14.6: Problems
1 . If a radioactive substance has a half-life of one year, does this mean that it will be completely decayed after two years? Explain.
2 . What is the probability of rolling a pair of dice and getting “snake eyes,” i.e., both dice come up with ones?
3 . Problem 3 has been deleted.
4 . Problem 4 has been deleted.
5
. Refer to the probability distribution for people's heights in figure
f
on page 828.
(a) Show that the graph is properly normalized.
(b) Estimate the fraction of the population having heights between 140 and 150 cm.(answer check available at lightandmatter.com)
6 . (a) A nuclear physicist is studying a nuclear reaction caused in an accelerator experiment, with a beam of ions from the accelerator striking a thin metal foil and causing nuclear reactions when a nucleus from one of the beam ions happens to hit one of the nuclei in the target. After the experiment has been running for a few hours, a few billion radioactive atoms have been produced, embedded in the target. She does not know what nuclei are being produced, but she suspects they are an isotope of some heavy element such as Pb, Bi, Fr or U. Following one such experiment, she takes the target foil out of the accelerator, sticks it in front of a detector, measures the activity every 5 min, and makes a graph (figure). The isotopes she thinks may have been produced are:
| isotope | half-life (minutes) |
| 211 Pb | 36.1 |
| 214 Pb | 26.8 |
214
Bi
|
19.7 |
| 223 Fr | 21.8 |
| 239 U | 23.5 |
Which one is it?
(b) Having decided that the original experimental conditions produced one specific isotope, she now tries using beams of ions traveling at several different speeds, which may cause different reactions. The following table gives the activity of the target 10, 20 and 30 minutes after the end of the experiment, for three different ion speeds.
| activity (millions of decays/s) after… | |||
| 10 min | 20 min | 30 min | |
| first ion speed | 1.933 | 0.832 | 0.382 |
| second ion speed | 1.200 | 0.545 | 0.248 |
| third ion speed | 7.211 | 1.296 | 0.248 |
Since such a large number of decays is being counted, assume that the data are only inaccurate due to rounding off when writing down the table. Which are consistent with the production of a single isotope, and which imply that more than one isotope was being created?
7 . Devise a method for testing experimentally the hypothesis that a gambler's chance of winning at craps is independent of her previous record of wins and losses. If you don't invoke the definition of statistical independence, then you haven't proposed a test.
8
. A blindfolded person fires a gun at a circular target of radius \(b\), and is allowed to continue firing until a shot actually hits it. Any part of the target is equally likely to get hit. We measure the random distance \(r\) from the center of the circle to where the bullet went in.
(a) Show that the probability distribution of \(r\) must be of the form \(D(r)=kr\), where \(k\) is some constant. (Of course we have \(D(r)=0\) for \(r>b\).)
(b) Determine \(k\) by requiring \(D\) to be properly normalized.(answer check available at lightandmatter.com)
(c) Find the average value of \(r\).(answer check available at lightandmatter.com)
(d) Interpreting your result from part c, how does it compare with \(b/2\)? Does this make sense? Explain.
9
. We are given some atoms of a certain radioactive isotope, with half-life \(t_{1/2}\). We pick one atom at random, and observe it for one half-life, starting at time zero. If it decays during that one-half-life period, we record the time \(t\) at which the decay occurred. If it doesn't, we reset our clock to zero and keep trying until we get an atom that cooperates. The final result is a time \(0\le t\le t_{1/2}\), with a distribution that looks like the usual exponential decay curve, but with its tail chopped off.
(a) Find the distribution \(D(t)\), with the proper normalization.(answer check available at lightandmatter.com)
(b) Find the average value of \(t\).(answer check available at lightandmatter.com)
(c) Interpreting your result from part b, how does it compare with \(t_{1/2}/2\)? Does this make sense? Explain.
10
. The speed, \(v\), of an atom in an ideal gas has a probability distribution of the form \(D(v) = bve^{-cv^2}\), where \(0\le v \lt \infty\), \(c\) relates to the temperature, and \(b\) is determined by normalization.
(a) Sketch the distribution.
(b) Find \(b\) in terms of \(c\).(answer check available at lightandmatter.com)
(c) Find the average speed in terms of \(c\), eliminating \(b\). (Don't try to do the indefinite integral, because it can't be done in closed form. The relevant definite integral can be found in tables or done with computer software.)(answer check available at lightandmatter.com)
11
. All helium on earth is from the decay of naturally occurring heavy radioactive elements such as uranium. Each alpha particle that is emitted ends up claiming two electrons, which makes it a helium atom. If the original \(^{238}\text{U}\) atom is in solid rock (as opposed to the earth's molten regions), the He atoms are unable to diffuse out of the rock. This problem involves dating a rock using the known decay properties of uranium 238. Suppose a geologist finds a sample of hardened lava, melts it in a furnace, and finds that it contains 1230 mg of uranium and 2.3 mg of helium. \(^{238}\text{U}\) decays by alpha emission, with a half-life of \(4.5\times10^9\) years. The subsequent chain of alpha and electron (beta) decays involves much shorter half-lives, and terminates in the stable nucleus \(^{206}\text{Pb}\). Almost all natural uranium is \(^{238}\text{U}\), and the chemical composition of this rock indicates that there were no decay chains involved other than that of \(^{238}\text{U}\).
(a) How many alphas are emitted per decay chain? [Hint: Use conservation of mass.]
(b) How many electrons are emitted per decay chain? [Hint: Use conservation of charge.]
(c) How long has it been since the lava originally hardened?(answer check available at lightandmatter.com)
12 . When light is reflected from a mirror, perhaps only 80% of the energy comes back. One could try to explain this in two different ways: (1) 80% of the photons are reflected, or (2) all the photons are reflected, but each loses 20% of its energy. Based on your everyday knowledge about mirrors, how can you tell which interpretation is correct? [Based on a problem from PSSC Physics.]
13 . Suppose we want to build an electronic light sensor using an apparatus like the one described in the section on the photoelectric effect. How would its ability to detect different parts of the spectrum depend on the type of metal used in the capacitor plates?
14 . The photoelectric effect can occur not just for metal cathodes but for any substance, including living tissue. Ionization of DNA molecules can cause cancer or birth defects. If the energy required to ionize DNA is on the same order of magnitude as the energy required to produce the photoelectric effect in a metal, which of the following types of electromagnetic waves might pose such a hazard? Explain.
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15
. (a) Rank-order the photons according to their wavelengths, frequencies, and energies. If two are equal, say so. Explain all your answers.
(b) Photon 3 was emitted by a xenon atom going from its second-lowest-energy state to its lowest-energy state. Which of photons 1, 2, and 4 are capable of exciting a xenon atom from its lowest-energy state to its second-lowest-energy state? Explain.
16 . Which figure could be an electron speeding up as it moves to the right? Explain.
17 . The beam of a 100-W overhead projector covers an area of \(1\ \text{m}\times1\ \text{m}\) when it hits the screen 3 m away. Estimate the number of photons that are in flight at any given time. (Since this is only an estimate, we can ignore the fact that the beam is not parallel.)(answer check available at lightandmatter.com)
18
. In the photoelectric effect, electrons are observed with virtually no time delay (\(\sim10\) ns), even when the light source is very weak. (A weak light source does however only produce a small number of ejected electrons.) The purpose of this problem is to show that the lack of a significant time delay contradicted the classical wave theory of light, so throughout this problem you should put yourself in the shoes of a classical physicist and pretend you don't know about photons at all. At that time, it was thought that the electron might have a radius on the order of \(10^{-15}\) m. (Recent experiments have shown that if the electron has any finite size at all, it is far smaller.)
(a) Estimate the power that would be soaked up by a single electron in a beam of light with an intensity of 1 \(\text{mW}/\text{m}^2\).(answer check available at lightandmatter.com)
(b) The energy, \(E_s\), required for the electron to escape through the surface of the cathode is on the order of \(10^{-19}\) J. Find how long it would take the electron to absorb this amount of energy, and explain why your result constitutes strong evidence that there is something wrong with the classical theory.(answer check available at lightandmatter.com)
19 . In a television, suppose the electrons are accelerated from rest through a voltage difference of \(10^4\) V. What is their final wavelength?(answer check available at lightandmatter.com)
20 . Use the Heisenberg uncertainty principle to estimate the minimum velocity of a proton or neutron in a \(^{208}\text{Pb}\) nucleus, which has a diameter of about 13 fm (1 fm=\(10^{-15}\) m). Assume that the speed is nonrelativistic, and then check at the end whether this assumption was warranted.(answer check available at lightandmatter.com)
21 . Find the energy of a particle in a one-dimensional box of length \(L\), expressing your result in terms of \(L\), the particle's mass \(m\), the number of peaks and valleys \(n\) in the wavefunction, and fundamental constants.(answer check available at lightandmatter.com)
22
. A free electron that contributes to the current in an ohmic material typically has a speed of \(10^5\) m/s (much greater than the drift velocity).
(a) Estimate its de Broglie wavelength, in nm.(answer check available at lightandmatter.com)
(b) If a computer memory chip contains \(10^8\) electric circuits in a 1 \(\text{cm}^2\) area, estimate the linear size, in nm, of one such circuit.(answer check available at lightandmatter.com)
(c) Based on your answers from parts a and b, does an electrical engineer designing such a chip need to worry about wave effects such as diffraction?
(d) Estimate the maximum number of electric circuits that can fit on a 1 \(\text{cm}^2\) computer chip before quantum-mechanical effects become important.
23
. In classical mechanics, an interaction energy of the form \(U(x)=\frac{1}{2}kx^2\) gives a harmonic oscillator: the particle moves back and forth at a frequency \(\omega=\sqrt{k/m}\). This form for \(U(x)\) is often a good approximation for an individual atom in a solid, which can vibrate around its equilibrium position at \(x=0\). (For simplicity, we restrict our treatment to one dimension, and we treat the atom as a single particle rather than as a nucleus surrounded by electrons). The atom, however, should be treated quantum-mechanically, not clasically. It will have a wave function. We expect this wave function to have one or more peaks in the classically allowed region, and we expect it to tail off in the classically forbidden regions to the right and left. Since the shape of \(U(x)\) is a parabola, not a series of flat steps as in figure
m
on page 869, the wavy part in the middle will not be a sine wave, and the tails will not be exponentials.
(a) Show that there is a solution to the Schrödinger equation of the form
and relate \(b\) to \(k\), \(m\), and \(\hbar\). To do this, calculate the second derivative, plug the result into the Schrödinger equation, and then find what value of \(b\) would make the equation valid for
all
values of \(x\). This wavefunction turns out to be the ground state. Note that this wavefunction is not properly normalized --- don't worry about that.(answer check available at lightandmatter.com)
(b) Sketch a graph showing what this wavefunction looks like.
(c) Let's interpret \(b\). If you changed \(b\), how would the wavefunction look different? Demonstrate by sketching two graphs, one for a smaller value of \(b\), and one for a larger value.
(d) Making \(k\) greater means making the atom more tightly bound. Mathematically, what happens to the value of \(b\) in your result from part a if you make \(k\) greater? Does this make sense physically when you compare with part c?
24
. (a) A distance scale is shown below the wavefunctions and probability densities illustrated in figure
e
on page 882. Compare this with the order-of-magnitude estimate derived in subsection
13.4.4
for the radius \(r\) at which the wavefunction begins tailing off. Was the estimate on the right order of magnitude?
(b) Although we normally say the moon orbits the earth, actually they both orbit around their common center of mass, which is below the earth's surface but not at its center. The same is true of the hydrogen atom. Does the center of mass lie inside the proton, or outside it?
25
. The figure shows eight of the possible ways in which an electron in a hydrogen atom could drop from a higher energy state to a state of lower energy, releasing the difference in energy as a photon. Of these eight transitions, only D, E, and F produce photons with wavelengths in the visible spectrum.
(a) Which of the visible transitions would be closest to the violet end of the spectrum, and which would be closest to the red end? Explain.
(b) In what part of the electromagnetic spectrum would the photons from transitions A, B, and C lie? What about G and H? Explain.
(c) Is there an upper limit to the wavelengths that could be emitted by a hydrogen atom going from one bound state to another bound state? Is there a lower limit? Explain.
26 . Find an equation for the wavelength of the photon emitted when the electron in a hydrogen atom makes a transition from energy level \(n_1\) to level \(n_2\).(answer check available at lightandmatter.com)
27 . Estimate the angular momentum of a spinning basketball, in units of \(\hbar\). Explain how this result relates to the correspondence principle.
28 . Assume that the kinetic energy of an electron the \(n=1\) state of a hydrogen atom is on the same order of magnitude as the absolute value of its total energy, and estimate a typical speed at which it would be moving. (It cannot really have a single, definite speed, because its kinetic and interaction energy trade off at different distances from the proton, but this is just a rough estimate of a typical speed.) Based on this speed, were we justified in assuming that the electron could be described nonrelativistically?
29 . Before the quantum theory, experimentalists noted that in many cases, they would find three lines in the spectrum of the same atom that satisfied the following mysterious rule: \(1/\lambda_1=1/\lambda_2+1/\lambda_3\). Explain why this would occur. Do not use reasoning that only works for hydrogen --- such combinations occur in the spectra of all elements. [Hint: Restate the equation in terms of the energies of photons.]
30 . The wavefunction of the electron in the ground state of a hydrogen atom is
where \(r\) is the distance from the proton, and \(a=5.3\times10^{-11}\) m is a constant that sets the size of the wave.
(a) Calculate symbolically, without plugging in numbers, the probability that at any moment, the electron is inside the proton. Assume the proton is a sphere with a radius of \(b=0.5\) fm. [Hint: Does it matter if you plug in \(r=0\) or \(r=b\) in the equation for the wavefunction?](answer check available at lightandmatter.com)
(b) Calculate the probability numerically.(answer check available at lightandmatter.com)
(c) Based on the equation for the wavefunction, is it valid to think of a hydrogen atom as having a finite size? Can \(a\) be interpreted as the size of the atom, beyond which there is nothing? Or is there any limit on how far the electron can be from the proton?
31 . Use physical reasoning to explain how the equation for the energy levels of hydrogen,
should be generalized to the case of an atom with atomic number \(Z\) that has had all its electrons removed except for one.
32
. A muon is a subatomic particle that acts exactly like an electron except that its mass is 207 times greater. Muons can be created by cosmic rays, and it can happen that one of an atom's electrons is displaced by a muon, forming a muonic atom. If this happens to a hydrogen atom, the resulting system consists simply of a proton plus a muon.
(a) Based on the results of section
13.4.4
, how would the size of a muonic hydrogen atom in its ground state compare with the size of the normal atom?
(b) If you were searching for muonic atoms in the sun or in the earth's atmosphere by spectroscopy, in what part of the electromagnetic spectrum would you expect to find the absorption lines?
33 . A photon collides with an electron and rebounds from the collision at 180 degrees, i.e., going back along the path on which it came. The rebounding photon has a different energy, and therefore a different frequency and wavelength. Show that, based on conservation of energy and momentum, the difference between the photon's initial and final wavelengths must be \(2h/mc\), where \(m\) is the mass of the electron. The experimental verification of this type of “pool-ball” behavior by Arthur Compton in 1923 was taken as definitive proof of the particle nature of light. Note that we're not making any nonrelativistic approximations. To keep the algebra simple, you should use natural units --- in fact, it's a good idea to use even-more-natural-than-natural units, in which we have not just \(c=1\) but also \(h=1\), and \(m=1\) for the mass of the electron. You'll also probably want to use the relativistic relationship \(E^2-p^2=m^2\), which becomes \(E^2-p^2=1\) for the energy and momentum of the electron in these units.
34 . Generalize the result of problem 33 to the case where the photon bounces off at an angle other than 180° with respect to its initial direction of motion.
35 . On page 869 we derived an expression for the probability that a particle would tunnel through a rectangular barrier, i.e., a region in which the interaction energy \(U(x)\) has a graph that looks like a rectangle. Generalize this to a barrier of any shape. [Hints: First try generalizing to two rectangular barriers in a row, and then use a series of rectangular barriers to approximate the actual curve of an arbitrary function \(U(x)\). Note that the width and height of the barrier in the original equation occur in such a way that all that matters is the area under the \(U\)-versus-\(x\) curve. Show that this is still true for a series of rectangular barriers, and generalize using an integral.] If you had done this calculation in the 1930's you could have become a famous physicist.
36 . Show that the wavefunction given in problem 30 is properly normalized.
37 . Show that a wavefunction of the form \(\Psi = e^{by} \sin ax \) is a possible solution of the Schrödinger equation in two dimensions, with a constant potential. Can we tell whether it would apply to a classically allowed region, or a classically forbidden one?
38 . Find the energy levels of a particle in a three-dimensional rectangular box with sides of length \(a\), \(b\), and \(c\).(answer check available at lightandmatter.com)
39 . Americium-241 is an artificial isotope used in smoke detectors. It undergoes alpha decay, with a half-life of 432 years. As discussed in example 18 on page 870, alpha decay can be understood as a tunneling process, and although the barrier is not rectangular in shape, the equation for the tunneling probability on page 870 can still be used as a rough guide to our thinking. For americium-241, the tunneling probability is about \(1\times10^{-29}\). Suppose that this nucleus were to decay by emitting a tritium (helium-3) nucleus instead of an alpha particle (helium-4). Estimate the relevant tunneling probability, assuming that the total energy \(E\) remains the same. This higher probability is contrary to the empirical observation that this nucleus is not observed to decay by tritium emission with any significant probability, and in general tritium emission is almost unknown in nature; this is mainly because the tritium nucleus is far less stable than the helium-4 nucleus, and the difference in binding energy reduces the energy available for the decay.
40 . As far as we know, the mass of the photon is zero. However, it's not possible to prove by experiments that anything is zero; all we can do is put an upper limit on the number. As of 2008, the best experimental upper limit on the mass of the photon is about \(1\times 10^{-52}\) kg. Suppose that the photon's mass really isn't zero, and that the value is at the top of the range that is consistent with the present experimental evidence. In this case, the \(c\) occurring in relativity would no longer be interpreted as the speed of light. As with material particles, the speed \(v\) of a photon would depend on its energy, and could never be as great as \(c\). Estimate the relative size \((c-v)/c\) of the discrepancy in speed, in the case of a photon with a frequency of 1 kHz, lying in the very low frequency radio range. \hwans{hwans:photon-mass}
41 . Hydrogen is the only element whose energy levels can be expressed exactly in an equation. Calculate the ratio \(\lambda_E/\lambda_F\) of the wavelengths of the transitions labeled E and F in problem 25 on p. 898. Express your answer as an exact fraction, not a decimal approximation. In an experiment in which atomic wavelengths are being measured, this ratio provides a natural, stringent check on the precision of the results.(answer check available at lightandmatter.com)
42 . Give a numerical comparison of the number of photons per second emitted by a hundred-watt FM radio transmitter and a hundred-watt lightbulb.(answer check available at lightandmatter.com)
43 . On pp. 884-885 of subsection 13.4.4 , we used simple algebra to derive an approximate expression for the energies of states in hydrogen, without having to explicitly solve the Schrödinger equation. As input to the calculation, we used the the proportionality \(U \propto r^{-1}\), which is a characteristic of the electrical interaction. The result for the energy of the \(n\)th standing wave pattern was \(E_n \propto n^{-2}\).
There are other systems of physical interest in which we have \(U \propto r^k\) for values of \(k\) besides \(-1\). Problem 23 discusses the ground state of the harmonic oscillator, with \(k=2\) (and a positive constant of proportionality). In particle physics, systems called charmonium and bottomonium are made out of pairs of subatomic particles called quarks, which interact according to \(k=1\), i.e., a force that is independent of distance. (Here we have a positive constant of proportionality, and \(r>0\) by definition. The motion turns out not to be too relativistic, so the Schrödinger equation is a reasonable approximation.) The figure shows actual energy levels for these three systems, drawn with different energy scales so that they can all be shown side by side. The sequence of energies in hydrogen approaches a limit, which is the energy required to ionize the atom. In charmonium, only the first three levels are known. 8
Generalize the method used for \(k=-1\) to any value of \(k\), and find the exponent \(j\) in the resulting proportionality \(E_n \propto n^j\). Compare the theoretical calculation with the behavior of the actual energies shown in the figure. Comment on the limit \(k\rightarrow\infty\). (answer check available at lightandmatter.com)
44
. The electron, proton, and neutron were discovered, respectively, in 1897, 1919, and 1932. The neutron was late to the party, and some physicists felt that it was unnecessary to consider it as fundamental. Maybe it could be explained as simply a proton with an electron trapped inside it. The charges would cancel out, giving the composite particle the correct neutral charge, and the masses at least approximately made sense (a neutron is heavier than a proton). (a) Given that the diameter of a proton is on the order of \(10^{-15}\ \text{m}\), use the Heisenberg uncertainty principle to estimate the trapped electron's minimum momentum.(answer check available at lightandmatter.com)
(b) Find the electron's minimum kinetic energy.(answer check available at lightandmatter.com)
(c) Show via \(E=mc^2\) that the proposed explanation fails, because the contribution to the neutron's mass from the electron's kinetic energy would be many orders of magnitude too large.
45 . Suppose that an electron, in one dimension, is confined to a certain region of space so that its wavefunction is given by
Determine the constant \(A\) from normalization.(answer check available at lightandmatter.com)
46 . In the following, \(x\) and \(y\) are variables, while \(u\) and \(v\) are constants. Compute (a) \(\partial(ux\ln (vy))/\partial x\), (b) \(\partial(ux\ln (vy))/\partial y\).(answer check available at lightandmatter.com)
47
. (a) A radio transmitter radiates power \(P\) in all directions, so that the energy spreads out spherically. Find the energy density at a distance \(r\).(answer check available at lightandmatter.com)
(b) Let the wavelength be \(\lambda\). As described in example
8
on p. 842, find the number of photons in a volume \(\lambda^3\) at this distance \(r\).(answer check available at lightandmatter.com)
(c) For a 1000 kHz AM radio transmitting station, assuming reasonable values of \(P\) and \(r\), verify, as claimed in the example, that the result from part b is very large.
Contributors and Attributions
Benjamin Crowell (Fullerton College). Conceptual Physics is copyrighted with a CC-BY-SA license.


15.2: Health Risks in Late Adulthood
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Learning Objectives
By the end of this section, you will be able to:
- Identify the major physical health concerns of late adulthood
- Identify the major mental health concerns of late adulthood
- Describe research and issues related to health-care expenses
Growing up, Jarvis listened to his older relatives complain about their aches and pains and thought, “I’ll never do that.” Now, as he enters his seventies, he realizes how wrong he was. Although his overall health is pretty good, years of serving in the military and working in construction have resulted in some hearing loss, as well as arthritis in his hands, wrists, and knees. These problems don’t significantly impair his functioning, but Jarvis takes more aspirin than he used to and has started to wear a hearing aid. Following a cancer scare, he gave up smoking and discovered that he was breathing better and had more stamina than he expected to have at his age. Jarvis worries less than he did when he was younger, although the cost of health care gives him some anxiety. He resolves to do his best to stay healthy so he can keep enjoying life.
This section describes different types of physical and mental health risks and how they relate to older adults. Policy issues related to health-care affordability and accessibility can also play a role in health risks in late adulthood.
Physical Health Concerns in Late Adulthood
In health research, good health is typically defined by omission. In other words, researchers typically view good health as the lack of any short- or long-term diseases or conditions that can have a negative effect on quality of life, functioning, or future.
Acute and Chronic Health Problems
An acute condition tends to occur quickly and last only a brief period. Examples include many short-term, common, and contagious conditions such as the common cold and flu. Acute conditions are not associated with age. On the other hand, a chronic condition often develops slowly over time and is positively correlated with age. The defining characteristic of a chronic condition is that it has a long duration, several months or longer, or even occurs on a constant basis and needs management to prevent worsening over time (National Council on Aging, 2020). Examples include osteoarthritis, heart disease, asthma, and diabetes. Poor lifestyle choices such as unhealthy diet, a sedentary lifestyle, and smoking increase the risk of developing some of these health conditions.
As we get older, the odds of facing chronic health conditions increase. Around 70 percent of U.S. adults aged fifty-five to sixty-four years manage at least one chronic health condition, and 14 percent have three or more. By age sixty-five years, those numbers have risen to 86 and 23 percent, respectively (Centers for Disease Control and Prevention, 2015). Due to the increasing population of older adults, however, the number of individuals with at least one chronic health problem is expected to double between 2020 and 2050 (Ansah & Chiu, 2023). This is particularly worrisome because of the high cost of medical care for chronic conditions; in the United States, 85 percent of health-care costs are spent on chronic conditions (Holman et al., 2020). In 2011, this cost was predicted to reach $47 trillion by 2030 (Hacker, 2024). Since the COVID-19 pandemic, the number of people needing care for chronic conditions has increased, and this figure is now predicted to be even higher (Cutler, 2022; Tene et al., 2023; Tufts et al., 2023).
Acute conditions, however, become less common with age. Preschool children on average experience six to eight colds per year, sometimes one per month, because of their limited immunity to different viruses. As we get older, the lifelong process of contracting these acute conditions builds immunity to specific infections, decreasing the odds of future acute illness (Pappas, 2022). This is not to say that acute conditions are not a problem for older adults. As mentioned earlier, many acute conditions such as the flu, pneumonia, RSV (respiratory syncytial virus), and COVID-19 can be more serious and potentially fatal for older adults because the immune system’s ability to respond to infections weakens with age. This is a main reason older adults have been more likely to experience serious illness from COVID-19 and why they are highly encouraged to get regularly vaccinated.
Sexual Functioning in Older Age
Sexual activity remains a common element of relationships in later life.  Several studies have indicated that 50 to 80 percent of adults aged sixty to seventy-four years and 15 to 30 percent of adults over age eighty years report being sexually active, with men reporting more sexual activity than women (Jackson et al., 2020; Lee et al., 2016).
Physical health may affect sexual functioning more in older adults than in other age groups. For example, estrogen and testosterone levels decline after middle age (Brotto et al., 2016; Hochberg & Konner, 2020; Hull et al., 2011; Ricoy-Cano et al., 2020), which can affect both sex drive and performance (including vaginal lubrication and the ability to achieve an erection). Joint pain and loss of flexibility, both of which increase in later life, may make certain sexual positions uncomfortable. Health problems such as high blood pressure and diabetes, also more likely in later life, are reliably associated with lower rates of sexual activity and desire (Brotto et al., 2016; Jackson et al., 2020; Lee et al., 2016). Some research proposes that loss of sex drive or performance may be an early indicator of health problems (Jackson et al., 2020), suggesting that health-care providers should ask older patients about their sexual health as a routine part of medical checkups.
Another issue is prevention of sexually transmitted infections (STIs). Over the past decade or so, rates of STIs have increased in many countries, and older adults aren’t exempt (Bourchier et al., 2020; Morgan et al., 2023; Smith et al., 2020). However, older adults are generally unlikely to use prevention methods such as condoms or PrEP (Morgan et al., 2023), and they demonstrate low knowledge of STI transmission, symptoms, and prevention (Smith et al., 2020). Therefore, health-care providers should screen for STIs and discuss safe sex methods with their older patients.
Specific Threats to Health in Older Age
Researchers examining data from the Centers for Disease Control and Prevention (Xu et al., 2022) found that seven of the ten leading causes of death for older adults were chronic health conditions that typically become more common in later years (Figure 15.10). The leading cause of death in the United States for 2020 and 2021 was heart disease , an example of a chronic condition associated with many lifestyle factors such as diet, exercise, and stress. The second leading cause of death was cancer, followed by COVID-19, injuries, stroke, respiratory disease, Alzheimer’s disease, diabetes, liver disease/cirrhosis, and kidney disease. The more common age-related causes of death, such as heart disease, some types of cancer, respiratory disease, and diabetes, are all at least partly caused by environmental and lifestyle characteristics (National Center for Health Statistics, 2021).
Many common causes of death are consistent across racial and ethnic groups, with subtle variations in their frequency. This is true for all the age-related causes of death as well, with heart disease and cancer leading across all racial and ethnic groups in the United States (Centers for Disease Control and Prevention, 2022). These variations are generally believed to be caused by environmental factors, particularly the chronic stress associated with poverty and institutionalized racism (Cunningham et al., 2017; Leitner et al., 2016). This chronic stress may accelerate the development of health problems (Forde, 2019; Simons, Lei, Klopack, Beach, et al., 2021; Simons, Lei, Kopack, Zhang, et al., 2021), and racism both within and outside of the health-care system can prevent people from receiving appropriate medical care (Leitner et al., 2016; Sim et al., 2021; Simons, Lei, Klopack, Beach, et al., 2021; Simons, Lei, Kopack, Zhang, et al., 2021). The end result is that medical problems may go untreated, thus leading to decreased life expectancy .
One study compared eleven of the highest-income countries—United States, United Kingdom, Germany, Sweden, France, the Netherlands, Switzerland, Denmark, Canada, Japan, and Australia—to better understand similarities and differences in health and health outcomes around the world. The United States appeared healthier than most countries on some high-risk factors such as smoking, and it ranked near the middle on alcohol consumption. On weight, however, the United States ranked worst, with 70.1 percent of the population overweight or obese; Australia was next at 63.4 percent; and Japan had the fewest problems related to weight, with 23.8 percent of the population considered overweight or obese. Overall, the United States had the shortest average life expectancy at 78.8 years, and Japan the highest at 83.9 years (Papanicolas et al., 2018) (Figure 15.11).
Health problems and outcomes in developing and low-income countries are vastly different, because they are often those associated with living in poverty. For example, data from the World Health Organization suggest that neonatal conditions, diarrhea, malaria, tuberculosis, and HIV are all top-ten causes of death in countries within the bottom 25 percent when ranked on income (World Health Organization, 2020).
An interesting variation on international differences in health and life expectancy comes from the study of “ Blue Zones ,” the areas of the world with an unusually high number of people older than their predicted life expectancy, in many cases 100 years of age or more (Buettner, 2008; Buettner & Skemp, 2016; Herbert et al., 2022; Kreouzi et al., 2022; Poulain & Herm, 2022). These places are Nicoya Peninsula, Costa Rica; Ikaria, Greece; Sardinia, Italy; Okinawa, Japan; and Loma Linda, California, in the United States. Their residents have longevity and good health in common, but they differ ethnically, economically, and geographically. What commonalities, then, are producing these good health outcomes?
Research has found several shared characteristics that appear to positively affect their health:
- Moving naturally —Blue Zone residents don’t get formal exercise but are physically active in their daily lives (gardening, doing housework, walking to get places).
- Having a sense of purpose —They are motivated and goal oriented.
- Managing stress —They have regular rituals, such as prayer and ancestor worship, that help them cope with stress.
- Monitoring their food portions and mealtimes —They eat until they’re only 80 percent full and also eat their largest meal at midday instead of in the evening.
- Following a plant-based diet —Most of their protein comes from beans and lentils; they may eat meat but not frequently.
- Drinking in moderation —Having one to two glasses of wine per day is associated with longer life in these groups; however, there’s controversy about the role of alcohol consumption in health.
- Participating in a faith-based community —Most residents attend some kind of religious services several times a month.
- Putting family first —Residents prioritize living with or near extended family and spending time with children and life partners.
- Surrounding themselves with support —They create networks of close friends who support their healthy behaviors and provide companionship (Buettner & Skemp, 2016; Herbert et al., 2022; Kreouzi et al., 2022; Poulain & Herm, 2022).
While it’s hard to say conclusively that these behaviors are causing the increased life expectancy, they’re consistent with other research associating them with positive health outcomes (Kreouzi et al., 2022). Many such behaviors, such as creating networks of close friends, consuming mainly plant-based protein, and being naturally active, can be achieved without spending a lot of money. This suggests that people in low-income situations may be able to incorporate some of these behaviors into their lives and increase their chances of living longer and healthier lives. However, other facets of SES may also limit awareness of best health practices. For example, some studies find that lower educational level and lower income are associated with less knowledge about how to promote good health (Svendsen et al., 2020). Research studies recommend public health strategies work to improve health equity by promoting better knowledge of health risk and health promotion behaviors across populations (Sorensen et al., 2015).
Mental Health Concerns in Late Adulthood
While some mental health problems are specific to later life, such as different types of dementia, other challenges such as anxiety and depression are less common for older adults than for younger age groups. A subfield of psychology called geropsychology focuses on enhancing the mental health of older adults (American Psychological Association, 2024). Geropsychologists are specially trained to address the way mental health problems may manifest themselves in late life.
Depression
Unfortunately, due to systemic ageism, people often negatively stereotype older adults as living lonely lives and struggling with depression as they face. However, according to research, these stereotypes are far from accurate. In fact, depression rates tend to decrease with age, and young adults are more than four times more likely to have depression than adults aged fifty years and older (Figure 15.12). Research in countries throughout Europe also suggests that older adults tend to defy stereotypes that they are a depressed group (Beller et al., 2021; Copeland et al., 1999). As it turns out, not only does the risk of depression decrease, but the experience of positive emotions increases (Cartensen & DeLiema, 2018).
These findings don’t suggest that depression is irrelevant to older populations. Nearly one in twenty older adults deal with major depression within a given year, and many more face less severe depressive symptoms. Research suggests that economic hardship, loneliness, serious chronic health problems, caregiving for a loved one, the loss of a loved one, and cognitive impairment all place older adults experiencing those events at a higher risk for depression than their peers (Alexopoulos, 2005; Işik et al., 2020). Adequate social support, exercise, and confidence in physical ability tend to reduce the risk of depression (Miller et al., 2019), as does setting and achieving realistic and personally meaningful goals (Samantaray & Kar, 2021). Being able to adapt to new circumstances is also beneficial; for example, looking at a move to a retirement community as a relief from the responsibilities of home ownership instead of a loss of independence may prevent someone from being distressed about this change (Regier & Parmalee, 2021).
Research shows that older adults are at a higher risk for suicide. Overall, men are much more likely to die by suicide, a finding consistent across race, ethnicity, and age. White people tend to have higher suicide rates than other racial and ethnic groups. Those in the earlier part of older adulthood (ages sixty-five to seventy-four years) have the lowest rate of suicide across adulthood, with around fifteen suicides per 100,000 individuals. The rate increases to nineteen suicides per 100,000 people for those aged seventy-five to eighty-four years, and more than twenty-two per 100,000 people for adults aged eighty-five years and older (American Foundation for Suicide Prevention, 2024). Having depression, being unmarried, and living alone make suicide more common in older age (Wiktorsson et al., 2010). Loss of a spouse has also been linked to suicide ideation in this population (Heuser & Howe, 2019).
Anxiety
Like depression, anxiety is more common in women than men; 23 percent of women reported experiencing an anxiety disorder in the past year compared to 14 percent of men (National Institute of Mental Health, n.d.). Lifetime prevalence of anxiety disorders is around 34 percent, but research generally suggests they typically first appear from childhood to early adulthood. They tend to peak in the early part of middle adulthood before decreasing during the later part of middle adulthood and throughout older age (Bandelow & Michaelis, 2022; Javaid et al., 2023) (Figure 15.13).
While anxiety is less common in later life, nearly one in ten older adults reported experiencing an anxiety disorder within the past year. Older adults with anxiety disorders frequently have comorbid disorders such as depression, serious medical issues, and cognitive decline. Late-onset anxiety is not common; most older adults dealing with anxiety also experienced it at younger ages. Anxiety in older adults can be successfully treated with either medication or cognitive behavioral therapy (Ando et al., 2023; Wolitzky-Taylor et al., 2010).
Health-Care Costs in Older Age
Paying for health care to treat the potential problems of later life can be both complicated and expensive in the United States. Despite having the lowest average life expectancy among high-income nations, the United States spent the highest percentage of gross domestic product (17.8 percent) and the highest amount per individual, including for individual health care, in addition to tax contributions toward public health-care programs (Figure 15.14) (Papanicolas et al., 2018). These statistics are concerning. They suggest that people in the United States are paying much more than those in comparable countries for health care, yet they are last in life expectancy among the nations studied.
One reason for the higher cost of health care in the United States may be the nation’s complex health-care system. Most other affluent countries have universal health care , a system that makes health-care services accessible to all citizens with little or no out-of-pocket costs for individuals because the government subsidizes the cost through taxes. In the United States, in contrast, health care can be paid for entirely out of pocket or through privately purchased health insurance, employer-provided health insurance, or a variety of federal and state programs. This complex system can be difficult to navigate, especially for older adults who may have limited income and greater health-care needs. Few of these options cover all costs or all types of health care, and individuals typically pay at least some expenses themselves, either directly or through the purchase of supplemental insurance. Thus, socioeconomic status is a factor in access to high-quality health care.
Medicare is a U.S. federal program that pays some health-care costs for many older adults, particularly related to hospital care and doctor visits. Because many expenses such as vision care and dental care are not covered, however, most older adults also purchase supplemental health insurance or a Medicare Advantage plan (Medicare, n.d.) if they don’t already have other insurance. Medicare does not automatically offer prescription drug coverage, a significant expense for older adults; this has to be added separately or covered through another form of insurance. Individuals who are unable to pay for health coverage may qualify for Medicaid , a program like Medicare but for those with limited income (Medicaid, n.d.).
Link to Learning
Read an article about older U.S. adults who either ration or skip taking their prescription drugs because they can’t afford the cost.
While older adults face several health risks such as those related to normative age-related changes and increased health care costs, there are many protective factors that can promote mental and physical health and well-being in older adulthood. For example, older adults often experience healthy sexual functioning and lower risks of depression and anxiety than in earlier adulthood. Finally, several health promoting behaviors, or protective factors, can be introduced to improve health and life expectancy in older adults, such as engaging in physical activity, eating well, having a good social support system, and managing stress.
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5.2: Communicate in online forums
Many online courses include forums, either as a required assignment, or to support your learning process. How can you use forums to support your learning in the best way possible?
Earlier in the workshop, you encountered the social presence and cognitive presence elements of the community of inquiry you are building in your course. Forums are a tool for creating collaborative learning relationships. They can also be a low-stakes way to express your developing ideas, and to get feedback on the ways in which you are learning the course material as you work towards larger assignments.
What makes a good forum post?
The rubric below outlines what distinguishes stronger online posts from those that are less successful. Review the chart below. What do you notice? If your instructor has provided a rubric for online posts, read the rubric, and identify your instructor’s criteria for success. [1]
Tips for Participating in Forums
- Develop a clear understanding of the expectations and ground rules for the forum. Review your course presentation (syllabus) to guidance on how often to post, the type of content to include in each post, and the best way to respond to others’ posts.
- Make connections between your posts and the content you are learning in the course. A forum post is often an excellent place to engage in critical reflection. Make connections between the course content and the ways that your growing understanding are shaping your present and future practice.
- Set a regular schedule for posting and commenting on forums. This prevents the amount of content from becoming overwhelming, and allows you to develop stronger relationships in the course by regularly engaging with classmates.
- Include resources that might be useful to other classmates or your instructor in your posts.
- Use language that is appropriate for an academic environment. Avoid writing in a way that is too informal (ie. writing that resembles a text message).
- Make sure that each post is clearly written and well structured. Take time to clarify the message you want to communicate in your post, and organize your content into clear and concise paragraphs. This is easier for your reader than a long or disorganized post.
- Respond to others’ posts in a supportive and challenging way. In writing, messages may be unintentionally misinterpreted. Be sure that your responses to others are respectful, positive in tone, and do not appear angry, even when you wish to disagree or present an alternative viewpoint.
- Participate in the community discussion. Read others’ comments before posting, and connect your ideas with what you are hearing from your classmates. [2]
For Reflection
Read through your course presentations and find any guidelines for posting to forums. Create a checklist for yourself based on these guidelines and the rubric above. What do you need to do to create successful forum posts in your course ? Keep this checklist, and refer to it regularly as you post to forums.
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Fenwick, T. J., & Parsons, J. (2009). The art of evaluation: a resource for educators and trainers (2nd ed.). Toronto: Thompson Educational Publishing.
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Pappas, C. (2015a, June 6). 10 netiquette tips for online discussions. Retrieved August 7, 2018, from
https://elearningindustry.com/10-netiquette-tips-online-discussions
; Pappas, C. (2015b, August 16). 7 tips on how to use forums in elearning. Retrieved August 7, 2018, from
https://elearningindustry.com/7-tips-use-forums-in-elearning
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4.3: Visual Summaries of Data
- Create and interpret frequency tables.
- Create and interpret bar plots and histograms.
Organizing Data
Once you have collected data, what will you do with it? Data can be described and presented in many different formats. For example, suppose you are interested in buying a house in a particular area. You may have no clue about the house prices, so you might ask your real estate agent to give you a sample data set of prices. Looking at all the prices in the sample often is overwhelming. A better way might be to look at the median price and the variation of prices. The median and variation are just two ways that you will learn to describe data. Your agent might also provide you with a graph of the data.
 This area of statistics is called "Descriptive Statistics." You will learn how to calculate, and even more importantly, how to interpret these measurements and graphs.
A statistical graph is a tool that helps you learn about the shape or distribution of a sample or a population. A graph can be a more effective way of presenting data than a mass of numbers because we can see where data clusters and where there are only a few data values. Graphs can be used to show trends and to enable readers to compare facts and figures quickly. Statisticians often graph data first to get a picture of the data. Then, more formal tools may be applied.
Some of the types of graphs that are used to summarize and organize data are the dot plot, the bar graph, the histogram, the stem-and-leaf plot, the frequency polygon (a type of broken line graph), the pie chart, and the box plot. In this chapter, we will briefly look at stem-and-leaf plots, line graphs, and bar graphs, as well as frequency polygons, and time series graphs. Our emphasis will be on histograms and box plots.
Frequency Tables
Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows:
5; 6; 3; 3; 2; 4; 7; 5; 2; 3; 5; 6; 5; 4; 4; 3; 5; 2; 5; 3.
Table lists the different data values in ascending order and their frequencies.
| DATA VALUE | FREQUENCY |
|---|---|
| 2 | 3 |
| 3 | 5 |
| 4 | 3 |
| 5 | 6 |
| 6 | 2 |
| 7 | 1 |
A frequency is the number of times a value of the data occurs. According to Table Table \(\PageIndex{A}\), there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample.
A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample–in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.
| DATA VALUE | FREQUENCY | RELATIVE FREQUENCY |
|---|---|---|
| 2 | 3 | \(\frac{3}{20}\) or 0.15 or 15% |
| 3 | 5 | \(\frac{5}{20}\) or 0.25 or 25% |
| 4 | 3 | \(\frac{3}{20}\) or 0.15 or 15% |
| 5 | 6 | \(\frac{6}{20}\) or 0.30 or 30% |
| 6 | 2 | \(\frac{2}{20}\) or 0.10 or 10% |
| 7 | 1 | \(\frac{1}{20}\) or 0.05 or 5% |
The sum of the values in the relative frequency column of Table \(\PageIndex{B}\) is \(\frac{20}{20}\), or 1, or 100%.
Because of rounding, the relative frequency column may not always sum to one, and the last entry in the cumulative relative frequency column may not be one. However, they should be close to one.
Table \(\PageIndex{C}\) represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.
| HEIGHTS (INCHES) | FREQUENCY | RELATIVE FREQUENCY |
|---|---|---|
| 60–62 | 5 | \(\frac{5}{100} = 0.05 = 5%\) |
| 62-64 | 3 | \(\frac{3}{100} = 0.03 = 3%\) |
| 64–66 | 15 | \(\frac{15}{100} = 0.15 = 15%\) |
| 66–68 | 40 | \(\frac{40}{100} = 0.40 = 40%\) |
| 68–70 | 17 | \(\frac{17}{100} = 0.17 = 17%\) |
| 70–72 | 12 | \(\frac{12}{100} = 0.12 = 12%\) |
| 72–74 | 7 | \(\frac{7}{100} = 0.07 = 7%\) |
| 74–76 | 1 | \(\frac{1}{100} = 0.01 = 1%\) |
| Total = 100 | Total = 1.00 |
In this sample, there are five players whose heights fall within the interval 60–62 inches, three players whose heights fall within the interval 62–64 inches, 15 players whose heights fall within the interval 64–66 inches, 40 players whose heights fall within the interval 66–68 inches, 17 players whose heights fall within the interval 68–70 inches, 12 players whose heights fall within the interval 70–72, seven players whose heights fall within the interval 72–74, and one player whose heights fall within the interval 74–76. All heights fall between the endpoints of an interval and not at the endpoints.
- From the Table \(\PageIndex{C}\), find the percentage of heights that are less than 66 inches.
- Find the percentage of heights that fall between 62 and 66 inches.
Answer
- If you look at the first, second, and third rows, the heights are all less than 66 inches. There are \(5 + 3 + 15 = 23\) players whose heights are less than 66 inches. The percentage of heights less than 66 inches is then \(\frac{23}{100}\) or 23%. This percentage is an example of the cumulative relative frequency .
- Add the relative frequencies in the second and third rows: \(0.03 + 0.15 = 0.18\) or 18%.
Table \(\PageIndex{D}\) shows the amount, in inches, of annual rainfall in a sample of towns.
- Find the percentage of rainfall that is less than 9.01 inches.
- Find the percentage of rainfall that is between 6.99 and 13.05 inches.
| Rainfall (Inches) | Frequency | Relative Frequency |
|---|---|---|
| 2.95–4.97 | 6 | \(\frac{6}{50} = 0.12\) |
| 4.97–6.99 | 7 | \(\frac{7}{50} = 0.14\) |
| 6.99–9.01 | 15 | \(\frac{15}{50} = 0.30\) |
| 9.01–11.03 | 8 | \(\frac{8}{50} = 0.16\) |
| 11.03–13.05 | 9 | \(\frac{9}{50} = 0.18\) |
| 13.05–15.07 | 5 | \(\frac{5}{50} = 0.10\) |
| Total = 50 | Total = 1.00 |
- Answer
-
- \(0.56\) or \(56%\)
- \(0.30 + 0.16 + 0.18 = 0.64\) or \(64%\)
Use the heights of the 100 male semiprofessional soccer players in Table \(\PageIndex{C}\). Fill in the blanks and check your answers.
- The percentage of heights that are from 68 to 72 inches is: ____.
- The percentage of heights that are from 68 to 74 inches is: ____.
- The percentage of heights that are more than 66 inches is: ____.
- The number of players in the sample who are between 62 and 72 inches tall is: ____.
- What kind of data are the heights?
- Describe how you could gather this data (the heights) so that the data are characteristic of all male semiprofessional soccer players.
Remember, you count frequencies . To find the relative frequency, divide the frequency by the total number of data values. To find the cumulative relative frequency, add all of the previous relative frequencies to the relative frequency for the current row.
Answer
- 29%
- 36%
- 77%
- 87
- quantitative continuous
- get rosters from each team and choose a simple random sample from each
From Table \(\PageIndex{D}\), find the number of towns that have rainfall between 2.95 and 9.01 inches.
- Answer
-
\(6 + 7 + 15 = 28\) towns
Nineteen people were asked how many miles, to the nearest mile, they commute to work each day. The data are as follows: 2; 5; 7; 3; 2; 10; 18; 15; 20; 7; 10; 18; 5; 12; 13; 12; 4; 5; 10. Table \(\PageIndex{E}\) was produced:
| DATA | FREQUENCY | RELATIVE FREQUENCY |
|---|---|---|
| 3 | 3 | \(\frac{3}{19}\) |
| 4 | 1 | \(\frac{1}{19}\) |
| 5 | 3 | \(\frac{3}{19}\) |
| 7 | 2 | \(\frac{2}{19}\) |
| 10 | 3 | \(\frac{3}{19}\) |
| 12 | 2 | \(\frac{2}{19}\) |
| 13 | 1 | \(\frac{1}{19}\) |
| 15 | 1 | \(\frac{1}{19}\) |
| 18 | 1 | \(\frac{1}{19}\) |
| 20 | 1 | \(\frac{1}{19}\) |
- Is the table correct? If it is not correct, what is wrong?
- True or False: Three percent of the people surveyed commute three miles. If the statement is not correct, what should it be? If the table is incorrect, make the corrections.
- What fraction of the people surveyed commute five or seven miles?
- What fraction of the people surveyed commute 12 miles or more? Less than 12 miles? Between five and 13 miles (not including five and 13 miles)?
Answer
- No. The frequency column sums to 18, not 19.
- False. The frequency for three miles should be one; for two miles (left out), two.
- \(\frac{5}{19}\)
- \(\frac{7}{19}\), \(\frac{12}{19}\), \(\frac{7}{19}\)
Table \(\PageIndex{D}\) represents the amount, in inches, of annual rainfall in a sample of towns. What fraction of towns surveyed get between 11.03 and 13.05 inches of rainfall each year?
- Answer
-
\(\frac{9}{50}\)
Table \(\PageIndex{F}\) contains the total number of deaths worldwide as a result of earthquakes for the period from 2000 to 2012.
| Year | Total Number of Deaths |
|---|---|
| 2000 | 231 |
| 2001 | 21,357 |
| 2002 | 11,685 |
| 2003 | 33,819 |
| 2004 | 228,802 |
| 2005 | 88,003 |
| 2006 | 6,605 |
| 2007 | 712 |
| 2008 | 88,011 |
| 2009 | 1,790 |
| 2010 | 320,120 |
| 2011 | 21,953 |
| 2012 | 768 |
| Total | 823,356 |
Answer the following questions.
- What is the frequency of deaths measured from 2006 through 2009?
- What percentage of deaths occurred after 2009?
- What is the relative frequency of deaths that occurred in 2003 or earlier?
- What is the percentage of deaths that occurred in 2004?
- What kind of data are the numbers of deaths?
- The Richter scale is used to quantify the energy produced by an earthquake. Examples of Richter scale numbers are 2.3, 4.0, 6.1, and 7.0. What kind of data are these numbers?
Answer
- 97,118 (11.8%)
- 41.6%
- 67,092/823,356 or 0.081 or 8.1 %
- 27.8%
- Quantitative discrete
- Quantitative continuous
Table \(\PageIndex{G}\) contains the total number of fatal motor vehicle traffic crashes in the United States for the period from 1994 to 2011.
| Year | Total Number of Crashes | Year | Total Number of Crashes |
|---|---|---|---|
| 1994 | 36,254 | 2004 | 38,444 |
| 1995 | 37,241 | 2005 | 39,252 |
| 1996 | 37,494 | 2006 | 38,648 |
| 1997 | 37,324 | 2007 | 37,435 |
| 1998 | 37,107 | 2008 | 34,172 |
| 1999 | 37,140 | 2009 | 30,862 |
| 2000 | 37,526 | 2010 | 30,296 |
| 2001 | 37,862 | 2011 | 29,757 |
| 2002 | 38,491 | Total | 653,782 |
| 2003 | 38,477 |
Answer the following questions.
- What is the frequency of deaths measured from 2000 through 2004?
- What percentage of deaths occurred after 2006?
- What is the relative frequency of deaths that occurred in 2000 or before?
- What is the percentage of deaths that occurred in 2011?
- What is the percentage of deaths that occurred between 1994 and 2006? Explain what this number tells you about the data.
Answer
- 190,800 (29.2%)
- 24.9%
- 260,086/653,782 or 39.8%
- 4.6%
- 75.1% of all fatal traffic crashes for the period from 1994 to 2011 happened from 1994 to 2006.
Visualizing Data
Below are tables comparing the number of part-time and full-time students at De Anza College and Foothill College enrolled for the spring 2010 quarter. The tables display counts (frequencies) and percentages or proportions (relative frequencies). The percent columns make comparing the same categories in the colleges easier. Displaying percentages along with the numbers is often helpful, but it is particularly important when comparing sets of data that do not have the same totals, such as the total enrollments for both colleges in this example. Notice how much larger the percentage for part-time students at Foothill College is compared to De Anza College.
| De Anza College | Foothill College | |||||
|---|---|---|---|---|---|---|
| Number | Percent | Number | Percent | |||
| Full-time | 9,200 | 40.9% | Full-time | 4,059 | 28.6% | |
| Part-time | 13,296 | 59.1% | Part-time | 10,124 | 71.4% | |
| Total | 22,496 | 100% | Total | 14,183 | 100% |
Tables are a good way of organizing and displaying data. But graphs can be even more helpful in understanding the data. There are no strict rules concerning which graphs to use. Two graphs that are used to display qualitative data are pie charts and bar graphs.
- In a pie chart , categories of data are represented by wedges in a circle and are proportional in size to the percent of individuals in each category.
- In a bar graph , the length of the bar for each category is proportional to the number or percent of individuals in each category. Bars may be vertical or horizontal.
- A Pareto chart consists of bars that are sorted into order by category size (largest to smallest).
Look at Figures \(\PageIndex{I}\) and \(\PageIndex{J}\) and determine which graph (pie or bar) you think displays the comparisons better.
It is a good idea to look at a variety of graphs to see which is the most helpful in displaying the data. We might make different choices of what we think is the “best” graph depending on the data and the context. Our choice also depends on what we are using the data for.
Bar Graphs
Bar graphs consist of bars that are separated from each other. The bars can be rectangles, or they can be rectangular boxes (used in three-dimensional plots), and they can be vertical or horizontal. The bar graphs have the possible values represented on the x -axis and proportions on the y -axis .
By the end of 2011, Facebook had over 146 million users in the United States. Table shows three age groups, the number of users in each age group, and the proportion (%) of users in each age group. Construct a bar graph using this data.
| Age groups | Number of Facebook users | Proportion (%) of Facebook users |
|---|---|---|
| 13–25 | 65,082,280 | 45% |
| 26–44 | 53,300,200 | 36% |
| 45–64 | 27,885,100 | 19% |
Answer
The population in Park City is made up of children, working-age adults, and retirees. Table shows the three age groups, the number of people in the town from each age group, and the proportion (%) of people in each age group. Construct a bar graph showing the proportions.
| Age groups | Number of people | Proportion of population |
|---|---|---|
| Children | 67,059 | 19% |
| Working-age adults | 152,198 | 43% |
| Retirees | 131,662 | 38% |
Answer
The columns in the following table contain: the race or ethnicity of students in U.S. Public Schools for the class of 2011, percentages for the Advanced Placement examine population for that class, and percentages for the overall student population. Create a bar graph with the student race or ethnicity (qualitative data) on the x -axis, and the Advanced Placement examinee population percentages on the y -axis.
| Race/Ethnicity | AP Examinee Population | Overall Student Population |
|---|---|---|
| 1 = Asian, Asian American or Pacific Islander | 10.3% | 5.7% |
| 2 = Black or African American | 9.0% | 14.7% |
| 3 = Hispanic or Latino | 17.0% | 17.6% |
| 4 = American Indian or Alaska Native | 0.6% | 1.1% |
| 5 = White | 57.1% | 59.2% |
| 6 = Not reported/other | 6.0% | 1.7% |
Solution
Park city is broken down into six voting districts. The table shows the percent of the total registered voter population that lives in each district as well as the percent total of the entire population that lives in each district. Construct a bar graph that shows the registered voter population by district.
| District | Registered voter population | Overall city population |
|---|---|---|
| 1 | 15.5% | 19.4% |
| 2 | 12.2% | 15.6% |
| 3 | 9.8% | 9.0% |
| 4 | 17.4% | 18.5% |
| 5 | 22.8% | 20.7% |
| 6 | 22.3% | 16.8% |
Answer
Histograms
For quantitative data, we frequently use histograms to display the data. 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.
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.
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.
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 suggest using 0.85 as the width of each bar or class interval. You can also use an interval with a width equal to one.
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:
\(\frac{6.5 - 0.5}{\text{number of bars}}\) = 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.
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
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.
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.
Percentages That Add to More (or Less) Than 100%
Sometimes percentages add up to be more than 100% (or less than 100%). In the graph, the percentages add to more than 100% because students can be in more than one category. A bar graph is appropriate to compare the relative size of the categories. A pie chart cannot be used. It also could not be used if the percentages added to less than 100%.
| Characteristic/Category | Percent |
|---|---|
| Full-Time Students | 40.9% |
| Students who intend to transfer to a 4-year educational institution | 48.6% |
| Students under age 25 | 61.0% |
| TOTAL | 150.5% |
Omitting Categories/Missing Data
The table displays Ethnicity of Students but is missing the "Other/Unknown" category. This category contains people who did not feel they fit into any of the ethnicity categories or declined to respond. Notice that the frequencies do not add up to the total number of students. In this situation, create a bar graph and not a pie chart.
| Frequency | Percent | |
|---|---|---|
| Asian | 8,794 | 36.1% |
| Black | 1,412 | 5.8% |
| Filipino | 1,298 | 5.3% |
| Hispanic | 4,180 | 17.1% |
| Native American | 146 | 0.6% |
| Pacific Islander | 236 | 1.0% |
| White | 5,978 | 24.5% |
| TOTAL | 22,044 out of 24,382 | 90.4% out of 100% |
The following graph is the same as the previous graph, but the “Other/Unknown” percent (9.6%) has been included. The “Other/Unknown” category is large compared to some of the other categories (Native American, 0.6%, Pacific Islander 1.0%). This is important to know when we think about what the data are telling us.
This particular bar graph in Figure \(\PageIndex{4}\) can be difficult to understand visually. The graph in Figure \(\PageIndex{5}\) is a Pareto chart . The Pareto chart has the bars sorted from largest to smallest and is easier to read and interpret.


6.3: Using SWOT for Strategic Analysis
- What is a SWOT analysis, and what can it reveal about a firm?
You may already have heard of one very common tool firms use to analyze their strategic and competitive situations: SWOT, which is an acronym for strengths, weaknesses, opportunities, and threats. Firms use SWOT analysis to get a general understanding of what they are good or bad at and what factors outside their doors might present chances for success or difficulty. Let’s take a look at SWOT analysis piece by piece (Figure \(\PageIndex{1}\)).
Strengths
A firm’s strengths are, to put it simply, what it is good at. Nike is good at marketing sports products, McDonald’s is good at making food quickly and inexpensively, and Ferrari is good at making beautiful fast cars. When a firm analyzes its strengths, it compiles a list of its capabilities and assets. Does the firm have a lot of cash available? That is a strength. Does the firm have highly skilled employees? Another strength. Knowing exactly what it is good at allows a firm to make plans that exploit those strengths. Nike can plan to expand its business by making products for a sport it doesn’t currently serve. Its sports marketing expertise will help it successfully launch that new product line.
Weaknesses
A firm’s weaknesses are what it is not good at—things that it does not have the capabilities to perform well. Weaknesses are not necessarily faults—remember that not all firms can be great at all things. When a firm understands its weaknesses, it will avoid trying to do things it does not have the skills or assets to succeed in, or it will find ways to improve its weaknesses before undertaking something new. A firm’s weaknesses are simply gaps in capabilities, and those gaps do not always have to be filled within the firm.
SWOT analysis alerts firms to the gaps in their capabilities so they can work around them, find help in those areas, or develop capabilities to fill the gaps. For example, Paychex is a firm that handles payroll for over 600,000 firms. 1 Paychex processes hours, pay rates, tax and benefits deductions, and direct deposit for firms that would rather not have to perform those tasks themselves. A large firm would need to have a team of employees dedicated to fulfilling that task and equip that team with software systems to do the job efficiently and accurately. For Paychex, these capabilities are a company strength—that’s what it does. Other companies that do not have the resources to develop this capability or may not be interested in doing so can hire Paychex to do the job for them.
Opportunities
While strengths and weaknesses are internal to an organization, opportunities and threats are always external. An opportunity is a potential situation that a firm is equipped to take advantage of. Think of opportunities in terms of things that happen in the market. Opportunities offer positive potential, however, sometimes a firm is not equipped to take advantage of an opportunity, which is why considering the entire SWOT is important before deciding what to do. For example, as cities are becoming more populated, parking is becoming scarcer. Younger consumers who live in cities are starting to question whether it makes sense to own a car at all, when public transportation is available and parking is not. Sometimes, however, a person might need a car to travel outside the city or transport a special purchase. Daimler, the manufacturer of Mercedes-Benz and Smart cars, started a car-sharing service in Europe, North America, and China called Car2Go to offer cars to this new market of part-time drivers. By establishing Car2Go, Daimler has found a way to sell the use of its products to people who would not buy them outright.
Threats
When a manager assesses the external competitive environment, she labels anything that would make it harder for her firm to be successful as a threat . A wide variety of situations and scenarios can threaten a firm’s chances of success, from a downturn in the economy to a competitor launching a better version of a product the firm also offers. A good threat assessment looks thoroughly at the external environment and identifies threats to the firm’s business so it can be prepared to meet them. Opportunities and threats can also be a matter of perspective or interpretation: the Car2Go service that Daimler developed to serve young urban customers who don’t own cars could also be cast as a defensive response to the trend away from car ownership in this customer group. Daimler could have identified decreasing sales among young urban professionals as a threat and developed Car2Go as an alternative way to gain revenue from these otherwise lost customers.
The Limitations of SWOT Analysis
Although a SWOT analysis can identify important factors and situations that affect a firm, it only works as well as the person doing the analysis. SWOT can generate a good evaluation of the firm’s internal and external environments, but it is more likely to overlook key issues because it is difficult to identify or imagine everything that could, for example, be a threat to the firm. That’s why the remainder of this chapter will present tools for developing a strategic analysis that is more thorough and systematic in examining both the internal and external environments that firms operate in.
- Explain the elements of a SWOT analysis.
- What information does a SWOT analysis provide managers? What information might it miss?
References
1. Paychex (2017). Company History. https://www.paychex.com/corporate/history.aspx Accessed July 28, 2017.