if and only if \(\Delta \in F\text{.}\)
- Determine the Galois groups of \(x^3 + 2 x - 4\) and \(x^3 + x -3\text{.}\)


8.6: Hypothesis Test of a Single Population Mean with Examples
Steps for performing Hypothesis Test of a Single Population Mean
Step 1:
State your hypotheses about the population mean.
Step 2:
Summarize the data. State a significance level. State and check conditions required for the procedure
- Find or identify the sample size, n, the sample mean, \(\bar{x}\) and the sample standard deviation, s .
The sampling distribution for the one-mean test statistic is, approximately, T- distribution if the following conditions are met
- Sample is random with independent observations .
- Sample is large. The population must be Normal or the sample size must be at least 30.
Step 3: Perform the procedure based on the assumption that \(H_{0}\) is true
- Find the Estimated Standard Error: \(SE=\frac{s}{\sqrt{n}}\).
- Compute the observed value of the test statistic: \(T_{obs}=\frac{\bar{x}-\mu_{0}}{SE}\).
- Check the type of the test (right-, left-, or two-tailed)
- Find the p-value in order to measure your level of surprise.
Step 4: Make a decision about \(H_{0}\) and \(H_{a}\)
- Do you reject or not reject your null hypothesis?
Step 5: Make a conclusion
- What does this mean in the context of the data?
The following examples illustrate a left-, right-, and two-tailed test.
\(H_{0}: \mu = 5, H_{a}: \mu < 5\)
Test of a single population mean. \(H_{a}\) tells you the test is left-tailed. The picture of the \(p\)-value is as follows:
\(H_{0}: \mu = 10, H_{a}: \mu < 10\)
Assume the \(p\)-value is 0.0935. What type of test is this? Draw the picture of the \(p\)-value.
Answer
left-tailed test
\(H_{0}: \mu \leq 0.2, H_{a}: \mu > 0.2\)
This is a test of a single population proportion. \(H_{a}\) tells you the test is right-tailed . The picture of the p -value is as follows:
\(H_{0}: \mu \leq 1, H_{a}: \mu > 1\)
Assume the \(p\)-value is 0.1243. What type of test is this? Draw the picture of the \(p\)-value.
Answer
right-tailed test
\(H_{0}: \mu = 50, H_{a}: \mu \neq 50\)
This is a test of a single population mean. \(H_{a}\) tells you the test is two-tailed . The picture of the \(p\)-value is as follows.
\(H_{0}: \mu = 0.5, H_{a}: \mu \neq 0.5\)
Assume the p -value is 0.2564. What type of test is this? Draw the picture of the \(p\)-value.
Answer
two-tailed test
Full Hypothesis Test Examples
Statistics students believe that the mean score on the first statistics test is 65. A statistics instructor thinks the mean score is higher than 65. He samples ten statistics students and obtains the scores 65 65 70 67 66 63 63 68 72 71. He performs a hypothesis test using a 5% level of significance. The data are assumed to be from a normal distribution.
Answer
Set up the hypothesis test:
A 5% level of significance means that \(\alpha = 0.05\). This is a test of a single population mean .
\(H_{0}: \mu = 65 H_{a}: \mu > 65\)
Since the instructor thinks the average score is higher, use a "\(>\)". The "\(>\)" means the test is right-tailed.
Determine the distribution needed:
Random variable: \(\bar{X} =\) average score on the first statistics test.
Distribution for the test: If you read the problem carefully, you will notice that there is no population standard deviation given . You are only given \(n = 10\) sample data values. Notice also that the data come from a normal distribution. This means that the distribution for the test is a student's \(t\).
Use \(t_{df}\). Therefore, the distribution for the test is \(t_{9}\) where \(n = 10\) and \(df = 10 - 1 = 9\).
The sample mean and sample standard deviation are calculated as 67 and 3.1972 from the data.
Calculate the \(p\)-value using the Student's \(t\)-distribution:
\[t_{obs} = \dfrac{\bar{x}-\mu_{\bar{x}}}{\left(\dfrac{s}{\sqrt{n}}\right)}=\dfrac{67-65}{\left(\dfrac{3.1972}{\sqrt{10}}\right)}\]
Use the T-table or Excel's t_dist() function to find p-value:
\(p\text{-value} = P(\bar{x} > 67) =P(T >1.9782 )= 1-0.9604=0.0396\)
Interpretation of the p -value: If the null hypothesis is true, then there is a 0.0396 probability (3.96%) that the sample mean is 65 or more.
Compare \(\alpha\) and the \(p-\text{value}\):
Since \(α = 0.05\) and \(p\text{-value} = 0.0396\). \(\alpha > p\text{-value}\).
Make a decision: Since \(\alpha > p\text{-value}\), reject \(H_{0}\).
This means you reject \(\mu = 65\). In other words, you believe the average test score is more than 65.
Conclusion: At a 5% level of significance, the sample data show sufficient evidence that the mean (average) test score is more than 65, just as the math instructor thinks.
The \(p\text{-value}\) can easily be calculated.
Put the data into a list. Press
STAT
and arrow over to
TESTS
. Press
2:T-Test
. Arrow over to
Data
and press
ENTER
. Arrow down and enter 65 for \(\mu_{0}\), the name of the list where you put the data, and 1 for
Freq:
. Arrow down to \(\mu\): and arrow over to \(> \mu_{0}\). Press
ENTER
. Arrow down to
Calculate
and press
ENTER
. The calculator not only calculates the \(p\text{-value}\) (p = 0.0396) but it also calculates the test statistic (
t
-score) for the sample mean, the sample mean, and the sample standard deviation. \(\mu > 65\) is the alternative hypothesis. Do this set of instructions again except arrow to
Draw
(instead of
Calculate
). Press
ENTER
. A shaded graph appears with \(t = 1.9781\) (test statistic) and \(p = 0.0396\) (\(p\text{-value}\)). Make sure when you use
Draw
that no other equations are highlighted in \(Y =\) and the plots are turned off.
It is believed that a stock price for a particular company will grow at a rate of $5 per week with a standard deviation of $1. An investor believes the stock won’t grow as quickly. The changes in stock price is recorded for ten weeks and are as follows: $4, $3, $2, $3, $1, $7, $2, $1, $1, $2. Perform a hypothesis test using a 5% level of significance. State the null and alternative hypotheses, find the p -value, state your conclusion, and identify the Type I and Type II errors.
Answer
- \(H_{0}: \mu = 5\)
- \(H_{a}: \mu < 5\)
- \(p = 0.0082\)
Because \(p < \alpha\), we reject the null hypothesis. There is sufficient evidence to suggest that the stock price of the company grows at a rate less than $5 a week.
- Type I Error: To conclude that the stock price is growing slower than $5 a week when, in fact, the stock price is growing at $5 a week (reject the null hypothesis when the null hypothesis is true).
- Type II Error: To conclude that the stock price is growing at a rate of $5 a week when, in fact, the stock price is growing slower than $5 a week (do not reject the null hypothesis when the null hypothesis is false).
The National Institute of Standards and Technology provides exact data on conductivity properties of materials. Following are conductivity measurements for 11 randomly selected pieces of a particular type of glass.
1.11; 1.07; 1.11; 1.07; 1.12; 1.08; .98; .98 1.02; .95; .95
Is there convincing evidence that the average conductivity of this type of glass is greater than one? Use a significance level of 0.05. Assume the population is normal.
Answer
Let’s follow a four-step process to answer this statistical question.
-
State the Question
: We need to determine if, at a 0.05 significance level, the average conductivity of the selected glass is greater than one. Our hypotheses will be
- \(H_{0}: \mu \leq 1\)
- \(H_{a}: \mu > 1\)
- Plan : We are testing a sample mean without a known population standard deviation. Therefore, we need to use a Student's-t distribution. Assume the underlying population is normal.
- Do the calculations : \(p\text{-value} ( = 0.036)\)
4. State the Conclusions : Since the \(p\text{-value} (= 0.036)\) is less than our alpha value, we will reject the null hypothesis. It is reasonable to state that the data supports the claim that the average conductivity level is greater than one.
Review
The hypothesis test itself has an established process. This can be summarized as follows:
- Determine \(H_{0}\) and \(H_{a}\). Remember, they are contradictory.
- Determine the random variable.
- Determine the distribution for the test.
- Draw a graph, calculate the test statistic, and use the test statistic to calculate the \(p\text{-value}\). (A t -score is an example of test statistics.)
- Compare the preconceived α with the p -value, make a decision (reject or do not reject H 0 ), and write a clear conclusion using English sentences.
Notice that in performing the hypothesis test, you use \(\alpha\) and not \(\beta\). \(\beta\) is needed to help determine the sample size of the data that is used in calculating the \(p\text{-value}\). Remember that the quantity \(1 – \beta\) is called the Power of the Test . A high power is desirable. If the power is too low, statisticians typically increase the sample size while keeping α the same.If the power is low, the null hypothesis might not be rejected when it should be.
References
- Data from Amit Schitai. Director of Instructional Technology and Distance Learning. LBCC.
- Data from Bloomberg Businessweek . Available online at www.businessweek.com/news/2011- 09-15/nyc-smoking-rate-falls-to-record-low-of-14-bloomberg-says.html.
- Data from energy.gov. Available online at http://energy.gov (accessed June 27. 2013).
- Data from Gallup®. Available online at www.gallup.com (accessed June 27, 2013).
- Data from Growing by Degrees by Allen and Seaman.
- Data from La Leche League International. Available online at www.lalecheleague.org/Law/BAFeb01.html.
- Data from the American Automobile Association. Available online at www.aaa.com (accessed June 27, 2013).
- Data from the American Library Association. Available online at www.ala.org (accessed June 27, 2013).
- Data from the Bureau of Labor Statistics. Available online at http://www.bls.gov/oes/current/oes291111.htm .
- Data from the Centers for Disease Control and Prevention. Available online at www.cdc.gov (accessed June 27, 2013)
- Data from the U.S. Census Bureau, available online at quickfacts.census.gov/qfd/states/00000.html (accessed June 27, 2013).
- Data from the United States Census Bureau. Available online at www.census.gov/hhes/socdemo/language/.
- Data from Toastmasters International. Available online at http://toastmasters.org/artisan/deta...eID=429&Page=1 .
- Data from Weather Underground. Available online at www.wunderground.com (accessed June 27, 2013).
- Federal Bureau of Investigations. “Uniform Crime Reports and Index of Crime in Daviess in the State of Kentucky enforced by Daviess County from 1985 to 2005.” Available online at http://www.disastercenter.com/kentucky/crime/3868.htm (accessed June 27, 2013).
- “Foothill-De Anza Community College District.” De Anza College, Winter 2006. Available online at research.fhda.edu/factbook/DA...t_da_2006w.pdf.
- Johansen, C., J. Boice, Jr., J. McLaughlin, J. Olsen. “Cellular Telephones and Cancer—a Nationwide Cohort Study in Denmark.” Institute of Cancer Epidemiology and the Danish Cancer Society, 93(3):203-7. Available online at http://www.ncbi.nlm.nih.gov/pubmed/11158188 (accessed June 27, 2013).
- Rape, Abuse & Incest National Network. “How often does sexual assault occur?” RAINN, 2009. Available online at www.rainn.org/get-information...sexual-assault (accessed June 27, 2013).
Glossary
- Central Limit Theorem
- Given a random variable (RV) with known mean \(\mu\) and known standard deviation \(\sigma\). We are sampling with size \(n\) and we are interested in two new RVs - the sample mean, \(\bar{X}\), and the sample sum, \(\sum X\). If the size \(n\) of the sample is sufficiently large, then \(\bar{X} - N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)\) and \(\sum X - N \left(n\mu, \sqrt{n}\sigma\right)\). If the size n of the sample is sufficiently large, then the distribution of the sample means and the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample means will equal the population mean and the mean of the sample sums will equal \(n\) times the population mean. The standard deviation of the distribution of the sample means, \(\frac{\sigma}{\sqrt{n}}\), is called the standard error of the mean.


13.1: Introduction
The U.S. spends more on healthcare than many other countries (OECD, 2023), and spends far more on healthcare than on public health (Pilar et al., 2020). Yet several of the leading causes of death are strongly influenced by lifestyle behaviors. This has led several researchers to quantify the actual causes of death in terms of behaviors and environmental factors. Prior to the COVID-19 pandemic, actual causes of death in 2017 were identified as the following list, with the percentage of total deaths associated in parenthesis. Note that these percentages will not add up to 100% because there are other causes of death not on this list, and many diseases have multiple causes.
- Poor nutrition (19%)
- Tobacco use (18%)
- Toxic agents (7%)
- Microbial agents (4%)
- Illicit drug use (4%)
- Alcohol use (3%)
- Physical inactivity (3%)
- Firearms (1%)
- Motor vehicles (1%)
- Sexual behavior (1%) (Pilar et al., 2020).
A recent study looked at trends in self-reported healthy behaviors between 1999-2000 and 2017-2020 using data from the National Health and Nutrition Examination Survey (NHANES). Results indicate that over the last few decades, more Americans are avoiding smoking tobacco (57.7%), being physically active at least 150 minutes per week (69.1%), and eating a healthy diet (24.5%). However, we are still gaining weight: those reporting a healthy weight status decreased from 33.1% to 24.6%, and our alcohol consumption has remained about the same. Those who reported 4 or more healthy lifestyle behaviors increased slightly from 15.7% to 20.3%. This study also showed consistent disparities in race and ethnicity, education level, and income level, suggesting that social determinants of health (SDOH) are still powerfully associated with healthy behaviors (Li et al., 2023).
It makes sense then that many community health interventions target changing lifestyle behaviors in a population in order to prevent a disease or injury from occurring in the first place (i.e. primary prevention). In order to effectively change human behavior on a large scale, we need to first understand what makes humans behave the way they do. In this chapter, we will go over a few (not all) of the behavioral change models used in public health practice, as well as community intervention models, and program planning models. At the end of this chapter we will discuss two examples of community interventions targeting physical activity and childhood obesity.


10.9: Summary
- 360 assessment
-
An evaluation tool that collects feedback from manager, peers, direct reports, and customers.
- 9-box
-
A matrix tool used to evaluate an organization’s talent pool based on performance and potential factors.
- Competencies
-
A set of defined behaviors that an organization might utilize to define standards for success.
- Employee life cycle
-
The various stages of engagement of an employee—attraction, recruitment, onboarding, development, retention, and separation.
- Employer-employee relationship
-
The employment relationship; the legal link between employers and employees that exists when a person performs work or services under specific conditions in return for payment.
- Human capital
-
The skills, knowledge, and experience of an individual or group and that value to an organization.
- Human resource management
-
The management of people within organizations, focusing on the touchpoints of the employee life cycle.
- Human resources compliance
-
The HR role to ensure adherence to laws and regulations that govern the employment relationship.
- Merit matrix
-
A calculation table that provides a framework for merit increases based on performance levels.
- Pay-for-performance model
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The process and structure for tying individual performance levels to rewards levels
- Performance management
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The process by which an organization ensures that its overall goals are being met by evaluating the performance of individuals within that organization.
- Society for Human Resource Management
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The world’s largest HR professional society, with more than 285,000 members in more than 165 countries. It is a leading provider of resources serving the needs of HR professionals.
- Succession planning
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The process of identifying and developing new leaders and high-potential employees to replace current employees at a future time.
- Talent acquisition
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The process of finding and acquiring skilled candidates for employment within a company; it generally refers to a long-term view of building talent pipelines rather than short-term recruitment.
- Talent development
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Integrated HR processes that are created to attract, develop, motivate, and retain employees.
- Talent review calibration process
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The meeting in which an organization’s 9-box matrix is reviewed and discussed, with input and sharing from organizational leadership.
- Total rewards strategy
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As coined by World at Work, includes compensation, benefits, work-life effectiveness, recognition, performance management, and talent development.
- Training, stretch assignments, individual assessments, individual development plans
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These are tools that may be used in talent development:
Training—a forum for learning in person or online
Stretch assignments—challenge roles for high-potential employees
Individual assessments—personality and work style inventories of employees
Individual development plans—documents that highlight an individual employee’s opportunities for growth and path of action
- War for talent
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Coined by McKinsey & Company in 1997, it refers to the increasing competition for recruiting and retaining talented employees.
Summary of Learning Outcomes
10.2 An Introduction to Human Resource Management
1. What has been the evolution of human resource management over the years, and what is the current value it provides to an organization?
Human resource management began in its first “wave” as a primarily compliance-type function, with the HR staff charged with enforcing compliance of employees and running the ongoing administrative processes. In the second wave, HR became focused on the design of HR practice areas, which could be built upon best-practice models. Wave 3 of HR brought with it the concept that HR should be a true partner to the business and should support the business strategy through its programs and services. Finally, in the fourth wave, HR is still a partner to the business, but it looks outside of the business to customers, investors, and communities to see how it can be competitive in terms of customer share, investor confidence, and community reputation.
Some key areas that HR supports within the employee life cycle process include human resources compliance, employee selection and hiring, performance management, compensation rewards, and talent development and succession planning.
10.3 Human Resource Management and Compliance
2. How does the human resources compliance role of HR provide value to a company?
Human resources help protect the company and its employees to ensure that they are adhering to the numerous regulations and laws that govern the employment relationship. The impact of noncompliance can be very costly and can be in the form of financial, legal, or reputational cost. Some of the key legislation that HR manages compliance around includes the Fair Labor Standards Act (FLSA), the Age Discrimination in Employment Act (ADEA), the Americans with Disabilities Act (ADA), and the Family and Medical Leave Act (FMLA), among others.
Some of the best practices for informing and holding employees accountable are to provide education and training to explain the regulations, to provide reference documentation for guidance with the regulations, and to schedule regular compliance audits to ensure that processes are being followed. Scheduling regular internal HR audits help the organization plan and feel comfortable with its level of preparedness and illustrates the value that a strong HR group can bring to the organization.
10.4 Performance Management
3. How do performance management practices impact company performance?
Performance management is a critical business process that the human resources group manages for the business. Performance management aligns the work of individual groups with the overall business objectives and enables the business to work toward its goals. Performance management should also help the company differentiate between different levels of employee performance through the management of feedback and a rewards structure.
Performance management also allows a company to identify its poor performers and provides a consistent process for tracking and managing poor performance in a manner that is fair and consistent with the law. There has been much discussion of best practices for a performance management process beyond a formal, annual process that often feels cumbersome to the business. However formal or informal, human resource management needs to ensure that the process helps to differentiate different levels of performance, manages the flow of feedback, and is consistent and fair for all employees.
10.5 Influencing Employee Performance and Motivation
4. How do companies use rewards strategies to influence employee performance and motivation?
Companies use rewards strategies to influence employee performance and motivation by differentiating between the various levels of performance. This strategy is called pay for performance, and it ties the employee’s performance level to a consistent framework of rewards at each level. Research indicates that the primary reason that companies implement pay-for-performance is to be able to recognize and reward their high performers.
To implement a pay-for-performance structure, HR and the organization first need to define a compensation philosophy, then perform a review of the financial implications of such a system. Gaps in the current system must be identified, and compensation practices should be updated in accordance with the determined pay-for-performance design. Finally, communication and training are key to helping employees understand the context and philosophy, as well as the specific methodology.
10.6 Building and Organization for the Future
5. What is talent acquisition, and how can it create a competitive advantage for a company?
Human resource management plays the important role of managing the talent processes for an organization, and it is critical in the process of acquiring talent from the outside. Talent acquisition is the process of determining what roles are still needed in the organization, where to find people, and whom to hire. Hiring top talent is a key source of competitive advantage for a company, and not all organizations are good at doing it.
The impact of hiring is especially magnified when you talk about top leadership talent. The right leadership candidate can make all the difference in an organization’s growth, performance, and trajectory over the years. HR should work with the business to assess the needs and specifics of the job, develop a pool of candidates, and then assess candidates for the right person to bring into the organization.
10.7 Talent Development and Succession Planning
6. What are the benefits of talent development and succession planning?
Talent development and succession planning processes provide organizations with the systems needed to assess and develop employees and to make the appropriate decisions on their internal movement and development. One important talent development process involves a talent review, in which leadership discusses the employees in its groups in terms of their performance and potential. Performance is based on current performance management evaluations of the current role. Potential is based on behavioral indications that would predict future high performance and promotability in an organization. There is then a discussion on the follow-up actions and development plans for the employees based on where they fall in the performance/potential matrix. The benefit of this process is that the organization gains a better understanding of where the top talent is within the organization and can make plans to manage the development of that talent.
Another key process for managing talent is succession planning. In this process, leadership and HR meet to identify leadership roles and other critical roles in the organization, and then they discuss a potential pipeline of internal and external successor candidates at different levels of readiness for the role. The output of succession planning is that an organization gets to understand the depth of its talent bench and knows the gap areas where it may need to focus on developing or acquiring additional candidates.
- What are the four “waves” of the human resource management evolution?
- What are some of the key regulations that human resources must manage compliance with?
- What are some of the unintended consequences of a forced ranking system?
- What are some of the performance management challenges that must be addressed, no matter what the system?
- Why are many companies interested in moving to a pay-for-performance strategy?
- What are the main process steps for implementing pay for performance?
- What are some best practices for recruiting new leadership candidates?
- Describe the steps of a talent review session.
- What is the difference between performance and potential?
- How can you tell if a candidate has potential?
Management Skills Application Exercises
- How has human resource management’s evolution over the years helped to make it a better partner to the business? In what way would you expect HRM to continue to evolve over the years?
- Do you believe that a formal, annual performance management process is necessary to help an organization reach its goals? Why or why not? What are the minimum process requirements that must be met to successfully evaluate performance?
- Is it possible for an organization to reward people fairly without implementing a pay-for-performance process? Why or why not? Do you see any pitfalls to a pay-for-performance process?
- How does the “war for talent” impact talent acquisition processes? How can HR be more successful working with the business to navigate the competitive talent landscape?
- What are the benefits of having talent review calibration processes? What is the downside of the process? Should an organization let employees know what their talent review “rating” is? Why or why not?
Managerial Decision Exercises
- You have been hired as a new Finance VP, and you oversee a team of almost 30 people. Your HR manager has recently informed you that there have been several employee relations in your group in the recent past, and you are concerned about the level of knowledge that your management team has around dealing with these issues. What could you do to close the gap in knowledge and mitigate the risk of issues in your group?
- Your company has decided to drop its formal, annual performance management process and move to a system based on ongoing feedback and communication with employees. You are concerned because you have always been careful to differentiate your employees by performance level, and you’re worried that this will hurt your stronger employees. How can use ensure that your feedback and communication with employees provides performance management, despite the lack of a formal system?
- Your company has recently implemented a pay-for-performance model for compensation. This worries you because you know that your employees will be even more upset with their performance ratings if they know that they are tied to compensation. What actions can you take to start to prepare for this change?
- You are the director of an engineering organization and have been fighting the “war for talent” for a while. It seems that whenever you have a role vacancy, you let HR know, but it takes forever to find someone—and the candidate often turns down the job. What are some ways to better partner with HR to get ahead of the curve for the next time?
- You are the VP of a line of business at an international manufacturing company. You and several of your long-time colleagues will be retiring over the next few years, and you need to start thinking about talent and succession planning. You are going into a talent review discussion next week, and you’re realizing that you have a dearth of potential within your organization. What are some actions you (and HR) can take now to ensure that your business unit isn’t floundering when you leave for retirement?
Critical Thinking Case
Zappos, Holacracy, and Human Resource Management
In 2013, Zappos was performing well under the leadership of Tony Hsieh and was getting ready to take on a new challenge that would, among other things, push the boundaries of traditional human resource management. Although business was booming, Tony Hsieh was not a man who wanted to be in status quo mode for too long, so he set out to implement an organizational and cultural change called a holacracy. Zappos was the largest and best known of the 300 companies worldwide that had adopted holacracy—a new form of hierarchy that is a “flexible, self-governing structure, where there are no fixed jobs but simply temporary functional roles.”
In a holacracy, the main unit is called the “circle,” a distinct yet fluid team. Leadership became similarly fluid with the changing circles. Circles are designed to meet certain goals and are created and disbanded as project needs change. The intent is that people self-select to work on projects that they want to work on and that they have the skills for. Tony also removed all previous titles. The role of manager went away and was replaced with three roles: “lead links” would focus on guiding the work in the circles; “mentors” would work on employee growth and development; and “compensation appraisers” would work on determining employees’ salaries. In 2015, he decided to further break down the divisions between many of the functions, changing them all to business-centric circles. There were changes to almost every human resource management structure that you can think of, and there were quite a few growing pains within the organization. Zappos began to look at employee pay, and Holacracy seemed to have a steep learning curve for many people, even though a “constitution” was created to provide guidance. Zappos was also facing 14% attrition, as some of the rapid and excessive changes were wearing on employees. Tony was a visionary, but for a lot of people, it was hard to catch up and see the same vision.
From a human resource management perspective, there could be some positive attributes of a holacracy if it were to succeed—such as building engagement and helping to build talent and skill sets. There were also a few risks that needed to be dealt with carefully. When you create an organization in which people don’t have set teams or projects but instead determine what they want to work on, one of the big challenges is going to be determining the level and nature of their role, as well as the compensation for that role. If holacracy is compared to a consulting organization, in which consultants are brought into different projects with different requirements, it is critical to first determine the level of their consultant role (based on their education, skills, experience, etc.) so that they can properly move from project to project yet maintain a role of a certain level. That level is then tied to a specific pay scale, so the same consultant will receive the same salary no matter which project he is on. If that consultant is “on the bench” or not placed on a project (or self-placed, in the case of holacracy), then after a certain defined period, that consultant may be at risk of termination.
Holacracy is, in some ways, a challenging concept to think about, and self-management may not be able to work in all environments. A company that is implementing a holacracy may find that they are able to master the process of self-selection of work in the “circles.” The “task” part of the equation may not be much of an issue once people figure out how to navigate the circles. However, the “people” part of the equation may need some work. The greatest challenge may lie in the structures and processes of human resource management that ultimately define the employer-employee relationship.
Critical Thinking Questions
- What are some of the human resource management processes that might be enhanced by a holacracy? What processes will be challenged?
- Do you think that a holacracy can be compared with a consulting company? How are they similar, and how are they different? Can you think of work areas or industries in which holacracy would be very difficult to implement?


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Iron deficiency anemia, characterized by low hemoglobin concentration and microcytic red blood cells, is the most common nutritional blood disorder in young children, pregnant women, and older adults.
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Nutrient-related hematologic disorders include vitamin B
12
deficiency, vitamins C, K, and D deficiency, and folate deficiency.
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Delays in diagnosing excessive or deficient nutritional intake can lead to hematologic abnormalities resulting in life-threatening conditions. Accurate assessment of the client’s nutritional status with early intervention gives clients the best chance for recovery.
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Acute and long-term illnesses such as trauma, burns, cancer, lung disease, and immune dysfunction require even higher protein and micronutrient intake to restore homeostasis during the healing stage.
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Aging adults are susceptible to nutrient-deficient blood abnormalities because of decreased production of hormones and red blood cells and a decreased appetite/food intake related to delayed gastric emptying, changes in papillae, and declining oral health.
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Use of the TTM for nutrition educational interventions can provide clients a degree of motivation for successful restoration of hematologic wellness.


3.4: The Nucleus and DNA Replication
By the end of this section, you will be able to:
- Describe the structure and features of the nuclear membrane
- List the contents of the nucleus
- Explain the organization of the DNA molecule within the nucleus
- Describe the process of DNA replication
The nucleus is the largest and most prominent of a cell’s organelles (Figure \(\PageIndex{1}\)). The nucleus is generally considered the control center of the cell because it stores all of the genetic instructions for manufacturing proteins. Interestingly, some cells in the body, such as muscle cells, contain more than one nucleus (Figure \(\PageIndex{2}\)), which is known as multinucleated. Other cells, such as mammalian red blood cells (RBCs), do not contain nuclei at all. RBCs eject their nuclei as they mature, making space for the large numbers of hemoglobin molecules that carry oxygen throughout the body (Figure \(\PageIndex{3}\)). Without nuclei, the life span of RBCs is short, and so the body must produce new ones constantly.
Interactive Link
View the University of Michigan WebScope to explore the tissue sample in greater detail.
Inside the nucleus lies the blueprint that dictates everything a cell will do and all of the products it will make. This information is stored within DNA. The nucleus sends “commands” to the cell via molecular messengers that translate the information from DNA. Each cell in your body (with the exception of germ cells) contains the complete set of your DNA. When a cell divides, the DNA must be duplicated so that the each new cell receives a full complement of DNA. The following section will explore the structure of the nucleus and its contents, as well as the process of DNA replication.
Organization of the Nucleus and Its DNA
Like most other cellular organelles, the nucleus is surrounded by a membrane called the nuclear envelope . This membranous covering consists of two adjacent lipid bilayers with a thin fluid space in between them. Spanning these two bilayers are nuclear pores. A nuclear pore is a tiny passageway for the passage of proteins, RNA, and solutes between the nucleus and the cytoplasm. Proteins called pore complexes lining the nuclear pores regulate the passage of materials into and out of the nucleus.
Inside the nuclear envelope is a gel-like nucleoplasm with solutes that include the building blocks of nucleic acids. There also can be a dark-staining mass often visible under a simple light microscope, called a nucleolus (plural = nucleoli). The nucleolus is a region of the nucleus that is responsible for manufacturing the RNA necessary for construction of ribosomes. Once synthesized, newly made ribosomal subunits exit the cell’s nucleus through the nuclear pores.
The genetic instructions that are used to build and maintain an organism are arranged in an orderly manner in strands of DNA. Within the nucleus are threads of chromatin composed of DNA and associated proteins (Figure \(\PageIndex{4}\)). Along the chromatin threads, the DNA is wrapped around a set of histone proteins. A nucleosome is a single, wrapped DNA-histone complex. Multiple nucleosomes along the entire molecule of DNA appear like a beaded necklace, in which the string is the DNA and the beads are the associated histones. When a cell is in the process of division, the chromatin condenses into chromosomes, so that the DNA can be safely transported to the “daughter cells.” The chromosome is composed of DNA and proteins; it is the condensed form of chromatin. It is estimated that humans have almost 22,000 genes distributed on 46 chromosomes.
DNA Replication
In order for an organism to grow, develop, and maintain its health, cells must reproduce themselves by dividing to produce two new daughter cells, each with the full complement of DNA as found in the original cell. Billions of new cells are produced in an adult human every day. Only very few cell types in the body do not divide, including nerve cells, skeletal muscle fibers, and cardiac muscle cells. The division time of different cell types varies. Epithelial cells of the skin and gastrointestinal lining, for instance, divide very frequently to replace those that are constantly being rubbed off of the surface by friction.
A DNA molecule is made of two strands that “complement” each other in the sense that the molecules that compose the strands fit together and bind to each other, creating a double-stranded molecule that looks much like a long, twisted ladder. This double helix can be constructed easily because the two strands are antiparallel, meaning the two strands run in opposite directions. Each side rail of the DNA ladder is composed of alternating sugar and phosphate groups (Figure \(\PageIndex{5}\)). The two sides of the ladder are not identical, but are complementary. These two backbones are bonded to each other across pairs of protruding bases, each bonded pair forming one “rung,” or cross member. The four DNA bases are adenine (A), thymine (T), cytosine (C), and guanine (G). Because of their shape and charge, the two bases that compose a pair always bond together. Adenine always binds with thymine, and cytosine always binds with guanine. The particular sequence of bases along the DNA molecule determines the genetic code. Therefore, if the two complementary strands of DNA were pulled apart, you could infer the order of the bases in one strand from the bases in the other, complementary strand. For example, if one strand has a region with the sequence AGTGCCT, then the sequence of the complementary strand would be TCACGGA.
DNA replication is the copying of DNA that occurs before cell division can take place. After a great deal of debate and experimentation, the general method of DNA replication was deduced in 1958 by two scientists in California, Matthew Meselson and Franklin Stahl. This method is illustrated in Figure \(\PageIndex{6}\) and described below.
Stage 1: Initiation. The two complementary strands are separated, much like unzipping a zipper. Special enzymes, including helicase , untwist and separate the two strands of DNA.
Stage 2: Elongation. Each strand becomes a template along which a new complementary strand is built. DNA polymerase brings in the correct bases to complement the template strand, synthesizing a new strand base by base. A DNA polymerase is an enzyme that adds free nucleotides to the end of a chain of DNA, making a new double strand. This growing strand continues to be built until it has fully complemented the template strand.
Stage 3: Termination. Once the two original strands are bound to their own, finished, complementary strands, DNA replication is stopped and the two new identical DNA molecules are complete.
Each new DNA molecule contains one strand from the original molecule and one newly synthesized strand. The term for this mode of replication is “semiconservative,” because half of the original DNA molecule is conserved in each new DNA molecule. This process continues until the cell’s entire genome , the entire complement of an organism’s DNA, is replicated. As you might imagine, it is very important that DNA replication take place precisely so that new cells in the body contain the exact same genetic material as their parent cells. Mistakes made during DNA replication, such as the accidental addition of an inappropriate nucleotide, have the potential to render a gene dysfunctional or useless. Fortunately, there are mechanisms in place to minimize such mistakes. A DNA proofreading process enlists the help of special enzymes that scan the newly synthesized molecule for mistakes and corrects them. Once the process of DNA replication is complete, the cell is ready to divide. You will explore the process of cell division later in the chapter.
Interactive Link
Watch this video to learn about DNA replication. DNA replication proceeds simultaneously at several sites on the same molecule. What separates the base pair at the start of DNA replication?