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

(a) Show that the vibrational part of the internal partition function is e−vib/kT =∞n=0 e−(n+1/2)hν/kT =exp(−hν/2kT) exp(hν/kT)exp(hν/kT) − 1. (4.72)
Explain why the factor after the sum is 1.56 4. Classical Statistics (Maxwell–Boltzmann)
(b) Show that the rotational part of the internal partition function is e−rot/kT =∞J=0(2J + 1)e−J(J+1)h2/2IkT = 8π2IkT. (4.73)
Show that the same result is obtained from the classical expression, i.e. fromdθdφdpθdpφ exp−p2θ2IkT −p2φ2I sin2 θkT. (4.74)
Example 4.12: Mean length of an elastic string An elastic string with a mass M at one end is hanging in the gravitational field of the earth. The string is to be considered as consisting of N equal elements, each of fixed lengtha, such that these can move up or down with no other forces of
Explain the result in the limit T = 0.
Example 4.13: Why doesn’t the atmosphere fall down?
Calculate the average value of the potential energy of a molecule of mass m in an isothermal atmosphere. How do you explain the result in contrast with the law of equipartition of energy?
Example 5.6: Maximum of entropy The entropy S (here SMB) and the canonical partition function Z are related as e.g. in Eq. (5.19)which can be rewritten in the form (with gi = 1)S = k(lnZ + E/kT), Z=i e−i/kT, Pi =1 Ze−i/kT . (5.61)
(a) Show that the probability Pi and S are connected by the following relation also called the Gibbs entropy [12]:S = −k iPi ln Pi, N i=1 Pi = 1. (5.62)
This formula describes the thermal entropy S of the second law of thermodynamics. Pi is the Boltzmann factor which represents the probability to find the system excited to its state i in a heat bath with temperature T with which it is in a state of equilibrium. In quantum mechanical operator form
(b) A certain system can occupy one of N states. The expression Pi is the probability for the system to occupy state i (i = 1, . . . ,N). Show with the help of the result of (a) that the maximum entropy is given by S = k ln N. (5.65)Note the difference between cases (a) and (b). The result (5.65)
(c) The accessible states of a particular system can be subdivided intoW sets, the i-th set consisting of Gi states of equal probabilities. The probability for the system to be in any one of these(equally probable) states is Pi. What is in this case the expression for the entropy?
Example 5.7: Density matrix equation Show with the help of the Schr¨odinger equation i∂∂tψ(x, t) = Hψ(x, t), (5.66)that the density operator ρN, i.e.ρN(β) = e−βH (5.67)(without normalization), satisfies in the configuration space matrix representation the equation∂ρN(x, x;
5.4 Problems without Worked Solutions 67
Example 5.8: Mean square deviation of energy Show that the mean square deviation of the energy E, i.e.(E)2 = (E − E)2 = E2 − E 2, (5.69)is given by(E)2 =∂2 lnZ∂β2= −∂E∂β, β=1 kT
. (5.70)
Example 5.9: Relative square of fluctuation of pressure Using the result (5.60), i.e. PV = 2E/3, show that the relative square of the fluctuation of the pressure P of an ideal gas of N particles is given by(P)2 P2 =2 3N. (5.71)
What do you conclude from this result for large N?
Example 5.10: Entropy of a bit
What is the entropy of a bit?(Answer: k ln 2).
Example 5.11: Entropy of a phase space volume V Show that if the probability density in phase space ρ(p, q) is ρ = 1/V inside a region Γ and zero outside, then the entropy is S = lnV .
Example 5.12: Entropy of a system of spins
Show that the maximum entropy of a lattice of discrete spins (lattice spacinga, volume V ) is V ln 2/a3.
Example 5.13: Negative specific heat Ordinarily the specific heat is positive and raises the temperature of a system. What happens if the specific heat happened to be negative (as in the case of a black hole)?
Example 6.12: Absence of alternate levels Prove the result (6.103), i.e. show that∞J=0,2,4,...e−J(J+1)/aT =1 2aT. (6.110)
Example 6.13: Number of states in a spherical volume V Consider free nonrelativistic motion of particles in a spherical volume of radius R. What is the number of states with momentum p in the interval (p, p + dp)?
Example 6.14: Classical diatomic perfect gas Consider a perfect gas consisting of diatomic molecules. Write down the Lagrangian function L of a single molecule of moment of inertia I. Let the harmonic vibrations of the two atoms along the axis and about an equilibrium position η0 be described by
Example 6.15: Pressure of a particle in a cube Consider a particle of mass m in a cube of volume V = L3 and calculate its pressure P on a wall.[Hint: From the Schr¨odinger equation (6.10) the energy E of the particle is given by the eigenvalue relation (6.13) as E =2π2 2mL2(l2 +m2 + n2).
In a change dL of a length L the particle of energy E performs work dW = −dE. Use (F meaning force) dW = FdL and P = F/L2]. (Answer: P = 2E/3V ).
Example 6.16: Number of photons
Consider thermal radiation at a temperature of 1000◦ K. What is the wavelength λ of the radiation?
What is the number N of photons in a volume V ignoring polarization?
Consider thermal radiation at a temperature of 1000◦ K. What is the wavelength λ of the radiation?What is the number N of photons in a volume V ignoring polarization?
Refer to the previous exercise. At the Inference for a Proportion applet at www.artofstat.com/webapps.html, enter 0 successes for n = 20. Click on Confidence Interval for type of inference.Why are the reported standard error and confidence interval not sensible?Clicking on the Agresti–Coull
5.76.* To encourage subjects to make responses on sensitive questions, the method of randomized response is often used. The subject is asked to flip a coin, in secret.If it is a head, the subject tosses the coin once more and reports the outcome, head or tails. If, instead, the first flip is a
5.75.* Let π be the probability that a randomly selected voter prefers the Republican candidate. You sample two people, and neither prefers theRepublican. Find the point estimate of π. Does this estimate seem sensible? Why?(The Bayesian estimator is an alternative one that uses a subjective
5.74.* Find the true standard error of the sample proportion when π = 0 or π = 1. What does this reflect?
5.73.* You know the sample mean ¯y of n observations.Once you know (n−1) of the observations, show that you can find the remaining one. In other words, for a given value of ¯y, the values of (n − 1) observations determine the remaining one. In summarizing scores on a quantitative variable,
5.72.* For a random sample of n subjects, explain why it is about 95% likely that the sample proportion has error no more than 1/√n in estimating the population proportion.(Hint: To show this “1/√n rule,” find two standard errors when π = 0.50, and explain how this compares to two standard
Refer to the previous exercise. Provide the proper interpretation.
A random sample of 50 records yields a 95% confidence interval for the mean age at first marriage of women in a certain county of 21.5 to 23.0 years. Explain what is wrong with each of the following interpretations of this interval.(a) If random samples of 50 records were repeatedly selected, then
Based on responses of 1467 subjects in General Social Surveys, a 95% confidence interval for the mean number of close friends equals (6.8, 8.0). Which of the following interpretations is (are) correct?(a) We can be 95% confident that ¯y is between 6.8 and 8.0.(b) We can be95% confident thatμis
Other things being equal, quadrupling the sample size causes the width of a confidence interval to (a) double,(b) halve, (c) be one quarter as wide, (d) stay the same.
Increasing the confidence level causes the width of a confidence interval to (a) increase, (b) decrease, (c) stay the same.
The reason we use a z-score from a normal distribution in constructing a confidence interval for a proportion is that(a) For large random samples, the sampling distribution of the sample proportion is approximately normal.(b) The population distribution is normal.(c) For large random samples, the
To use the large-sample confidence interval for a proportion, you need at least 15 outcomes of each type.Show that the smallest value of n for which the method can be used is (a) 30 when πˆ = 0.50, (b) 50 when πˆ = 0.30,(c) 150 when πˆ = 0.10.That is, the overall nmust increase as πˆ moves
The publication Attitudes towards European Union Enlargement from Eurobarometer states, “The readers are reminded that survey results are estimations, the accuracy of which rests upon the sample size and upon the observed percentage. With samples of about 1000 interviews, the real percentages
You would like to find the proportion of bills passed by Congress that were vetoed by the President in the last congressional session. After checking congressional records, you see that for the population of all 40 bills passed, 2 were vetoed. Does it make sense to construct a confidence interval
Explain the reasoning behind the following statement:Studies about more diverse populations require larger sample sizes. Illustrate for the problem of estimating mean income for all medical doctors in the United States compared to estimating mean income for all entrylevel employees at McDonald’s
How does population heterogeneity affect the sample size required to estimate a population mean? Illustrate with an example.
Give an example of a study in which it would be important to have(a) A high degree of confidence.(b) A high degree of precision.
Why would it be unusual to see a (a) 99.9999%,(b) 25% confidence interval?
Explain why confidence intervals are wider with(a) larger confidence levels, (b) smaller sample sizes.
An interval estimate for a mean is more informative than a point estimate, because with an interval estimate you can figure out the point estimate, but with the point estimate alone you have no idea about the width of the interval estimate.(a) Explain why this statement is correct, illustrating
What is the purpose of forming a confidence interval for a parameter? What can you learn from it that you could not learn from a point estimate?
(a) Explain what it means for an estimator to be unbiased.(b) Explain why the sample range is a biased estimator of the population range.
The observations on daily TV watching for the 17 subjects in the 2014 GSS who were raised Jewish were 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 5. A 95% confidence interval for the mean of the corresponding population is(1.5, 2.9). Suppose the observation of 5 was incorrectly recorded as 50.
When subjects in a recent GSS were asked whether they agreed with the following statements, the (yes, no)counts under various conditions were as follows:• Women should take care of running their homes and leave running the country up to men: (275, 1556).• It is better for everyone involved if
A recent General Social Survey asked married respondents,“Did you live with your husband/wife before you got married?” The responses were 57 yes and 115 no for those who called themselves politically liberal, and 45 yes and 238 no for those who called themselves politically conservative.
In 2014, the GSS asked about the number of hours a week spent on the World Wide Web, excluding e-mail (variable denoted WWWHR). State a research question you could address about this response variable and a relevant explanatory variable. Go to sda.berkeley.edu/GSS and analyze the data. Prepare a
Refer to the data file created in Exercise 1.12(page 10). For variables chosen by your instructor, pose a research question, and conduct inferential statistical analyses using basic estimation methods. Summarize and interpret your findings, and explain how you could use them to answer the research
Refer to the Students data file (Exercise 1.11 on page 9). Using software, construct and interpret a 95%confidence interval for (a) the mean weekly number of hours spent watching TV, (b) the proportion believing in life after death. Interpret.
Refer to the previous exercise. Using this applet, let’s check that the confidence interval for a proportion may work poorly with small samples. Set the population proportion π = 0.10, with n = 10. Draw 100 random samples, each of size 10, forming 95% confidence intervals forπ for each one.(a)
Use the Explore Coverage applet at www.artofstat.com/webapps.html to repeatedly generate random samples and construct confidence intervals for a proportion, to illustrate their behavior when used for many samples. Set the population proportion value (labeled as p) to 0.50, the sample size to 200,
Refer to Example 5.9 (page 129). Construct a 95%confidence interval for the median time since a book was last checked out. Interpret.Concepts and Applications
Refer to Exercise 5.33.Use the bootstrap to construct a 95% confidence interval for the median annual income of the public housing residents. Interpret.
You randomly sample five students at your school to estimate the proportion of students who like tofu. None of the five students say they like it.(a) Find the sample proportion who like it and its standard error. Does the usual interpretation of se make sense?(b) Why is it not appropriate to use
An anthropologist wants to estimate the proportion of children in a tribe in the Philippines who die before reaching adulthood. For families she knew who had children born between 1990 and 1995, 3 of 30 children died before reaching adulthood. Can you use the ordinary largesample formula to
How large a sample size is needed to estimate the mean annual income of Native Americans correct to within $1000 with probability 0.99? There is no prior information about the standard deviation of annual income of Native Americans, but we guess that about 95% of their incomes are between $5000 and
A social scientist plans a study of adult South Africans living in townships on the outskirts of Cape Town, to investigate educational attainment in the black community. Many of the study’s potential subjects were forced to leave Cape Town in 1966 when the government passed a law forbidding
An estimate is needed of the mean acreage of farms in Manitoba, Canada. The estimate should be correct to within 100 acres with probability 0.95.A preliminary study suggests that 500 acres is a reasonable guess for the standard deviation of farm size.(a) How large a sample of farms is required?(b)
In 2006, the Pew Global Attitudes Project(www.people-press.org) reported that the percentage of people in Europe who reported a lot of, or some confidence (instead of not too much, or no confidence)in President George W. Bush was 14% in France, 30% in Britain, 24% in Germany, and 7% in Spain. In
A 2011 poll in Canada indicated that 41% of Canadians favored bringing back the death penalty for convicted murderers. (The United States is the only Western democracy that has it.) A report by Amnesty International on this and related polls (www.amnesty.ca)did not report the sample size but
A public health unit wants to sample death records for the past year in Toronto to estimate the proportion of the deaths that were due to accidents. They want the estimate to be accurate to within 0.02 with probability 0.95.(a) Find the necessary sample size if, based on data published by
A television network plans to predict the outcome of an election between Jacalyn Levin and Roberto Sanchez. They will do this with an exit poll on election day. They decide to use a random sample size for which the margin of error is 0.04 for 95% confidence intervals for population proportions.(a)
To estimate the proportion of traffic deaths in California last year that were alcohol related, determine the necessary sample size for the estimate to be accurate to within 0.06 with probability 0.90.Based on results of studies reported by the National Highway Traffic Safety Administration
In repeated random samples of this size, the sample mean could be expected to fall within 1.0 of the true mean about 95% of the time.”(a) Construct and interpret a 95% confidence interval for the mean.(b) The administrator decides that this interval is too wide, and she prefers one of only half
Ahospital administrator wants to estimate the mean length of stay for all inpatients in that hospital. Based on a systematic random sample of 100 records of patients for the previous year, she reports that “The sample mean was
A study estimates the mean annual family income for families living in public housing in Chicago. For a random sample of 30 families, the annual incomes (in hundreds of dollars) are in the Chicago data file at the text website and here:133 140 127 150 133 114 128 142 123 172 146 110 135 136 158 120
At sda.berkeley.edu/GSS, consider responses to the question “On how many days in the past 7 days have you felt lonely” (coded LONELY) for the most recent survey in which this was asked.(a) Find a point estimate of the population mean.(b) Construct the 95% confidence interval, and interpret.
The General Social Survey asks respondents to rate their political views on a seven-point scale, where 1 = extremely liberal, 4 = moderate, and 7 = extremely conservative.A researcher analyzing data from the 2014 GSS gets software output:----------------------------------------------------Mean
For the Students data file mentioned in Exercise 1.11, software reports the results for responses on the number of times a week the subject reads a newspaper:-------------------------------------------------------Variable N Mean Std Dev SE Mean 95.0% CI News 60 4.1 3.0 0.387 (3.32,
The 2014 General Social Survey asked respondents how many sex partners they had in the previous 12 months. Software reports-------------------------------------------------------Variable N Mean StDev SE Mean 95.0% CI partners 2296 0.996 0.918 0.0192 (0.96,
A recent GSS asked, “How many days in the past 7 days have you felt sad?” The 816 women who responded had a median of 1, mean of 1.81, and standard deviation of 1.98.The 633 men who responded had a median of 1, mean of 1.42, and standard deviation of 1.83.(a) Find a 95% confidence interval for
A General Social Survey asked subjects, “How long have you lived in the city, town, or community where you live now?” The responses of the 1415 subjects had a median of 16 years, a mean of 20.3, and a standard deviation of 18.2.(a) Do you think that the population distribution is normal?Why or
In response to the GSS question in 2014 about the number of hours daily spent watching TV, the responses by the seven subjects who identified themselves as having been raised Islamic were 1, 1, 1, 2, 2, 3, 6.(a) Estimate the mean, standard deviation, and standard error.(b) Construct a 95%
The 2014 GSS asked, “On the average day about how many hours do you personally watch television?”Stata software reports-----------------------------------------------------Mean estimation Number of obs = 1669 Mean Std. Err. [95.0% Conf. Interval]TVHOURS 2.98 0.063 2.86
Example 5.5 (page 117) analyzed data from a study that compared therapies for anorexia. For the 17 girls who received the family therapy, the changes in weight during the study (which are in the data file Anorexia at the text website) were 11, 11, 6, 9, 14,−3, 0, 7, 22,−5,−4, 13, 13, 9, 4, 6,
Refer to the previous exercise. For the 397 males in the sample, the mean was 2.89 and the standard deviation was 1.77.(a) Show that the standard error of the sample mean is 0.089.(b) Show that the 95% confidence interval for the population mean is (2.7, 3.1), and explain what “95%
AGeneral Social Survey asked, “What do you think is the ideal number of children for a family to have?” The 497 females who responded had a median of 2, mean of 3.02, and standard deviation of 1.81.(a) Find and interpret the standard error of the sample mean.(b) The 95% confidence interval is
The 2014 General Social Survey asked male respondents how many female partners they have had sex with since their 18th birthday. For the 131 males in the sample between the ages of 23 and 29, the median = 6 and mode = 1 (16.8% of the sample). Software summarizes other
Find and interpret the 95% confidence interval forμ, if ¯y = 70 and s = 10, based on a sample size of (a) 5,(b) 20.
Report the t-score that multiplies by the standard error to form a(a) 95% confidence interval for μ with 15 observations.(b) 95% confidence interval for μ with 25 observations.(c) 95% confidence interval for μ with df = 25.(d) 99% confidence interval for μ with df = 25.
In 2013 the Harris Poll reported results of a survey about religious beliefs. Of 2252 American adults surveyed, 24% believed in reincarnation. Treating this as a random sample, a 95% confidence interval for the population proportion believing in reincarnation is (0.22, 0.26).Without doing any new
For an exit poll of people who voted in a gubernatorial election, 40% voted for Jones and 60% for Smith.Assuming this is a random sample of all voters, construct a 99% confidence interval for the proportion of votes that Jones received, if the sample size was (a) 400, (b) 40. In each case, indicate
Of the 1824 voters sampled in the exit poll discussed in the previous chapter (page 80), 60.5% said they voted for Jerry Brown. Is there enough evidence to predict the winner of the election? Base your decision on a 95%confidence interval, stating needed assumptions for that decision.
A 2012 report by the Centers for Disease Control provided a point estimate of 18.1% for the percentage of adult Americans who currently smoke cigarettes. The sample size was 34,525. Assuming that this sample has the characteristics of a random sample, construct and interpret a 99% confidence
When the GSS most recently asked whether human beings developed from earlier species of animals, 53.8%of 1095 respondents answered that this was probably or definitely not true. Find a 99% confidence interval for the corresponding population proportion, and indicate whether you can conclude that a
The General Social Survey has asked respondents,“Do you think the use of marijuana should be made legal or not?” View results for all years at sda.berkeley.edu/GSS by entering the variables GRASS and YEAR.(a) Of the respondents in 2014, what proportion said legal and what proportion said not

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