Questions and Answers of Modern Mathematical Statistics With Applications

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
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
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
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
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
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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.
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
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
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
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.
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
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
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
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.
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
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
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
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
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
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
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
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.
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
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
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-
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
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
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
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
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”
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
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
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
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
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?
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
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
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
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.
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
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
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
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
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
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 =
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
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
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
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
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
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
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
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
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].
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
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
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
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
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
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
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
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
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
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
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),
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
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
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.
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 =
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
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.
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
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.
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
It can be shown that if Yn converges in probability to a constant τ, then h(Yn) converges to h(τ) for any function h(·) that is continuous at τ. Use this to obtain a consistent estimator for the
Apply the Law of Large Numbers to show that χv2 / v approaches 1 as v becomes large.

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