Questions and Answers of Modern Mathematical Statistics With Applications

Dental composites used to fill cavities will decay over time. The article “In vitro Aging Behavior of Dental Composites Considering the Influence of Filler Content, Storage Media and Incubation
Many people suffer back or neck pain due to bulging discs in the lumbar or cervical spine, but the thoracic spine (the section inbetween) is less well-studied. The article “Kinematic analysis of
Is that Yelp review real or fake? The article “A Framework for Fake Review Detection in Online Consumer Electronics Retailers” (Information Processing and Management 2019: 1234–1244) tested
Sustainability in corporate culture is typically described as having three dimensions: economic, environmental, and social. The article “Development of Indicators for the Social Dimension of
Image segmentation is a key tool in computer vision (i.e., helping computers “see” the meaning in pictures). The article “Efficient Quantum Inspired Meta-Heuristics for Multi-Level True Colour
In the context of a randomized block experiment, let Wi denote the sample rank sum associated with the ith population treatment. Show that the Friedman’s test statistic can be re-expressed asUse a
In a study described in the article “Hyaluronan Impairs Vascular Function and Drug Delivery in a Mouse Model of Pancreatic Cancer” (Gut 2013: 112–120), hyaluronan was depleted from mice using
The article “Long Telomeres are Associated with Clonality in Wild Populations of … C. tenuispina” (Heredity 2015: 437– 443) reported the following telomere measurements for (1) a normal arm
Adding mileage information to roadside amenity signs (“Motel 6 in 1.2 miles”) can be helpful but might also increase accidents as drivers strain to read the detailed information at a distance.
The accompanying observations on axial stiffness index resulted from a study of metal-plate connected trusses in which five different plate lengths—4 in., 6 in., 8 in., 10 in., and 12 in.—were
Physicians use a variety of quantitative sensory testing (QST) tools to assess pain in patients, but there is concern about the consistency of such tools. The article “Test-Retest Reliability of
The article “Physiological Effects During Hypnotically Requested Emotions” (Psychosomatic Med. 1963: 334–343) reports the following data (xij) on skin potential in millivolts when the emotions
A certain type of novelty coin is manufactured so that 80% of the coins are fair while the rest have a .75 chance of landing heads. Let θ denote the probability of heads for a novelty coin randomly
The number of customers arriving during a one-hour period at an ice cream shop is modeled by a Poisson distribution with unknown parameter μ. Based on past experience, the owner believes that the
Three assembly lines for the same product have different nonconformance rates: p = .1 for Line A, p = .15 for Line B, and p = .2 for Line C. One of the three lines will be selected at random (but you
A study in the Journal of Marketing Research investigated the relationship between facility conditions at gas stations and aggressiveness in the pricing of gasoline. The accompanying data is based on
Qualifications of male and female head and assistant college athletic coaches were compared in the article “Sex Bias and the Validity of Believed Differences Between Male and Female Interscholastic
A Gallup survey (August 9, 2019) asked adults who consume alcoholic beverages for their favorite type. The following table shows responses separated by the region where each respondent lives:Does the
The Associated Press (Dec. 7, 2005) reported on an international survey about the treatment of terrorist suspects. Random samples of 1000 adults from each of several nations were asked, “Do you
Example 1.4 presented data on the starting salaries of n = 38 civil engineering graduates. Use this data to construct a 95% confidence interval for the population lower quartile (25th percentile) of
The following Houston, TX house prices($1000’s) were extracted from zillow.com in August 2019:Treating these 25 houses as a random sample of all available homes in Houston, calculate a 90% upper
Based on a random sample of 40 observations from any continuous population, construct a confidence interval formula for the population median that has confidence level (at least) 95%.
Let η.3 denote the 30th percentile of a population. Find the smallest sample size n for which P(Y1 ≤ η.3 ≤ Yn) is at least .99.In other words, determine the smallest sample size
At children’s party, kids toss ping-pong balls into the swimming pool hoping to land inside a plastic ring (picture a small hula hoop). Let p denote the probability of successfully tossing a ball
Consider a random sample X1, X2, …, Xn from the Poisson distribution with mean µ. If the prior distribution for µ is a gamma distribution with hyperparameters α0 and β0, show that the
Wait times for Uber rides as people exit a certain sports arena are uniformly distributed on the interval [0, θ] with h unknown. Suppose the following Pareto prior distribution is assigned to
Refer back to Exercise 3.a. Calculate the Bayes estimate of the Poisson mean parameter µ.b. Calculate and interpret a 95% credible interval for µ.Data From Exercise 3 The number of customers
Refer back to Exercise 4.a. Calculate the Bayes estimate of the probability p.b. Calculate and interpret a 95% credible interval for p.Data from in Exercise 4 At children's party, kids toss
Laplace’s rule of succession says that if all n Bernoulli trials have been successes, then the probability of a success on the next trial is (n + 1)/(n + 2). For the derivation, Laplace used a
In a study of 70 restaurant bills, 40 of the 70 were paid using cash. Let p denote the population proportion paying cash.a. Assuming a beta prior distribution for p with α0 = 2 and β0 =
Consider a random sample X1, . . ., Xn from an exponential distribution with parameter λ, and assign to λ a Gamma(α0, β0) prior distribution.a. What is the prior expectation of λ?b.
Consider a random sample X1, . . ., Xn from a Poisson distribution with unknown mean µ, and assign to µ Gamma(α0, β0) prior distribution.a. What is the prior expectation of µ?b. Determine the
Two political scientists wish to forecast the proportion of votes that a certain U.S. senator will earn in her upcoming reelection contest. The first political scientist assigns a Beta(3, 2) prior to
If α and β are large, then the beta distribution can be approximated by the normal distribution using the beta mean and variance given. This is useful in case beta distribution software is
Here are the IQ scores for the 15 first-grade girls from the study mentioned in Example 15.9.Assume that the data is a random sample from a normal distribution with mean µ and σ = 15, and assign to
Consider again the rv X = hazardous flood rate given in Example 4.5. What is the probability that the flood rate isa. At most 40 m3/s?b. More than 40 m3/s? At least 40 m3/s?c. Between 40 and 60
Answer parts (a)–(d) of Exercise 14 for the random variable X, lecture time past the hour, given in Exercise 5.Data From Exercise 14Data From Exercise 5 Let X denote the amount of space occupied by
Recall the distribution of hazardous flood rate used in Example 4.5.a. Obtain the mean and standard deviation of this distribution.b. What is the probability that the flood rate is within 1 standard
Consider the rv X = hazardous flood rate in Example 4.5.a. Find the moment generating function and use it to find the mean and variance.b. Now consider a random variable whose pdf isFind the moment
Consider independent and identically distributed random variables X1, X2, X3, … where each Xi has a discrete uniform distribution on the integers 0, 1, 2, …, 9; that is, P(Xi = k) = 1/10
If two bulbs are randomly selected from the box of lightbulbs described in Exercise 42 and at least one of them is found to be rated 75 W, what is the probability that both of them are 75-W bulbs?
Chebyshev’s inequality, introduced in Chapter 3 Exercise 45, is valid for continuous as well as discrete distributions. It states that for any number k ≥ 1, P(lX – μl ≥kσ) ≤ 1/k2 (see
Consider the pdf for the rv X = magnitude (in newtons) of a dynamic load on a bridge, given in Example 4.7:Write a program to simulate values from this distribution using the inverse cdf method.Data
a. Use Table A.5 to find χ2 .05;2.b. Verify the answer to (a) by integrating the pdf.c. Verify the answer to (a) by using software.Table A.5
Use Table A.6 to find t.102;1.b. Verify the answer to part (a) by integrating the pdf.c. Verify the answer to part (a) using software.Table A.6
a. Use Table A.6 to find t.005,10.b. Use Table A.8 to find F.01,1,10 and relate this to the value you obtained in part (a).c. Verify the answer to part (b) using software.Table A.6Table A.8
a. Use Table A.8 to find F.1,2,4.b. Verify the answer to part (a) using the pdf.c. Verify the answer to part (a) using software.Table A.8
a. Use Table A.6 to find t.25,10.b. Use (a) to find the median of the F1,10 distribution.c. Verify the answer to part (b) using software.Table A.6
Refer back to the previous exercise. Consider the following alternative estimator of the parameter λ:a. Determine the mean, variance, and MSE of b. Which of the two estimators, oris
Refer to Exercise 25. Instead of selecting n = 20 helmets to examine, suppose we examine helmets in succession until we have found r = 3 flawed ones. If the 20th helmet is the third flawed one, what
In Example 7.30, we started with U = I(X1 = 0) and used a conditional expectation argument to obtain an unbiased estimator of the zero-defect probability based on the sufficient statistic. Consider
Exercise 17 provided alcohol percentage data for a sample of 16 beers. The sample standard deviation of those measurements was s = .8483. Construct a 90% CI for the population variance σ2 of
Exercise 24 gave a random sample of 20 ACT scores from students taking college freshman calculus. Calculate a 99% CI for the standard deviation of the population distribution. Is this interval valid
Consider the following core wood density measurements (g/mm3) from a sample of 25 canopy trees in western Thailand (“Radial Variation of Wood Functional Traits Reflect Size-Related Adaptations of
Repeat the analysis of Exercise 40 using the P-value method. Do you arrive at the same conclusion?Data From Exercise 40 Natural cork in wine bottles is subject to deterioration, and as a result
Consider a random sample of n component lifetimes, where the distribution of lifetime is exponential with parameter λ.a.b. Obtain a most powerful test for Ho: λ = 1 versus Ha: λ = .5, and express
Reconsider the one-sample t test of Example 9.27.a. b. Apply part (a) to test the hypotheses of Exercise 59, using the data given there.Example 9.27 show that With t = (x - Ho)/(s/√n), the
Exercise 35 presented data on the pour size of two groups of experienced bartenders, one group pouring into tumblers and the other into highball glasses. Test the hypothesis that the variability in
Refer back to the pooled t procedures described at the end of Section 10.2. The test statistic for testing H0: μ1 – μ2 = Δ0 isShow that when μ1 – μ2 = Δ0 (some
Consider testing the hypothesis of equal population means based on the data in the previous exercise.Data From previous exercisea. Carry out a two-tailed test using the method of Section 10.3. Is the
Consider the box compression data presented in Example 11.1. Carry out an analysis of variance F test at significance level .01, and summarize the results in an ANOVA table.Example 11.1. Example 11.1
The article referenced in Example 11.3 also looked at the time (min) to charge a 4.2-V battery.a. Check the ANOVA assumptions with a normal plot and a test for equal variances.b. Does mean charge
Of the five modules in Example 11.3, the first two are existing commercial devices and the last three are existing commercial devices and the last three are prototypes constructed by the study’s
Repeat the previous exercise supposing that x̅2 = 502.8 in addition to x̅3 = 427:5.Data From previous exerciseData From previous Exercise the previous exercise,
Refer back to the bike helmet data of Exercise 5.a. Apply Tukey’s procedure at the .05 level to determine which bike helmets have honestly significantly different mean peak linear acceleration
Derive the fundamental identity SST = SSTr + SSE by squaring both sides of Equation (11.1) and summing over all i and j.Equation (11.1) Xij - X.. = (Xij — X₁.) + (xį. — X..) )+(−
The article “From Dark to Light: Skin Color and Wages among African Americans” (J. Human Res. 2007: 701–738) includes the following information on hourly wages ($) for representative samples of
The article “Can You Test Me Now? Equivalence of GMA Tests on Mobile and Non-Mobile Devices” (Int. J. Select Assess. 2017: 61–71) describes a study in which 1769 people were recruited through
Subacromial impingement syndrome (SIS) refers to shoulder pain resulting from a particular impingement of the rotator cuff tendon. The article “Evaluation of the Effectiveness of Three
Power for the F test in two-factor ANOVA is calculated using a similar method to the one shown in Section 11.3. For fixed values of α1, α2, …, αI, power calculations are based on the noncentral
For unequal sample sizes, the noncentrality parameter for F test power calculations is Referring to Exercise 27, what is the power of the test when μ2 = μ3, μ1 = μ2 – r,
Many studies have been conducted to measure mercury (Hg) levels in fish, but little information exists on Hg concentrations in marine mammals. The article “Factors Influencing Exposure of North
Check the normality and constant variance assumptions graphically for the data of Example 11.17.Example 11.17 Example 11.17 A consumer product-testing organization wished to compare the annual
The article “Step-Counting Accuracy of Activity Monitors in Persons with Down Syndrome” (J. Intellect. Disabil. Res. 2019: 21–30) describes a study in which 17 people with DS walked for a set
a. Show thatb. E (X₁..X...) = ₁, so that - X₁.. - X... is an unbiased estimator for %₁ in the fixed effects model.
If you go to a major league baseball game, how long do you expect the game to be? From the 2430 games played in 2018, here is a random sample of 25 times (min):This is one of those rare instances in
Refer to the baseball game times in Exercise 31. Calculate an upper confidence bound with confidence level 95% for the population standard deviation of game time. Interpret your interval. Explore the
A study to assess the capability of subsurface flow wetland systems to remove biochemical oxygen demand (BOD, a measure of organic matter in sewage) and various other chemical constituents resulted
Verapamil is used to treat certain heart conditions, including high blood pressure and arrhythmia. Studies continue on the factors that affect the drug’s absorption into the body. The article
Obesity is associated with higher foot load that can potentially increase pain and discomfort, but little research has been done on this relationship in children and its possible effects. A graph in
The article “Effects of Pulmonary Rehabilitation on Exercise Capacity and Disease Impact in Patients with Chronic Obstructive Pulmonary Disease and Obesity” (Physiotherapy 2018: 248–250)
The article “The Effectiveness of Virtual and Augmented Reality in Health Science and Medical Anatomy” (Anatom. Sc. Educ. 2017: 549–559) describes an experiment in which 59 health science
The article “Antimicrobial Activities of Essential Oil of Eight Plant Species from Different Families Against Some Pathogenic Microorganisms” (Res. J. Microbiol. 2016: 28–34) reported on an
The article “Is It Really Good to Talk? Testing the Impact of Providing Concurrent Verbal Protocols on Driving Performance” (Ergonomics 2017: 770–779) reported an experiment in which 20 drivers
Several factors can impact the structural soundness of 3D-printed objects, including the “struts” that connect various pieces. The following data appears in the article “Analyzing the Effects
A 2005 article in Issues in Accounting Education described an experiment in which students in an introductory accounting course were randomly assigned to one of two computer-based learning (CBL)
In a study reported in the article “Can ‘Low-Fat’ Nutrition Labels Lead to Obesity?” (J. Market. Res. 2006: 605–617), students and parents attending a university open house were offered one
Students Philip Hurst and Yuan Chao Jiang investigated the accuracy of three different brands of .22 caliber ammunition at two different distances. The brands were Remington, Winchester, and Federal,
Artificial human joints are usually secured with acrylic bone cement. The following data on the force (in Newtons) required to break an acrylic cement bond under different temperatures and media is
In an experiment described in the article “The Impact of SMS Advertising on Members of a Virtual Community” (J. Advert. Res. 2008: 363–374), researchers worked with an online gaming forum to
The article “Effect of Face Value on Product Valuation in Foreign Currencies” (J. Consum. Res. 2002) describes a classroom experiment involving 97 business students to see whether they could
An experiment was conducted to assess the effect of current and voltage on the tensile strength (ksi) of welds made using a tungsten inert gas (TIG) welding tool, which resulted in the following
The efficiency (%) of a 0.7 L Daihatsu diesel engine at 3100 rpm was determined at various torques (N-m) and coolant temperatures (°C), resulting in the following data kindly provided by the
In a 2018 class activity, 54 students measured how much time (sec) it took to melt each of the following in their mouths: (1) A butterscotch chip, (2) A chocolate chip, (3) A white
The following data is presented in the article “Influence of Cutting Parameters on Drill Bit Temperature” (Ind. Lubr. Tribol. 2007: 186–193); values in the table are temperatures in °C.a.
The number of miles of useful tread wear (in 1000’s) was determined for tires of each of five different makes of subcompact car (factor A, with I = 5) in combination with each of four different
The following data on number of cycles to failure (106) appears in the article “An Experimental Study of Influence of Lubrication Methods on Efficiency and Contact Fatigue Life of Spur Gears” (J.
The authors of the article “Exploring the Impact of Social Media Practices on Wine Sales in US Wineries” (J. Direct Data, Digital Market. Pract. 2016: 272–283) interviewed 361 winery managers.
A study at Bentley College, a large business school in the eastern United States, examined students’ anxiety levels toward the subject of accounting (“Determinants of Accounting Anxiety in
The article “Iron and Manganese Present in Underground Water Promote Biochemical, Genotoxic, and Behavioral Alterations in Zebrafish” (Environ. Sci. Pollut. Res. 2019: 23555–23570) reports the

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