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modern mathematical statistics with applications

Questions and Answers of Modern Mathematical Statistics With Applications

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
Consider the data on maximum load for cracked wood beams presented in Exercise 4. Would it make sense to apply Tukey’s method to this data? Why or why not?Data From Exercise 4The article
Does music affect memorization skills, and does it matter if the music includes vocals/ lyrics? In one experiment (P. Ramos, “The Impact of Music on Short Term Memory and Cognitive Processes,”
One popular approach to lower back pain is to apply a piece of durable tape along the lower spine. In one study, 108 women with lower back pain were randomly assigned to receive one of four
Plastic waste, particularly microplastics in oceans and waterways, has become an increasing global environmental concern. The article “Environmentally Relevant Concentrations of Microplastic
The article “Differences in Impact Performance of Bicycle Helmets During Oblique Impacts” (J. Biomech. Engr. 2018) describes an experiment in which 10 different bicycle helmet brands (4 of each
The article “Load-Carrying Capacity of Lengthwise Cracked Wood Beams Retrofitted by Self-Tapping Screws” (J. Struct. Engr. 2017) provides data on the maximum load (kN) of J = 5 specimens of I = 4
Freezing and thawing out food can adversely affects its texture. In one experiment described in the article “Effects of Freezing Treatments Before Convective Drying on Quality Parameters” (J. of
An experiment to compare I = 5 brands of golf balls involved using a robotic driver to hit J = 7 balls of each brand. The resulting between-sample and within-sample estimates of σ2 were MSTr =
For the data of Exercise 29 find a 90% confidence interval for the ratio of population standard deviations, and relate your CI to the test of Exercise 74.Data From Exercise 29Data From Exercise 74 As
In this exercise, we develop a CI for that is valid whatever the shape of the population distribution as long as it is continuous.Let X1,. . . , Xn be a random sample from the distribution and Y1<
Because of extra-inning games, the median might be a more meaningful statistic for the length-of-game data in the previous exercise. The median length of all 2430 MLB games in 2018 was min.a. Use
Exercise 16 of Chapter 1 presented data on the noise level (dBA) experienced by a sample of 77 individuals working at a particular office.a. Construct a 95% confidence interval for the true average
If X has the pdf ƒX(x) = x/8, 0 < x < 4, find a transformation Y ∼ g(X) such that Y ∼ Unif[0, 1]. The target is to achieve ƒY(y) = 1 for 0 ≤ y ≤ 1. The Transformation Theorem will
For θ > 0 consider a random sample from a uniform distribution on the interval from θ to 2θ, and use the factorization theorem to determine a sufficient statistic for θ. The Factorization
Assume that the number of alpha particles emitted in one second by a particular radioactive source has a Poisson distribution with parameter μ. Consider estimating μ based on a random sample X1,
The number of attempts required to successfully transmit a message across a noisy channel can be modeled by a geometric distribution, whose pmf is (1 – p)x–1p for x = 1, 2, 3, …. To estimate
Here are the alcohol percentages for a random sample of 16 beers (light beers excluded): a. Construct a normal probability plot of the data. Is it plausible that these values represent a sample
Here are the lengths (in minutes) of the 63 nine-inning games from the first week of the 2001 major league baseball season:Assume that this is a random sample of nineinning games (the mean differs by
A random sample of 50 patients who had been seen at an outpatient clinic was selected, and the waiting time to see a physician was determined for each one, resulting in a sample mean time of 40.3 min
The article “Evaluating Tunnel Kiln Performance” (Amer. Ceramic Soc. Bull., Aug. 1997: 59–63) gave the following summary information for fracture strengths (MPa) of n = 169 ceramic bars fired
As health care costs rise and health care systems worldwide become overburdened, patients are made to wait longer for critical procedures. One Canadian study of 539 cardiac patients waiting for
A sample of 66 obese adults was put on a low-carbohydrate diet for a year. The average weight loss was 11 lb and the standard deviation was 19 lb. Calculate a 99% lower confidence bound for the true
A study was done on 41 first-year medical students to see if their anxiety levels changed during the first semester. One measure used was the level of serum cortisol, which is associated with stress.
University administrators wish to estimate the mean time to graduation, for the population of students who actually graduate, to within ± 3 months (one-quarter). It is known that the maximum time to
Young people may feel they are carrying the weight of the world on their shoulders, when what they are actually carrying too often is an excessively heavy backpack. The article “Effectiveness of a
According to Oklahoma State University’s 2015 Food Demand Survey, 859 of 1044 randomly selected adults support mandatory labels on foods produced with genetic engineering (popularly called GMO
In a survey of 1100 drivers, 90% admitted to careless or aggressive driving during the previous six months (“Nine out of Ten Drivers Admit in Survey to Having Done Something Dangerous,” Knight
A June, 2019 Gallup survey of 1018 randomly selected US adults found that 53% supported the government sponsoring a manned mission to Mars.a. Construct and interpret a 95% lower confidence bound for
The article “Repeatability and Reproducibility for Pass/Fail Data” (J. Testing Eval. 1997: 151–153) reported that in n = 48 trials in a particular laboratory, 16 resulted in ignition of a
The article “Teens and Young Adults Embrace Online Multiplayer and Competitive Video Games” (Washington Post, Apr. 3, 2018) reported that, in a survey of 522 Americans age 14–21, 38% said they
As reported by CNBC (Dec. 11, 2018), 57% of people surveyed admitted to shopping online while at work. The survey was based on a random sample of n = 2020 US adults. Construct and interpret a 90%
There is increasing concern within the health sciences community over the use of electronic tobacco products. The article “Exposure to Tobacco and Nicotine Product Advertising: Associations with
The article “Broad Agreement on Most Ideas to Curb School Shootings” from the Gallup.com website reported on a survey carried out from March 5–11, 2018. There was overwhelming support for more
Here are the names of 12 orchestra conductors and their performance times in minutes for Beethoven’s Ninth Symphony:a. Check to see that normality is a reasonable assumption for the performance
A mortgage company wishes to estimate the proportion of all borrowers who default on their home loans, to within a margin of error of 2 percentage points (±.02). What sample size is required to
A recent student project asked students at Cal Poly (where one of the authors teaches) whether they have ever been arrested on alcohol- or drug-related charges, including drunk driving. Out of 57
In a survey, students gave their study time per week (h), and here are the 22 values:A 95% confidence interval for the population mean μ is desired.a. Compute the t-based confidence interval of
Here are 68 weight gains (lb) for pregnant women from conception to delivery (“Classifying Data Displays with an Assessment of Displays Found in Popular Software,” Teach. Statist., Autumn 2002:
Nine Australian soldiers were subjected to extreme conditions, which involved a 100- min walk with a 25-lb pack when the temperature was 40°C (104°F). One of them overheated (above 39°C) and was
Before opening a new location, franchise companies conduct market research to determine if sufficient demand exists for their products. A national sandwich chain recently conducted a survey to
There were 12 first-round heats in the men’s 100-m race at the 1996 Atlanta Summer Olympics. Here are the reaction times in seconds (time to first movement) of the top four finishers of each heat.
Let θ1 and θ2 denote the mean weights for animals of two different species. A biologist wishes to estimate the ratio θ1/θ2. Unfortunately the species are extremely rare, so the estimate will be
The industry standard for the amount of alcohol poured into many types of drinks (e.g., gin for a gin and tonic, whiskey on the rocks) is 1.5 oz. Each individual in a sample of 8 bartenders with at
Let the test statistic T have a t distribution when H0 is true. Give the significance level for each of the following situations:a.b.c. Ha: > Mo, df = 15, rejection region t> 3.733
Consider 95% CIs for two different parameters θ1 and θ2, and let Ai (i = 1, 2) denote the event that the value of θi is included in the random interval that results interval that results in
The one-sample CI for a normal mean and PI for a single observation from a normal distribution were both based on the central t distribution. A CI for a particular percentile (e.g., the 1st
A sample of 12 radon detectors of a certain type was selected, and each was exposed to 100 pCi/L of radon. The resulting readings were as follows:a. Does this data suggest that the population mean
In an experiment designed to measure the time necessary for an inspector’s eyes to become used to the reduced amount of light necessary for penetrant inspection, the sample average time for n = 9
The target thickness for silicon wafers used in a type of integrated circuit is 245 μm. A sample of 50 wafers is obtained and the thickness of each one is determined, resulting in a sample mean
On the label, Pepperidge Farm bagels are said to weigh four ounces each (113 g). A random sample of six bagels resulted in the following weights (in grams):a. Based on this sample, is there any
The article “Development of Novel Industrial Laminated Planks from Sweetgum Lumber” (J. Bridge Engr. 2008: 64–66) provides the following data on the modulus of rupture (psi) for a sample of
Although the U.S. Food and Drug Administration recommends against using kitchen utensils to dose liquid medicines, many people still do so, resulting in dosing errors and even pediatric poisonings.
Exercise 46 in Chapter 1 gave n = 26 observations on escape time (sec) for oil workers in a simulated exercise, from which the sample mean and sample standard deviation are 370.69 and 24.36,
Let F(x, ν, δ) denote the cdf of the noncentral t distribution.a. Determine the power function of an upper-tailed one-sample t test in terms of F.b. Repeat part (a) for the two-tailed one sample t
An aspirin manufacturer fills bottles by weight rather than by count. Since each bottle should contain 100 tablets, the average weight per tablet should be 5 grains. Each of 100 tablets taken from a
Let μ denote true average serum receptor concentration for all pregnant women. The average for all women is known to be 5.63. The article “Serum Transferrin Receptor for the Detection of Iron
Scientists have recently become concerned about the safety of Teflon cookware and various food containers because perfluorooctanoic acid (PFOA) is used in the manufacturing process. An An article in
The authors of the article “Luck of the Draw: Creating Chinese Brand Names” (J. of Advertising Res. 2008: 523–530) counted the number of “strokes” in the characters for the names of 1202
A university library ordinarily has a complete shelf inventory done once every year. Because of new shelving rules instituted the previous year, the head librarian believes it may be possible to save
Natural cork in wine bottles is subject to deterioration, and as a result wine in such bottles may experience contamination. The article “Effects of Bottle Closure Type on Consumer Perceptions of
State DMV records indicate that of all vehicles undergoing emissions testing during the previous year, 70% passed on the first try. A random sample of 200 cars tested in a particular county during
Show that when μ = μ' the one-sample t statistic (9.2) has a noncentral t distribution with n – 1 df and noncentrality parameter d given by (9.4).Equation 9.2 and 9.4 T= X - Ho S/√n (9.2)
Suppose that X, the fraction of a container a.b.c. If n = 50, what is the (approximate) value of c for which the test has significance level .05? that is filled, has pdf f(x;0) = 0x1 for 0 < x < 1
Suppose that each of n randomly selected individuals is classified according to his/her genotype with respect to a particular genetic characteristic and that the three possible genotypes are AA, Aa,
The error in a measurement is normally distributed with mean μ and standard deviation 1. Consider a random sample of n errors, and show that the likelihood ratio test for Ho: μ = 0 versus Ha:
Measurement error in a particular situation is normally distributed with mean value l and standard deviation 4. Consider testing H0: μ = 0 versus Ha: μ ≠ 0 based on a sample of n = 16
This exercise establishes the “duality” between confidence intervals/bounds and hypothesis tests for the one-sample t procedures. a.b.c. Consider the lower-tailed t test of Ho: Mo versus Ha:
The test statistic in the Neyman–Pearson Lemma and the likelihood ratio test statistic K are intimately related. Consider testing H0: θ = θ0 versus Ha: θ = θa, and let Λ* denote the test
A test of whether a coin is fair will be based on n = 50 tosses. Let X be the resulting number of heads. Consider two rejection regions: R1 = {x: either x ≤ 17 or x ≥ 33} and R2 = {x: either
Use the bootstrap hypothesis-testing method described in this section to test H0: μ = 1.5 versus Ha: μ ≠ 1.5 for the alcohol content data presented in Exercise 22.Data From Exercise 22 The
When a drug is recalled for safety concerns (e.g., too many people having serious adverse reactions), the pharmaceutical company making the drug can only re-issue it by convincing the FDA that the
To test the ability of auto mechanics to identify simple engine problems, an automobile with a single such problem was taken in turn to 72 different car repair facilities. Only 42 of the 72 mechanics
After a period of apprenticeship, an organization gives an exam that must be passed to be eligible for membership. Let p = P(randomly chosen apprentice passes). The organization wishes an exam that
A service station has six gas pumps. When no vehicles are at the station, let pi denote the probability that the next vehicle will select pump i (i = 1,2,…, 6). Based on a sample ofa. Show that the
Consider testing a pair of simple hypotheses Ho: 0= 0o versus Ha: 0 = 0₁. Rather than prescribing the significance level and minimizing P(type II error), imagine trying minimize the linear
Standard-size boxes for a particular brand of cereal indicate a net weight of 14 oz. A consumer group purchases a random sample of 50 such cereal boxes and weighs their contents. If the average of
The error X in a measurement has a normal distribution with mean value 0 and variance σ2. Consider testing H0: σ2 = 2 versus Ha: σ2 = 3 based on a random sample X1, …, Xn of
The article “Return Migration, Investment in Children, and Intergenerational Mobility: Comparing Sons of Foreign- and Native- Born Fathers” (J. Hum. Res. 2008: 299–324) presented the following
The accompanying time-to-repair (min) data for both high rail and low rail breaks on curved track appeared in the article “Uncertainty Estimation in Railway Track Life-Cycle Cost” (J. Rail Rapid
Due to recent concerns about player concussions, football helmets have recently increased in both size and mass. Have these changes made a difference? The article “The Effects of Helmet Weight on
Much research has focused on comparing business environment cultures across several countries. The article “Perception of Internal Factors for Corporate Entrepreneurship: A Comparison of Canadian
Returning to the previous exercise, the mean and standard deviation of the number of credit cards for the no-involvement group were 2.22 and 1.58, respectively, whereas the mean and standard
Credit card spending and resulting debt pose very real threats to consumers in general, and the potential for abuse is especially serious among college students. It has been estimated that about
Expert and amateur pianists were compared in a study “Maintaining Excellence: Deliberate Practice and Elite Performance in Young and Older Pianists” (J. Exp. Psychol. Gen. 1996: 331–340). The
Imaging of the colon with a contrast dye to evaluate for injury requires that the colon first be distended by pumping in carbon dioxide. The article “Determination of Normal Distribution of
A 2008 study in the J. Family Econ. Issues compared the work and home habits of female and male lawyers in Canada. All participants in the survey had at least one child; single parents and couples
To decide whether chemistry or physics majors have higher starting salaries in industry, n B.S. graduates of each type are surveyed, yielding x̅ = $61;500 for chemistry and y̅ = $61;000 for
Let μ1 and μ2 denote true average tread lives (miles) for two competing brands of size P205/65R15 tires.a.b. Test Ho: ₁-₂ = 0 versus Ha: M₁-M₂ 70 at level .05 using the following
What affects the time a consumer spends looking at a product on the shelf prior to selection? The following data summarized elapsed time (in seconds) for purchasers of fabric softener and washing-up
A study seeks to compare hospitals based on the performance of their intensive care units. The response variable variable is the mortality ratio, the ratio of the number of deaths over the predicted
According to a 2018 report by the CDC, the mean body mass index (BMI) for American adult men is 29.1 kg/m2, while the mean for women is 29.6 kg/m2. Suppose these are population averages.a. Let X̅ be
The article “Characterization of Bearing Strength Factors in Pegged Timber Connections” (J. Struct. Engr. 1997: 326–332) gave the following summary data on proportional stress limits for
Exercise 22 from Chapter 9 gave the following data on amount (oz) of alcohol poured into a short, wide tumbler glass by a sample of experienced bartenders 2.00, 1.78, 2.16, 1.91, 1.70, 1.67, 1.83,
Which factors are relevant to the time a consumer spends looking at a product on the shelf prior to selection? The article “Effects of Base Price upon Search Behavior of Consumers in a
The article “Gender Differences in Individuals with Comorbid Alcohol Dependence and Post-Traumatic Stress Disorder (Am. J. Addict. 2003: 412–423) reported the accompanying data on total score on
Is the incidence of head or neck pain among video display terminal users related to the monitor angle (degrees from horizontal)? The paper, “An Analysis of VDT Monitor Placement and Daily Hours of
For the data of the previous two exercises, we want a 95% confidence interval for the ratio of population standard deviations. Previous Exercise a. Recall that this method requires the data to
For the data of the previous exercise, now consider testing the hypothesis of equal population variances.a. Carry out a two-tailed test using the method of Section 10.5. Recall that this method

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