Statistics With Common Sense 1st Edition David Kault - Solutions

Unlock the full potential of your learning with the "Statistics With Common Sense 1st Edition" by David Kault. Whether you're looking for an online answers key or a comprehensive solution manual, we've got you covered. Our solutions PDF is packed with solved problems and detailed questions and answers. Delve into the test bank and chapter solutions with step-by-step answers for a complete understanding. Perfect for students and instructors alike, our instructor manual and textbook solutions are available for free download. Enhance your study sessions and master statistics effortlessly!

Evaluate explicitly the fourth cumulants of the multinomial distribution for the five distinct index patterns.
Show that the multinomial distribution with index m and probability vector π1, …,πk has cumulant generating function m {k (θ + ξ) − k (θ)}, where k(θ) = log[Σexp(θj)] andπi = exp (θi)/∑exp (θj).Hence show that the first four cumulants are where, for example, δijk = 1 if i = j = k
Show that κij,kl − κi,j,rκk,l,sκr,s, regarded as a p 2 × p 2symmetric matrix, is nonnegative definite. By examining the trace of this matrix in the case where κi = 0, κi,j = δij, show thatρ4 ≥ ρ2 23 − p − 1, with equality if and only if the joint distribution is concentrated on
Show that ρ2 13 ≥ 0 with equality only if κi,j,k= 0 identically.μ2r =1.3…(2r−1)νr(ν−2)…(ν−2r) 2r < νρ4 = 6/ (ν − 4) for ν > 4¯
By considering the expression var {κi,jXiXj − ciXi}with ci = κi,rκs,tκr,s,t, show that ρ4 ≥ ρ2 13 − 2, where 4 and ρ2 13 are defined by (2.14)and (2.16). In addition, by examining the expression∫ (a + aix i + aijx ix j)2 fX (x)dx, show that ρ4 = ρ2 13 − 2 if and only if the joint
Verify that 4 = p−1κi,j,k,lκi,jκk,l is invariant under affine non-singular transformation of X.
Let a = α1,…,αp and b = b 1,…,b p, where a j ≤ b j, be the c∞rdinates of two points in Rp and denote by (a,b) the Cartesian product in R p of the intervals (a j,bj). Let f(x) =f(x l,…. xp) be a p-dimensional joint density function and define F (a,b) = ∫x∈(a,b)f (x)dx where dx = dx1
Let a ij be the elements of a square matrix, not necessarily symmetrical, and let its inverse, aij, satisfy a ijakj = δi k = a jiajk, being careful regarding the positions of the indices. Show that the derivatives satisfy
Show that if ai are the components of a vector of coefficients satisfying aiajκi,j = 0, then aiκi,j = 0, aiκi,j,k = 0, aiκi,j,k,l = 0 and so on. Hence deduce that the choice of generalized inverse has no effect on the scalars derived in Section 2.8.
By taking X to be a two-dimensional standardized random variable with zero mean and identity covariance matrix, interpret Q3(w), defined in the previous exercise, as a directional standardized skewness. [Take w to be a unit vector.] Show that, in the polar representation, ϵ3 − ϵ1 is invariant
In the two-dimensional case, show that the homogeneous cubic form κi,j,kwiwjwk can be written, using power notation, in the form Q3 (w) = κ30w3 1 + κ03w3 2 + 3κ21w2 1w2 + 3κ12w1w2 2.By transforming to polar coordinates, show thatκr = − (r − 1)!∑i ar i = − (r − 1)! < r > (r < n)Xj+1
Let {ϵj}, {ϵ′j} and {Zj}, j = … − 1,0,1,… be three doubly infinite, mutually independent sequences of independent unit exponential random variables. The bivariate sequence {Xj,Yj}, j = … − 1,0,1,…, defined by is known as a bivariate exponential recurrence process. Show that(i) Xj
Let X be a scalar random variable and write M(n)X(ξ) = 1 + μ′1ξ + μ′2ξ2/2! + … + μ′nξn/n!for the truncated moment generating function. The zeros of this function, a−1 1,…, a−1 n, not necessarily real, are defined by M(n)X(ξ) = (1 − a1ξ)(1 − a2ξ)…(1 − anξ).Show that
Let X1 and X2 be independent and identically distributed p-dimensional random variables with zero mean and identity covariance matrix. The spherical polar representation of X is written (R, θ) where θ has p − 1 components and R = |X| is the Euclidean norm of X. Show that where R1R2 cos θ12 =
Using (2.18), show that if X and Y are independent real-valued random variables, then the variance of the product is var(XY ) = κ2 10κ02 + κ2 01κ20 + κ20κ02.
Using the notation of Exercises 2.28 and 2.30, show that if the eigenvalues decrease exponentially fast, say λj = λj, with |λ| < 1, then nR has a non-degenerate limiting distribution for large n, with cumulantsκr (nR) ≃ 2 r−1 (r − 1)!λr/ (1 − λr).Show that this result is false if λ
In the notation of Exercise 2.28, let R = Y2/Y1. Show that the first four cumulants of this ratio are and explain what is meant by the notation κr(λ).κ1 (R) = κ1 (λ)κ2 (R) = 2κ2 (λ)/ (n + 2)κ3 (R) = 8κ3 (λ)/ {(n + 2)(n + 4)}κ4 (R) = 48κ4 (λ)/ {(n + 2)(n + 4)(n + 6)}+ 48nκ2 2 (λ)/{(n
Show that if the ratio R = Y/X is independent of X, then the moments of the ratio are the ratio of the moments, i.e.μ′r (R) = μ′r (Y )/μ′r (X).
Let X1,…,Xn be independent X 2 1random variables. Show that the joint cumulant generating function is −1 2 ∑i log (1 − 2ξi). Show also that the joint cumulant generating function of Υ1 = Σ Xj and −1 2 ∑i log (1 − 2ξ1 − 2λiξ2). Hence show that the joint cumulants of (Y1, Y2)
Show that if ϒ = {υ1,…, υν} is a partition of the indices j1, …,jn and ω = e2πi/n is a primitive nth root of unity, then∑jωj1+…+jn δ (υ1)…δ (υν) = {where the sum extends over all positive integer vectors having components in the range(l,n).∂aij/∂a rs = −aisarj∂a
Let X1,…,Xn be independent and identically distributed scalar random variables having cmnulants κ1, κ2, κ3, …. Show that where i2= −1. Hence, by writing ω = e2πi/r and rκr = limn→∞n−1[X1 + ωX2 + ω2X3 + … + ωnr−1Xnr]r, give an interpretation of cumulants as coefficients in
In the two-dimensional case, show that the homogeneous quartic formκi,j,k,lwiwjwkwl can be written, using power notation, in the form Q4 (w) = κ40w4 1 + κ04w4 2 + 4κ31w 31w2 + 4κ13w1w 32 + 6κ22w2 1w2 2.By transforming to polar c∞rdinates, show that Q4 (w) = r 4 {τ0 + τ2 cos(2θ − 2ϵ2)
Derive expressions (2.7) for cumulants in terms of moments (i) directly from (2.6) and(ii) by expansion of log ΜX(ξ).
In the notation of the previous exercise, letωij = γi rγj s ϵrs, where ϵrs = ϵrs and γi rτr j = δi j Show that ωij is the Moore-Penrose inverse of ωij.
Show that the identity matrix is a generalized inverse of any projection matrix. Show also that a projection matrix, not necessarily symmetric, is its own generalized inverse satisfying (1.9) and (1.11). Under what conditions is a projection matrix self-inverse in the Moore-Penrose sense?
Let ωij, with inverse ωij, be the components of a p × p symmetric matrix of rank p. Show thatγrs,ij = ωriωsj + ωrjωsi, regarded as a p 2 × p 2 matrix with rows indexed by (r, s) and columns by (i, j), is symmetric with rank p(p+l)/2. Show also that ωriωsj/2 is a generalized inverse. Find
Let π1,…,πk be positive numbers adding to unity and define the multinomial covariance matrixωij = { .Show that ωij has rank k − 1 and that is a generalized inverse of rank k. Find the Moore-Penrose generalized inverse.
Let the p × q matrix A, with components a ri, be considered as defining a linear transformation from the domain, R q, to the range in R p. Interpret the singular values of A as invariants under independent orthogonal transformation of the domain and range spaces. For the definition of singular
Let A, A−1 and X be symmetric matrices with components aij, a ij and xij respectively. Show that the Taylor expansion for the log determinant of A + X about the origin may be written Describe the form of the general term in this expansion. Compare with Exercise 1.6 and generalize to asymmetric
Justify the claim that Kronecker’s delta, δj i, is a tensor.
Show that δij is a tensor under the orthogonal group but not under any larger group.
Show that δijk, δijkl, … are tensors under the symmetric group but not under any larger group.[The symmetric group, which is most conveniently represented by the set of permutation matrices, arises naturally in the study of sample moments based on simple random samples, where the order in which
Show that the symmetric group is a subgroup of the orthogonal group, which, in turn, is a subgroup of the general linear group.
Hooke’s Law: In the mechanics of deformable solids, the components of the stress tensor, pij, measure force per unit area in the following sense.Let ei be the unit vector in the ith coordinate direction and let ēi be the orthogonal plane. Then pii is the force per unit area normal to the plane,
The array ϵijk of order 3×3×3 defined by is known as the alternating tensor (Ames & Mumaghan, 1929, p. 440). For any 3 × 3 matrix a ir, show thatϵijka ira jsa kt = ϵrst det(A).Hence show that ϵijk is an isotropic tensor under O+, the orthogonal group with positive determinant (Jeffreys &
Consider the decomposition ωij = τr iτs jϵrs of the symmetric covariant tensor ωij, where the notation is that used in (1.14). Defineω+ij = τr iτs j|ϵrs|.Show that ω+ij is a covariant tensor and that the scalar s = ωijω+ij is independent of the choice of τr iand also of the choice of
Show that there exists a linear transformation Y r = a riXi from X to Y such that the quadratic form ωijX iX j may be written as ϵrsY rY s where ϵii =±1 and ϵij= 0 if i ≠ j.
Let a ij be the elements of a square matrix, not necessarily symmetrical, and let its inverse, b ji satisfy a ij bj k = δi k = a jk bi j. Show that the derivatives satisfy 1.7 Show that the spectral decomposition of the the symmetric matrix A A = QΛQT, QQT = I, Λ = diag {λ1,…, λp}is unique
Show that if X 1is independent of (X 2,X 3), then the cumulant κrst of order r + s + t with r, s, t ≥ 1 is equal to zero.
Prove directly using (2.7) that if X iis independent of the pair (X j,Xk), then κi,j,k = 0.
The density function of Student’s distribution on ѵ degrees of freedom is(1+t 2/ν)−(ν+1)/2ν1/2B(1 2,ν/2) − ∞ < t < ∞, r − 1 j − 1 r − 1 j − 1 r − 1 j − 1μ′r =r ∑j=0( )μr−j(μ′1)jμr =r ∑j=0( )μ′r−j(−μ′1)j.r jr jwhere B(.,.) is the beta function. Show
Show that the central and non-central moments satisfy(Kendall and Stuart, 1977, p. 58).
If μ(ϒ) and κ(ϒ) denote the ordinary moment and the ordinary cumulant corresponding to the indices in ϒ = {i1,…,tp}, show thatμ (ϒ) = κ (ϒ) + ∑{υ1,υ2}κ (υ1)μ (υ2)where {υ1,υ2} is a partition of ϒ into two non-empty blocks and the sum extends over all partitions such that i1
Let X be a scalar random variable. Prove by induction on r that the derivatives ofΜX(ξ) and KX(ξ), if they exist, satisfy M(r)X(ξ) =r ∑j=1( )M(r−j)X(ξ)K(j)X(ξ) r ≥ 1.Hence show thatμ′r = κr +r−1∑j=1( )κjμ′r−j and, for r ≥ 4, thatμr = κr +r−2∑j=2(
Let X be a standard normal random variable and set Y = exp(X). Show that the rth moment of Y about the origin is μ′r = exp(r2/2). Hence find expressions for the first four cumulants of Y. Show that the series expansions for ΜY(ξ) and KY(ξ) about ξ = 0 are divergent for all real ξ> 0 even
Let X have density function given by fX (x) = {Show that the mean of X is finite but that all higher-order moments are infinite. Find an expression for the density of Y = 1/X and show that all moments and cumulants are finite.Let μ′rs = E(X rY s) be the joint moment of order r+s (using the power
Show that if ωijk is a contravariant tensor and ωijk is a covariant tensor, then ωijkωijk is an invariant.
Show directly, using the notation in (1.7), that UrUs Irs is invariant under the group of invertible transformations acting on the parameter space.
Let irs = −E{Urs;θ} and let i rs be the matrix inverse. Under which group of transformations is UrUa irs an invariant?
Show, using the notation in (1.7), that Vij = Uij − κij,ki klUl is a covariant tensor, where κrs,t is the covariance of Urs and Ut.
Derive the transformation laws (1.7) for log likelihood derivatives.
By taking X to be a two-dimensional standardized random variable with zero mean and identity covariance matrix, interpret Q4(W), defined in the previous exercise, as a directional standardized kurtosis. Taking ϵ1 as defined in Exercises 2.36 and 2.37, show, using the polar representation, that
In the one-dimensional case, the functions of the cumulants of X that are invariant under affine transformation areκ2rκr 2,κ2 2r+1κ2r+1 2, r = 1, 2,…using power notation. All other invariants can be expressed as functions of this sequence.By extending the results described in the previous
Discuss the difficulties encountered in extending the above argument beyond the bivariate case.
Spherically symmetric random variables: A random variable X is said to be spherically symmetric if its distribution is unaffected by orthogonal transformation.Equivalently, in spherical polar coordinates, the radial vector is distributed independently of the angular displacement. Show that the odd
For the previous exercise, show thatτ6 ≥ −τ2τ4 − τ3 2 +3(τ4+τ2 2 )2(p+4)τ2¯.
A number of farms are selected at random from a list of farms. Thirty are family farms, and of these, ten are classified as being severely eroded. Twenty farms are owned by absentee landlords, and of these, fifteen are classified as being severely eroded. Does this data provide reasonably
The careers of 50 males who at the age of eighteen years were over 180 cm tall were compared with those of 100 males who at the age of eighteen years were under 170 cm tall. It was found that 12 men in both groups became administrators. Do you believe that this study provides reasonably convincing
A retrospective study is set up to examine the risk factors for parenting a mentally retarded child. The parents of all children with mental retardation in a given region all agree to cooperate in answering an extensive questionaire covering a large range of factors that could possibly be relevant
A sample of 2,000 people had their serum cholesterol level measured and were then followed up for five years. At the end of the five years it was found that 20 of the 2,000 had met a violent death. Seventeen of the twenty were those who had a below median level of cholesterol. Discuss the p value
A prospective study of heart attacks in a certain city enrolled 250 randomly selected men aged fifty to fifty-five with no history of heart trouble. In other words, these men were identified as living in the city in which the study was carried out and a check was made that they fell into the
A retrospective study is set up to examine the risk factors for suffering a broken right leg, A control group is obtained that consists of those who have suffered a broken left leg. Would this study reveal that motor bike accidents are the main cause of suffering a broken right leg? P-963
Think of an example where all three possible directions of causality—A ==• B, B =» A, and C =• both A and B—are likely to be operating simultaneously. P-963
A sample of chocolate bars from a factory are weighed and the results are 201.3, 201.7, 202.0, 202.4, 202.8, and 203.0 grams. Finda. the 90 percent confidence interval for the population mean of the weight of a chocolate bar.b. the 95 percent confidence interval for the population mean of the
Consider each of the following null hypotheses and decide whether in testing each of these hypotheses you would use a one-tail test or a two-tail test. In each case give brief reasons for your choice.a. Students who are provided with a private tutor visiting their home are as likely to get good
In each of the following cases decide whether the hypothesis listed is appropriate as a null hypothesis that could be tested by collecting appropriate data. If you decide the hypothesis is not appropriate as a null hypothesis, state why not. If you believe that the null hypothesis is appropriate,
It is suspected that watching violence on television may have an adverse impact on human emotional development and social interaction. As part of a study on the issue, a number of young children are shown a nonviolent cartoon on one occa- sion and a cartoon containing violence on another occasion.
The time from purchase until major repairs are required is recorded for eight cheap bicycles and five expensive bicycles with the following results in days: cheap bikes—39, 117, 561, 57, 3, 27, 8, 2; expensive bikes—289, 641, 105, 111, 903.a. Is it reasonable to perform a parametric test (test
Two different brands of racing bicycles are tested by five cyclists. The top speeds attained are given in the following list. Is there convincing evidence that one brand is superior?cyclist 1 cyclist 2 cyclist 3 cyclist 4 cyclist 5 cyclist 6 Brand A 30.5 41.2 29.8 35.9 25.7 40.3 Brand B 30.3 46.3
A drug thought to increase strength was given to three men, with nine other men acting as controls. The results in arbitrary units were 19, 20, and 21 for the men with the drug and 10, 11, 12, 13, 14, 15, 16, 17, and 18 for the men without the drug. The drug was also tried on nine women, with three
A survey was conducted in which people were asked their religious affiliations and also asked whether they regarded preservation of the natural environment as a top-priority issue. Of thirty-three protestants surveyed, ten rated the environment as a top priority; the same was true of nine out of
A researcher is interested in a possible association between having a carpeted bedroom and asthma. Accordingly, she identifies 200 families, each with two children where just one of the children has asthma. She records the information that in 130 families both children's bedrooms are carpeted and
Consider a survey on an issue in which the American population is thought to be approximately evenly divided.a. How many people would we need to survey for us to have a 99 percent chance of obtaining a value that differs from the true percentage by no more than 1 percentage point?b. After taking
On a test in a statistics class the mean mark was 69.71 and the standard deviation was 11.65. However, only twenty-one of the twenty-two students now enrolled took the test, as one student was unable to take it because of ill health.a. Assume this one student has been attending class as much as the
Say that the entire adult population of northern hairy-nose wombats consists of 180 individuals and that we survey a randomly selected sample of 100 of these individuals and measure their weights. If the results are that the mean is 27 kg and the standard deviation is 2 kg, find the 95 percent
The rabbit populations of 100 fields were estimated and the 9 fields with the highest estimated rabbit populations were selected for a rabbit inhibition experiment.The inhibition experiment consisted of erecting gruesome pictures in the middle of each field portraying farmers killing rabbits. After
A researcher looking for evidence of climate change notes the ten localities in America with the highest rainfall totals for the twenty-four hours ending at midnight, January 2-3, 2001. The following year he checks the rainfall in the same ten localities over the same twenty-four-hour period. He
Six people are questioned by a researcher and are ranked according to the amount of daily exercise that they undertake. The same six people also have a blood cholesterol test. It turns out that the person who undertakes the most exercise also has the lowest cholesterol level. The person who
In the following list the row marked x gives the prices paid by the previous owners to purchase houses in a particular town in 1990; the row marked y gives the price paid by the current owners to purchase the same houses in 2000. P-963 x 33 37 38 39 41 36 y 67 73 74 75 97 84 Comment on the null
The following list gives the rainfall in centimeters in Brisbane, Australia, in April of each year and the number of books imported into the United States in December of the same year. Data from the years 1995 to 2000 are given. Use an appropriate statistical method to test the null hypothesis that
In a study of geographical temperature variation, research assistants in Townsville(latitude 19°30'), Brisbane (latitude 27°30'), Sydney (latitude 34°00'), and Melbourne (latitude 37°45') were each asked to choose, at random, two days in May and record the maximum temperatures on those days.
A skeptical farmer wonders whether there is really any benefit in terms of yield in the various high-yield varieties of wheat produced by agricultural researchers.Accordingly, he keeps a record over the years of the yield with various varieties.His results are as follows:Traditional variety 2.7 1.9
The day of the week on which a number of statistics students were born was recorded along with their marks for a statistics course. The results are displayed here:Monday Tuesday Wednesday Thursday Friday Saturday Sunday 58 6 21 71 39 62 81 97 19 58 86 58 31 94 43 55 23 51 26 37 65 Perform an
To test if a die is fair, it is thrown 600 times. It lands on the numbers one to six, 93, 98, 107, 102, 91, and 109 times, respectively. Perform an appropriate test and discuss whether you believe that the die is fair. P-963
For each of the following scenarios state the most appropriate statistical test.a. People are classified into two groups depending on whether they were abused as children and the recorded outcome is whether they have had a conviction for theft.b. Groups of people in four different industries are
All senior high school students in a certain state take a test on which the statewide results are approximately normally distributed with mean 60 and standard deviation 10. A random selection of sixty-four test results is chosen.a. What is the probability that the average of the sixty-four test
The sensitivity of a medical test is the probability of it giving a positive result when the patient actually has the disease. The specificity of a medical test is the probability of it giving a negative result when the patient doesn't have the disease.The proportion of people who have the disease
Assume that you are told that 2 percent of children under the age of twelve are anemic, 10 percent of females in the reproductive years twelve to fifty are anemic, 1 percent of males in this age group are anemic, and 3 percent of people of both sexes over fifty are anemic. Also assume that 15
According to a news report, one-half of all women and one-third of all men over the age of seventy suffer from osteoporosis (unduly weak bones). Men make up one-quarter of this age group. Find the overall prevalence of osteoporosis in people over seventy (take "one-half to mean a proportion of
Human blood can be grouped in several different ways. The rhesus blood groups(D+ and D-) and the ABO blood groups are inherited independently of each other.About 14 percent of the population has the rhesus blood group D-, the remaining having D+, and 3 percent of the population has ABO blood group
There are eighty women and sixty men in the class. Thirty women and thirty men own dogs. The lecturer shuts his or her eyes, spins around, and points to a person at random.a. What is the probability that the person picked:i. Is a woman?ii. Owns a dog?iii. Is a female dog owner?iv. Is either a woman
In terms of a measure of belief. For example, we can state a subjective probability about the chance of a nuclear war over the next ten years. A person might believe that this probability is one-sixth. This would be a valid statement of belief, even though it is clearly not possible to view
In terms of the long-run proportion of the time that something happens. For example, the proportion of time "6" appears on throwing a fair die many times is one-sixth. We say the probability of throwing a "6" is one-sixth. The expressions"the proportion of the time something happens" and "the
An opinion poll is taken by phoning people whose names are chosen at random from the telephone book. Why might this sample not be representative? POLI967
Measure the heights of the people in your class. Make a stem and leaf plot to help you both order and display the data. Then create a frequency table, a histogram, and a boxplot to display your results (for the boxplot you will need to specify the rules you use in its construction; in particular,
Consider the same scenario as in question 3, but this time you are going to get the self-declared clairvoyant to perform the number guessing twice in a row and you will believe in her powers only if she is correct both times. Again, how big would n have to be in order to convince you (assuming you
Imagine your neighbor claimed to be a clairvoyant and asked you to verify her powers by getting you to write down a number between 1 and n (where n stands for a number like 10, 20, 100, or 500) that she then correctly guessed. How big would n have to be in order to convince you (assuming you can
The roulette wheel in a casino can stop in thirty-eight different positions. The casino is known to operate the roulette wheel fairly. You notice that the roulette wheel hasn't stopped on "36" even once in the last 500 goes, so you place a bet on"36." What is the chance that you will lose your
A judge initially believes that there is a 50-50 chance that the man before him is guilty. Forensic evidence is then produced that shows that the guilty man has blood group O. It is known that 50 percent of people in this region have blood group O and so does the man on trial. What should the judge
A plane has crashed in an unknown location, but it was initially thought that there was a 40-percent chance (probability 0.4) that it had crashed over the sea and a 60-percent chance it had crashed on land. If it crashed on land there is a probability of 0.8 that a search will find it. If it

Showing 6200 - 6300 of 6613

First
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67

Statistics With Common Sense 1st Edition David Kault - Solutions

Step by Step Answers