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

16.3. Lax pair Consider the pair of first-order differential operators (called Lax pair):L(x, t | θ) = d dx+iβ4∂tφσ3 +msinhθ,cosβφ2σ1 +mcoshθ sinβφ2σ2 M(x, t | θ) = d dt+iβ4∂xφσ3 +mcoshθ cosβφ2σ1 +msinhθ sinβφ2σ2 where σi are the Pauli matrices and θ the rapidity
16.4. Derrick theorem The aim of this exercise is to show that the static solitonic solution of finite energy can only exist for 1+1 dimensional theories. Consider, in the (d +1) dimensional Minkowski space, the Lagrangian L = 1 2∂μφ∂μφ −U(φ), where U(φ) is a non-negative function, that
16.5. Liouville theory and minimal modelsa. In the quantization scheme of the Sine–Gordon model in terms of the Liouville theory, determine the quantized values of the coupling constant g that reproduce the central charges of the minimal models. Prove that the conformal dimension of the vertex
16.6. Conserved currents Using the algebra of the operators Dn and the null vector equation at the level 2 satisfied by 1,2 and 2,1, find the linear combination T6 of the basis vectors T(1)6 and T(2)6 that satisfies∂¯ zT6 = ∂z4.Determine the density 4.
17.1. Causality and analiticity Consider a linear system in which the output b(t) depends on the input a(t) as b(t) = t−∞G(t −t) a(t) dt.If the system is causal, the Green function G(t −t) vanishes when t < t. LetˆG(ω) = ∞−∞eiωτ G(τ )dτ = ∞0 eiωτ G(τ )dτits
17.2. Decay process A particle of mass M and three-dimensional momentum P decays in two particles of masses m1 and m2.a. Use the conservation of the energy and the momentum to prove that the total energy of the first particle in the reference frame of the centre of mass is E1 = M2 +m21−m22 2M and
17.3. Physical region of the amplitudes Determine the physical region of the s-channel process when the mass of the particles are different.
17.4. Yang–Baxter equations Prove that the Yang–Baxter equations given in eqn. (17.3.52) of the text can be obtained as a consequence of the associativity condition of the Faddev–Zamolodchikov algebra.
17.5. Reflection amplitude Consider the following scattering amplitudes of a particle A and its anti-particle A| A(θ1)A(θ2) = S(θ) | A(θ2)A(θ1),| A(θ1)A(θ2) = t(θ) | A(θ2)A(θ1)+r(θ) | A(θ2)A(θ1).a. Prove that it holds S(θ)S(−θ) = t(θ) t(−θ)+r(θ) r(−θ) = 1 t(θ)
17.6. Bootstrap equations Derive the bootstrap equations (17.4.68) imposing the commutativity of the processes shown in Figure (17.13).Hint. Note that the line of the particle Ai in the second graph is parallel to the same line of the first graph. Identify the angles in the two figures and use the
17.7. Scattering in a potential with two delta functions Consider a one-dimensional system of quantum mechanics with Hamiltonian given by H = p2 2m+V(x)with V(x) = −g1 δ(x+a)−g2 δ(x+a)(g1 and g2 positive).a. Compute the phase shifts δ0 and δ1 and the corresponding S-matrix elements.b.
17.8. Interpretation of the two-dimensional S-matrix.The non-relativistic S-matrix of a particle of mass m = 1 relative to the potential V(x) =−2aπδ(x) is given by˜S(k) = k+iπa k−iπa.If we would like to generalize this result to the relativistic case, we must use the rapidity variable θ.
17.9. S-matrix with resonances.Consider an S-matrix for a neutral scalar particle.a. Show that the unitarity and crossing invariance equations S(θ)S(−θ) = 1, S(θ) = S(iπ −θ), imply that that S(θ) is a periodic function along the imaginary axis of the rapidity variable, i.e.S(θ) = S(θ
18.1. Bootstrap equations Prove that the most general solution of the bootstrap equation relative to a particle bound state of itself SAA(θ) = SAAθ − iπ3 SAAθ + iπ3is given by SAA(θ) = f23(θ)i fxi (θ) f 2 3−xi(θ).Study the motion of the poles of the function SAA(θ) under the
18.2. Analytic structure of the S-matrix of the Bullogh–Dodd modela. Study the structure of the poles and zeros of the S-matrix of the Bullogh–Dodd model S(θ) = f23(θ) f−B3(θ) f B−2 3(θ)with B(λ) = λ2 2π1 1+ λ2 4πby varying the coupling constant λ.b. Make the analytic continuation
18.3. Multiple poles Prove that the amplitude S11 of the fundamental particle cannot have higher-order poles by showing that the resonance angle of two heavier masses is larger than 2π/3. This makes it impossible to draw a diagram the one in Figure 18.2.
18.4. Double poles Use the values of the resonance angles of the S-matrix of the thermal TIM to explain the double poles that enter the amplitude S1,6 in terms of multi-scattering processes.
18.5. Non-relativistic scattering of distinguishable particles Consider the non-relativistic Hamiltonian H =− ∂2 x2ma−∂2 y2mb+2λδ(x−y) (18.12.33)and let χ(x,y) be its scattering wavefunction, where the coordinate x is referred to the first particle of mass ma and y to the second
18.6. S-matrix of the Gross–Neveu model The Gross–Neveu model is a model of n-component neutral Fermi–field ψk(x); k =1, 2, . . . ,n (n ≥ 3) with four-fermion interaction L = i 2nk=1¯ψkγ μ∂μψk + g 8 nk=1¯ψkψk2 where ¯ψk = ψkγ 0 and the 2×2 γ μ matrices satisfy the
18.7. Integral representation Use the expansions 1coshx= 2∞k=0(−1)k e−(2k+1)x, 1 sinhx= 2∞k=0 e−(2k+1)x the infinite-product(α) (β)(α +γ ) (β −γ )=∞k=01+ γα +k1− γβ +kand the integral∞0 dt te−βt sin(αt) = 1 2i log1+iα/β1−iα/βto prove the integral
18.8. Sine–Gordona. Study the analytic structure of the S-matrix of the solitons of the Sine–Gordon model, identifying all the sequences of the poles in the amplitudes.b. Using the following definition of the breathers Bn Bnθ1 +θ2 2 = limθ1−θ2→inξ
18.9. Reflectionless points At ξ = π/n (n = 1, 2, . . .), the amplitude SR of the Sine–Gordon vanishes and the scattering of the soliton–anti-soliton reduces to a pure transmission. Use the properties of the function to prove that at these values of the coupling constant the transmission
18.10. Bound states and semi-classical limit It can be proved that the renormalized coupling constantξ = β2 81 1− β2 8π.of the Sine–Gordon model comes from the quantum correction of the classical action.By the same token, it is possible to prove that the exact mass of the
18.11. Sine–Gordon and non-unitary modelsa. Find the value of ξ for which the S-matrix element S(1,1)(θ) of the Sine–Gordon model coincides with the S-matrix of the Yang–Lee model. Explain why the restriction of the Sine–Gordon model produces a negative residue at the poleθ = 2πi/3.b.
19.1. Form factors of a free theory Consider the theory of a free bosonic field φ(x) associated to a particle A of mass m.a. Compute the form factors of φ(x) and prove that 0|φ(0)|A = 1/sqrt2. Show that the Euclidean correlation function is given byφ(x)φ(0) = 1πK0(mr).b. Show that the
19.2. Feynman gasa. Derive the equation of state of the Feynman gas associated to the form factors of the magnetization operators in the nearest-neighbour approximation. Prove that the pressure p(z) satisfies the integral equation (19.9.19).b. Justify the accuracy of the approximation of the
19.3. Infinite products Using the integral dt te−βt sin2 αt 2= 1 4logα2 +β2β2 , and the identity satisfied by the functions(α)(β)(α +γ )(β −γ )=∞k=01+ γα +k1− γβ +k, to derive the expression of Fmin(θ) of the Sinh–Gordon model.
19.4. Cluster properties Consider the form factors of a scattering theory based on the functions fx(θ) =tanh 12(θ +iπx)tanh 12(θ −iπx)that have the property limθ→∞fx(θ) = 1.a. Using the Watson equation satisfied by the form factors FOa n (θ1, . . . ,θn) of an operator Oa, prove that
19.5. Correlation functions of the Ising model Use the fermionic representation of the energy operator of the Ising model, = i ¯ ψψ, and the mode expansion of the fermionic field in terms of the creation and annihilation operators, to compute the matrix elements of (x) and its two-point
19.6. Form factors of the Yang–Lee model Using the form factors of the Sinh–Gordon model, obtain the form factors of the Yang–Lee by using the analytic continuation B→−23
20.1. Non-relativistic gas Consider a one-dimensional gas of N non-relativistic bosons on an interval of length L, with two-body repulsive interaction given by a delta-function. The Hamiltonian of such a system is H = −Ni=1∂2∂x2i+2ci>jδ(xi −xj) c > 0.a. Find the phase shift of the
20.2. Simple TBA system Consider the TBA equations for a relativistic system made of one massive particle and with kernelϕ(θ) = 1 2πδ(θ).a. Solve explicitly the equation for the pseudo-energy (θ) and show that it is given by (θ) = logemRcoshθ −1#.b. Plot the scaling function c(R) = 6
20.3. L-channel for Majorana fermions Consider the Dirac action of a Majorana massive fermion on a finite volume S = dt R2−R2 dx ¯ψ i γ μ ∂μ −m ψ.Quantize this system in the canonical way and show that the finite volume ground state energy E0(R) can be written as E0(R) = −π
21.1. Boundary States for a bosonic field Consider the two analytic and anti-analytic U(1) currents j(z) =n jnz−n−1 and¯j( ¯ z) =n¯jn ¯ z−n−1 related to a massless bosonic field ϕ(z, ¯ z) by the relations j(z) = i∂zϕand ¯j( ¯ z) = i∂¯ zϕ. Let us use the conformal map z =
21.2. Bogoliubov transformation and boundary state Consider a free massive scalar field ϕ(x, t) in (1+1) dimension with mass m = m− t < 0 m+ t ≥ 0.Therefore it admits two different mode expansions, with two sets of annihilation and creation operators (A−,A†−) and (A+,A†+), which refer
22.1. Tricritical Ising model with even perturbations Consider the TIM deformed by its even fields (x) (energy density field) and t(x)(vacancy density field) with conformal weights given by 2 = 1/10 and 4 = 3/5 respectively. The action of the perturbed model can be formally written as A = A0 +g2
23.1. Decay processes Consider the decay processes of the higher particles in the spectrum of 4 theory, Bk→rB1 +sB2 where r and s are all those integers which satisfy mk ≥ rm1 +sm2, r +s = n.Take as an explicit example ξ < 15. The decay width of these processes is given by the Fermi golden
23.2. Short and long kinks Consider a simplified version of a potential with three vacua configurations, realized by the potential shown in the Figure below V(ϕ) = m2 2⎧⎨⎩(ϕ +2b)2, ϕ ≤−b;ϕ2, −b ≤ ϕ ≤ a;(ϕ −2a)2, ϕ >a.Shape of V (ϕ) with a = b (left hand side) and with a >
23.3. Instantons Instantons are finite action classical solutions of Euclidean equations of motion and are closely related to tunnelling phenomena among degenerate vacua. Consider a unit-mass particle in one dimension, under the potential V(q) = 1−cosq.a. Write down the Euclidean action of this
24.1. Supersymmetric quantum mechanics Consider two quantum mechanics Hamiltonians H+ = −¯h2 2m d2ψdx2+V+(x), H− = −¯h2 2m d2ψdx2+V−(x).where the potential terms are obtained in terms of a superpotential W(x) as V±(x) = W2(x)∓¯h √2m W(x).a. Show that H± can be written as H+ =
24.2. Exponential behaviour of the fermionic zero mode The poles present in the Fourier transform of ψ(0)ab (x) are determined by the exponential behaviour at x→−∞of this function. This behaviour depends on the interaction term V(ϕ(x)), which we assume can be expanded nearby the vacuum
24.3. Asymmetric well potential It is interesting to see what happens when the bosonic potential has two asymmetric wells| a and | b: in this case the asymptotic behaviours of the kink at x±∞are different and therefore we should expect to find two different spectra piling up on the two vacua.
In a General Social Survey, in response to the question“Do you believe in heaven?” 1127 people answered“yes” and 199 answered “no.”(a) Estimate the probability that a randomly selected adult in the United States believes in heaven.(b) Estimate the probability that an American adult does
Software for statistical inference methods often sets the default probability of a correct inference to be 0.95.Suppose we make an inference about the population proportion of people who support legalization of marijuana, and we consider this separately for men and for women.Let A denote the
A recent GSS asked subjects whether they are a member of an environmental group and whether they would be very willing to pay much higher prices to protect the environment. Table 4.4 shows results.(a) Estimate the probability that a randomly selected American adult is a member of an environmental
Let y = number of languages in which a person is fluent. According to Statistics Canada, for residents of Canada y has probability distribution P(0) = 0.02, P(1) =0.81, and P(2) = 0.17, with negligible probability for higher values of y.(a) Is y a discrete, or a continuous, variable?Why?(b)
Let y denote the number of people known personally who were victims of homicide within the past 12 months. According to results from recent General Social Surveys, for a randomly chosen person in the United States the probability distribution of y is approximately P(0) = 0.91,P(1) = 0.06,P(2) =
A ticket for a statewide lottery costs $1. With probability 0.0000001, you win a million dollars ($1,000,000), and with probability 0.9999999 you win nothing. Let y denote the winnings from buying one ticket. Construct the probability distribution for y. Show that the mean of the distribution
Let y be the outcome of selecting a single digit using a random number generator.(a) Construct the probability distribution for y. (This type of distribution is called a uniform distribution, because of the uniform spread of probabilities across the possible outcomes.)(b) Find the mean of this
For a normal distribution, find the probability that an observation falls (a) at least one standard deviation above the mean; (b) at least one standard deviation below the mean.
For a normal distribution, verify that the probability between(a) μ − σ and μ + σ equals 0.68.(b) μ − 1.96σ and μ + 1.96σ equals 0.95.(c) μ − 3σ and μ + 3σ equals 0.997.(d) μ − 0.67σ and μ + 0.67σ equals 0.50.
Find the z-value for which the probability that a normal variable exceeds μ + zσ equals (a) 0.01, (b) 0.025,(c) 0.05, (d) 0.10, (e) 0.25, (f) 0.50.
Find the z-value such that for a normal distribution the interval from μ − zσ to μ + zσ contains (a) 50%,(b) 90%, (c) 95%, (d) 99% of the probability.
Find the z-values corresponding to the (a) 90th, (b)95th, (c) 99th percentiles of a normal distribution.
If z is the number such that the interval from μ−zσto μ+zσ contains 90% of a normal distribution, then explain why μ + zσ is the 95th percentile.
If z is the positive number such that the interval fromμ − zσ to μ + zσ contains 50% of a normal distribution, then(a) Which percentile is (i) μ + zσ? (ii) μ − zσ?(b) Find this value of z. Using this result, explain why the upper quartile and lower quartile of a normal distribution are
What proportion of a normal distribution falls(a) above a z-score of 2.10?(b) below a z-score of −2.10?(c) between z-scores of −2.10 and 2.10?
Find the z-score for the number that is less than only 1% of the values of a normal distribution.
Mensa is a society of high-IQ people whose members have a score on an IQ test at the 98th percentile or higher.(a) How many standard deviations above the mean is the 98th percentile?(b) For the normal IQ distribution with mean 100 and standard deviation 16, find the IQ score for the 98th percentile.
According to a recent Current Population Reports, self-employed individuals in the United States work an average of 45 hours per week, with a standard deviation of 15. If this variable is approximately normally distributed, what proportion averaged more than 40 hours per week?
The Mental Development Index (MDI) of the Bayley Scales of Infant Development is a standardized measure used in studies with high-risk infants. It has approximately a normal distribution with a mean of 100 and a standard deviation of 16.(a) What proportion of children have an MDI of at least
For a study in Aarhus University Hospital(Denmark), 5459 pregnant women who reported information on length of gestation until birth had mean=281.9 days and standard deviation = 11.4 days. A baby is classified as premature if the gestation time is 258 days or less.(a) If gestation times are normally
Suppose that the weekly use of gasoline for motor travel by adults in Canada is approximately normally distributed, with a mean of 16 gallons and a standard deviation of 5 gallons.(a) What proportion of adults use more than 20 gallons per week?(b) Assuming that the standard deviation and the normal
On the midterm exam in introductory statistics, an instructor always gives a grade of B to students who score between 80 and 90. One year, the scores have approximately a normal distribution with mean 83 and standard deviation 5. About what proportion of the students get a B?
For a SAT distribution (μ = 500, σ = 100) and an ACT distribution (μ = 21, σ = 4.7), which score is relatively higher, SAT = 600 or ACT = 29? Explain.
Suppose that property taxes on homes in Iowa City, Iowa, have an approximately normal distribution with a mean of $4500 and a standard deviation of $1500. The property tax for one particular home is $7000.(a) Find the z-score corresponding to that value.(b) What proportion of the property taxes
An energy study in Gainesville, Florida, found that in March 2015, household use of electricity had a mean of 673 and a standard deviation of 556kWh (kilowatt-hours).(a) If the distribution were normal, what percentage of the households had use above 1000 kWh?(b) Do you think the distribution is
Five students—the females Ann and Betty and the males Clint, Douglas, and Edward—are rated equally qualified for admission to law school, ahead of other applicants.However, all but two positions have been filled for the entering class. The admissions committee can admit only two more students,
Construct the sampling distribution of the sample proportion of heads, for flipping a balanced coin(a) Once.(b) Twice. (Hint: The possible samples are (H, H), (H, T),(T, H), (T, T).)(c) Three times. (Hint: There are 8 possible samples.)(d) Four times. (Hint: There are 16 possible samples.)(e)
The probability distribution associated with the outcome of rolling a balanced die has probability 1/6 attached to each integer, {1, 2, 3, 4, 5, 6}. Let (y1, y2) denote the outcomes for rolling the die twice.(a) Enumerate the 36 possible (y1, y2) pairs (e.g., (2, 1)represents a 2 followed by a
An exit poll of 1126 voters in the 2014 New York gubernatorial election indicated that 55% voted for the Democratic candidate, Andrew Cuomo, with most of the rest voting for the Republican candidate, Rob Astorino.(a) If actually 50% of the population voted for Cuomo, find the standard error of the
According to Current Population Reports, the population distribution of number of years of education for self-employed individuals in the United States has a mean of 13.6 and a standard deviation of 3.0.Find the mean and standard error of the sampling distribution of ¯y for a random sample of (a)
Refer to Exercise 4.6.The mean and standard deviation of the probability distribution for the lottery winnings y are μ = 0.10 and σ = 316.23.Suppose you play the lottery 1 million times. Let ¯y denote your average winnings.(a) Find the mean and standard error of the sampling distribution of
According to a General Social Survey, in the United States the distribution of y = number of good friends (not including family members) has a mean of 5.5 and a standard deviation of 3.9.Suppose these are the population mean and standard deviation.(a) Does y have a normal distribution? Explain.(b)
The scores on the Psychomotor Development Index(PDI), a scale of infant development, are approximately normal with mean 100 and standard deviation 15.(a) An infant is selected at random. Find the probability that PDI is below 90.(b) A study uses a random sample of 25 infants. Specify the sampling
A study plans to sample randomly 100 government records of farms in Ontario to estimate the mean acreage of farms in that province. Results from an earlier study suggest that 200 acres is a reasonable guess for the population standard deviation of farm size.(a) Approximate the probability that the
According to the U.S. Census Bureau, the number of people in a household has a mean of 2.6 and a standard deviation of 1.5.Suppose the Census Bureau instead had estimated this mean using a random sample of 225 homes, and that sample had a mean of 2.4 and standard deviation of 1.4.(a) Identify the
At a university, 60% of the 7400 students are female.The student newspaper reports results of a survey of a random sample of 50 students about various topics involving alcohol abuse, such as participation in binge drinking.They report that their sample contained 26 females.(a) Explain how you can
The distribution of family size in a particular tribal society is skewed to the right, with μ = 5.2 and σ = 3.0.These values are unknown to an anthropologist, who samples families to estimate mean family size. For a random sample of 36 families, she gets a mean of 4.6 and a standard deviation of
Use the applet for the Sampling Distribution for the Sample Proportion at www.artofstat.com/webapps.html to illustrate this concept. Set the population proportion as 0.50 and sample size n = 100.(a) Simulate once (setting the number of samples to 1 and clicking on Draw Sample) and report the counts
Use the applet for the Sampling Distribution for the Sample Mean for continuous variables at www.artofstat.com/webapps.html to investigate the sampling distribution of ¯y.(a) Select the skewed population distribution and set the skewness = 2. Take 10,000 samples of size 50 each. How does the
Go to the applet for the Sampling Distribution for the Sample Mean for discrete variables at www.artofstat.com/webapps.html.(a) Construct a population distribution that you think is plausible for y = number of alcoholic drinks in the past day.(b) Draw a single sample of size n = 1000 to reflect
For a single toss of a coin, let y = 1 for a head and y = 0 for a tail, to simulate the vote in an election with two equally preferred candidates.(a) Construct the probability distribution for y, and find its mean.(b) The coin is flipped 10 times, yielding six heads and four tails. Construct the
(Class Exercise) Refer to Exercises 1.11 and 1.12(pages 9 and 10). Using the population defined by your class or using the Students data file, the instructor will select a variable, such as weekly time watching television.(a) Construct a histogram or stem-and-leaf plot of the population
Sunshine City was designed to attract retired people.Its current population of 50,000 residents has a mean age of 60 years and a standard deviation of 16 years. The distribution of ages is skewed to the left, reflecting the predominance of older individuals. A random sample of 100 residents of
(Class Exercise) Table 4.5 provides the ages of all 50 heads of households in a small Nova Scotian fishing village. The data are in the data file Ages at the text website. The distribution of these ages is characterized byμ = 47.18 and σ = 14.74.(a) Construct a stem-and-leaf plot or histogram of
(a) Which distribution does the sample data distribution tend to resemble more closely—the sampling distribution or the population distribution? Explain.(b) Explain carefully the difference between a sample data distribution and the sampling distribution of ¯y. Illustrate your answer for a
The Palestinian Central Bureau of Statistics(www.pcbs.gov.ps) asked mothers of age 20–24 about the ideal number of children. For those living on the Gaza Strip, the probability distribution is approximately P(1) = 0.01, P(2) = 0.10, P(3) = 0.09, P(4) = 0.31, P(5) = 0.19, and P(6 or more) =
For a normal distribution, show that(a) The upper quartile equals μ + 0.67σ.(b) According to the 1.5(IQR) criterion, an outlier is an observation falling more than 2.7 standard deviations below or above the mean, and this happens for only 0.7% of the data.
In an exit poll of 2696 voters in the 2014 gubernatorial election in Florida, 50.5% said they voted for Rick Scott and 49.5% said they voted for Charlie Crist. Based on this information, would you be willing to predict the winner of the election? Explain your reasoning.
For an election exit poll that uses random sampling, find the standard error of the sample proportion voting for a candidate for whom the population proportion is 0.50, when n = 100, when n = 1000, and when n = 10, 000.In each case, predict an interval within which the sample proportion is almost
The standard error of a statistic describes(a) The standard deviation of the sampling distribution of that statistic.(b) The standard deviation of the sample data.(c) How close that statistic is likely to fall to the parameter that it estimates.(d) The variability in the values of the statistic for
The Central Limit Theorem implies that(a) All variables have bell-shaped sample data distributions if a random sample contains at least about 30 observations.(b) Population distributions are normal whenever the population size is large.(c) For large random samples, the sampling distribution of ¯y
True or False: As the sample size increases, the standard error of the sampling distribution of ¯y increases. Explain your answer.
4.54.* LakeWobegon Junior College admits students only if they score above 400 on a standardized achievement test. Applicants from group A have a mean of 500 and a standard deviation of 100 on this test, and applicants from group B have a mean of 450 and a standard deviation of 100. Both
4.55.* From the formula on page 72, the standard deviation of a discrete probability distribution is(a) When y can equal only 0 and 1, letting π = P(y = 1)and 1 − π = P(y = 0), show that μ = π and that σ = π(1 − π).(b) Show that the standard error of a sample proportion for a random
4.56.* Refer to the formula for the normal distribution curve shown in the footnote on page 72. Show that this curve is symmetric, by showing that for any constantc, the curve has the same value at y = μ+ c as at y = μ− c.(The integral of f (y) for y between μ + zσ and∞equals the tail
4.57.* The standard error formula σ¯y = σ/√n treats the population size as infinitely large relative to the sample size n. The formula for σ¯y for a finite population size denoted by N isThe term (N − n)/(N − 1) is called the finite population correction.(a) When n = 300 students are
4.58.* A general rule states that for independent observations, the variance of yi is the sum of the variances, which is nσ2 for n observations.(a) Explain intuitively why yi would have a larger variance than a single observation y. (b) Since the variance of a probability distribution is σ2 =
4.59.* Ellenberg (2014) noted that when you use sample data to rank states by brain cancer rates, the highest ranking state (South Dakota) and the nearly lowest ranking state (North Dakota) had relatively small sample sizes.Also, when schools in North Carolina were ranked by their average

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