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modern mathematical statistics with applications

Modern Mathematical Statistics With Applications 3rd Edition Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton - Solutions

A student project by Heather Kral studied students on “lifestyle floors” of a dormitory in comparison to students on other floors. On a lifestyle floor the students share a common major, and there are a faculty coordinator and resident assistant from that department. Here are the GPAs of 30
Let X1,......, Xm ∼ N(μ1, σ)and Y1, . . .,Yn ∼ N(μ2, σ) be independent random samples from the specified normal population distributions (note that the population sd’s are equal). Let S21 and S22 denote the sample variance of the two samples, and define a pooled
The Weaver–Dunn procedure with a fiber mesh tape augmentation is commonly used to treat AC joint (a joint in the shoulder) separations requiring surgery. The article “TightRope Versus Fiber Mesh Tape Augmentation of Acromioclavicular Joint Reconstruction” (Am. J. Sport Med. 2010: 1204–1208)
The article “Effect of Wearable Technology Combined With a Lifestyle Intervention on Long-term Weight Loss” (JAMA 2017: 1161–1171) describes a study in which adults in a large cohort were provided the same weight-loss regimen for six months. Then, participants were randomly assigned to either
Management at a large retail appliance chain required all full-time sales staff to attend a one-day training session on improving sales staff to attend a one-day training session on improving sales technique. To evaluate the effectiveness of this rather expensive training, the number of sales in
The article discussed in the previous exercise also describes an experiment in which 35 students were asked to price two boxes of silverware: a 24-piece box with all 24 pieces intact, and a 40-piece box with only 31 pieces of silverware intact. Each student indicated the amount s/he would be
The article “Less Is Better: When Lowvalue Options Are Valued More Highly than High-value Options” (J. Behav. Decis. Making 1998 107–121) describes several experiments pertaining to consumer behavior. In one experiment, 46 students were split into two groups: 23 who were shown 7 oz of ice
The 33 children were each observed in about 100 ten-second intervals in the first grade, and again in the second and third grades. Because private speech occurs more in challenging circumstances, the children were observed while doing their mathematics. The speech was classified as on task (about
Can people operate touch screen devices more quickly with their index finger or their thumb? Holding the device in landscape or portrait position? The article “Evaluation of a Psychomotor Vigilance Task (PVT) for Touch Screen Devices” (Hum. Factors 2017: 661–670) describes a study in which 13
As integrated circuits operate at eversmaller resolutions, the clean handling of wafers in the manufacturing process has become even more important. The article “Particle Free Handling of Substrates” (IEEE Trans. Semicond. Manuf. 2016: 314–319) provides the following data on the number of
A sample of 300 urban adult residents of a particular state revealed 63 who favored increasing the highway speed limit from 55 to 65 mph, whereas a sample of 180 rural residents yielded 75 who favored the increase. Does this data indicate that the sentiment for increasing the speed limit is
It has been estimated that between 1945 and 1971, as many as 2 million children were born to mothers treated with diethylstilbestrol (DES), a nonsteroidal estrogen recommended for pregnancy maintenance. The FDA banned this drug in 1971 because research indicated a link with the incidence of
It is thought that the front cover and the nature of the first question on mail surveys influence the response rate. The article “The Impact of Cover Design and First Questions on Response Rates for a Mail Survey of Skydivers” (Leisure Sci. 1991: 67–76) tested this theory by experimenting
Independent random samples of 237 African-Americans and 396 Caucasian- Americans were asked to identify their “favorite” category of television programming (“Television Types and TV Attitudes of African-Americans, Latinos, and Caucasians,” J. Advert. Res. 2008: 235–246). In the survey,
Several states have an annual “sales tax holiday” to encourage spending. A survey of 695 randomly selected shoppers at a large retail center in Texas asked how important people felt the tax holiday was (Am. J. Bus. 2007). 565 shoppers indicated that the tax holiday was important in their
Air travelers often complain that recirculated air in the cabin leads to the spread of colds, while the airline industry generally disputes this claim. A 2002 study in the J. Am. Med. Assoc. reported the following information for passengers on flights with and without recirculated air:Assume these
The authors of the article “Adjuvant Radiotherapy and Chemotherapy in Node- Positive Premenopausal Women with Breast Cancer” (New Engl. J. Med. 1997: 956–962) reported on the results of an experiment designed to compare treating cancer patients with only chemotherapy to treatment with a
The article “Luck of the Draw: Creating Chinese Brand Names” (J. Advertising Res. 2008: 523–530) explores the use of “lucky” brand names in China, and whether that use varies by the uncertainty in a brand’s business environment (its market sector). In a sample of 654 brands from sectors
In medical investigations, the ratio h = p1/p2 is often of more interest than the difference p1 − p2 (e.g., individuals given a. Use these facts to obtain a large-sample 95% CI formula for estimating ln(θ), and then a CI for θ itself.b. The article “Low-Dose Aspirin for
In a 2017 class project, two team members each approached 50 students on campus (randomly selected using systematic sampling). Each student was asked to participate in a survey, but the survey itself was a ruse: the real goal was to see who would agree to be surveyed by Melissa (who has a British
Refer to the railway repair time data in Exercise 16. Carry out a test at significance level .01 to see whether the population standard deviation for time-to-repair is larger for high rail breaks than for low rail breaks.Data From Exercise 16 The accompanying time-to-repair (min) data for both high
Statin drugs are used to decrease cholesterol levels and therefore hopefully to decrease the chances of a heart attack. In a British study (“MRC/BHF Heart Protection Study of Cholesterol Lowering with Simvastin in 20,536 High-Risk Individuals: A Randomized Placebo-Controlled Trial,” Lancet
Return to the fat loss experiment described in Exercise 24. Calculate a 95% CI for the ratio of the population standard deviations for the experimental and control groups.Data From Exercise 24 24. The article "Supervised Exercise Versus Non-Supervised Exercise for Reducing Weight in Obese Adults"
For the data of the previous exercise we want to compare population medians for the runs in Denver versus the runs in Phoenix.Data From Previous Exercisea. Use software to generate a bootstrap sample of differences of medians. Check the bootstrap distribution for normality using a normal
In this application from major league baseball, the populations represent an abstraction of what the players can do, so the populations will vary from year to year. The Colorado Rockies and the Arizona Diamondbacks played nine games in Phoenix and ten games in Denver in 2001. The The thinner air in
Can the right diet help us cope with diseases associated with aging such as Alzheimer’s disease? A study (“Reversals of Age-Related Declines in Neuronal Signal Transduction, Cognitive, and Motor Behavioral Deficits with Blueberry, Spinach, or Strawberry Dietary Supplement,” J. Neurosci. 1999:
For the data of the previous exercise, we now want to test the hypothesis of equal population means.Data from in previous exercise a. Carry out a two-tailed test using the two sample t test. Although this test requires normal data, it will still work pretty well for moderately nonnormal data.
Recall the scenario from Example 10.8 about the experiment in the low-level college mathematics course. Here are the 85 final exam scores for those in the experimental group (E) and the 79 final exam scores for those in the control group (C):a. Determine a 95% confidence interval for the difference
Researchers at the University of Alaska have been trying to find inexpensive feed sources for Alaska reindeer growers (“Effects of Two Barley-Based Diets on Body Mass and Intake Rates of Captive Reindeer During Winter,” Poster Presentation: School of Agriculture and Land Resources Management,
Return to the data of Example 10.8.a. Carry out a two-tailed permutation test for the hypothesis of equal population means.b. Compare the results for (a) and Example 10.8. Why should you have expected (a) and Example 10.8 to give similar results? Example 10.8 A study was carried out in an attempt
Two students, Miguel Melo and Cody Watson, compared textbook prices at the campus bookstore and Amazon.com. To be fair, they included the sales tax for the local store and added shipping for Amazon. Here are the prices for a sample of 27 books.a. Determine a 95% confidence interval for the
For the data of Example 10.8 it might be more appropriate to compare medians.a. Find the medians for the two groups. With the help of a stem-and-leaf display for each group, explain why the medians are much closer than the means.b. Carry out a two-tailed permutation test to compare population
Compare bootstrapping with approximate permutation tests in which random permutations are used. Discuss the similarities and differences.
Assume that X is uniformly distributed on [−1, 1] and the Y distribution is uniform on the two intervals [–101, –100] and [100, 101]. Thus the means are both 0, but the variances differ substantially. We take random samples of size three from each distribution and apply a permutation test for
Consider the pooled t variable Tp from part (b) of the previous exercise.a. Use this t variable to obtain a pooled t confidence interval formula for μ1 – μ2.b. The article “Effect of Welding on a High-Density Polyethylene Liner” (J. Mater. Civil Engr. 1996: 94–100) reported the
Refer back to the scenario of the previous exercise.a.b. Show that the pooled t variable.Data From Pervious Exercise Verify that the standardized variable [(X - Y) - (μ₁ has a standard normal distribution. −μ₂)]/√√0² (1/m+1/n)
Let R1 be a rejection region with significance level a for testing a. Show that the UIT is a level α test.b. H01:0₁ versus Hal: 02₁, and let R₂ be a level a rejec- tion region for testing Ho2:02 versus Ha2: 092, where ₁ and ₂ are two disjoint sets of possible values of 0.
Is touching by a coworker sexual harassment? This question was included on a survey given to federal employees, who responded on a scale of 1–5, with 1 meaning a strong negative and 5 indicating a strong yes. The table summarizes the results.Of course, with 1–5 being the only possible values,
The insulin-binding capacity (pmol/mg protein) was measured for four different groups of rats: (1) Nondiabetic, (2) Untreated diabetic, (3) Diabetic treated with a low dose of insulin, (4) Diabetic treated with a high dose of insulin. The accompanying table gives sample sizes and sample standard
Are male college students more easily bored than their female counterparts? This question was examined in the article “Boredom in Young Adults—Gender and Cultural Comparisons” (J. Cross-Cult. Psych. 1991: 209–223). The authors administered a scale called the Boredom Proneness Scale to 97
To assess the tendency of people to rationalize poor performance, 246 college students were randomly assigned to one of two groups: a negative feedback group and a positive feedback group. All students took a test which asked them to identify people’s emotions based on photographs of their faces.
The accompanying data on the alcohol content of wine is representative of that reported in a study in which wines from the years 1999 and 2000 were randomly selected and the actual content was determined by laboratory analysis (London Times August 5, 2001).The two-sample t test gives a test
The accompanying summary data on the ratio of strength to cross-sectional area for knee extensors is taken from the article “Knee Extensor and Knee Extensor and Knee Flexor Strength: Cross- Sectional Area Ratios in Young and Elderly Men” (J. Gerontol. 1992: M204–M210).Does this data suggest
After the Enron scandal in the fall of 2001, faculty in accounting began to incorporate ethics more into accounting courses. One study looked at the effectiveness of such educational intervention “pre-Enron” and “post-Enron.” The data below shows students’ improvement in score on the
The authors of the article “Dynamics of Canopy Structure and Light Interception in Pinus elliotti, North Florida” (Ecol. Monogr. 1991: 33–51) planned an experiment to determine the effect of fertilizer on a measure of leaf area. A number of plots were available for the study, and half were
The article “Two Parameters Limiting the Sensitivity of Laboratory Tests of Condoms as Viral Barriers” (J. Test. Eval. 1996: 279– 286) reported that, in brand A condoms, among 16 tears produced by a puncturing needle, the sample mean tear length was 74.0 lm, whereas for the 14 brand B tears,
An experiment was carried out to compare various properties of cotton/polyester spun yarn finished with softener only and yarn finished with softener plus 5% DP-resin (“Properties of a Fabric Made with Tandem Spun Yarns,” Textile Res. J. 1996: 607– 611). One particularly important
Is the number of export markets in which a firm sells its products related to the firm’s return on sales? The article “Technology Industry Success: Strategic Options for Small and Medium Firms” (Bus. Horizons, Sept.–Oct. 2003: 41–46) gave the accompanying information on the number of
Suppose when using a two-sample t procedure that m < n, and show that m > m − 1. (This is why some authors suggest using min(m − 1, n − 1) as df in place of Welch’s formula). What impact does this have on the CI and test procedure?
A group of 115 University of Iowa students was randomly divided into a build-up condition group (m = 56) and a scale down condition group (n = 59). The task for each subject was to build his or her own pizza from a menu of 12 ingredients. The build-up group was told that a basic cheese pizza costs
Reconsider the “regression through the origin” situation presented in the previous exercise. Consider the following three estimators for the slope parameter β (one of which is the mle obtained in the previous exercise):a. Show that all three of these estimators are unbiased for β.b. Determine
Regression through the origin. Let x denote the number of items in an order and y denote time (min) necessary to process the order. Processing time may be determined by various factors other than order size. So for any particular value of x, we now regard the value of total production time as
Let X, the payoff from playing a certain game, have pmfa. Verify that p(x; θ) is a legitimate pmf, and determine the expected payoff.b.c. What is the approximate variance of the mle when n is large? p(x; 0) = = 0 (1 - 0) ²0x x = -1 - 0)²0x x=0,1,2,...
The principle of unbiased estimation has been criticized on the grounds that in some situations the of fruit has a Poisson distribution with parameter μ. You are going to purchase two such pieces of fruit and wish to estimate 0=e-2", the probability that neither of these has any blemishes. But
Here is a result that allows for easy identification of a minimal sufficient statistic: Suppose there is a function t(x1, …, xn) such that for any two sets of observations a. Identify a minimal sufficient statistic when the Xi’s are a random sample from a Poisson distribution.b. Identify a
The fraction of a bottle that is filled with a particular liquid is a continuous random variable X with pdf ƒ(x; θ) = θxθ–1 for 0 < x < 1 (where θ > 0).a. Obtain the method of moments estimator for θ.b. Is the estimator of (a) a sufficient statistic? If not, what is a sufficient
Let p denote the proportion of all individuals who are allergic to a particular medication. An investigator tests individual after individual to obtain a group of r individuals who have the allergy. Let Xi = 1 if the ith individual tested has the allergy and Xi = 0 otherwise (i = 1, 2, 3,…).
For 0 < θ < 1 consider a random sample from a uniform distribution on the interval from θ to 1/θ. Identify a sufficient statistic for h.
Survival times have the exponential distribution with pdf ƒ(x; λ) = λe –λx for x ≥ 0, where λ > 0 is unknown. However, we wish to estimate the mean μ = 1/λ based on the random sample X1, X2, …, Xn, so let’s re-express the pdf in the form (1/μ)e–x/μ.a. Find the
In this section, it was established that and X̅ are both sufficient statistics for estimating the parameter μ of a Poisson distribution. We know that E(X̅) = μ, but it is also true that E(S2) = σ2 = μ for Poisson data. So, another unbiased estimator for μ is Which of these three
The probability that any particular component of a certain type works in a satisfactory manner is p. If n of these components are independently selected, then the statistic X, the number among the selected components that perform in a satisfactory manner, is sufficient for p. You must purchase two
Suppose waiting time for delivery of an item is uniform on the interval from θ1 to θ2. Consider a random sample of n waiting times, and use the factorization theorem to show that the sample minimum and maximum are a pair of jointly sufficient statistics for θ1 and θ2.
Messages are sent repeatedly across a noisy communication system until rarrive successfully. Let X = the number of transmission a required, so that X has a negative binomial distribution with parameters r (known) and p (unknown). Determine a sufficient statistic for p based on a random sample X1, .
Identify a pair of jointly sufficient statistics for the two parameters of a beta distribution based on a random sample of size n from that distribution.
Identify a pair of jointly sufficient statistics for the two parameters of a gamma distribution based on a random sample of size n from that distribution.
The long run proportion of vehicles that pass a certain emissions test is p. Suppose that three vehicles are independently selected for testing. Let Xi = 1 if the ith vehicle passes the test and Xi = 0 otherwise (i = 1, 2, 3), and let T = X1 + X2 + X3. Use the definition of sufficiency to show
Six Pepperidge Farm bagels were weighed, yielding the following data (grams):a. Assuming that the six bagels are a random sample and the weight is normally distributed, estimate the true average weight and standard deviation of the weight using maximum likelihood.b. Again assuming a normal
Two different computer systems are monitored for a total of n weeks. Let Xi denote the number of breakdowns of the first system during the ith week, and suppose the Xi’s are independent and drawn from a Poisson distribution with parameter μ1. Similarly, let Yi denote the number of breakdowns of
A random sample of n bike helmets manufactured by a company is selected. Let X = the number among the n that are flawed, and let p = P(flawed). Assume that only X is observed, rather than the sequence of S’s and F’s.a. Derive the maximum likelihood estimator of p. If n = 20 and x = 3, what is
Show that is a consistent estimator of h (in the mean-square sense) if and only if both 0
Show that MSE (the mean squared error proposition from earlier in this section). Write μ for Expand the two quadratic expressions, and use the variance shortcut formula to rewrite (Ô) = E[(Ô – 0)²] = V(Ô) + [E(Ô) — 0]²
Why should χv2 be approximately normal for large v? What theorem applies here, and why?
Show that the t pdf approaches the standard normal pdf for large df values. T(x+1/2)/[√x (x)] → 1 and (1+a/x)* → ea asx→ ∞.
An insurance policy issued to a boat owner has a deductible amount of $1000, so the amount of damage claimed must exceed this deductible before there will be a payout. Suppose the amount (1000s of dollars) of a randomly selected claim is a continuous rv with pdf ƒ(x) = 3/x4 for x > 1. Consider
The cdf of checkout duration X as described in Exercise 1 isUse this to compute the following:a. P(X ≤ 1)b. P(.5 ≤ X ≤ 1)c. P(X > .5)d. The median checkout duratione. F′(x) to obtain the density function ƒ(x)Data From Exercises 1Let X denote the amount of time for which a book on 2-h
Suppose the diameter at breast height (in.) of trees of a certain type is normally distributed with μ = 8.8 and σ = 2.8, as suggested in the article “Simulating a Harvester-Forwarder Softwood Thinning” (Forest Products J., May 1997: 36–41).a. What is the probability that the diameter of a
A store will order q gallons of a liquid product to meet demand during a particular time period. This product can be dispensed to customers in any amount desired, so demand during the period is a continuous random variable X with cdf F(x). There is a fixed cost c0 for ordering the product plus a
Consider randomly selecting two points A and B on the circumference of a circle by selecting their angles of rotation, in degrees, independently from a uniform distribution on the interval [0, 360]. Connect points A and B with a straight line segment. What is the probability that this random chord
Consider the following technique, called the accept–reject method, for simulating values from a continuous distribution ƒ. Identify a distribution g from which values can already be simulated and a constant c ≥ 1 such that ƒ(x) ≤ cg(x) for all x. Proceed as follows: (1) Generate Y ∼ g
Suppose 51% of the individuals in a certain population have brown eyes, 32% have blue eyes, and the remainder have green eyes. Consider a random sample of 10 people from this population.a. What is the probability that 5 of the 10 people have brown eyes, 3 of 10 have blue eyes, and the other 2 have
At a certain university, 20% of all students are freshmen, 18% are sophomores, 21% are juniors, and 41% are seniors. As part of a promotion, the university bookstore is running a raffle for which all students are eligible. Ten students will be randomly selected to receive prizes (in the form of
Suppose that X and Y are independent rvs with moment generating functions MX(t) and MY(t), respectively. If Z = X + Y, show that MZ(t) = MX(t) · MY(t).
In practice, it is often desired to predict the value of a variable Y from the known value of some other variableX. For example, a doctor might wish to predict the lifespan Y of someone who smokes X cigarettes a day, or an engineermay require predictions of the tensile strength Y of steel made with
Let X1, X2, and X3 represent the times A necessary to perform three successive repair tasks at a service facility. Suppose they are independent, normal rvs with expected values μ1, μ2, and μ3 and variances σ21 ; σ22 ; and σ23 , respectively.a.b.c.d.e. If μ₁ = μ₂ =
For men, pulse rates (in beats per minute) are normally distributed with mean 70 and standard deviation 10. Women’s pulse rates are normally distributed with mean 77 and standard deviation 12. Let X̅ = the sample average pulse rate for a random sample of 40 men and let Y̅ = the sample average
A class has 10 mathematics majors, 6 computer science majors, and 4 statistics majors. A committee of two is selected at random to work on a problem. Let X be the number of mathematics majors, and let Y be the number of computer science majors chosen.a. Determine the joint probability mass function
A one-foot-long stick is broken at a point X (measured from the left end) chosen randomly uniformly along its length. Then the left part is broken at a point Y chosen randomly uniformly along its length. In other words, X is uniformly distributed between 0 and 1 and, given X = x, Y is uniformly
Suppose that X is uniformly distributed between 0 and 1. Given X = x, Y is uniformly distributed between 0 and x2.a. Determine E(Y|X = x) and then V(Y|X = x).b. Determine ƒ(x, y) using ƒX(x) and ƒYΙX(yΙx).c. Determine ƒY(y).
Consider three ping-pong balls numbered 1, 2, and 3. Two balls are randomly selected with replacement. If the sum of the two resulting numbers exceeds 4, two balls are again selected. This process continues until the sum is at most 4. Let X and Y denote the last two numbers selected. Possible (X,
Let X be a random digit (0, 1, 2, …, 9 are equally likely), and let Y be a random digit not equal to X. That is, the nine digits other than X are equally likely for Y.a. Determine pX(x), pY|X(y|x), and pX,Y(x, y).b. Determine a formula for E(Y|X = x).
This week the number X of claims coming into an insurance office is Poisson with mean 100. The probability that any particular claim relataes to automobile insurance is .6, independent of any other claim. If Y is the number of automobile claims, then Y is binomial with X trials, each with
In the previous exercise, show that the distribution of Y is Poisson with mean 60. [You will need to recognize the Maclaurin series expansion for the exponential function.] Use the knowledge that Y is Poisson with mean 60 to find E(Y) and V(Y).Data from exercise 85:This week the number X of claims
For a few years, the SAT consisted of three components: writing, critical reading, and mathematics. Let W = SAT Writing score and X = SAT Critical Reading score for a randomly selected student. According to the College Board, in 2012 W had mean 488 and standard deviation 114, while X had mean 496
Use the intuitive argument sketched in this section to obtain the general formula for the joint pdf of two order statistics given in the last proposition.
A sample of 20 students who had recently taken introductory statistics yielded the following information on brand of calculator owned (T = Texas Instruments, H = Hewlett- Packard, C = Casio, S = Sharp):a. Estimate the true proportion of all such students who own a Texas Instruments calculator.b. Of
The number of dirt specks on a randomly selected square yard of polyethylene film of a certain type has a Poisson distribution with a mean value of 2 specks per square yard. Consider a random sample of n = 5 film specimens, each having area 1 square yard, and let X̅ be the resulting sample mean
Let X represent the amount of gasoline (gallons) purchased by a randomly selected customer at a gas station. Suppose that the mean value and standard deviation of X are 11.5 and 4.0, respectively.a. In a sample of 50 randomly selected customers, what is the approximate probability that the sample
The first assignment in a statistical computing class involves running a short program. If past experience indicates that 40% of all students will make no programming errors, compute the (approximate) probability that in a class of 50 studentsa. At least 25 will make no errors.b. Between 15 and 25
The Central Limit Theorem says that X is approximately normal if the sample size is large. More specifically, the theorem states that the standardized X̅ has a limiting standard normal distribution. That is, the rv (X̅ – μ) / (σ / √n) has a distribution approaching the standard normal. Can
A friend commutes by bus to and from work 6 days/week. Suppose that waiting time is uniformly distributed between 0 and 10 min, and that waiting times going and returning on various days are independent of each other. What is the approximate probability that total waiting time for an entire week is

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