Questions and Answers of Modern Mathematical Statistics With Applications

The Bonett CI for a population variance σ2 mentioned at the end of Section 8.4, unlike the chi-squared method, does not hinge on population normality. This interval involves a transformation along
Consider again the weight gain data from the previous exercise.a. Use the method of Section 8.4 to obtain a 95% confidence interval for σ. Discuss normality for the weight gain data: do you have
Consider the situation described in the previous exercise.a.b. What is the probability that exactly one of the n original observations is smaller than the median?c.d.e. Determine the confidence level
Exercise 4 gave data on x = BOD mass loading and y = BOD mass removal. Values of relevant summary quantities areData From Exercise 4 a. Obtain the equation of the least squares line.b. Predict
The scientist Francis Galton, an early developer of regression methodology, used “midparent height,” the average of the father’s and mother’s heights, in order to predict children’s
Data from the EPA’s Fuel Efficiency Guide suggests an approximate linear relationship between y = highway fuel efficiency (mpg) and x = weight (lbs) for midsize cars. Suppose the equation of the
The cetane number is a critical property in specifying the ignition quality of a fuel used in a diesel engine. Determination of this number for a biodiesel fuel is expensive and time-consuming The
A number of studies have shown lichens (certain plants composed of an alga and a fungus) to be excellent bioindicators of air pollution. The article “The Biomonitor of Atmospheric Nitrogen and
A regression of y = calcium content (g/L) on x = dissolved material (mg/cm2) was reported in the article “Use of Fly Ash or Silica Fume to Increase the Resistance of Concrete to Feed Acids” (Mag.
Visual and musculoskeletal problems associated with the use of visual display terminals (VDTs) have become rather common in recent years. Some researchers have focused on vertical gaze direction as a
For many years, rubber powder has been used in asphalt cement to improve performance. The article “Experimental Study of Recycled Rubber-Filled High-Strength Concrete” (Mag. Concrete Res. 2009:
Show that under the assumptions of the simple linear regression model, the mles of β0 and β1 are identical to the least squares estimates. (1) The pdf of Yi is normal with mean li =
Verify Property 2 of the correlation coefficient: the value of r is independent of the units in which x and y are measured; that is, if x = ax + c and y by; +d, a > 0, b>0, then r for the (x,y)
It is time-consuming and costly to have trucks stop in order to be weighed on a static scale. The Minnesota Department of Transportation considered using a scale that would weigh trucks while they
The body armor report introduced in Example 12.1 also reported studies in which two different methods were used to measure the same body armor deformations from bullet impact. The goal of these
The wicking properties of certain fabrics were investigated in the article “Thermal and Water Vapor Transport Properties of Selected Lofty Nonwoven Products” (Textile Res. J. 2017: 1413–1424).
Hydrogen content is conjectured to be an important factor in porosity of aluminum alloy castings. The article “The Reduced Pressure Test as a Measuring Tool in the Evaluation of Porosity/Hydrogen
The article “Optimization of HVAC Control to Improve Comfort and Energy Performance in a School” (Energy Engr. 2008: 6–22) gives an analysis of the electrical and gas costs for a high school in
Many parts of the USA are experiencing increased erosion along waterways due to increased flow from global climate change. The report “Evaluation and Assessment of Environmentally Sensitive Stream
Global warming is a major issue, and CO2 emissions are an important part of the discussion The article “Effects of Atmospheric CO2 Enrichment on Biomass Accumulation and Distribution in
The power for the t test for H0: β1 = β10 can be computed in the same manner as it was computed for the t tests, using the noncentral t distribution. If the alternative value of
Let x be number of hours per week of studying and y be grade point average. Suppose we have one sample of (x, y) pairs for females and another for males. Then we might like to test the hypothesis H0:
Athletic footwear is a multibillion-dollar industry, and manufacturers need to understand which features are most important to customers. The article “Overall Preference of Running Shoes Can Be
The accompanying data on sale price (thousands of dollars), size (thousands of sq. ft.), and land-to-building ratio for 10 large industrial properties appeared in the paper “Using Multiple
Roads in Egypt are notoriously haphazard, often lacking proper engineering consideration. The article “Effect of Speed Hump Characteristics on Pavement Condition” (J. Traffic Transp. Engr. 2017:
There has been a shift recently away from using harsh chemicals to dye textiles in favor of natural plant extracts. The article “Ecofriendly Dyeing of Silk with Extract of Yerba Mate” (Textile
Carbon nanotubes (CNTs) are used for everything from structural reinforcement to communication antennas. The article “Fast Mechanochemical Synthesis of Carbon Nanotube-Polyanaline Hybrid
Electric vehicles have greatly increased in popularity recently, but their short battery life (with a few exceptions) continues to be of concern. The article “Design of Robust Battery Capacity
Steel microfibers are an alternative to conventional rebar reinforcement of concrete structures. The article “Measurement of Average Tensile Force for Individual Steel Fiber Using New Direct
The article “Selling Prices/Sq. Ft. of Office Buildings in Downtown Chicago – How Much Is It Worth to Be an Old but Class-A Building?” (J. Real Estate Res. 2010: 1–22) considered a regression
Many studies have researched how traffic load affects road asphalt, but fewer have examined the effect of extreme cold weather. The article “Effects of Large Freeze-Thaw Cycles on Stiffness and
Cerium dioxide (also called ceria) is used in many applications, including pollution control and wastewater treatment. The article “Mechanical Properties of Gelcast Cerium Dioxide from 23 to 1500
Sometimes an investigator wishes to decide whether a group of m predictors (m > 1) can simultaneously be eliminated from the model. The null hypothesis says that all β’s associated with these m
Ethyl vanillin is used in food, cosmetics, and pharmaceuticals for its vanilla-like scent. The article “Determination and Cor Correlation of Ethyl Vanillin Solubility in Different Binary Solvents
In the construction industry, a “project labor agreement” (PLA) between clients and contractors stipulates that all bidders on a project will use union labor and abide by union rules. The article
The following data set has ratings from ratebeer.com along with values of IBU (international bittering units, a measure of bitterness) and ABV (alcohol by volume) for 25 beers. Notice which beers
Consider fitting the model Y = β0 + β1x1 + b2x2 + ε to the following data:a. Determine X and y, and express the normal equations in terms of matrices.b. Determine the β̂
The article “Promoting Healthy Choices: Information versus Convenience” (Amer. Econ. J.: Appl. Econ. 2010: 164–178) reported on a field experiment at a fast-food sandwich chain to see whether
Consider the model Y = β0 + β1x1 + ε for the following data:a. Determine the X and y matrices and express the normal equations in terms of matrices.b. Determine the β̂
Suppose that we have Y1,…, Ym ∼ N(μ1,σ), Ym+1, …, Ym+n ∼ N(μ2, σ), and all m + n observations are independent. These are the assumptions of the pooled t procedure in Section 10.2.
a. Prove that the hat matrix H satisfies H2 = H.b. Prove Equation (12.19). Look at the derivation of Equation (12.18).Equation 12.19Equation 12.18 Cov(YY) = ² (I-H)
a. Show that the normal equations are equivalent to X'e = 0.b. Use part (a) to prove the ANOVA identity SST = SSE + SSR by showing that (ŷ - y̅)'e = 0. Part (a) also implies that each row of X' is
Use Eqs. (12.18) and (12.19) to show that each of the leverages is between 0 and 1, and therefore the variances of the predicted values and residuals are between 0 and σ2.Equation 12.18Equation
The measurements here are similar to those in Example 12.31, except that here the students did the measuring at home, and the results suffered in accuracy.a. Regress wingspan on the other two
Refer back to the centered simple regression model in Exercise 102.a. Show that the leverage hii, the ith diagonal entry of the hat matrix H, is given byb. Show that the sum of the leverages in
In Example 12.32, the probability of cancer metastasizing was given by the logistic regression model with β0 = −2 and β1 = 0.5.Example 12.32a.b. Explain what happens to the odds when x is
Adolescents are getting less sleep that ever before, and this can have serious behavioral repercussions. The article “Dose-Dependent Associations Between Sleep Duration and Unsafe Behaviors Among
A major electronics retailer sensibly believes that customers are more likely to redeem an emailed coupon if it’s worth more money. With x = coupon discount amount ($), and Y = 1 if a customer
The pharmaceutical industry has increasingly developed “nanoformulations” for drug delivery, but quality control at such a small scale is tricky. The article “Quality by Design Approach Using
The article “Consumer Attitudes Toward Genetic Modification and Other Possible Production Attributes for Chicken” (J. Food Distr. Res. 2005: 1–11) reported a survey of 498 randomly selected
Road trauma is the leading cause of death and injury among young people in Australia. The article “The Journey from Traffic Offender to Severe Road Trauma Victim: Destiny or Preventive
Whale-watching is big business in Alaska, particularly around salmon release sites where whales tend to congregate. The article “Humpback Whales Feed on Hatchery-Released Juvenile Salmon” (Roy.
The article “Developing Coal Pillar Stability Chart Using Logistic Regression” (J. Rock Mech. Mining Sci. 2013: 55–60) includes the following data on x1 = height– width ratio, x2 =
Benford’s Law, introduced in Chapter 3 Exercise 19, postulates that the lead digits (1, 2, …, 9) in a large data set should follow the rule (So, for example, the proportion of numbers with a
A retail bookstore manager is re-evaluating weekday staffing by looking at recent sales. The accompanying table summarizes a sample of 92 recent weekday sales.Assuming these sales are representative
A statistics department at a large university maintains a tutoring center for students in its introductory service courses The center has been staffed with the expectation that 40% of its
Modeling the proliferation of E. coli in farm animals is critical to a safe food supply. The article “Ecological and Genetic Determinants of Plasmid Distribution in E. coli” (Environ. Biol. 2016:
An article in Annals of Mathematical Statistics reports the following data on the number of borers (i.e., insects that bore into wood) in each of 120 groups of borers. Does the Poisson pmf provide a
The following data on X = number of corrosion defects per segment is consistent with data on one of the pipelines described in the article “The Negative Binomial Distribution as a Model for
Reconsider the tax holiday data of Exercise 60(b) in Chapter 10. Use a χ2 statistic to test the hypothesis of equal population proportions. The χ2 statistic should be the square of the z statistic
Should illegal downloading of intellectual property (music, images, etc.) be punished? This question was asked of 501 teenagers in a study published by KRC Research (January, 2008). The teenagers
The authors of the article cited in the previous exercise also considered the relationship between children’s dental health and their parents’ education level. In the accompanying table,
Brushing your teeth helps prevent cavities, doesn’t it? Consider the following data from a survey and subsequent dental exam of Italian 12-year-olds (“Influence of Occlusal Disorders, Food Intake
Do vacation habits vary by sex? The 2006 Expedia Vacation Deprivation Survey interviewed 968 Canadian adults (Ipsos Insight May 18, 2006). The accompanying table shows each person cross-classified by
How universal is the notion of “green light good, red light bad”? The article “Effects of Personal Experiences on the Interpretation of the Meaning of Colours Used in the Displays and Controls
A random sample of 100 faculty at a university gives the results shown below for professorial rank versus sex.a. Test for a relationship at the 5% level using a chi-squared statistic.b. Test for a
The article “Student-Faculty Interaction in Research Universities” (Res. High. Educ. 2009: 437–459) reported that 20.4% of 3168 students from lower-class families said they frequently talked
The report “Majoring in Money: How American College Students Manage Their Finances” (Sallie Mae 2016) includes the following data on whether students in different age groups have at least one
The accompanying data on x = true density (kg/mm3) and y = moisture content (% d. b.) was read from a plot in the article “Physical Properties of Cumin Seed” (J. Agric. Engr. Res. 1996:
Nickel-based alloys are especially difficult to machine due to characteristics including high hardness and low thermal conductivity. The article “Multi-response Optimization Using ANOVA and
The report cited in the previous exercise also asked students with credit cards how much they pay off each month.Perform a χ2 test, and report your results at the .05 significance level. Be clear
Show that the two formulas for the Kruskal– aWallis test statistic in Expression (14.7) are identical and both equal the original formula for H.Equation 14.7 H 12 n(n + 1) i=1 1 2 - J₁ . " +
Refer back to Exercise 2. The median home value in the state of Texas in August 2019 was $197,000. Use the data in Exercise 2 to test the hypothesis that the true median home price in Houston exceeds
The following data on grip strength (N) for 42 individuals was read from a graph in the article “Investigation of Grip Force, Normal Force, Contact 2008: 734–744):a. What does the data suggest
Child development specialists widely believe that diverse recreational activities can improve the social and emotional conduct of children. The article “Influence of Physical Activity on the Social
Use the Wilcoxon test to analyze the data given in Example 9.12. Example 9.12 Particulate matter from roads contributes to pollution when those particles are washed into nearby waterways by rain. The
Consider the following scene: an actor watches a man in a gorilla suit hide behind haystack A. The actor leaves the area, and the “gorilla” moves to behind haystack B, after which the actor
Consider the following data on resting energy expenditure (REE, in calories per day) for eight subjects both while on an intermittent fasting regimen and while on a standard diet (“Intermittent
The article “Hitting a High Note on Math Tests: Remembered Success Influences Test Preferences” (J. of Exptl. Psych. 2016: 17–38) reported a study in which 130 participants were administered
The following pH measurements at a proposed water intake site appear in the 2011 report “Sacramento River Water Quality Assessment for the Davis Woodland Water Supply Project”:Use the Wilcoxon
The article “Punishers Benefit from Third- Party Punishment in Fish” (Science, 8 Jan 2010: 171) describes an experiment meant to simulate behavior of cleaner fish, so named because they eat
The signed-rank statistic can be represented as S+ = 1 · U1 + 2 · U2+.....+ n · Un where Ui = 1 if the sign of the (xi − µ0) with the ith largest absolute magnitude is
Fifty-three participants performed a series of tests in which a small “zap” was delivered to one compass point, selected at random, on a joystick in their hand. In one setting, subjects were told
Reconsider the calorie-burning data in Exercise 10.Data From Exercise 10a. Utilize the Wilcoxon signed-rank procedure to test H0: μD = 0 versus Ha: μD < 0 for the population of REE
Obtain a 95% signed-rank interval for true average diagnostic time using the data in Exercise 14.Data From Exercise 14 A random sample of 15 automobile mechanics certified to work on a certain type
The accompanying data shows the alcohol content (percent) for random samples of 7 German beers and 8 domestic beers. Does the data suggest that German beers have a different average alcohol content
Obtain a CI for lD of Exercise 17 using the data given there; your confidence level should be roughly 95%.Data From Exercise 17The article “Punishers Benefit from Third- Party Punishment in Fish”
A modification has been made to one assembly line for a particular automobile chassis. Because the modification involves extra cost, it will be implemented throughout all lines only if sample data
The following observations are copper contents (%) for a sample of Bidri artifacts (a type of ancient Indian metal handicraft) at the Victoria and Albert Museum in London (“Enigmas of Bidri,”
Can video games improve balance among the elderly? The article “The Effect of Virtual Reality Gaming on Dynamic Balance in Older Adults” (Age and Ageing 2012: 549– 552) reported an experiment
The article “Mutational Landscape Determines Sensitivity to PD-1 Blockade in Non-Small Cell Lung Cancer” (Science, 3 April 2015) described a study of 16 cancer patients taking the drug Keytruda.
Refer to Exercise 32. Sort the ranks of the Xi’s, so that R1 < R2 < ........< Rm.a.b.c.Use the mean and variance of W to determine E(U) and V(U).Data From Exercise 32 In terms of the
The article “Opioid Use and Storage Patterns by Patients after Hospital Discharge following Surgery” (PLoS ONE 2016) reported a study of 30 women who just had Caesarian sections. The women were
The article “Controlled Clinical Trial of Canine Therapy Versus Usual Care to Reduce Patient Anxiety in the Emergency Department” (PLoS ONE 2019) reported on an experiment in which 80 adult
Obtain the 90% rank-sum CI for µ1 − µ2 using the data in Exercise 25.Data From Exercise 25 In an experiment to compare the bond strength of two different adhesives, each adhesive was
The Wilcoxon rank-sum statistic can be represented as W = R1 + R2 + Rm, where Ri is the rank of Xi − Δ0 among all m + n such differences. When H0 is true, each Ri is equally
The article cited in Example 14.10 also reported the fasting C-peptide levels (FCP, nmol/L) for the children in the study.(Sample sizes are different here than in Example 14.10 due to missing data in
Obtain a 95% CI for µ1 − µ2 using the data in Exercise 27. Is your interval consistent with the result of the hypothesis test in that exercise?Data From Exercise 27 A modification has been
The article “Analyses of Phenotypic Differentiations among South Georgian Diving Petrel Populations Reveal an Undescribed and Highly Endangered Species from New Zealand” (PLoS ONE, June 27, 2018)
The following data on the fracture load (kN) of Plexiglas at three different loading point locations appeared in the article “Evaluating Fracture Behavior of Brittle Polymeric Materials Using an

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