Statistical Theory A Concise Introduction Texts In Statistical Science 2nd Edition Felix Abramovich, Ya'acov Ritov - Solutions

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An experiment is conducted to determine whether a new computer program will speed up the processing of credit card billing at a large bank. The mean time to process billing using the present program is 12.3 min. with a standard deviation of 3.5 min. The new program is tested with 100 billings and
The manufacturer in Exercise 11 tried another, less expensive adjustment on another machine. A sample of 25 windshields was measured yielding a sample mean thickness of 3.4. Calculate the p value resulting from this mean using the same hypothesis and assumptions as in Exercise 11. LO.1
The manufacturer of auto windows discussed in Exercise 18 of Chapter 2 has developed a new plastic material that can be applied much thinner than the conventional material. To use this material, however, the production machinery must be adjusted. A trial adjustment was made on one of the 10
A drug company is testing a drug intended to increase heart rate. A sample of 100 yielded a mean increase of 1.4 beats per minute, with a standard deviation known to be 3.6. Since the company wants to avoid marketing an ineffective drug, it proposes a 0.001 significance level. Should it market the
An experiment designed to estimate themean reaction time of a certain chemical process has ¯y = 79.6 s, based on 144 observations. The standard deviation isσ = 8.(a) What is the maximum error of estimate at 0.95 confidence?(b) Construct a 0.95 confidence interval on μ.(c) How large a sample must
three units of the element Analyze for trend using only linear and quadratic terms. Perform a lack of fit test.6.10 Chapter Exercises 315 LO.1
Suppose that for a given population with σ = 7.2 we want to test H0:μ = 80 against H1:μ < 80 based on a sample of n = 100.(a) If the null hypothesis is rejected when ¯y < 76, what is the probability of a type I error?(b) What would be the rejection region if we wanted to have a level of
Suppose in Example 3.2 we were to reject H0 if all the jelly beans in a sample of size four were red.(a) What is α?(b) What is β? LO.1
The family incomes in a certain city in 1970 had amean of $14,200with a standard deviation of $2600. A random sample of 75 families taken in 1975 produced y¯ =$15,300 (adjusted for inflation).(a) Assume σ has remained unchanged and test to see whether mean income has changed using a 0.05 level of
A class of 30 students has a mean grade of 92.(a) Test the null hypothesis that the grades from this class are a random sample from the stated distribution. (Use α = 0.05.)(b) What is the p value associatedwith this test?(c) Discuss the practical uses of the results of this statistical test. LO.1
Two orthogonal comparisons are independent. LO.1
A sum of squares is a measure of dispersion. LO.1
A standardized test for a specific college course is constructed so that the distribution of grades should have μ = 100 and σ = LO.1
A study of the effect of different types of anesthesia on the length of postoperative hospital stay yielded the following for cesarean patients:Group A was given an epiduralMS.Group B was given an epidural.Group C was given a spinal.Group D was given general anesthesia.The data are presented in
Repeat Exercise 3 using n = 100. What principles of hypothesis testing do these exercises illustrate? LO.1
Assume that a random sample of size 25 is to be taken from a normal population with μ = 10 and σ = 2. The value of μ, however, is not known by the persontaking the sample.(a) Suppose that the person taking the sample tests H0:μ = 10.4 against H1:μ =10.4. Although this null hypothesis is not
Suppose that in Example 3.3, σ was 0.15 instead of 0.2 and we decided to adjust the machine if a sample of 16 had a mean weight below 7.9 or above 8.1 (same as before).(a) What is the probability of a type I error now?(b) Draw the operating characteristic curve using the rejection region obtained
The following pose conceptual hypothesis test situations. For each situation define H0 and H1 so as to provide control of the more serious error. Justify your choice and comment on logical values for α.(a) You are deciding whether you should take an umbrella to work.(b) You are planning a
You are reading a research article that states that there is no significant evidence that the median income in the two groups differs, at α = 0.05. You are interested in this conclusion, but prefer to use α = 0.01.(1) You would also say there is no significant evidence that the medians differ.(2)
If the value of the test statistic falls in the rejection region, then:(1) We cannot commit a type I error.(2) We cannot commit a type II error.(3) We have proven that the null hypothesis is true.(4) We have proven that the null hypothesis is false.(5) None of the above is correct. LO.1
If we decrease the confidence level, the width of the confidence interval will:(1) increase(2) remain unchanged(3) decrease(4) double(5) none of the above LO.1
Failure to reject the null hypothesismeans:(1) acceptance of the alternative hypothesis(2) rejection of the null hypothesis(3) rejection of the alternative hypothesis(4) absolute acceptance of the null hypothesis(5) none of the above LO.1
For a particular sample, the 0.95 confidence interval for the population mean is from 11 to 17. You are asked to test the hypothesis that the population mean is 18 against a two-sided alternative. Your decision is:(1) Fail to reject the null hypothesis, α = 0.05.(2) Reject the null hypothesis, α
If the value of any test statistic does not fall in the rejection region, the decision is:(1) Reject the null hypothesis.(2) Reject the alternative hypothesis.(3) Fail to reject the null hypothesis.(4) Fail to reject the alternative hypothesis.(5) There is insufficient information to make a
If the null hypothesis is really false, which of these statements characterizes a situationwhere the value of the test statistic does not fall in the rejection region?(1) The decision is correct.(2) A type I error has been committed.(3) A type II error has been committed.(4) Insufficient
If the null hypothesis is really false, which of these statements characterizes a situation where the value of the test statistic falls in the rejection region?(1) The decision is correct.(2) A type I error has been committed.(3) A type II error has been committed.(4) Insufficient information has
Three sets of five mice were randomly selected to be placed in a standard maze but with different color doors. The response is the time required to complete the maze as seen in Table 6.29.Table 6.29 Data for Exercise 2 Color Time Red 9 11 10 9 15 Green 20 21 23 17 30 Black 6 5 8 14 7a) Perform the
A research report states: The differences between public and private school seventh graders’ attitudes towardminority groupswas statistically significant at theα = 0.05 level. This means that:(1) It has been proven that the two groups are different.(2) There is a probability of 0.05 that the
The following sample was taken from a normally distributed population with a known standard deviation σ = 4. Test the hypothesis that the mean μ = 20 using a level of significance of 0.05 and the alternative that μ > 20:23, 32, 22, 31, 27, 25, 21, 24, 20, 18. LO.1
What sample size is required for a maximum error of estimation of 10 for a population whose standard deviation is 40 using a confidence interval of 0.95? How much largermust the sample size be if themaximumerror is to be 5? LO.1
Using the data in Exercise 1 and using a 0.05 level of significance, test the null hypothesis that the population sampled has a mean of μ = 171. Use a two-tailed alternative. LO.1
From extensive research it is known that the population of a particular species of fish has a mean length μ = 171mm and a standard deviation σ = 44mm. The lengths are known to have a normal distribution. A sample of 100 fish from such a population yielded a mean length ¯y = 167 mm. Compute the
A research article reports that a 95% confidence interval for mean reaction time is from 0.25 to 0.29 seconds.About 95% of individualswill have reaction times in this interval. LO.1
If the sample size is increased and the level of confidence is decreased, the width of the confidence interval will increase. LO.1
If a 95% confidence interval on μ was from 50.5 to 60.6, we would reject the null hypothesis that μ = 60 at the 0.05 level of significance. LO.1
If we decrease the confidence coefficient for a fixed n, we decrease the width of the confidence interval. LO.1
If the null hypothesis is false, increasing the level of significance (α)for a specified sample size will increase the probability of rejecting the null hypothesis. LO.1
A manufacturer of air conditioning ducts is concerned about the variability of the tensile strength of the sheet metal among the many suppliers of this material.Four samples of sheet metal from four randomly chosen suppliers are tested for tensile strength. The data are given in Table 6.30.Table
If the null hypothesis is true, increasing only the sample size will increase the probability of rejecting the null hypothesis. LO.1
The risk of a type II error is directly controlled in a hypothesis test by establishing a specific significance level. LO.1
If the test statistic falls in the rejection region, the null hypothesis has been proven to be true. LO.1
If a null hypothesis is rejected at the 0.01 level of significance, itwill also be rejected at the 0.05 level of significance. LO.1
If the null hypothesis is rejected by a one-tailed hypothesis test, then it will also be rejected by a two-tailed test. LO.1
In a hypothesis test, the p value is 0.043. This means that the null hypothesiswould be rejected at α = 0.05. LO.1
A local bank has three branch offices. The bank has a liberal sick leave policy, and a vice-president was concerned about employees taking advantage of this policy. She thought that the tendency to take advantage depended on the branch at which the employee worked. To see whether there were
The set of artificial data shown in Table 6.32 is used in several contexts to provide practice in implementing appropriate analyses for different situations. The use of the same numeric values for the different problems will save computational effort.Table 6.32 Data for Exercise 5 Treatment 1 2 3 4
additive type A, made by manufacturer I 2. no additive 3. additive type B, made by manufacturer I 4. additive type A, made by manufacturer II LO.1
additive type B, made by manufacturer II Construct three orthogonal contrasts to test meaningful hypotheses about the effects of the additives.(c) Assume the data represent battery life resulting from different amounts of a critical element used in the manufacturing process.The treatments are: 1.
To judge the extent of damage from Hurricane Ivan, an Escambia County official randomly selects addresses of 30 homes from the county tax assessor’s roll and then inspects these homes for damage.Identify each of the following by writing the appropriate letter into the blank.Target population (a)
A university published the following distribution of students enrolled in the various colleges:College Enrollment College Enrollment Agriculture 1250 Liberal arts 2140 Business 3675 Science 1550 Earth sciences 850 Social sciences 2100 Construct a bar chart of these data.
Table 1.21 shows the times in days from remission induction to relapse for 51 patients with acute nonlymphoblastic leukemia who were treated on a common protocol at university and private institutions in the Pacific Northwest. This is a portion of a larger study reported by Glucksberg et al.
The scores of eight persons on the Stanford–Binet IQ test were:95 87 96 110 150 104 112 110 The median is:(1) 107(2) 110(3) 112(4) 104 (5) none of the above
A sample of 100 IQ scores produced the following statistics:mean = 95 lower quartile = 70 median = 100 upper quartile = 120 mode = 75 standard deviation = 30 Which statement(s) is (are) correct?(1) Half of the scores are less than 95.(2) The middle 50% of scores are between 100 and 120.(3)
A sample of 100 IQ scores produced the following statistics:mean = 100 lower quartile = 70 median = 95 upper quartile = 120 mode = 75 standard deviation = 30 Which statement(s) is (are) correct?(1) Half of the scores are less than 100.(2) The middle 50% of scores are between 70 and 120.(3)
A small sample of automobile owners at Texas A & M University produced the following number of parking tickets during a particular year: 4, 0, 3, 2, 5, 1, 2, 1,
The mean number of tickets (rounded to the nearest tenth) is:(1) 1.7(2) 2.0(3) 2.5(4) 3.0(5) none of the above
In Problem 15, the implied sampling unit is:(1) an individual automobile(2) an individual automobile owner(3) an individual ticket
A sample of pounds lost in a given week by individual members of a weightreducing clinic produced the following statistics:mean = 5 pounds first quartile = 2 pounds median = 7 pounds third quartile = 8.5 pounds mode = 4 pounds standard deviation = 2 pounds Identify the correct statement:(1)
Calculate the mean and standard deviation of the following sample:−1, 4, 5, 0.
The following is the distribution of ages of students in a graduate course:Age (years) Frequency 20–24 11 25–29 24 30–34 30 35–39 18 40–44 11 45–49 5 50–54 1(a) Construct a bar chart of the data.(b) Calculate the mean and standard deviation of the data.
Table 1.19 gives data on population (in thousands) and expenditures on criminal justice activities (in millions of dollars) for the 50 states and the District of Columbia as obtained from the 2005 Statistical Abstract of the United States.(a) Describe the distribution of states’ criminal justice
The data in Table 1.17 consist of 25 values for four computer-generated variables called Y1, Y2, Y3, and Y4. Each of these is intended to represent a particular distributional shape. Use a stem and leaf and a box plot to ascertain the nature of each distribution and then see whether the empirical
Someonewants to knowwhether the direction of pricemovements of the general stock market, as measured by the New York Stock Exchange (NYSE) Composite Index, can be predicted by directional pricemovements of the New York Futures Contract for the next month. Data on these variables have been collected
Becausewaterfowl are an important economic resource,wildlife scientists study how waterfowl abundance is related to various environmental variables. In such a study, the variables shown in Table 1.15 were observed for a sample of 52 ponds.Table 1.15 Waterfowl Data OBS WATER VEG FOWL OBS WATER VEG
Most of the problems in this and other chapters deal with “real” data for which computations are most efficiently performedwith computers. Since a little experience in manual computing is healthy, here are 15 observations of a variable having no particular meaning:12 18 22 17 20 15 19 13 23 8
The percentage change in the consumer price index (CPI) is widely used as a gauge of inflation. The following numbers show the percentage change in the average CPI for the years 1993 through 2007:3.0 2.6 2.8 3.0 2.3 1.6 2.2 3.4 2.8 1.6 2.3 2.7 3.4 3.2 2.8(a) Using time as the horizontal axis and
The data set in Table 1.20 lists all cases of Down syndrome in Victoria, Australia, from 1942 through 1957, as well as the number of births classified by the age of the mother (Andrews and Herzberg, 1985).(a) Construct a relative frequency histogram for total number of births by age group.(b)
Exercise4.3 Continue Example4.8fortesting H0 : μ = μ0 vs. H1 : μ = μ1 based onasample Y1, ...,Yn ∼N (μ,σ2), where σ2 is known,andnow μ0 > μ1.1. Showthatthetestthatrejects H0 if ¯Y ≤ μ0−zα√σn is theMPtestatlevel α.2. What isthepowerofthetest?3. What
Exercise3.3 Let Y1, ...,Yn be i.i.d. U (0,θ).1. Verifythat Ymax/θ is apivot.(Hint: thedistributionof Ymax wasderivedinExercise2.12.)2. Showthat (Ymax,α−1 nYmax) is a (1−α)100% confidenceintervalfor θ.
Exercise3.2 1. Let Y be acontinuousrandomvariablewithacdf F(y). Definearandomvariable Z = F(Y).Showthat Z ∼ U (0,1) for any F(·).2. Let Y1, ...,Yn be arandomsamplefromadistributionwithacdf Fθ (y), where θ is anunknown parameter.Showthat Ψ(Y;θ) = −Σni=1 lnFθ (Yi) ∼ 1 2 χ2 2n and,
Exercise3.1 Let Y1, ...,Yn be arandomsamplefromthelog-normaldistribution fμ,σ (y) =√ 1 2πσy e−(lny−μ)2 2σ2 , whereboth μ and σ are unknown.Find100(1−α)% confidenceintervalsforμ and σ.(Hint: thedistributionof X = lnY may beuseful.)
Exercise2.22 ∗Let Y1, ...,Yn ∼ fθ (y). ShowthattheM-estimatorcorrespondingto ρ(θ,y)=α(y−θ)++(1−α)(θ −y)+ for agiven0 < α < 1 isthesample α · 100%-quantile Y(α).Exercise2.23 ∗ProveProposition2.2fortheFisherinformationmatrix I(θ).
Exercise2.21 A qualitycontrollaboratoryrandomlyselected n batches of k components produced by afactoryandinspectedthemfordefectives.Itisassumedthattheproducedcomponentsare independent, eachwiththesameprobabilityofbeingdefective.Thenumbersofdetecteddefectives were respectively Y1, ...,Yn.
Exercise2.20 Let Y1, ...,Yn ∼ Geom(p).1. Find theMLEof p.2. Find anunbiasedestimatorof p.(Hint: firstfindatrivialunbiasedestimatorof p and thenimproveitby“Rao–Blackwellization”;recall that Σni=1Yi ∼ NB(n, p).)3. Is theunbiasedestimatorfromthepreviousparagraphUMVUE?
Exercise2.19 Consider asimplelinearregressionmodelwithoutanintercept:Yi = βxi+εi, i = 1, ...,n, where εi are i.i.d N (0,σ2).1. Find theMLEfor β. Isitunbiased?FinditsMSE.2. Consider an estimator β˜ =Σni=1YiΣni=1 xi. Isitunbiased?FinditsMSEandcompareitwiththatofthe MLE.3. Find theMLEfor
Exercise2.18 A largetelephonecompanywantstoestimatetheaveragenumberofdailytelephone calls madebyitsprivateclients.Itisknownthatontheaveragewomenmake a times morecalls than men.ItisreasonabletoassumethatthenumberofdailycallshasaPoissondistribution.During a
Exercise3.4 Let Y1, ...,Yn ∼ N (μ,a2μ2), where a is known.Findapivotanduseittoconstructa 100(1−α)% confidenceintervalfor μ.
Exercise3.5 Continue Example2.6.Find100(1−α)% confidenceintervalsfortheexpectedser-vice timesperclientsofAliceandKate.(Hint: recallthat exp(θ) = 1 2θ χ2 2 to findthedistributionofasumofexponentialrandomvariables.)
Exercise3.6 A reactiontimeforacertainstimulusforbothcatsanddogsisknowntobeexpo-nentially distributed.Inaddition,biologistsbelievethatanaveragereactiontimeforcatsis ρ times fasterthanthatfordogs.Abiologistperformedaseriesofexperimentsandmeasuredthereaction times X1, ...,Xn of n dogs and Y1, ...,Ym
Exercise4.2 Continue Example4.7andderivethe25%-levelMPtesttodeterminetheoriginofthe man byaskingabouthisdrinkofchoice,where H0 :UK vs. H1 : France. Comparetheresultswith those obtainedinExample4.7.
Exercise4.1 The advertisementofthefastfoodchainofrestaurants“FastBurger”claimsthatthe averagewaitingtimeforfoodinitsbranchesis30secondsunlike50secondsoftheircompetitors.Mr.Skepticdoesnotbelievemuchinadvertisinganddecidedtotestitstruthbythefollowingtest:he
Exercise3.13 ∗Givenasampleofvectors Y1, ...,Yn from a p-dimensional multinormaldistribu-tion withanunknownmeanvector μ and aknownvariance-covariancematrix V, findBonferroni and exact100(1−α)% confidenceregionsfor μj, j = 1, ..., p.(Hint: recallthat (Yi−μ)⊤V−1(Yi−μ) ∼ χ2 p.)
Exercise3.11 Let t1, ..., tn ∼ exp(θ). InExample3.6,weconstructed95%confidenceintervalsforθ, μ = Et = 1/θ and p = P(t > 5) = e−5θ based onthepivot2θ Σni=1 ti ∼ χ2 2n. Showthatanother pivotis θtmin ∼ exp(n). Useittoconstruct95%confidenceintervalsfor θ, μ, and p, andcompare them
Exercise3.10 Continue Exercise2.3.Find100(1−α)% confidenceintervalsfor θ, theexpected lifetime ofadevice μ and fortheprobability p that adevicewillstillbeoperatingafter a hours.
Exercise3.9 (Proposition3.1) Let (L,U) be a (1−α)100% confidenceintervalfor θ, and g(·) a strictly increasingfunction.Showthat (g(L),g(U)) is a (1−α)100% confidenceintervalfor g(θ)).
Exercise3.8 Continue Example3.4.Showthatthe χ2-confidence intervalfortheunknownvarianceσ2 with aminimalexpectedlengthis(n−1)s2 b , (n−1)s2 a, where a and b satisfy a2χ2 n−1(a) = b2χ2 n−1(b) and Z b aχ2 n−1(t)dt = 1−α, where χ2 n−1(·) is thedensityof χ2 n−1 distribution.
Exercise3.7 Showthatthe (1−α)100% confidenceinterval(3.4)fortheunknownnormalmeanμ has aminimalexpectedlengthwhenitissymmetric,thatis, α1 = α/2.
Exercise2.17 An aircraftlaunchedamissiletowardsagroundtarget.Let X and Y be south–north and east–westrandomdeviationsofthemissilefromthetarget,respectively.Itmaybeassumedthat X and Y are i.i.d N (0,σ2). Define R =√X2+Y2 to bethedistancebetweenthetargetandthe actual
Exercise2.16 An annualnumberofearthquakesinacertainseismicallyactiveregionisdistributed Poisson withanunknownmean λ. Let Y1, ...,Yn be thenumberofearthquakesintheareaduringthe last n years (itispossibletoassumethatthe Yi are independent).1. Find theMLE ˆλ for λ.2. Is ˆλ unbiased?3. Does ˆλ
Exercise2.4 A start-upcompanyrecruitsnewemployeesamonggraduates.Hiringrequirements include passinganIQtestwithascorelargerthan120.Withinapopulationofgraduates,IQis normally distributedwithanunknownmean μ and astandarddeviationof5.Accordingtothe
Exercise2.3 An electronicdevicecontains k identical, independentlyworkingsensitiveelements whose lifetimesareexponentiallydistributedexp(θ). Evenifoneelementdoesnotoperate,the entire deviceisrenderedoutoforderandmustbesenttoalaboratoryforrepair. n deviceswith lifetimes t1, ..., tn hours
Exercise2.2 Let Y1, ...,Yn ∼ Gamma(α,β).1. Can youobtaintheMLEof α and β in theclosedform?2. Estimate α and β by themethodofmoments.
Exercise2.1 Let Y1, ...,Yn be arandomsamplefromthefollowingdistributionswiththeunknown parameter(s). Estimatethembymaximumlikelihoodandbythemethodofmoments.1. doubleexponentialdistribution fθ (y) = θ2 e−θ|y|, −∞ < y < ∞, θ > 0 2. shifted exponentialdistribution fθ,τ (y) =
Exercise1.5 ∗Let Y1, ...,Yn ∼ N (μ,a2μ2), where a > 0 isknown(seeExample1.11).Showthat T(Y) =(Σni=1Yi,Σni=1Y2 i ) is aminimalsufficientstatisticfor μ.
Exercise1.4 Consider apopulationwiththreekindsofindividualslabeled,say,1,2,and3occur-ring inthefollowingproportions: p1 = p2, p2 = 2p(1−p), p3 = (1−p)2 (0 < p < 1). Inarandom sample ofsize n chosen fromthispopulation, n1 individualshavebeenlabeled1, n2—2 and n3—3 respectively(n1+n2+n3 =
Exercise1.3 Which ofthedistributionsinExercise1.2belongtotheexponentialfamily?Findtheir natural parameters.Inaddition,consideralsothefollowingdistributions:1. Poisson distribution Pois(λ)2. Cauchydistribution fθ (y) = 1π(1+(y−θ)2) , −∞ < y < ∞3. fθ (y) = θ(1+y)θ+1 , y ≥ 0, θ > 0
Exercise1.2 Find sufficientstatisticsfortheparametersofthefollowingdistributionsbasedona random sampleofsize n:1. geometric distribution Geom(p)2. uniform distribution U (θ,θ +1)3. uniform distribution U (−θ,θ)4. uniform distribution U (θ1,θ2)5. shifted exponentialdistribution fθ,τ (y) =
Exercise2.5 A counterrecordsadailynumberofcarspassingacrossajunctionthatisknown to haveaPoissondistributionwithanunknown λ. Thecountertransmitsitsdailyrecordtothe control centereverymidnightandresetstorestarttocountdataforthenextday.However,withan(unknown)probability p the
Exercise2.6 Let Y1, ...,Yn ∼ exp(θ), whereitisknownthat θ ≤ θ0. FindtheMLEof θ.
Exercise2.7 Tworandomsamplesof n men and m womenparticipatedinapoliticalpolltoevaluate the supportforCandidateAinthepopulation.Outofthoseparticipatedinthepoll, X men and Y womensupportedhiscandidacy.Inaddition,itisalsoknownthatintheentirepopulation,heis more

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Statistical Theory A Concise Introduction Texts In Statistical Science 2nd Edition Felix Abramovich, Ya'acov Ritov - Solutions

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