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Statistical Methods For The Social Sciences 5th Edition Alan Agresti - Solutions

4.5 One mole of superheated water at 110 °C and at 1 atm is evaporated to steam at the same temperature and pressure. Calculate the change in entropy given that:(i) In the pressure range 1 atm to 1.4 atm the entropy of liquid water at 110 °C can be considered constant.(ii) At 110 °C liquid water
4.6 Phosphine exists in three allotropic forms, known as the α, β andγ forms. The α and β forms are in equilibrium at 49.43 K, the α andγ forms at 30.29 K. Obtain the molar heat of transformation for the γform changing to the α form at 30.29 K from the following data:(i) The entropy of the
4.7 What is the maximum useful work which can be obtained by cooling 1 mole of a perfect gas at constant volume from a temperature T to the temperature T0 of the surrounding?
4.8 What is the maximum useful work which can be obtained by expanding 1 mole of a perfect gas from pressure P1 to pressure P2 at a constant temperature T0 equal to that of the surrounding? The surrounding is at pressure P0.
5.1 The efficiency of a heat engine operating between temperatures T1 and T2 (
5.2 A Carnot engine using a perfect gas as working substance operates between the temperatures and (< ), measured on the ideal gas temperature scale. Show that the efficiency of the engine is ( – )/ .
5.3 Derive the maximum efficiency of the standard Otto cycle, Eq.(5.9).
5.4 CP and CV are the atomic heat capacities of lead at constant pressure and constant volume. Find (CP–CV)/3R at 300K given the following data: atomic volume = 18.27 cmVmol, linear coefficient of thermal expansion = 2.89x 10–5K–1, isothermal compressibility= 2.237 × 10–11 m2/N.
5.5 Derive Eq. (5.43).
5.6 Consider the adiabatic Joule–Thomson expansion of one mole of a perfect gas from pressure P1 to pressure P2. Obtain the change of entropy of the gas for this expansion.
5.7 Near STP argon may be described by the virial expansion with B = – 22 cm3/mol and dB/dT=0.25 cm3 K–1 mol–1. Obtain an approximate value of the Joule–Thomson coefficient of argon at STP.
5.8 In the Joule–Thomson expansion of a gas from temperature T1 and pressure P1 to temperature T2 and pressure P2, T1 and P2 are in practice essentially fixed:T1 by the liquid used to precool the gas before expansion, while P2 is atmospheric pressure. What initial pressure P1 leads to the lowest
5.9 For a gas obeying van der Waals’ equation(i) obtain the isobaric coefficient of thermal expansion α;(ii) show that the energy E satisfies(iii) show that the molar heat capacity at constant volume is a function of the temperature only;(iv) obtain the difference of the molar heat capacities at
5.10 A paramagnetic salt whose magnetic ions have spin is at a temperature of 0.6 K in a magnetic field of 4 tesla. The field is slowly reduced to 6×10–3 tesla, the salt being thermally isolated during the process. What is the final temperature of the salt?
*5.11 The Gibbs free energy for a body in an applied uniform magnetic intensity is defined by , where is the component of the magnetic moment of the body in the direction of . Show that , and hence derive the Maxwell relation
6.1 Qualitatively the melting point of a crystalline solid can be defined as the temperature Tm at which the amplitude of simple harmonic vibrations of the individual atoms about their lattice sites is about ten per cent of the interatomic spacinga. Show that on this picture the frequency v of the
6.2 Calculate the entropy of the lattice vibrations of a monatomic crystal as described by the Einstein theory.Obtain limiting expressions for the entropy valid at low and at high temperatures.
6.3 The same as the last problem but for the Debye theory.
6.4 The relation between the frequency ν and the wavelength λ for surface tension waves on the surface of a liquid of density ρ and surface tension σ is Use a method analogous to the Debye theory of specific heats to find a formula, analogous to the Debye T3 law, for the temperature dependence
7.1 According to quantum mechanics, the molecules of a diatomic gas possess rotational energy levels(where I is a constant), the level εr being (2r+ l)-fold degenerate. Write down an expression for the partition function of the rotational motion. Hence find the molar rotational heat capacity of
7.2 A diatomic molecule may to a good approximation be considered as vibrating in one-dimensional simple harmonic motion with circular frequency w. According to quantum mechanics such a system possesses an infinite set of nondegenerate energy levels with energy ħω(r + ), r= 0,1,2, . . . ∞.
7.3 At its normal boiling point (N.B.P.) of 630 K, saturated mercury vapour is monatomic and satisfies the perfect gas laws. Calculate the entropy per mole of saturated mercury vapour at its N.B.P.The latent heat of vaporization of mercury at its N.B.P. is 5.93 x 104 J mol–1. Hence find the
7.4 Consider a perfect gas of particles in the extreme relativistic range (i.e. the kinetic energy E of a particle and its momentum p are related by E=c|p|). Find the Helmholtz free energy and hence the heat capacity at constant volume for this gas.
7.5 For a gas obeying the Maxwell velocity distribution obtain:(i) the most probable speed of the molecules,(ii) the most probable kinetic energy of the molecules,(iii) the mean speed of the molecules,(iv) the rms speed of the molecules,(v) the mean kinetic energy of the molecules.
7.6 Derive Eqs. (7.74) and (7.75). (Hint: Use Eqs. (A.33) and(A.36) from Appendix A.)
7.7 An imperfect gas has an intermolecular potential If ξ is small so that the partition function may be expanded in a series in ξ, calculate the change in the equation of state to orderξ.
7.8 For a real gas at low density obtain the internal energy of the gas up to terms proportional to its density, in terms of the intermolecular potential u(r).The radial distribution function g(r, p, T) is defined as follows.Consider a gas at temperature Tin a volume F and let p = N/V be its mean
7.9 A diatomic molecule consists of atoms of masses M1 and M2, whose position coordinates are rl and r2. The atoms interact through an attractive potential K(|r1–r2|). Show how the energy of the molecule decomposes into translational, vibrational and rotational parts.Apply the equipartition
7.10 A polyatomic molecule consists of n atoms, each possessing three degrees of freedom (i.e. we treat each atom as a point mass neglecting its internal degrees of freedom). Divide the degrees of freedom of the molecule into translational, rotational and vibrational degrees: (i) for a nonlinear
7.11 Carbon dioxide (CO2) is a linear molecule and it possesses 4 vibrational modes. The vibrational temperatures of these modes are 3360, 1890, 954 and 954 K (i.e. there are two distinct modes with the same frequency). Use Fig. 6.5 to estimate the molar heat capacity at constant volume of CO2 at
8.1 Derive Eqs. (8.9) and (8.10) from Eq. (8.8).
8.2 The transition temperature of grey and white tin at a pressure of one atmosphere is 291 K, grey tin being the stable modification below this temperature. The change in enthalpy for this transition is 2238 J/mol. The densities of grey and white tin are 5.75 and 7.30 g/cm3 respectively, and the
8.3 In the temperature range 700 to 730 K, the vapour pressure of magnesium can be approximately represented by(P in mmHg, T in K). What is the molar latent heat of sublimation of magnesium in this temperature range?
8.4 The latent heat of fusion and the latent heat of vaporization of neon at the triplepoint have the values 335 J/mol and 1803 J/mol respectively. The temperatureof the triple point is 24.57 K. What is the latent heat of sublimation of neon at the triple point? What is the change in entropy when
8.5 The vapour pressure of water at 298.15 K is 23.75 mmHg. What is the vapour pressure of water at 273.16 K, given that the latent heat of evaporation of water at 298.15 and 273.16 K is 43,991 and 45,049 J/mol respectively?
8.6 The liquid and solid phases of helium 4 may coexist in equilibrium at 0 K, the density of the solid being greater than that of the liquid. What is the slope dP/dT of the phase equilibrium curve at the absolute zero of temperature?
8.7 Derive an expression for the vapour pressure of a solid monatomic substance given that the vapour may be treated as a perfect monatomic gas and that the solid can be described by the Einstein model, the temperature of the system being large compared with the Einstein temperature of the
8.8 Obtain the critical constants for Dieterici’s equation of state
10.1 Let (λ,T )dλ be the energy density per unit volume in the wavelength range λ to λ+ dλ of black-body radiation at temperature T. At what wavelength does the wavelength spectrum (λ, T) have its maximum value?The radiation emitted by the sun has very nearly the spectral distribution of
11.1 Due to its spin, the electron possesses a magnetic moment μB.Treating the conduction electrons in a metal as a free electron gas, obtain an expression (involving integrals over Fermi–Dirac distribution functions) for the magnetization due to the magnetic moments of the conduction electrons,
11.2 Derive Eq. (11.88) for the chemical potential of a free electron gas at temperatures T very small compared with the Fermi temperature TF.(For T TF and any function ϕ(ε)(11.177)See Landau and Lifshitz,7 pp. 169–170 for the derivation of this result.)
11.3 Derive Eq. (11.95) for the heat capacity of a free electron gas at low temperatures, i.e. T TF. (Use Eq. (11.177).)
11.4 The molar heat capacity at constant volume of a perfect gas of bosons of mass m at a temperature T below the condensation temperature Tc is given by For this gas at a temperature T< Tc obtain: (i) the internal energy per mole, (ii) the entropy per mole, iii) the pressure.
11.5 Obtain the pressure of a free electron gas, of density N/V electrons per unit volume, at the absolute zero of temperature.
11.6 In order to escape from a metal the conduction electrons, whose motion within the metal is described by the free electron model, must overcome a potential barrier of height χ at the surface of the metal, i.e. an electron will escape only if its component of velocity normal to the surface
11.7 Consider the equilibrium of an isolated single-phase system in which the chemical reaction(11.138)occurs. Obtain the equilibrium condition for this system by maximizing its entropy.
11.8 Obtain the equilibrium constant Kc(T) for the ionization of atomic hydrogen which occurs at very high temperatures:Assume that the electrons, protons and hydrogen atoms can be treated as perfect classical gases.
11.9 At very high temperatures, molecular hydrogen dissociates into hydrogen atoms:Obtain the equilibrium constant Kc(T) for this reaction, expressing your answer in terms of the internal partition function of molecular hydrogen.
11.10 Consider the dissociative gaseous reaction Let f be the fraction of N2O4 molecules dissociated into NO2 molecules, when the system is in equilibrium. Derive the pressure dependence off, at a fixed temperature.
11.11 A system of s components, Al, A2, …As, consists of p phases in contact, so that components can pass between them. The Gibbs free energy of the system is given by(11.178)where the index α( = 1, 2, …p) labels the p phases of the system, Ni(α) is the number of molecules of component i in
11.12 A semiconductor possesses n donor levels, whose energy is–ε0 . A donor level can be occupied either by a ‘spin up’ electron or a ‘spin down’ electron but it cannot be simultaneously occupied by two electrons. Obtain the grand partition function for the electrons in the donor levels
11.13 For a system of s components, existing in a single phase, the Gibbs free energy G(T, P, N1, … Ns) has the property that(11.179)where μi is the chemical potential of the ith component of the system.Derive this relation, which is the generalization of Eq. (11.118) to a multi-component phase.
11.14 Consider the chemical reaction (11.138) between perfect gases, the temperature and the volume of the system being kept constant. Show that the molar heat of reaction at constant volume for this process is given by(11.181)where Kc(T) is the equilibrium constant (11.156).
11.15 Consider a given region of volume v inside a large volume V of a perfect classical gas at temperature T. Show that the probability of finding N particles in this region of the gas is given by the Poisson distribution
3.80.* The Russian mathematician Tchebysheff proved that for any k > 1, the proportion of observations that fall more than k standard deviations from the mean can be no greater than 1/k2. This holds for any distribution, not just bell-shaped ones.(a) Find the upper bound for the proportion of
3.79.* Show that (yi − ¯y) must equal 0 for any collection of observations y1, y2, . . . , yn.
For a sample with mean ¯y, adding a constant c to each observation changes the mean to ¯y+c, and the standard deviation s is unchanged. Multiplying each observation by c changes the mean to c¯y and the standard deviation to |c|s.(a) Scores on a difficult exam have a mean of 57 and a standard
For Table 3.2 (page 31), explain why the mean of these 50 observations is not necessarily the same as the violent crime rate for the entire U.S. population.
Ten people are randomly selected in Florida and another 10 people are randomly selected in Alabama. Table 3.20 provides summary information on mean income.The mean is higher in Alabama both in rural areas and in urban areas. Which state has the larger overall mean income?(The reason for this
A teacher summarizes grades on the midterm exam by Min = 26, Q1 = 67, Median = 80, Q3 = 87, Max = 100, Mean = 76, Mode = 100, Standard dev. = 76, IQR = 20.She incorrectly recorded one of these. Which one do you think it was? Why?
In a study of graduate students who took the Graduate Record Exam (GRE), the Educational Testing Service recently reported that for the quantitative exam, U.S.citizens had a mean of 529 and standard deviation of 127, whereas the non-U.S. citizens had a mean of 649 and standard deviation of 129.(a)
The 2014 GSS asked whether having sex before marriage is (always wrong, almost always wrong, wrong only sometimes, not wrong at all). The response counts in these four categories were (324, 111, 257, 956). This distribution is(a) Skewed to the right.(b) Approximately bell shaped.(c) Somewhat
Statistics Canada reported from a recent census that for the categories (Catholic, Protestant, Other Christian, Muslim, Jewish, None, Other) for religious affiliation, the relative frequencies were (42%, 28%, 4%, 2%, 1%, 16%, 7%).(a) The median religion is Protestant.(b) Only 2.7% of the subjects
Answer true or false to the following:(a) The mean, median, and mode can never all be the same.(b) The mean is always one of the data points.(c) The median is the same as the second quartile and the 50th percentile.(d) For 67 sentences for murder recently imposed using U.S. Sentencing Commission
To measure variability, why is(a) The standard deviation s usually preferred over the range?(b) The IQR sometimes preferred to s?
To measure center, why is the (a) median sometimes preferred over the mean? (b) mean sometimes preferred over the median? In each case, give an example to illustrate your answer.
Give an example of a variable having a distribution that you expect to be (a) approximately symmetric, (b)skewed to the right, (c) skewed to the left, (d) bimodal,(e) skewed to the right, with a mode and median of 0 but a positive mean.
Give an example of a variable for which the mode applies, but not the mean or median.
According to a report from theU.S. National Center for Health Statistics, for males with age 25–34 years,2%of their heights are 64 inches or less, 8% are 66 inches or less, 27% are 68 inches or less, 39% are 69 inches or less, 54%are 70 inches or less, 68% are 71 inches or less, 80% are 72 inches
The fertility rate (mean number of children per adult woman) varies in European countries between a low of 1.2 (Poland and Portugal) and a high of 1.9 (Ireland and France). For each woman, the number of children is a whole number, such as 0 or 1 or 2. Explain why it makes sense to measure a mean
According to the U.S. Federal Reserve, between 1989 and 2013 the median family net worth (in 2013 dollars)changed from $85,060 to $81,400 whereas the mean family net worth changed from $342,300 to $528,420.What do these statistics suggest about the change in the distribution of family net worth
A Credit Suisse Global Wealth report in 2014 summarized adult wealth in the United States by the numbers$44,900 and $301,500. One of these was the mean, and one was the median.Which do you think was the median?Why?
The Internet site www.artofstat.com/webapps.html has useful applets for illustrating data analyses and properties of statistical methods.(a) Using the Explore Quantitative Data applet, construct a sample of 20 observations on y = number of hours of physical exercise in the past week having ¯y < s.
For Table 3.19, construct a scatterplot to investigate the association between gun deaths and number of firearms. Identify the effect of any outlier.
Use software to load the UN data file shown in Table 3.9.(a) Conduct a descriptive statistical analysis of the prison rates. Summarize your conclusions, highlighting any unusual observations.(b) Use scatterplots and correlations to investigate the association between prison rate and the other
Zagat restaurant guides publish ratings of restaurants for many large cities around the world (see www.zagat.com). The review for each restaurant gives a verbal summary as well as a 0-to-30-point rating of the quality of food, decor, service, and the cost of a dinner with one drink and tip. Figure
For Table 3.9, pose a research question for two variables relating to the direction of their association, identifying the response variable and explanatory variable.Using software, construct a scatterplot and find the correlation. Interpret, and indicate what they suggest about the research
In 2014, UNICEF reported child poverty rates7 for many nations in 2012 and in 2008. In 2012, values in Scandinavia were Norway 5.3%, Finland 8.8%, Denmark 10.2%, and Sweden 12.1%, and values in North America were Canada 20.8%, United States 32.2%, and Mexico 34.3%.(a) Use descriptive statistical
Table 3.19, the Guns data file at the text website, shows annual homicide rates (including homicide, suicide, and accidental deaths) per million population in advanced industrialized nations. The values in parentheses are the nation’s number of firearms, per 100 people. Prepare a report in which
Refer to the data file your class created for Exercise 1.12 (page 10). For variables chosen by your instructor, conduct descriptive statistical analyses. In your report, give an example of a research question that could be addressed using your analyses, identifying response and explanatory
For the Students data file at the text website(see Exercise 1.11 on page 9), use software to conduct graphical and numerical summaries for (a) distance from home town, (b) weekly hours of TV watching. Describe the shapes of the distributions, and summarize your findings.
What is the difference between the descriptive measures symbolized by (a) ¯y and μ? (b) s and σ?Concepts and Applications
Using national surveys, the Pew Research Center has estimated the percent of people who say that religion plays a very important role in their lives.Values forOECD nations (with GDP values in parentheses from the OECD data file) include Australia 21% (43,550), Canada 25%(43,247), France 14%
For the data for OECD nations in Table 3.13 in Exercise 3.6 and in the UN data file, use software to construct a scatterplot relating x = carbon dioxide emissions (CO2)and y = fertility rate.(a) Based on this plot, would you expect the correlation between these variables to be positive, or
Refer to the previous exercise. Using regression, the prediction equation relatingGDP(in thousands of dollars per capita) to carbon dioxide emissions (in metric tons per capita) is Predicted CO2 = 1.93 + 0.178(GDP).(a) What type of pattern would you expect for the points in a scatterplot for these
For 2014WorldBank data (data.worldbank.org)for several nations, a prediction equation relating fertility(the mean number of children per adult woman) and percentage of people using the Internet is Predicted fertility = 3.20 − 0.02(Internet use).(a) Compare the predicted fertility of a nation with
Table 3.18 shows reported happiness for those subjects in the 2014 GSS who attend religious services rarely and for those who attend frequently.(a) Identify the response variable and the explanatory variable.(b) At each level of religious attendance, find the percentage who reported being very
A study compares Democrats and Republicans on their opinions about national health insurance (favor or oppose).(a) Identify the response variable and the explanatory variable.(b) Explain how the data could be summarized in a contingency table.
The United Nations publication Energy Statistics Yearbook lists consumption of energy (unstats.un.org/unsd/energy). For the 27 nations that made up the European Union (EU) in 2011, the energy values(in kilowatt-hours per capita) had a mean of 5963 and a standard deviation of 2292.(a) France had a
According to values from Table 3.9, for the nations in the European Union (EU) excluding Luxembourg, carbon dioxide emissions (metric tons per capita) had a mean of 7.4 and standard deviation of 1.7.(a) How many standard deviations above the mean was the value of 20.9 for Luxembourg?(b) Sweden’s
Areport by the OECD6 indicated that annual water consumption for nations in the OECD (see Exercise 3.6)was skewed to the right, with values (in cubic meters per capita) having a median of about 500 and ranging from about 200 in Denmark to 1700 in the United States Consider the possible values for
Using software, analyze the murder rates from the Crime data file at the text website.(a) Find the five-number summary.(b) Construct a box plot, and interpret.(c) Repeat the analyses, adding the D.C. murder rate of 15.9 to the data file, and compare results.
In 2014, the National Center for Educational Statistics reported that high school graduation rates in the United States had minimum = 59 (DC), lower quartile =75, median = 80, upper quartile = 83, and maximum = 88(Iowa).(a) Report and interpret the range and the interquartile range.(b) Are there
In 2013, the five-number summary for the U.S.statewide percentage of people without health insurance had minimum = 4% (Massachusetts), Q1 = 10%, Med =12%, Q3 = 15%, and maximum = 20% (Texas).(a) Sketch a box plot.(b) Do you think that the distribution is symmetric, skewed to the right, or skewed to
Infant mortality rates (number of infant deaths, per 1000 live births) are reported by theWorld Bank. In their report for 2010–2014, the five-number summaries (min, Q1, median, Q2, max) were (19, 55, 74, 86, 107) for Africa and (2, 3, 4, 4, 6) for Europe. Sketch side-by-side box plots, and use
For the number of times a week reading a newspaper, from the Students data file referred to in Exercise 1.11, Figure 3.18 shows software output (rather crude)of the stem-and-leaf plot and the box plot.(a) From the box plot, identify the minimum, lower quartile, median, upper quartile, and
The January 2014 unemployment rates of adults of age 24 or less in the 28 countries in the European Union ranged from 7.9 (Germany) to 57.3 (Greece), with lower quartile = 18.9, median = 23.7, upper quartile = 33.5, mean = 26.0, and standard deviation = 13.0.Sketch a box plot, labeling which of
For parts (a), (b), and (f) of the previous exercise, sketch box plots that would be plausible for the variable.
For each of the following variables, indicate whether you would expect its relative frequency histogram to be bell shaped, U-shaped, skewed to the right, or skewed to the left.(a) Exam score of easy exam (with ¯y = 88, s = 10, minimum= 65, Q1 = 77, median = 85, Q3 = 91, and maximum= 100).(b) IQ
For each of the following, sketch what you expect a histogram to look like, and explain whether the mean or the median would be greater.(a) The selling price of new homes in 2008.(b) The number of children ever born per woman age 40 or over.(c) The score on an easy exam (mean = 88, standard

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