2.6: Vesicles
The most basic definition of a vesicle is a compartment composed of many phospholipids with some form of head group. In a biological context, vesicles are typically formed by cells to uptake, excrete, or otherwise transport materials between membranous compartments in the cell. A synthetic vesicle, called a liposome, can be created by mixing phospholipid molecules in an aqueous environment. This article outlines some fundamental properties of vesicles, then goes on to discuss them in a biological context.
Formation of a Vesicle
Firstly, curvature must be established in a bilayer. Eventually, the indentation must protrude farther and farther, until a neck can be formed. Scission of the neck occurs to free the newly formed vesicle from the membrane. [8]
The initial indentation into the bilayer necessitates a difference in the radius of curvature between the inner and outer leaflet. Compensation for this physical requirement can happen in several ways. One method involves leaflet asymmetry, where more phospholipids are shifted to the outside leaflet to compensate for the larger radius. If different types of phospholipids with varying geometries are used, these may also be incorporated in a fashion that compensates for the difference in radii equally well. The same balance can be struck using proteins incorporated into the membrane. [8,17]
In an in vitro setting, phospholipids that are introduced to an aqueous environment can form several structures to optimize the arrangement of its hydrophobic and hydrophilic portions. Among these are micelles, liposomes, and multilamellar vesicles. Sonication of multilamellar vesicles can yield unilamellar vesicles and giant unilamellar vesicles (GUVs). Using a single type of phospholipid under these conditions yields a closed pure lipid bilayer membrane, which has been a useful model for understanding phase shifts among the lipids of the membrane, the mechanics of membrane formation and deformation, the dynamic behavior of elements that reside within the membrane, and the interactions between membranes and their environments. [2,4]
Biological Context
Vesicles mainly serve to hold some biological element apart from the surroundings. This may include biological molecules that must be transported from site to site or housing a particular chemical reaction until the products can be released into the cell.
Vesicles are most notably involved in taking nutrients in and excreting waste products out of the cell, called endocytosis and exocytosis, respectively. Vesicles serve as the chief mode of transport between the extracellular environment and various compartments in the cell such as the exosome, endosome, the trans golgi network (TGN), and the endoplasmic reticulum. Extracellular messaging works along this same system. As signals diffuse throughout the extracellular environment, they may interact with receptors that are then taken inside the cell in a vesicle or taken up inside a vesicle themselves. A number of secondary messengers may come into play to transduce the signal, or the signal may have an immediate intracellular target that elicits some change – expression of a particular gene or initiation of some post-translational modification, for example. Neuronal vesicles can carry signals between cells across the synaptic cleft or within the neuron itself.
In the cell, formation of any vesicle begins with a lipid bilayer. Perturbation and deformation at a section of this bilayer creates the proper combination of negative and positive curvature to develop the vesicle, a structure with wholly positive curvature.[8] The range of vesicles studied in a cell stretches anywhere between 30nm to 1.2µm. [6]
Vesicle Formation in vivo
The formation, or budding, of a vesicle in a cell involves deforming the membrane and creating specificity for a certain cargo. Early electron microscopy studies detected electron dense ‘coats’ in budding intermediates. The electron dense areas were later discovered to be coat proteins involved in creation of a vesicle.[10] These proteins coat a membrane section to create and stabilize curvature, then select the proper cargo. The vesicle then must be pinched off at the neck to create a membraneous structure separate from the larger membrane. These processes are typically mediated by groups of proteins. The best characterized mechanisms involve the clathrin protein, co at p rotein I (COPI), and co at p rotein II (COPII). [1, 9, 10]
Clathrin proteins are trimers, called triskellions, best represented as a central domain composed of three heavy chains (~190kDa) interacting with each other at their C-termini, bound with three light chains (~25kDa) extending outwards. Distal to the central domain are the N-termini, seven-bladed β-propeller structures with binding sites for a multitude of adaptor proteins to transfer a variety of cargo types in the potential vesicle. In solution, triskelia form flat hexagonal rings with relative ease. For the formation of a vesicle, triskelia must also form pentagonal rings, in addition to hexagonal rings, to generate enough curvature for the encapsulation of a vesicle. At least 12 pentagons must be present in the lattice to form a full vesicle. N-terminal repeat domains of a triskelion bind different adaptor proteins that can then in turn bind cargo as well as membrane directly. A single triskelion will bind others, forming the aforementioned shapes and eventually a cage that can encapsulate a vesicle. [9,10]
COPI is a heptameric protein complex associated with shuttling substrates from the cis-golgi to the rough endoplasmic reticulum, called retrograde traffic. It is composed of seven subunits (α, β, β’, δ, γ, ε and ζ), comprising the F subcomplex (a heteromer of β, δ, γ and ζ subunits) and B subcomplex (a heteromer of α and β’ subunits). The α and β’ subunits contain WD40 repeats, much like that of clathrin. COPII coat proteins are associated with the transportation of materials outward from the ER to the TGN and the cell membrane, or anterograde transfer. COPII proteins might be considered as subcomplexes of Sec13/31p and Sec23/Sec24p. Portions of the Sec13/31p have WD40 repeats, again like that of clathrin. [1,9]
Other Mechanisms of Vesicle Formation
Cytoskeletal branching, treadmilling and catastrophe affect membrane shape in the development of pseudopodia and lamellopodia, and can conceivably contribute to vesicle formation. Proteins with a BAR (Bin/amphiphysin/Rvs) domain specifically insert helices into membranes with curvature and play an important role in stabilizing curvature established by coat proteins. [8, 11, 17]
Multivesicular Bodies and Intralumenal vesicles
In the endocytic pathway, multivesicular bodies (MVB) are organelles that have vesicles inside their lumen, called intralumenal vesicles (ILT). They typically carry ubiquinated cargo to the lysosome, but can perform other functions as well. These vesicles require the e ndosomal s orting c omplex r equired for t ransport (ESCRT) proteins to form. ESCRTI, -II, and –III work alongside other associated proteins like vacuolar sorting protein 4 (Vsp4) to choose cargo, create a small bud, eventually perform scission on the bud, and target the vesicle to a final destination. A complex interplay between ESCRT proteins, associated targeting proteins that work with the ESCRT proteins, and post-translational modifications of the cargo governs these processes. [7] There is evidence that ESCRT proteins also participate in viral budding, cytokinesis, cell signaling, and development through its membrane interactions. [13,14,15,16]
References
- Aridor, M., & Traub, L. M. (2002). Cargo selection in vesicular transport: the making and breaking of a coat. Traffic, 3(8), 537-546.
- Basu, S. C., & Basu, M. (Eds.). (2002). Liposome methods and protocols (No. 199). Springer Science & Business Media.
- Chenette, E. J. (2014). ESCRTing intraluminal vesicle formation. Nature cell biology, 16(5), 400-400.
- Dua, J. S., Rana, A. C., & Bhandari, A. K. (2012). Liposome: methods of preparation and applications. Int J Pharm Stud Res, 3, 14-20.
- Hanson, P. I., & Cashikar, A. (2012). Multivesicular body morphogenesis. Annual review of cell and developmental biology, 28, 337-362.
- Jena, B. P. (2008). Intracellular Organelle Dynamics at nm Resolution. Methods in cell biology, 90, 19-37.
- Jouvenet, N. (2012). Dynamics of ESCRT proteins. Cellular and Molecular Life Sciences, 69(24), 4121-4133.
- McMahon, H. T., & Gallop, J. L. (2005). Membrane curvature and mechanisms of dynamic cell membrane remodelling. Nature, 438(7068), 590-596.
- McMahon, H. T., & Mills, I. G. (2004). COP and clathrin-coated vesicle budding: different pathways, common approaches. Current opinion in cell biology, 16(4), 379-391.
- Pearse BM (April 1976). "Clathrin: a unique protein associated with intracellular transfer of membrane by coated vesicles". Proceedings of the National Academy of Sciences of the United States of America 73 (4): 1255–9.
- Peter, B. J., Kent, H. M., Mills, I. G., Vallis, Y., Butler, P. J. G., Evans, P. R., & McMahon, H. T. (2004). BAR domains as sensors of membrane curvature: the amphiphysin BAR structure. Science, 303(5657), 495-499.
- Piper, R. C., & Katzmann, D. J. (2007). Biogenesis and function of multivesicular bodies. Annual review of cell and developmental biology, 23, 519.
- Rusten, T. E., Vaccari, T., & Stenmark, H. (2012). Shaping development with ESCRTs. Nature Cell Biology, 14(1), 38-45.
- Slagsvold, T., Pattni, K., Malerød, L., & Stenmark, H. (2006). Endosomal and non-endosomal functions of ESCRT proteins. Trends in cell biology, 16(6), 317-326.
- Tu, C., Ahmad, G., Mohapatra, B., Bhattacharyya, S., Ortega-Cava, C., Chung, B. M., et. al & Band, H. (2011). ESCRT proteins: Double-edged regulators of cellular signaling. Bioarchitecture, 1(1), 45-48.
- Wegner, C. S., Rodahl, L. M., & Stenmark, H. (2011). ESCRT proteins and cell signalling. Traffic, 12(10), 1291-1297.
- Zimmerberg, J., & Kozlov, M. M. (2006). How proteins produce cellular membrane curvature. Nature Reviews Molecular Cell Biology, 7(1), 9-19.


14.1.3: Building the Entrepreneurial Dream Team
Learning Objectives
By the end of this section, you will be able to:
- Understand key resource needs for entrepreneurial ventures: accountant, attorney, banker/financial institution, insurance agent, and industry expert
- Understand the value of developing a cross-functional team
Over the weekend of July 4, 1970, Casey Kasem started American Top 40 , a radio broadcast that played songs listed in Billboard magazine’s top 100 singles. What started as a simple compilation of popular songs that were played in ascending order of popularity ended thirty-nine years later on the July 4th weekend of 2009. When Kasem signed off for the final time, he gave credit to those with whom he worked. “Success doesn’t happen in a vacuum. You’re only as good as the people you work with and the people you work for. I’ve been lucky. I’ve worked for and with the very best.”
Jay Samit, a vice chairman at Deloitte Consulting LLP, specializes in exponential technologies, a revolutionary method of creating and manipulating information and data for learning, planning, and communicating. Working with multiple clients who each have their own unique limitations and face a wide variety of challenges, Samit and his colleagues at Deloitte devise tailored solutions that resolve each client’s problems while overcoming each client’s constraints.
Two professionals, two eras, two industries, and two vastly different technologies, but one concept: Entrepreneurial success is sustained by those around you. The concept that teamwork leads to individual success is evident in many other areas. All the great National Football League quarterbacks will tell you that they depend as much on their linemen as on their receivers. Pitchers in Major League Baseball need a very close relationship with their catchers, but the fielders are the ones who make most of the outs in the game and can make a pitcher look very good. Surgeons need nurses and anesthesiologists, police officers need good partners as well as dispatchers, ground troops need air support, and airline pilots need fantastic ground crews and maintenance crews, and so on.
In reality, no one works alone. As an entrepreneur, you have the luxury of searching, soliciting, and selecting your own team (Figure <IP_ADDRESS>). Entrepreneurial success depends on who is included on that team, and who is excluded from the team. In this section, we discuss advisors and cross-disciplinary teams.
Advisor: Accountant
One of the most important decisions that a business owner will make before beginning a new venture is hiring a good accountant (Figure <IP_ADDRESS>). Businesses and their owners must be in sound financial health, or the company risks being closed because of financial difficulties. Just as doctors are trained to diagnose and treat health issues and symptoms, accountants are trained to recognize and treat symptoms and issues associated with the financial health of a company. And just as doctors can determine the health of an infant before birth, accountants can determine the likely financial health of a business before it begins.
In the early stages of planning a business, the entrepreneur’s personal financial history is the only financial picture that investors, creditors, vendors, or lenders can review. Therefore, it is essential to have a professionally prepared tax return in hand before you approach anyone about opening a new business. Having a professional accountant prepare and file personal tax returns establishes credibility and confidence in an aspiring business owner’s financial decision-making. Furthermore, when a business owner is willing to let someone else see all of their finances, it indicates to other professionals that the owner is willing to expose a very personal and sensitive realm—money management.
A highly skilled accountant will help any small business owner set up proper procedures that track cash transactions and record financial activities. Accountants assist in creating a pro forma , a financial statement that shows how much and where monies will be collected and spent in the early stages of business creation. For an income statement, many of the figures used in a pro forma are hypothetical, a mere guess, but still are loosely based on historical industry or market data. Documenting debt service (explained in the following text) and other fixed expenses can be a simple task. However, projecting future sales with variable costs can clearly become complicated. Many times, approval for a loan or a building lease will depend on reasonable and sound projections of future sales and expenses. Too often, entrepreneurs overestimate income and underestimate expenses. An accountant frequently brings a reality check to the entrepreneur’s wishful thinking and keeps income and expense projections credible.
Tax payments, especially payroll taxes, must be made on time to avoid large, costly penalties. Annual income tax for both a business—whether a sole proprietor, a partnership, or a corporation—and the individuals involved can become very complicated. Tax laws constantly change, but a good accountant will know how these changes will affect the company and how the company should respond. Often, the accountant’s fees are lower than any penalty and fine assessed by tax authorities.
Besides tracking tax obligations, accountants can help identify trends within the financial picture. Many new business owners focus on profitability of the company and not cash flow. However, cash flow is far more important to the existence of a new company. Cash flow includes money collected by the business through revenue and funding versus money disbursed by the business through expenses. In other words, cash flow equals money in minus money out.
It’s true that profitability can be affected by depreciation of assets, and an accountant can apply different depreciation rules to reduce profits and tax liabilities. However, the business must maintain positive cash flows to pay its weekly or monthly obligations. Although today’s accounting software can easily generate cash flow reports, the expense of paying a professional pales in comparison to the money and time a business owner would spend in purchasing an accounting package, maintaining yearly upgrades, and learning how to run reports.
When a small business needs to expand or make capital improvements , major repairs or improvements that add to the life of an asset by improving its longevity, increasing its value or productivity, or extending its usefulness, it may be necessary for the business to take out a loan. Banks expect and require the company to have a professional accounting service prepare and file federal and state tax returns. Without the impartial eye of a professional accountant reviewing the financial health of the company, lenders may consider the business to be high risk and therefore would be unlikely to offer any loans. Expansion may be delayed or halted because lenders want to know if the company has properly managed its existing business before they are willing to finance additional operations and larger financial transactions.
Bankers will want to see not only income statements, balance sheets, and tax returns, normally for the past three years, but also cash flow statements over the same time period. If the company is functioning as a sole proprietor, LLC, or S-corporation, the bank may require all individuals with management authority or ownership to submit copies of both federal and state individual tax returns plus personal net worth statements. Any investments outside the entrepreneurial company will need to be disclosed as well. All of these requests from lending officers at the bank indicate that they are looking for something. An experienced accountant will be able to interpret their requests, create documents that provide the information they ask for, and advise the entrepreneur on how to address the bankers’ concerns.
Inventory management and debt service are other key financial areas where accountants can quickly identify problems that need attention and then offer advice to the entrepreneur. Inventory management consists of business operations that include managing the quantity of inventory units in stock, scheduling and cataloguing replacement inventory, organizing shelf location for unit storage, paying for inventory received, and processing, fulfilling, packaging, and shipping customer orders. Debt service is repayment of a loan, including principal and interest.
Some vendors will require the same professionally prepared financial documents as bank loan officers do. Creditworthiness , or a lender’s assessment of how reliable a company is in repaying a debt, is a valuable asset for a business. Being able to buy inventory on credit can greatly improve a company’s marketing and overall performance. However, creditors have a different concern about a company’s financial health. A vendor that sells inventory on credit may not be able to recoup a loss once the inventory is delivered to the new business. Banks can seize a building, equipment, land, and cash deposits, but creditors are not afforded that luxury. Therefore, creditors are skittish about releasing inventory to financially unstable businesses. Additionally, some businesses may owe inventory or personal property tax on inventory that is in stock.
Advisor: Attorney
If hiring an attorney to keep you out of trouble is expensive, hiring one to get you out of trouble will be exorbitant. Getting an attorney involved with a business in the very early stages, even in the idea development stage, can be a very good investment that will save a lot of legal expenses and protect the company’s income. Areas of expertise and the ability to practice in certain areas of the law will vary among attorneys. Entrepreneurs must first determine what legal issue they need help with. Then, they would determine the cost of getting legal help as well as the cost of not getting legal help.
Hiring an attorney can be similar to hiring someone to do construction work on your home. Hiring a general contractor who can do most of the work will be cheaper than hiring a general contractor who subcontracts everything out to specialists. Some attorneys are generalists, with a practice that spans many areas in which the entrepreneur or small business owner will need help. Other attorneys are specialists who limit their practice to a few specific areas of expertise and refer clients with needs outside those areas to other specialists. Generalists are frequently less expensive than specialists, who charge higher fees for their in-depth knowledge of particular areas. In many situations, the entrepreneur or small business owner will not need a high level of legal expertise. When in doubt about hiring an attorney, the new entrepreneur can visit with a few members of their established network to get their input before making a decision.
One of the important questions to ask up front is how the attorney bills for services. Some may charge a flat rate for specific services, whereas others will bill at an hourly rate. Attorneys may add any additional costs and expenses to the client’s bill. For example, when filing papers at the county courthouse to register a new company, the attorney may charge for copying, tolls to drive to the courthouse, parking, and mailing fees in addition to completing the actual registration papers. Knowing how the attorney calculates the bill—what fees will be charged and what additional costs will be added—is very important in deciding which attorney to hire.
Attorneys may be associates or partners of a firm. A large firm may have several equity partners and junior associates who are supported by paralegals. Larger firms offer a wider range of expertise and support for your attorney, but that extra help and larger resource pool are available at a higher price. On the other hand, an attorney who is in a stand-alone or small boutique firm may be less expensive but has fewer resources to tap into if necessary and may be unavailable for extended periods of time when actively working for other clients.
An attorney in your specific geographic area or community could be a valuable source of uncommon knowledge. Another consideration to think about when selecting an attorney is the location of their office. Office location may seem trivial, but attorneys in large office buildings probably have higher fees because they have higher office expenses. However, they are more likely to know other business professionals in their building, and those connections may be helpful at some point. Moreover, attorneys are often keenly aware of major developments in markets, such as real estate transactions and development, new roadways, and regulations.
Law practices can be divided into broad categories and specific areas. If you look at the landing pages of many of the country’s prominent law schools, you will find only one commonality among all of them—they include specialized areas in their programs. Five broad categories of law that are relevant to the entrepreneur are administrative, civil, corporate, criminal, and family law. For the entrepreneur, corporate and family law are probably the most important, because entrepreneurs may have family investors and personal assets at risk. An attorney who is skilled in family law would know how to protect the entrepreneur as well as family members if the startup business fails. If you are considering starting a business in a regulated profession, such as electrical repair, HVAC, or childcare, you will want an attorney familiar with the state’s administrative code. Hopefully, a startup entrepreneur will not need a criminal defense lawyer. One of the first areas where you should consult an attorney is in determining the type of business structure. Any business structure has both legal and tax ramifications. A business attorney should help the entrepreneur select the legal structure that is best for the business, the industry, and the entrepreneur. Each legal structure—sole proprietorship, partnership, or corporation—has its advantages and disadvantages. The attorney, in collaboration with the accountant, should help advise the entrepreneur about which legal form the business should take and why, not only for tax purposes but also for liability protection for the entrepreneur’s personal assets and for ease of operations.
Other issues may arise during the startup stage that would require legal advice. These include branding, patents, trademark registration, and contracts. Will the entrepreneur be leasing a space or purchasing a stand-alone building? If leasing, who is responsible for the safety and well-being of employees and for protection of the equipment and inventory? In the excitement of getting started and making money, entrepreneurs may consider such questions to be unimportant. However, discussing these issues and making decisions before an adverse incident occurs can save the new business a lot of time and money. A good attorney will help keep the focus on important issues before they become catastrophic ones.
Another volatile area for new businesses involves employees and employment law. New business owners may also be new to management. Ignorance of employment laws can be very costly, especially if a disgruntled ex-employee seeks protection from state and federal agencies. Formulating company and employee policies is very important. Having an attorney help develop and use appropriate phrases will be a blessing that you will recognize long after the attorney has cashed your check.
LINK TO LEARNING
Visit the Society of Human Resource Management website with extensive resources for small business owners on many topics related to employment law, including their Workplace Compliance Newsletter and articles on current news at their site.
 Having a good attorney who confronts and debunks the naive beliefs and behaviors of the nascent entrepreneur can help that entrepreneur mature into a professional. As a new entrepreneur, you may not find it easy to leave behind the “employee” mentality and enter the “owner” mindset. The entrepreneurial lifestyle, responsibility, position, status, authority, and so on are foreign to the employee population. Therefore, you may need to undergo a trial by fire. A good attorney, like a good marine drill sergeant, has to destroy the old mindset and develop a new mindset within the new entrepreneur: It’s all mine.
Although a good attorney is necessary to many aspects of a business, the entrepreneur also needs to set limits and not allow the attorney to tread into the owner’s domain. Attorneys should focus on legal issues, not operational, strategic, or—under any circumstances—financial issues. Attorneys may be tempted to go beyond their realm to help out a client, but in reality, they should stay within the areas of law and regulation, and out of operational or strategic decisions. A good attorney will provide information, insight, and recommendations for managing risks, but will not bully a client by insisting on eliminating risks. An entrepreneur should always be ready to terminate the relationship with anyone, including an attorney, who seems to be interfering in areas beyond their domain.
Hourly rates for experienced accountants and attorneys can be frightening. Even if you readily acknowledge that the information and guidance provided by these two professionals is valuable, having the funds to pay for their work in the pre-startup stage of the business can be impractical. Successful entrepreneurs are known for solving problems, and at the very beginning of your own business, you can start solving your own business problems. The question is: How can I get professional advice at discounted rates?
Often, young lawyers and accountants who have just completed their education or received their license to practice will have very few clients.  Those two scenarios are ripe for seeking discounted billing rates. Both the entrepreneur and the budding professional have a common need but few, if any, paying customers. Therefore, other business professionals or small business owners whose businesses are in the fledgling stage may be willing to offer special rates or payment terms.
Advisor: Banker or Financial Institution
All banks are not the same. Entrepreneurs need to select a bank or financial institution that can meet their current and future needs (Figure <IP_ADDRESS>).  Officers of the local bank may personally know local business owners, employees of the local businesses, and other key members of the local community. When a small business has a financial need, officers of the company may make decisions based on the reputation of the entrepreneur and the business. Sometimes a local bank will make loans and provide financial assistance with less scrutiny than an entrepreneur would face at a larger bank. For the small entrepreneurial business, banking is personal. Bankers like to see businesses in their backyard succeed.
Large banks with multiple branches in numerous cities, states, or countries may be a better choice for banking services if your company will have broad geographic and financial needs. With employees, customers, and vendors scattered over a large market, a large company is better off having a bank that mirrors that broad reach. For example, if an entrepreneur starts a business in Hutchinson, Kansas, and banks with a local bank in Hutchinson, that intimate relationship between the business and its employees in Hutchinson probably works great. However, as the business grows into the Wichita and Kansas City markets, its banking services should expand too. Likewise, expanding a business across state lines should initiate a thorough review of banking arrangements to ensure that banking services will match the needs of the expanded business.
Large companies with multiple branches or employees over a larger area have bigger demands in banking products and services. They will be better off with larger banks that can respond faster and more effectively to market shifts or individual needs.
Advisor: Insurance Agent
Having insurance is a necessity for every business. Entrepreneurs, however, should not overlook the advantages of an insurance agent’s knowledge and industry insight. Insurance agents ought to be in the entrepreneur’s portfolio of industry professionals. Furthermore, insurance agents likely have other clients who have been successful business owners for many years and may well introduce a new entrepreneur to seasoned veterans.
The insurance industry is a trove of data regarding almost every aspect of any industry or profession. All of that information is at the fingertips of your insurance agent. Agents can obtain information about any industry by running liability reports according to numbers compiled by the North American Industry Classification System (NAICS) , a standard used by US federal agencies to collect, analyze, and report statistical information about businesses; the government provides a searchable database related to the codes as well ( https://www.census.gov/eos/www/naics/ ). Every business in the United States is assigned a NAICS number, and insurance premiums are determined by the risk associated within each classification. You can obtain your NAICS number from your federal income tax return form 1120S or 1040 Schedule C. Knowing your business classification and the risks associated with it, your agent can assist you in reclassifying your business and lowering your insurance premiums, a potentially big financial savings.
Advisor: Industry Expert
Studies show industry expertise and skills are vital to successfully launching and operating one’s own business. However, a lack of industry skills is not an impenetrable barrier to entrepreneurship. In fact, about 15 percent to 20 percent of successful entrepreneurs have no industry experience or have limited knowledge about an industry before entering entrepreneurship. In the service professions, the entrepreneur’s lack of skill is hardly noticeable, but technical industries have a higher success rate for entrepreneurs who already possess the skills needed in the industry. Even if you do have those skills, you are more likely to succeed if you select industry experts who know the ins and outs of your profession (Figure <IP_ADDRESS>).
With the industry professional doing the work, the new entrepreneur can focus on the business aspects of the operation while gaining industry experience and insight through hands-on exposure. As many first-time entrepreneurs find out in their first year, they will spend less time on the technical aspects of working in the business and more time managing and working on the business itself. Good collaboration between a technician and an entrepreneur increases the chances of long-term success.
As you begin searching for and interviewing professional advisors such as attorneys, accountants, or industry experts to make your team (Figure <IP_ADDRESS>), you should follow a few fundamental principles. First and most important, you—the business owner—are the decision-maker. Business owners, not members of the advisory team, reap the benefits of good decisions and bear the cost of poor decisions. Advisors inform, explain options and risks, share knowledge about others with similar situations and their outcomes, and make recommendations. An advisor’s role is neither to make decisions nor to “sell” one particular viewpoint. Second, expectations and fees should be discussed before any actual work is done. Often though, a brief visit with a professional may be conducted at no charge so that each party can determine what is needed, whether the professional has the time and skills to meet the client’s needs, and whether the client can afford the fees.
Another consideration is that all professionals have limitations. Licensed professionals such as attorneys and accountants are experts in their respective fields, but they are not necessarily experts in general business operations or strategy. Industry experts may be very knowledgeable in many aspects of the business, but they too have boundaries. Last, shopping for experts to advise you on crucial matters should be based mostly on competence and compatibility, not strictly on fees. Securing the services of a professional is a serious matter and should be treated as such. Yet, it is a business arrangement subject to traditional market factors such as service quality, affordability, accessibility, and change over time.

Who Am I? Who Am I Not?
Knowing what type of person you are can help you make good career choices. Knowing what type of person your potential employee is can help you make a good hiring decision. Create a list of personality traits that you think you possess. Ask close friends, family members, instructors, or others in your life who will give you their input about your personality. People with all types of personalities and traits can be successful entrepreneurs. The difference is in how you manage yourself and what you do to compensate for areas of weakness.
A reliable tool to identify certain personalities traits is the Sixteen Personality Factor Questionnaire (16PF) developed by Raymond B. Cattell. Visit the Sixteen Personality Factors Questionnaire site to take the 16PF questionnaire.
- What factors are reflected that you expected?
- What factors are revealed that you did not expect?
- If your results show that you are timid, what might you do to overcome that when you become an entrepreneur?
- If you tend to be a perfectionist, how might you manage the uncertainties of entrepreneurship?
- Which traits or skills on your list match or are similar to those listed by your acquaintances? Which responses do not match?
Research the functional skills that you will need to start a business. Create a list of functional skills that you believe you possess. Ask close friends, family members, instructors, or others in your life who will give you their input about your business knowledge and skill. Then:
- Make a list of the functional skills you need for your new business. Add your name next to the functional skills that you have.
- Make a list of people you know who have different functional skills than you. Add their names next to the functional skills you do not have.
- Which of your friends and acquaintances appear on both the personality and functional lists? What do you think this means?
Building a Cross-Disciplinary Team
Rarely does an entrepreneurial venture start or function due to the effort of only one person. “Birds of a feather flock together” may be a popular saying, but it is a very poor organizational strategy for building a team to start a successful business. Diversity is a key feature of successfully managed organizations. Compatibility and collaboration are also important, as each employee is duty-bound to work with, support, and assist other employees when necessary. Having staff with complementary skills and who get along improves the likelihood of success for a new business.
A new restaurant provides a great example of how employees with assorted talents, expertise, and responsibilities are assembled to make a bustling organization profitable. For a new owner, the first hire is a manager. Hiring a good manager with experience means a larger payroll expense but produces better financial dividends over the long term. A good manager oversees all staff as well as all operational functions such as scheduling, buying, pricing, marketing, health code compliance, and business support functions. Perhaps the second key hire is the chef, who is responsible for creating the menu, distinguishing the restaurant from its competitors, and creating repeat customers who want high-quality, tasteful meals.
Front-house employees—the hosts/hostesses, servers, and bussers—play critical roles as the faces and voices of a restaurant. The first experience in a restaurant will leave a lasting impression, so those at the front are obliged to appear and act professional at all times. Servers, who have the most direct contact with customers, are the sales force of the business and the liaison with the chef. Servers’ incomes depend on tips and turning tables, so it is essential for them to have tables cleared quickly and properly prepared for the next group. Servers, therefore, rely heavily on the bussers for those important tasks. In many restaurants, bussers receive a portion of the tips left for the servers, establishing a codependency between those two key positions.
Other positions in a restaurant are the bartender, dishwasher, custodian, payroll clerk, bookkeeper, and so on who must perform their duties accurately and efficiently. Subpar service in any one of these functions jeopardizes the viability of the restaurant. Every employee at each and every level is crucial, ind ividually as well as collectively.

Make a List, Check It Twice
Building a team is a skill that you can learn. One of the first steps is to identify what tasks need to be completed and what skills are necessary to complete those tasks. People in all types of leadership positions build teams. Whether they are in government, large businesses, individual retail stores, small businesses, local athletic teams, or schools, leaders go through the same process of identifying tasks and the skills necessary to accomplish those tasks and then searching for people with those skills.
A key to learning is practice. You can practice building a team and then checking with people in charge of various organizations on how well you did. For example, select a local organization that you are somewhat familiar with—a local school, a community athletic league, a church, or a scout group. List the tasks and skills you think are needed to run the organization successfully. Then observe the organization in action. Make adjustments to your list. Ask to consult the manager in charge to see how well you did. How does your list compare with the actual operational structure? How is your list different? What did you overlook? How many people are doing more than one task?
After doing this a few times, you will begin to see organizations from a functional viewpoint. This is a skill any entrepreneur needs to have. What does my business do? What skills are needed to do those activities? Which person will I select to do those activities? If my first choice declines, who will be the backup?
Not only does a business need people to perform functional activities and day-to-day operations, but it also needs people to advise in other areas such as strategy, finances, management, staff, or legal. Should I have sales? Special promotions? Expand my product lines? Raise my prices? Get another investor for expansion? What are your long-term objectives? How will you achieve them? Having individuals you can talk to about your long-term goals is important. Surrounding yourself with those who can ask the right questions, confront you on weak areas, make you consider topics that you had not considered—all without judging you—is important if you have any plans to grow your business.
Create a second list of people you know and trust, a list solely for advising purposes. Members on this list can be from any industry as these are strategic questions, relevant across all industries and markets. They will help with your business strategy and structure, not operations.
A very common organizational structure for a new venture is the flat organization, consisting of family members, friends, or professional colleagues who take responsibility for different tasks. The bond that brings this group together in launching this new business is unlikely to bring to the table all of the skills, talents, personalities, perspectives, and viewpoints that can lead to long-term success. Therefore, expanding the team’s human resources beyond the founding members who also manage the business is crucial. Although they do not have to become employees, access to them is as vital as having key personnel on your payroll.
An entrepreneur with a creative or big-picture mindset may not want to be bothered by day-to-day activities. If that is the case, then someone else in the business needs to be the analytical, linear-thinking individual who can process information and data to make sound decisions. After carefully considering a situation, collecting information, and studying all relevant facts affecting the business, a problem solver can recommend what action the entrepreneur should take, to whom should the tasks be assigned, when to implement the solution, and how much money to dedicate to solving the problem. In other words, the problem solver becomes a lead advisor to the entrepreneur, the manager. If the creative entrepreneur is one side of the coin, the problem solver is the other side. When those two minds work in tandem, good things can happen.
In contrast, an entrepreneur may be a functional expert or licensed professional who is obliged to perform the tasks personally—for example, an HVAC technician, dentist, or professional driver. In that case, a business manager is needed to run the business side of the company. Rules, regulations, and deadlines for business activities are beyond the functional entrepreneur’s scope of interest, but they must be complied with accurately and in a timely manner, or the business may close. Like the creative founder who hires a day-to-day manager, a performance entrepreneur needs to hire someone dedicated to business functions.
Successful business owners keep careful track of metrics. They categorize and track expenses and analyze profit margins, production performance improvements or declines, employee attendance, and other measurable activities. Accurately interpreting the financial and operational performance of the company by the numbers provides the management team the information they need to make sound decisions. Having someone on the team with an aptitude for working with numbers is critical. The numbers must speak for themselves. Personnel must remain inside the box when they draw conclusions from data.
However, solutions to problems are not always inside the box. Nonlinear thinking, also known as creativity, or “thinking outside of the box,” is sometimes needed to solve problems. Creativity is the source of many new ideas, products, and processes. With companies facing shorter times of competitive advantage, the entrepreneur needs to be constantly reinventing both self and company.
Over time, as the business grows, the entrepreneur makes the transition from owner-operator of a startup through the small-business phase to being the owner-operator of a mature business. Entrepreneurs eventually need to make the cognitive shift from working in a state of ambiguity to performing methodically in a predictable environment. A business model where routine, repetitiveness, and predictability occur is more appropriate for established businesses because it brings stability and confidence to employees, customers, lenders, and investors alike. Using time-tested business methods and learning from previous experiences, an entrepreneur may avoid pitfalls that could doom a startup company in the early stages.
Every organization—whether for-profit, not-for-profit, political, religious, or social organization—relies on revenue. For-profit businesses rely on sales as their main source of revenue. Nonprofit organizations, such as community organizations, political groups, or religious groups, depend upon donations, contributions, gifts, and grants. Some nonprofits, such as youth sports leagues, community performing arts groups, or scouting groups, bring in cash through fundraisers or through ticket sales for events. Governments customarily depend on various taxes as sources of revenue. Some local governments receive funds through block grants from their state and federal governments. Having multiple revenue streams improves cash flow and increases the prospects of long-term success.
Recruiting the person who will generate income for the organization should be a high priority during the earliest stages, perhaps even before formal operations, of the business. For a salesperson, grant writer, donor coordinator, or any other title referring to an income-generating position, a startup organization may have to offer a sweeter-than-normal compensation package. If the person can produce revenue and generate cash flow in excess of their total cost of employment, then he or she is worth the costs of higher commissions and bigger bonuses.
Trying to improve short-term financial performance and increase profits by reducing a key employee’s compensation is likely to be detrimental over the long term. Losing a good sales rep may make the company vulnerable to losing very important and profitable customers. Failing to retain an effective grant writer could be the beginning of the demise, or at least a major downfall, for a nonprofit. Smart entrepreneurs pay their revenue producers appropriately, even if it means paying them more than the entrepreneurs pay themselves. As the old proverb states, “Never bite the hand that feeds you.”
WORK IT OUT
Design the Business? Build the Product!
Stan enjoys building wooden patio furniture. He started his craft while in high school and worked with a cabinet maker during his college years. For the past twenty-five years, he has been a marketing and sales representative and sales division manager for a restaurant equipment manufacturer. Recently, his company was bought by an international company, and Stan’s entire division was eliminated.
Stan was offered a new position with his current employer, but he would have to move to a new city. His family is against moving. His wife would have to quit her job teaching art in elementary school, his son does not want to miss his senior year of playing baseball, and his daughter has just started a new job at a nearby restaurant.
Stan decided to start his own business making customized outdoor patio furniture. He already owns most of the equipment and tools needed to make the furniture. He has long-term relationships with his suppliers. However, his workshop in the backyard is not large enough to store raw materials or support an assistant. Besides, if he wants to make a living wage, he has to have a large place to handle the volume necessary to be successful yet safe.
Stan is comfortable with the technical aspects of making furniture. He knows how to do marketing and sales, including pricing and financing. What he is not comfortable with is everything else he needs to get the business started. Stan does not want to spend a lot of his retirement funds, so getting started with minimal expense is his goal.
What does Stan need to start his business? What type of professional help does he need and why? Who can help him?


2.3: Histograms, Frequency Polygons, and Time Series Graphs
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For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of 100 values or more.
A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data.
The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample.(Remember, frequency is defined as the number of times an answer occurs.) If:
- \(f\) is frequency
- \(n\) is total number of data values (or the sum of the individual frequencies), and
- \(RF\) is relative frequency,
then:
\[RF=\dfrac{f}{n} \label{2.3.1}\]
For example, if three students in Mr. Ahab's English class of 40 students received from 90% to 100%, then, f = 3, n = 40, and RF = fn = 340 = 0.075. 7.5% of the students received 90–100%. 90–100% are quantitative measures.
To construct a histogram, first decide how many bars or intervals, also called classes, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval to be less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is 6.1 and this is the smallest value, a convenient starting point is \(6.05 (6.1 – 0.05 = 6.05)\). We say that 6.05 has more precision. If the value with the most decimal places is 2.23 and the lowest value is 1.5, a convenient starting point is \(1.495 (1.5 – 0.005 = 1.495)\). If the value with the most decimal places is 3.234 and the lowest value is 1.0, a convenient starting point is \(0.9995 (1.0 – 0.0005 = 0.9995)\). If all the data happen to be integers and the smallest value is two, then a convenient starting point is \(1.5 (2 - 0.5 = 1.5)\). Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data.
Example \(\PageIndex{1}\)
The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are continuous data, since height is measured.
60; 60.5; 61; 61; 61.5
63.5; 63.5; 63.5
64; 64; 64; 64; 64; 64; 64; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5
66; 66; 66; 66; 66; 66; 66; 66; 66; 66; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5
68; 68; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69.5; 69.5; 69.5; 69.5; 69.5
70; 70; 70; 70; 70; 70; 70.5; 70.5; 70.5; 71; 71; 71
72; 72; 72; 72.5; 72.5; 73; 73.5
74
The smallest data value is 60. Since the data with the most decimal places has one decimal (for instance, 61.5), we want our starting point to have two decimal places. Since the numbers 0.5, 0.05, 0.005, etc. are convenient numbers, use 0.05 and subtract it from 60, the smallest value, for the convenient starting point.
60 – 0.05 = 59.95 which is more precise than, say, 61.5 by one decimal place. The starting point is, then, 59.95.
The largest value is 74, so 74 + 0.05 = 74.05 is the ending value.
Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars.
\[\dfrac{74.05−59.95}{8}=1.76\]
We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals.
The boundaries are:
- 59.95
- 59.95 + 2 = 61.95
- 61.95 + 2 = 63.95
- 63.95 + 2 = 65.95
- 65.95 + 2 = 67.95
- 67.95 + 2 = 69.95
- 69.95 + 2 = 71.95
- 71.95 + 2 = 73.95
- 73.95 + 2 = 75.95
The heights 60 through 61.5 inches are in the interval 59.95–61.95. The heights that are 63.5 are in the interval 61.95–63.95. The heights that are 64 through 64.5 are in the interval 63.95–65.95. The heights 66 through 67.5 are in the interval 65.95–67.95. The heights 68 through 69.5 are in the interval 67.95–69.95. The heights 70 through 71 are in the interval 69.95–71.95. The heights 72 through 73.5 are in the interval 71.95–73.95. The height 74 is in the interval 73.95–75.95.
The following histogram displays the heights on the x -axis and relative frequency on the y -axis.
Exercise \(\PageIndex{1}\)
The following data are the shoe sizes of 50 male students. The sizes are discrete data since shoe size is measured in whole and half units only. Construct a histogram and calculate the width of each bar or class interval. Suppose you choose six bars.
9; 9; 9.5; 9.5; 10; 10; 10; 10; 10; 10; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5
11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5
12; 12; 12; 12; 12; 12; 12; 12.5; 12.5; 12.5; 12.5; 14
Answer
Smallest value: 9
Largest value: 14
Convenient starting value: 9 – 0.05 = 8.95
Convenient ending value: 14 + 0.05 = 14.05
\(\frac{14.05-8.95}{6}\) = 0.85
The calculations suggests using 0.85 as the width of each bar or class interval. You can also use an interval with a width equal to one.
Example \(\PageIndex{2}\)
The following data are the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data , since books are counted.
1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1
2; 2; 2; 2; 2; 2; 2; 2; 2; 2
3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3
4; 4; 4; 4; 4; 4
5; 5; 5; 5; 5
6; 6
Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.
Because the data are integers, subtract 0.5 from 1, the smallest data value and add 0.5 to 6, the largest data value. Then the starting point is 0.5 and the ending value is 6.5.
Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers 1, 2, 3, 4, 5, 6, and the starting point is 0.5, a width of one places the 1 in the middle of the interval from 0.5 to 1.5, the 2 in the middle of the interval from 1.5 to 2.5, the 3 in the middle of the interval from 2.5 to 3.5, the 4 in the middle of the interval from _______ to _______, the 5 in the middle of the interval from _______ to _______, and the _______ in the middle of the interval from _______ to _______ .
Answer
Calculate the number of bars as follows:
\(\dfrac{6.5 - 0.5}{\text{number of bars}}\) = 1
where 1 is the width of a bar. Therefore, bars = 6.
The following histogram displays the number of books on the x -axis and the frequency on the y -axis.
Note
Go to [link] . There are calculator instructions for entering data and for creating a customized histogram. Create the histogram for Example .
- Press Y=. Press CLEAR to delete any equations.
- Press STAT 1:EDIT. If L1 has data in it, arrow up into the name L1, press CLEAR and then arrow down. If necessary, do the same for L2.
- Into L1, enter 1, 2, 3, 4, 5, 6.
- Into L2, enter 11, 10, 16, 6, 5, 2.
- Press WINDOW. Set Xmin = .5, Xscl = (6.5 – .5)/6, Ymin = –1, Ymax = 20, Yscl = 1, Xres = 1.
- Press 2 nd Y=. Start by pressing 4:Plotsoff ENTER.
- Press 2 nd Y=. Press 1:Plot1. Press ENTER. Arrow down to TYPE. Arrow to the 3 rd picture (histogram). Press ENTER.
- Arrow down to Xlist: Enter L1 (2 nd 1). Arrow down to Freq. Enter L2 (2 nd 2).
- Press GRAPH.
- Use the TRACE key and the arrow keys to examine the histogram.
Exercise \(\PageIndex{2}\)
The following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted.
1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1
2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2
3; 3; 3; 3; 3; 3; 3; 3
20 student athletes play one sport. 22 student athletes play two sports. Eight student athletes play three sports.
Fill in the blanks for the following sentence. Since the data consist of the numbers 1, 2, 3, and the starting point is 0.5, a width of one places the 1 in the middle of the interval 0.5 to _____, the 2 in the middle of the interval from _____ to _____, and the 3 in the middle of the interval from _____ to _____.
Answer
1.5
1.5 to 2.5
2.5 to 3.5
Example \(\PageIndex{3}\)
Using this data set, construct a histogram.
| 9.95 | 10 | 2.25 | 16.75 | 0 |
| 19.5 | 22.5 | 7.5 | 15 | 12.75 |
| 5.5 | 11 | 10 | 20.75 | 17.5 |
| 23 | 21.9 | 24 | 23.75 | 18 |
| 20 | 15 | 22.9 | 18.8 | 20.5 |
Answer
Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.
Exercise \(\PageIndex{3}\)
The following data represent the number of employees at various restaurants in New York City. Using this data, create a histogram.
22; 35; 15; 26; 40; 28; 18; 20; 25; 34; 39; 42; 24; 22; 19; 27; 22; 34; 40; 20; 38 and 28
Use 10–19 as the first interval.
Count the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think is appropriate. You may want to experiment with the number of intervals.
Frequency Polygons
Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons. To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the x -axis and y -axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.
Example \(\PageIndex{4}\)
A frequency polygon was constructed from the frequency table below.
| Lower Bound | Upper Bound | Frequency | Cumulative Frequency |
|---|---|---|---|
| 49.5 | 59.5 | 5 | 5 |
| 59.5 | 69.5 | 10 | 15 |
| 69.5 | 79.5 | 30 | 45 |
| 79.5 | 89.5 | 40 | 85 |
| 89.5 | 99.5 | 15 | 100 |
The first label on the x -axis is 44.5. This represents an interval extending from 39.5 to 49.5. Since the lowest test score is 54.5, this interval is used only to allow the graph to touch the x -axis. The point labeled 54.5 represents the next interval, or the first “real” interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point 104.5 representing the interval from 99.5 to 109.5. Again, this interval contains no data and is only used so that the graph will touch the x -axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.
Exercise \(\PageIndex{4}\)
Construct a frequency polygon of U.S. Presidents’ ages at inauguration shown in the Table.
| Age at Inauguration | Frequency |
|---|---|
| 41.5–46.5 | 4 |
| 46.5–51.5 | 11 |
| 51.5–56.5 | 14 |
| 56.5–61.5 | 9 |
| 61.5–66.5 | 4 |
| 66.5–71.5 | 2 |
Answer
The first label on the x -axis is 39. This represents an interval extending from 36.5 to 41.5. Since there are no ages less than 41.5, this interval is used only to allow the graph to touch the x -axis. The point labeled 44 represents the next interval, or the first “real” interval from the table, and contains four scores. This reasoning is followed for each of the remaining intervals with the point 74 representing the interval from 71.5 to 76.5. Again, this interval contains no data and is only used so that the graph will touch the x -axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.
Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.
Example \(\PageIndex{5}\)
We will construct an overlay frequency polygon comparing the scores from Example with the students’ final numeric grade.
| Lower Bound | Upper Bound | Frequency | Cumulative Frequency |
|---|---|---|---|
| 49.5 | 59.5 | 5 | 5 |
| 59.5 | 69.5 | 10 | 15 |
| 69.5 | 79.5 | 30 | 45 |
| 79.5 | 89.5 | 40 | 85 |
| 89.5 | 99.5 | 15 | 100 |
| Lower Bound | Upper Bound | Frequency | Cumulative Frequency |
|---|---|---|---|
| 49.5 | 59.5 | 10 | 10 |
| 59.5 | 69.5 | 10 | 20 |
| 69.5 | 79.5 | 30 | 50 |
| 79.5 | 89.5 | 45 | 95 |
| 89.5 | 99.5 | 5 | 100 |
Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.
One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don‘t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.
Constructing a Time Series Graph
To construct a time series graph, we must look at both pieces of our paired data set . We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.
Example \(\PageIndex{6}\)
The following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only.
| Year | Jan | Feb | Mar | Apr | May | Jun | Jul |
|---|---|---|---|---|---|---|---|
| 2003 | 181.7 | 183.1 | 184.2 | 183.8 | 183.5 | 183.7 | 183.9 |
| 2004 | 185.2 | 186.2 | 187.4 | 188.0 | 189.1 | 189.7 | 189.4 |
| 2005 | 190.7 | 191.8 | 193.3 | 194.6 | 194.4 | 194.5 | 195.4 |
| 2006 | 198.3 | 198.7 | 199.8 | 201.5 | 202.5 | 202.9 | 203.5 |
| 2007 | 202.416 | 203.499 | 205.352 | 206.686 | 207.949 | 208.352 | 208.299 |
| 2008 | 211.080 | 211.693 | 213.528 | 214.823 | 216.632 | 218.815 | 219.964 |
| 2009 | 211.143 | 212.193 | 212.709 | 213.240 | 213.856 | 215.693 | 215.351 |
| 2010 | 216.687 | 216.741 | 217.631 | 218.009 | 218.178 | 217.965 | 218.011 |
| 2011 | 220.223 | 221.309 | 223.467 | 224.906 | 225.964 | 225.722 | 225.922 |
| 2012 | 226.665 | 227.663 | 229.392 | 230.085 | 229.815 | 229.478 | 229.104 |
| Year | Aug | Sep | Oct | Nov | Dec | Annual |
|---|---|---|---|---|---|---|
| 2003 | 184.6 | 185.2 | 185.0 | 184.5 | 184.3 | 184.0 |
| 2004 | 189.5 | 189.9 | 190.9 | 191.0 | 190.3 | 188.9 |
| 2005 | 196.4 | 198.8 | 199.2 | 197.6 | 196.8 | 195.3 |
| 2006 | 203.9 | 202.9 | 201.8 | 201.5 | 201.8 | 201.6 |
| 2007 | 207.917 | 208.490 | 208.936 | 210.177 | 210.036 | 207.342 |
| 2008 | 219.086 | 218.783 | 216.573 | 212.425 | 210.228 | 215.303 |
| 2009 | 215.834 | 215.969 | 216.177 | 216.330 | 215.949 | 214.537 |
| 2010 | 218.312 | 218.439 | 218.711 | 218.803 | 219.179 | 218.056 |
| 2011 | 226.545 | 226.889 | 226.421 | 226.230 | 225.672 | 224.939 |
| 2012 | 230.379 | 231.407 | 231.317 | 230.221 | 229.601 | 229.594 |
Answer
Exercise \(\PageIndex{5}\)
The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO 2 emissions for the United States.
| Ukraine | United Kingdom | United States | |
|---|---|---|---|
| 2003 | 352,259 | 540,640 | 5,681,664 |
| 2004 | 343,121 | 540,409 | 5,790,761 |
| 2005 | 339,029 | 541,990 | 5,826,394 |
| 2006 | 327,797 | 542,045 | 5,737,615 |
| 2007 | 328,357 | 528,631 | 5,828,697 |
| 2008 | 323,657 | 522,247 | 5,656,839 |
| 2009 | 272,176 | 474,579 | 5,299,563 |
Uses of a Time Series Graph
Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.
Review
A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually goes on y -axis with the frequency being graphed on the x -axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time.Glossary
References
- Data on annual homicides in Detroit, 1961–73, from Gunst & Mason’s book ‘Regression Analysis and its Application’, Marcel Dekker
- “Timeline: Guide to the U.S. Presidents: Information on every president’s birthplace, political party, term of office, and more.” Scholastic, 2013. Available online at www.scholastic.com/teachers/a...-us-presidents (accessed April 3, 2013).
- “Presidents.” Fact Monster. Pearson Education, 2007. Available online at http://www.factmonster.com/ipka/A0194030.html (accessed April 3, 2013).
- “Food Security Statistics.” Food and Agriculture Organization of the United Nations. Available online at http://www.fao.org/economic/ess/ess-fs/en/ (accessed April 3, 2013).
- “Consumer Price Index.” United States Department of Labor: Bureau of Labor Statistics. Available online at http://data.bls.gov/pdq/SurveyOutputServlet (accessed April 3, 2013).
- “CO2 emissions (kt).” The World Bank, 2013. Available online at http://databank.worldbank.org/data/home.aspx (accessed April 3, 2013).
- “Births Time Series Data.” General Register Office For Scotland, 2013. Available online at www.gro-scotland.gov.uk/stati...me-series.html (accessed April 3, 2013).
- “Demographics: Children under the age of 5 years underweight.” Indexmundi. Available online at http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en (accessed April 3, 2013).
- Gunst, Richard, Robert Mason. Regression Analysis and Its Application: A Data-Oriented Approach . CRC Press: 1980.
- “Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013).
- Frequency
- the number of times a value of the data occurs
- Histogram
- a graphical representation in \(x-y\) form of the distribution of data in a data set; \(x\) represents the data and \(y\) represents the frequency, or relative frequency. The graph consists of contiguous rectangles.
- Relative Frequency
- the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes


How do we come to understand how our mind works? The
theory of mind
is the understanding that the mind holds people’s beliefs, desires, emotions, and intentions. One component of this is understanding that the mind can be tricked or that the mind is not always accurate.
A two-year-old child does not understand very much about how their mind works. They can learn by imitating others, they are starting to understand that people do not always agree on things they like, and they have a rudimentary understanding of cause and effect (although they often fall prey to transitive reasoning). By the time a child is four, their theory of the mind allows them to understand that people think differently, have different preferences, and even mask their true feelings by putting on a different face that differs from how they truly feel inside.
To think about what this might look like in the real world, imagine showing a three-year-old child a bandaid box and asking the child what is in the box. Chances are, the child will reply, “bandaids.” Now imagine that you open the box and pour out crayons. If you now ask the child what they thought was in the box before it was opened, they may respond, “crayons.” If you ask what a friend would have thought was in the box, the response would still be “crayons.” Why?
Before about four years of age, a child does not recognize that the mind can hold ideas that are not accurate, so this three-year-old changes their response once shown that the box contains crayons. The child’s response can also be explained in terms of egocentrism and irreversibility. The child’s response is based on their current view rather than seeing the situation from another person’s perspective (egocentrism) or thinking about how they arrived at their conclusion (irreversibility). At around age four, the child would likely reply, “bandaids” when asked after seeing the crayons because by this age a child is beginning to understand that thoughts and realities do not always match.
Watch It
Watch as researchers demonstrate several versions of the false belief test to assess the theory of mind in young children.
A link to an interactive elements can be found at the bottom of this page.
This awareness of the existence of mind is part of social intelligence and the ability to recognize that others can think differently about situations. It helps us to be self-conscious or aware that others can think of us in different ways, and it helps us to be able to be understanding or empathic toward others. This developing social intelligence helps us to anticipate and predict the actions of others (even though these predictions are sometimes inaccurate). The awareness of the mental states of others is important for communication and social skills. A child who demonstrates this skill is able to anticipate the needs of others.
Impaired Theory of Mind in Individuals with Autism
People with autism or an autism spectrum disorder (ASD) typically show an impaired ability to recognize other people’s minds. Under the DSM-5,
autism
is characterized by persistent deficits in social communication and interaction across multiple contexts, as well as restricted, repetitive patterns of behavior, interests, or activities. These deficits are present in early childhood, typically before age three, and lead to clinically significant functional impairment. Symptoms may include lack of social or emotional reciprocity, stereotyped and repetitive use of language or idiosyncratic language, and persistent preoccupation with unusual objects.
About half of parents of children with ASD notice their child’s unusual behaviors by age 18 months, and about four-fifths notice by age 24 months, but often a diagnoses comes later, and individual cases vary significantly. Typical early signs of autism include:
No babbling by 12 months.
No gesturing (pointing, waving, etc.) by 12 months.
No single words by 16 months.
No two-word (spontaneous, not just echolalic) phrases by 24 months.
Loss of any language or social skills, at any age.
Children with ASD experience difficulties with explaining and predicting other people’s behavior, which leads to problems in social communication and interaction. Children who are diagnosed with an autistic spectrum disorder usually develop the theory of mind more slowly than other children and continue to have difficulties with it throughout their lives.
For testing whether someone lacks the theory of mind, the
Sally-Anne
test is performed. The child sees the following story: Sally and Anne are playing. Sally puts her ball into a basket and leaves the room. While Sally is gone, Anne moves the ball from the basket to the box. Now Sally returns. The question is: where will Sally look for her ball? The test is passed if the child correctly assumes that Sally will look in the basket. The test is failed if the child thinks that Sally will look in the box. Children younger than four and older children with autism will generally say that Sally will look in the box.
Watch It
Watch this video to see the Sally-Anne test in action.
A link to an interactive elements can be found at the bottom of this page.
a developmental disorder affecting communication and behavior
theory of mind
the understanding that the mind holds people’s beliefs, desires, emotions, and intentions. One component of this is understanding that the mind can be tricked or that the mind is not always accurate
Contributors and Attributions
CC licensed content, Original
Modification, adaptation, and original content.
Authored by
: Stephanie Loalada for Lumen Learning.
Provided by
: Lumen Learning.
License
:
CC BY: Attribution
The False Belief Test: Theory of Mind.
Provided by
: 007IceWeasel.
Located at
:
https://www.youtube.com/watch?v=8hLubgpY2_w
.
License
:
Other
.
License Terms
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The theory of mind test.
Provided by
: The Globe and the Mail.
Located at
:
https://www.youtube.com/watch?v=YGSj2zY2OEM
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:
Other
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License Terms
: Standard YouTube License


1.10: Becoming a Better Employee
Discuss how effective communication improves you as an employee
Communication is something we often take for granted but not often something we think to improve. And yet, being a good communicator can open doors for you as an employee, make you more valuable to your employer and help you get ahead.
Think about the tree swing in Figure 1. The creation of that swing started out all wrong and then got worse from there.
Let’s approach that process as though you were in charge. How could you have been a better employee and improved that result with better communication? We’ll look at it step by step.
- Meeting with the project sponsor: As the sales associate of this tree swing, you met with the project sponsor, Mark, and heard his translation of the customer’s wishes. Mark very clearly told you he wanted a swing with three seats, one on top of another. Now, you’re an expert on tree swings. Does his request make sense? This is an effective communication opportunity. You could have asked questions, clarified and repeated back what Mark was telling you. You could have told him that this kind of swing design was highly unusual and not generally embraced by tree swing aficionados. An opportunity to engage in effective communication with the customer was missed.
- Specifying the project request: This is where you, as swing sales person, made your first mistake. Rather than requesting three seats, you requested three ropes to secure the swing to the tree. This is another effective communication opportunity. You could have proofread and double-checked to make sure your request matched Mark’s request, but you did not. An opportunity to ensure effective communication via reinforcement and repetition was missed.
- Designing, production and installation: Here’s where the project went from wrong to wrong-er. Your swing production team not only didn’t question your request for three ropes, they went off and did their own thing with the concept and design of the product. The architect misunderstood your request completely. The production team reviewed the architect’s request and knew his design wasn’t functional, and so made their own changes. Finally, the installation team got there and the product wouldn’t work without additional reinforcements, so they did what was needed to make the product functional. This is another effective communication opportunity Not only did they miss opportunities to communicate with each other, but you missed an opportunity for follow-up, reinforcement and repetition.
Your manager, Gloria, is going to get calls from customers, and when she’s done hearing their complaints, is she going to think you’re employee of the year? Probably not. You didn’t deliver good results.
Employees who communicate effectively by listening, repeating, reinforcing and following up avoid all these issues. They are presented with a problem, they take in all the necessary information, and then they direct their teams with messaging that is
- Clear
- Concise
- Objective
- Consistent
- Complete
- Relevant
- Considerate of the knowledge his audience possesses
Because of better communication, they become better team members, co-workers, and employees.


9.4: Reproductive Healthcare and Family Planning
Reproductive healthcare includes regular screenings and testing for sexually transmitted infections (STIs), reproductive cancers, and treatment of reproductive disorders such as infertility, pelvic inflammatory diseases, endometriosis, polycystic ovarian syndrome, and others. Reproductive healthcare also includes counseling and education to help those who want to become pregnant, as well as those who wish to prevent pregnancy. Many families rely on Title X funded clinics (discussed in Chapter 14) for access to reproductive care and contraception. Family planning allows for individuals and couples to make important decisions on when to have children, to ensure they are healthy enough to have children, and to determine the spacing between children if they wish to have more than one child. It is important to note that hormonal contraceptives are also used for purposes beyond preventing pregnancy - they can be used to treat reproductive disorders like dysmenorrhea (painful periods), hormonal imbalances, and even acne (ACOG, 2023a).
Planning for pregnancy and parenthood helps to ensure appropriate resources and support systems are in place, and the health of the whole family is taken into consideration. This process may include considerations to improve fertility in the parents, preventing exposure to teratogens, ensuring adequate nutrition (including folic acid supplementation), and supporting the physical and mental health of the entire family. It can also include planning for parental leave from work, financial commitments (such as physician visits, labor and delivery, daycare, school, etc.), and studying parenting styles. Yet, nearly half of pregnancies in the U.S. are unintended, and have an increased risk of exposure to teratogens, delayed prenatal care (or lack thereof), and risk factors from other social determinants of health (Seabert et al., 2021).


8.E: Relativity (Exercises)
1 . The figure illustrates a Lorentz transformation using the conventions employed in section 7.2 . For simplicity, the transformation chosen is one that lengthens one diagonal by a factor of 2. Since Lorentz transformations preserve area, the other diagonal is shortened by a factor of 2. Let the original frame of reference, depicted with the square, be A, and the new one B. (a) By measuring with a ruler on the figure, show that the velocity of frame B relative to frame A is \(0.6c\). (b) Print out a copy of the page. With a ruler, draw a third parallelogram that represents a second successive Lorentz transformation, one that lengthens the long diagonal by another factor of 2. Call this third frame C. Use measurements with a ruler to determine frame C's velocity relative to frame A. Does it equal double the velocity found in part a? Explain why it should be expected to turn out the way it does.(answer check available at lightandmatter.com)
2
. Astronauts in three different spaceships are communicating with each other. Those aboard ships A and B agree on the rate at which time is passing, but they disagree with the ones on ship C.
(a) Alice is aboard ship A. How does she describe the motion of her own ship, in its frame of reference?
(b) Describe the motion of the other two ships according to Alice.
(c) Give the description according to Betty, whose frame of reference is ship B.
(d) Do the same for Cathy, aboard ship C.
3 . What happens in the equation for \(\gamma\) when you put in a negative number for \(v\)? Explain what this means physically, and why it makes sense.
4
. The Voyager 1 space probe, launched in 1977, is moving faster relative to the earth than any other human-made object, at 17,000 meters per second.
(a) Calculate the probe's \(\gamma\).
(b) Over the course of one year on earth, slightly less than one year passes on the probe. How much less? (There are 31 million seconds in a year.)(answer check available at lightandmatter.com)
5 . In example 2 on page 391, I remarked that accelerating a macroscopic (i.e., not microscopic) object to close to the speed of light would require an unreasonable amount of energy. Suppose that the starship Enterprise from Star Trek has a mass of \(8.0\times10^7\) kg, about the same as the Queen Elizabeth 2. Compute the kinetic energy it would have to have if it was moving at half the speed of light. Compare with the total energy content of the world's nuclear arsenals, which is about \(10^{21}\) J.(answer check available at lightandmatter.com)
6 . The earth is orbiting the sun, and therefore is contracted relativistically in the direction of its motion. Compute the amount by which its diameter shrinks in this direction.(answer check available at lightandmatter.com)
7 . In this homework problem, you'll fill in the steps of the algebra required in order to find the equation for \(\gamma\) on page 389. To keep the algebra simple, let the time \(t\) in figure k equal 1, as suggested in the figure accompanying this homework problem. The original square then has an area of 1, and the transformed parallelogram must also have an area of 1. (a) Prove that point P is at \(x=v\gamma\), so that its \((t,x)\) coordinates are \((\gamma,v\gamma)\). (b) Find the \((t,x)\) coordinates of point Q. (c) Find the length of the short diagonal connecting P and Q. (d) Average the coordinates of P and Q to find the coordinates of the midpoint C of the parallelogram, and then find distance OC. (e) Find the area of the parallelogram by computing twice the area of triangle PQO. [Hint: You can take PQ to be the base of the triangle.] (f) Set this area equal to 1 and solve for \(\gamma\) to prove \(\gamma=1/\sqrt{1-v^2}\).(answer check available at lightandmatter.com)
8 . (a) A free neutron (as opposed to a neutron bound into an atomic nucleus) is unstable, and undergoes beta decay (which you may want to review). The masses of the particles involved are as follows:
| neutron | 1.67495×10 − 27 kg |
| proton | 1.67265×10 − 27 kg |
| electron | 0.00091×10 − 27 kg |
| antineutrino | < 10 − 35 kg |
Find the energy released in the decay of a free neutron. (answer check available at lightandmatter.com)
(b) Neutrons and protons make up essentially all of the mass of the ordinary matter around us. We observe that the universe around us has no free neutrons, but lots of free protons (the nuclei of hydrogen, which is the element that 90% of the universe is made of). We find neutrons only inside nuclei along with other neutrons and protons, not on their own.
If there are processes that can convert neutrons into protons, we might imagine that there could also be proton-to-neutron conversions, and indeed such a process does occur sometimes in nuclei that contain both neutrons and protons: a proton can decay into a neutron, a positron, and a neutrino. A positron is a particle with the same properties as an electron, except that its electrical charge is positive (see chapter 7). A neutrino, like an antineutrino, has negligible mass.
Although such a process can occur within a nucleus, explain why it cannot happen to a free proton. (If it could, hydrogen would be radioactive, and you wouldn't exist!)
9
. (a) Find a relativistic equation for the velocity of an object in terms of its mass and momentum (eliminating \(\gamma\)).(answer check available at lightandmatter.com)
(b) Show that your result is approximately the same as the classical value, \(p/m\), at low velocities.
(c) Show that very large momenta result in speeds close to the speed of light.
10
. (a) Show that for \(v=(3/5)c\), \(\gamma\) comes out to be a simple fraction.
(b) Find another value of \(v\) for which \(\gamma\) is a simple fraction.
11
. An object moving at a speed very close to the speed of light is referred to as ultrarelativistic. Ordinarily (luckily) the only ultrarelativistic objects in our universe are subatomic particles, such as cosmic rays or particles that have been accelerated in a particle accelerator.
(a) What kind of number is \(\gamma\) for an ultrarelativistic particle?
(b) Repeat example
18
on page 418, but instead of very low, nonrelativistic speeds, consider ultrarelativistic speeds.
(c) Find an equation for the ratio \(\mathcal E/p\). The speed may be relativistic, but don't assume that it's ultrarelativistic.(answer check available at lightandmatter.com)
(d) Simplify your answer to part c for the case where the speed is ultrarelativistic.(answer check available at lightandmatter.com)
(e) We can think of a beam of light as an ultrarelativistic object --- it certainly moves at a speed that's sufficiently close to the speed of light! Suppose you turn on a one-watt flashlight, leave it on for one second, and then turn it off. Compute the momentum of the recoiling flashlight, in units of \(\text{kg}\!\cdot\!\text{m}/\text{s}\).(answer check available at lightandmatter.com)
(f) Discuss how your answer in part e relates to the correspondence principle.
12
. As discussed in chapter
6
, the speed at which a disturbance travels along a string under tension is given by \(v=\sqrt{T/\mu}\), where \(\mu\) is the mass per unit length, and \(T\) is the tension.
(a) Suppose a string has a density \(\rho\), and a cross-sectional area \(A\). Find an expression for the maximum tension that could possibly exist in the string without producing \(v>c\), which is impossible according to relativity. Express your answer in terms of \(\rho\), \(A\), and \(c\). The interpretation is that relativity puts a limit on how strong any material can be.(answer check available at lightandmatter.com)
(b) Every substance has a tensile strength, defined as the force per unit area required to break it by pulling it apart. The tensile strength is measured in units of \(\text{N}/\text{m}^2\), which is the same as the pascal (Pa), the mks unit of pressure. Make a numerical estimate of the maximum tensile strength allowed by relativity in the case where the rope is made out of ordinary matter, with a density on the same order of magnitude as that of water. (For comparison, kevlar has a tensile strength of about \(4\times10^9\) Pa, and there is speculation that fibers made from carbon nanotubes could have values as high as \(6\times10^{10}\) Pa.)(answer check available at lightandmatter.com)
(c) A black hole is a star that has collapsed and become very dense, so that its gravity is too strong for anything ever to escape from it. For instance, the escape velocity from a black hole is greater than \(c\), so a projectile can't be shot out of it. Many people, when they hear this description of a black hole in terms of an escape velocity greater than \(c\), wonder why it still wouldn't be possible to extract an object from a black hole by other means than launching it out as a projectile.  Why might this not work?
13
. (a) A charged particle is surrounded by a uniform electric field. Starting from rest, it is accelerated by the field to speed \(v\) after traveling a distance \(d\). Now it is allowed to continue for a further distance \(3d\), for a total displacement from the start of \(4d\). What speed will it reach, assuming classical physics?
(b) Find the relativistic result for the case of \(v=c/2\).
14 . Problem 14 has been deleted.
15 . Expand the equation \(K = m(\gamma-1)\) in a Taylor series, and find the first two nonvanishing terms. Show that the first term is the nonrelativistic expression for kinetic energy.
16 . Expand the relativistic equation for momentum in a Taylor series, and find the first two nonvanishing terms. Show that the first term is the classical expression.
17 . (solution in the pdf version of the book) As promised in subsection 7.2.8 , this problem will lead you through the steps of finding an equation for the combination of velocities in relativity, generalizing the numerical result found in problem 1 . Suppose that A moves relative to B at velocity \(u\), and B relative to C at \(v\). We want to find A's velocity \(w\) relative to C, in terms of \(u\) and \(v\). Suppose that A emits light with a certain frequency. This will be observed by B with a Doppler shift \(D(u)\). C detects a further shift of \(D(v)\) relative to B. We therefore expect the Doppler shifts to multiply, \(D(w)=D(u)D(v)\), and this provides an implicit rule for determining \(w\) if \(u\) and \(v\) are known. (a) Using the expression for \(D\) given in section 7.2.8 , write down an equation relating \(u\), \(v\), and \(w\). (b) Solve for \(w\) in terms of \(u\) and \(v\). (c) Show that your answer to part b satisfies the correspondence principle.
18 . The figure shows seven four-vectors, represented in a two-dimensional plot of \(x\) versus \(t\). All the vectors have \(y\) and \(z\) components that are zero. Which of these vectors are congruent to others, i.e., which represent spacetime intervals that are equal to one another?
19 . Four-vectors can be timelike, lightlike, or spacelike. What can you say about the inherent properties of particles whose momentum four-vectors fall in these various categories?
20 . The following are the three most common ways in which gamma rays interact with matter:
Photoelectric effect: The gamma ray hits an electron, is annihilated, and gives all of its energy to the electron.
Compton scattering: The gamma ray bounces off of an electron, exiting in some direction with some amount of energy.
Pair production: The gamma ray is annihilated, creating an electron and a positron.
Example a href="#eg:no-pair-prod-in-vacuum">23 on p. 420 shows that pair production can't occur in a vacuum due to conservation of the energy-momentum four-vector. What about the other two processes? Can the photoelectric effect occur without the presence of some third particle such as an atomic nucleus? Can Compton scattering happen without a third particle?
21 . Expand the relativistic equation for the longitudinal Doppler shift of light \(D(v)\) in a Taylor series, and find the first two nonvanishing terms. Show that these two terms agree with the nonrelativistic expression, so that any relativistic effect is of higher order in \(v\).
22 . Prove, as claimed in the caption of figure a on p. 424, that \(S-180°=4(s-180°)\), where \(S\) is the sum of the angles of the large equilateral triangle and \(s\) is the corresponding sum for one of the four small ones.(solution in the pdf version of the book)
23 . If a two-dimensional being lived on the surface of a cone, would it say that its space was curved, or not?
24 . (a) Verify that the equation \(1-gh/c^2\) for the gravitational Doppler shift and gravitational time dilation has units that make sense. (b) Does this equation satisfy the correspondence principle?
25 . (a) Calculate the Doppler shift to be expected in the Pound-Rebka experiment described on p. 429. (b) In the 1978 Iijima mountain-valley experiment (p. 384), analysis was complicated by the clock's sensitivity to pressure, humidity, and temperature. A cleaner version of the experiment was done in 2005 by hobbyist Tom Van Baak. He put his kids and three of his atomic clocks in a minivan and drove from Bellevue, Washington to a lodge on Mount Rainier, 1340 meters higher in elevation. At home, he compared the clocks to others that had stayed at his house. Verify that the effect shown in the graph is as predicted by general relativity.
26 . The International Space Station orbits at an altitude of about 350 km and a speed of about 8000 m/s relative to the ground. Compare the gravitational and kinematic time dilations. Over all, does time run faster on the ISS than on the ground, or more slowly?
27 .  Although this estimate is not really right, it turns out to give the right answer to within about a factor of 2. To roughly what size would the earth have to be compressed in order to become a black hole?
28 . Clock A sits on a desk. Clock B is tossed up in the air from the same height as the desk and then comes back down. Compare the elapsed times. \hwhint{hwhint:tossed-clock} (solution in the pdf version of the book)
29 . The angular defect \(d\) of a triangle (measured in radians) is defined as \(s-\pi\), where \(s\) is the sum of the interior angles. The angular defect is proportional to the area \(A\) of the triangle. Consider the geometry measured by a two-dimensional being who lives on the surface of a sphere of radius \(R\). First find some triangle on the sphere whose area and angular defect are easy to calculate. Then determine the general equation for \(d\) in terms of \(A\) and \(R\).(answer check available at lightandmatter.com)
Contributors and Attributions
Benjamin Crowell (Fullerton College). Conceptual Physics is copyrighted with a CC-BY-SA license.


9.4: Using Sources Creatively
Figure \(\PageIndex{1}\)
Using Sources Creatively
Heather Logan
(
printable version here
)
When writing papers that require the use of outside source material, it is often tempting to cite only direct quotations from your sources. If, however, this is the only method of citation you choose, your paper will become nothing more than a series of quotations linked together by a few connecting words. Your paper will seem to be a collection of others’ thoughts and will contain little thinking on your part.
To avoid falling into this trap, follow a few simple pointers:
- Avoid using long quotations merely as space-fillers . While this is an attractive option when faced with a ten-page paper, the overuse of long quotations gives the reader the impression you cannot think for yourself.
- Don’t use only direct quotations . Try using paraphrases in addition to your direct quotations. To the reader, the effective use of paraphrases indicates that you took the time to think about the meaning behind the quote’s words. (For further assistance see our materials on “ Using Paraphrases .”)
- When introducing direct quotations, try to use a variety of verbs in your signal phrases . Don’t always rely on stock verbs such as “states” or “says.” Think for a little while about the purpose of your quotation and then choose a context-appropriate verb.
Also, when using direct quotations try qualifying them in a novel or interesting manner. Depending on the system of documentation you’re using, the signal phrases don’t always have to introduce the quotation.
For example, instead of saying:
“None of them knew the color of the sky” is the opening line of Stephen Crane’s short story, “The Open Boat” (339). This implies the idea that “all sense of certainty” in the lives of these men is gone (Wolford 18).
Try saying:
“None of them knew the color of the sky,” the opening line of Stephen Crane’s, “The Open Boat,” implies that “all sense of certainty” in the lives of these men is gone (Crane 339; Wolford 18).
The combination of these two sentences into one is something different. It shows thought on the writer’s part in how to combine direct quotations in an interesting manner.