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Probability And Statistics For Engineering And The Sciences 9th Edition Jay L. Devore - Solutions
The article "Optimization of Surface Roughness in Drilling Using Vegetable-Based Cutting Oils Developed from Sunflower Oil" (Industrial Lubrication and Tribology, 2011: 271-276) gave the following data on x1 = spindle speed (rpm), x2 = feed rate (mm/rev), x3 = drilling depth (mm), and y = surface
Utilization of sucrose as a carbon source for the production of chemicals is uneconomical. Beet molasses is a readily available and low-priced substitute. The article "Optimization of the Production of b-Carotene from Molasses by Blakeslea Trispora" (J. of Chem. Tech. and Biotech., 2002: 933-943)
Snowpacks contain a wide spectrum of pollutants that may represent environmental hazards. The article "Atmospheric PAH Deposition: Deposition Velocities and Washout Ratios" (J. of Environmental Engineering, 2002: 186-195) focused on the deposition of polyaromatic hydrocarbons. The authors proposed
In the accompanying table, we give the smallest SSE for each number of predictors k (k = 1, 2, 3, 4) for a regression problem in which y = cumulative heat of hardening in cement, x1 = % tricalcium aluminate, x2 = % tricalcium silicate, x3 = % aluminum ferrate, and x4 = % dicalcium silicate.In
Reconsider the wood specific gravity data referred to in Exercise 56.a. Minitab's Best Regression option was used resulting in the accompanying output. Which model(s) would you recommend for investigating in more detail?b. The accompanying Minitab output resulted from applying both the backward
The accompanying scatterplot is based on data provided by authors of the article "Spurious Correlation in the USEPA Rating Curve Method for Estimating Pollutant Loads" (J. of Envir. Engr., 2008: 610-618); here discharge is in ft3/s as opposed to m3/s used in the article. The point on the far right
Pillar stability is a most important factor to ensure safe conditions in underground mines. The authors of "Developing Coal Pillar Stability Chart Using Logistic Regression" (Intl. J. of Rock Mechanics & Mining Sci., 2013: 55-60) used a logistic regression model to predict stability. The article
A study carried out to investigate the relationship between a response variable relating to pressure drops in a screenplate bubble column and the predictors x1 = superficial fluid velocity, x2 = liquid viscosity, and x3 = opening mesh size resulted in the accompanying data ("A Correlation of
Multiple regression output from Minitab for the PAH data of Exercise 53 in the previous section included the following information:Unusual ObservationsWhat does this suggest about the appropriateness of using the previously given fitted equation as a basis for inferences? The investigators actually
Curing concrete is known to be vulnerable to shock vibrations, which may cause cracking or hidden damage to the material. As part of a study of vibration phenomena, the paper "Shock Vibration Test of Concrete" (ACI Materials J., 2002: 361-370) reported the accompanying data on peak particle
The article "Validation of the Rockport Fitness Walking Test in College Males and Females" (Research Quarterly for Exercise and Sport, 1994: 152-158) recommended the following estimated regression equation for relating y = VO2 max (L/min, a measure of cardio respiratory fitness) to the predictors
Air pressure (psi) and temperature (°F) were measured for a compression process in a certain piston-cylinder device, resulting in the following data (from Introduction to Engineering Experimentation, Prentice-Hall, Inc., 1996, p. 153):a. Would you fit the simple linear regression model to the
Composite honeycomb sandwich panels are widely used in various aerospace structural applications such as ribs, flaps, and rudders. The article "Core Crush Problem in Manufacturing of Composite Sandwich Structures: Mechanisms and Solutions" (Amer. Inst. of Aeronautics and Astronautics J., 2006:
An aeronautical engineering student carried out an experiment to study how y = lift drag ratio related to the variables x1 = position of a certain forward lifting surface relative to the main wing and x2 = tail placement relative to the main wing, obtaining the following data (Statistics for
An ammonia bath is the one most widely used for depositing Pd-Ni alloy coatings. The article "Modelling of Palladium and Nickel in an Ammonia Bath in a Rotary Device" (Plating and Surface Finishing, 1997: 102-104) reported on an investigation into how bath composition characteristics affect coating
The accompanying data on x1 = card cylinder speed (rpm), card production rate (kg/h), x3 = number of draw frame doubling, and y = tenacity (RKM) appeared in the article "Impact of Carding Parameters and Draw Frame Doubling on the Properties of Ring Spun Yarn" (J. of Engineered Fibers and Fabrics,
The article "Sensitivity Analysis of a 2.5 kW Proton Exchange Membrane Fuel Cell Stack by Statistical Method" (J. of Fuel Cell Sci. and Tech., 2009: 1-6) used regression analysis to investigate the relationship between fuel cell power (W) and the independent variables x1 = H2 pressure spsid, x2 =
Coir fiber, derived from coconut, is an eco-friendly material with great potential for use in construction. The article "Seepage Velocity and Piping Resistance of Coir Fiber Mixed Soils" (J. of Irrig. and Drainage Engr., 2008: 485-492) included several multiple regression analyses. The article's
Continuous recording of heart rate can be used to obtain information about the level of exercise intensity or physical strain during sports participation, work, or other daily activities. The article "The Relationship Between Heart Rate and Oxygen Uptake During Non-Steady State Exercise"
A sample of n = 20 companies was selected, and the values of y = stock price and k = 15 variables (such as quarterly dividend, previous year's earnings, and debt ratio) were determined. When the multiple regression model using these 15 predictors was fit to the data, R2 = .90 resulted. a. Does the
Does exposure to air pollution result in decreased life expectancy? This question was examined in the article "Does Air Pollution Shorten Lives?" (Statistics and Public Policy, Reading, MA, Addison-Wesley, 1977). Data on y = total mortality rate (deaths per 10,000) x1 = mean suspended particle
Given that R2 = .723 for the model containing predictors x1, x4, x5, and x8 and R2 = .689 for the model with predictors x1, x3, x5, and x6, what can you say about R2 for the model containing predictors a. x1, x3, x4, x5, x6, and x8? Explain. b. x1 and x4? Explain.
An article in Lubrication Engr. ("Accelerated Testing of Solid Film Lubricants," 1972: 365-372) reported on an investigation of wear life (y, in hr) for solid film lubricant. Three sets of journal bearing tests were run on a Mil-L-8937-type film at each combination of three speeds (x1, in rpm), and
Consider the following four (x, y) data sets; the first three have the same x values, so these values are listed only once (Frank Anscombe, "Graphs in Statistical Analysis," Amer. Statistician, 1973: 17-21):For each of these four data sets, the values of the summary statistics Σxi,
What conclusion would be appropriate for an uppertailed chi-squared test in each of the following situations? a. a = .05, df = 4, x2 = 12.25 b. a = .01, df = 3, x2 = 8.54 c. a = .10, df = 2, x2 = 4.36 d. a = .01, k = 6, x2 = 10.20
a. Show that another expression for the chi-squared statistic isWhy is it more efficient to compute x2 using this formula? b. When the null hypothesis is H0: p1 = p2 =... = pk = 1/k (i.e., pi0 = 1/k for all i), how does the formula of part (a) simplify? Use the simplified expression to calculate x2
a. Having obtained a random sample from a population, you wish to use a chi-squared test to decide whether the population distribution is standard normal. If you base the test on six class intervals having equal probability under H0, what should be the class intervals?b. If you wish to use a
Consider a large population of families in which each family has exactly three children. If the genders of the three children in any family are independent of one another, the number of male children in a randomly selected family will have a binomial distribution based on three trials.a. Suppose a
A study of sterility in the fruit fly ("Hybrid Dysgenesis in Drosophila melanogaster: The Biology of Female and Male Sterility," Genetics, 1979: 161-174) reports the following data on the number of ovaries developed by each female fly in a sample of size 1388. One model for unilateral sterility
The article "Feeding Ecology of the Red-Eyed Vireo and Associated Foliage-Gleaning Birds" (Ecological Monographs, 1971: 129-152) presents the accompanying data on the variable X = the number of hops before the first flight and preceded by a flight. The author then proposed and fit a geometric
A certain type of flashlight is sold with the four batteries included. A random sample of 150 flashlights is obtained, and the number of defective batteries in each is determined, resulting in the following data:Let X be the number of defective batteries in a randomly selected flashlight. Test the
Let X = the number of adult police contacts for a randomly selected individual who previously had at least one such contact prior to age 18. The following frequencies were calculated from information given in the article "Examining the Prevalence of Criminal Desistance" (Criminology, 2003:
In a genetics experiment, investigators looked at 300 chromosomes of a particular type and counted the number of sister-chromatid exchanges on each ("On the Nature of Sister-Chromatid Exchanges in 5-Bromodeoxyuridine- Substituted Chromosomes," Genetics, 1979: 1251- 1264). A Poisson model was
The article "A Probabilistic Analysis of Dissolved Oxygen-Biochemical Oxygen Demand Relationship in Streams" (J. Water Resources Control Fed., 1969: 73-90) reports data on the rate of oxygenation in streams at 20°C in a certain region. The sample mean and standard deviation were computed as =
Each headlight on an automobile undergoing an annual vehicle inspection can be focused either too high (H), too low (L), or properly (N). Checking the two headlights simultaneously (and not distinguishing between left and right) results in the six possible outcomes HH, LL, NN, HL, HN, and LN. If
The article "Compatibility of Outer and Fusible Interlining Fabrics in Tailored Garments (Textile Res. J., 1997: 137-142) gave the following observations on bending rigidity smN ? md for medium-quality fabric specimens, from which the accompanying Minitab output was obtained:Would you use a
The article from which the data in Exercise 20 was obtained also gave the accompanying data on the composite mass/outer fabric mass ratio for high-quality fabric specimens.Minitab gave r = .9852 as the value of the Ryan-Joiner test statistic and reported that P-value . .10. Would you use the
The article "A Method for the Estimation of Alcohol in Fortified Wines Using Hydrometer Baumé and Refractometer Brix" (Amer. J. of Enol. and Vitic., 2006: 486-490) gave duplicate measurements on distilled alcohol content (%) for a sample of 35 port wines. Here are averages of those duplicate
The article "Nonbloated Burned Clay Aggregate Concrete" (J. of Materials, 1972: 555-563) reports the following data on 7-day flexural strength of nonbloated burned clay aggregate concrete samples (psi):Test at level .10 to decide whether flexural strength is a normally distributed variable.
The accompanying two-way table was constructed using data in the article "Television Viewing and Physical Fitness in Adults" (Research Quarterly for Exercise and Sport, 1990: 315-320). The author hoped to determine whether time spent watching television is associated with cardiovascular fitness.
In an investigation of alcohol use among college students, each male student in a sample was categorized both according to age group and according to the number of heavy drinking episodes during the previous 30 days ("Alcohol Use in Students Seeking Primary Care Treatment at University Health
Contamination of various food products is an ongoing problem all over the world. The article "Prevalence and Quantitative Detection of Salmonella in Retail Raw Chicken in Shaanxi, China" (J. of Food Production, 2013) reported the following data on the occurrence of salmonella in chicken of three
The article "Human Lateralization from Head to Foot: Sex-Related Factors" (Science, 1978: 1291-1292) reports for both a sample of right-handed men and a sample of right-handed women the number of individuals whose feet were the same size, had a bigger left than right foot (a difference of half a
A random sample of 175 Cal Poly State University students was selected, and both the email service provider and cell phone provider were determined for each one, resulting in the accompanying data. State and test the appropriate hypotheses................................................Cell Phone
The accompanying data on degree of spirituality for samples of natural and social scientists at research universities as well as for a sample of non-academics with graduate degrees appeared in the article "Conflict Between Religion and Science Among Academic Scientists" (J. for the Scientific Study
Three different design configurations are being considered for a particular component. There are four possible failure modes for the component. An engineer obtained the following data on number of failures in each mode for each of the three configurations. Does the configuration appear to have an
A random sample of smokers was obtained, and each individual was classified both with respect to gender and with respect to the age at which he/she first started smoking. The data in the accompanying table is consistent with summary results reported in the article "Cigarette Tar Yields in Relation
Eclosion refers to the emergence of an adult insect from an egg. The following data on eclosion rates when nymphs were exposed to heat for various durations was extracted from the article "High Temperature Determines the Ups and Downs of Small Brown Planthopper Laodelphax Striatellus Population"
Show that the chi-squared statistic for the test of independence can be written in the formWhy is this formula more efficient computationally than the defining formula for x2?
Suppose that each student in a sample had been categorized with respect to political views, marijuana usage, and religious preference, with the categories of this latter factor being Protestant, Catholic, and other. The data could be displayed in three different two-way tables, one corresponding to
Suppose that in a particular state consisting of four distinct regions, a random sample of nk voters is obtained from the k th region for k 5 1, 2, 3, 4. Each voter is then classified according to which candidate (1, 2, or 3) he or she prefers and according to voter registration (1 = Dem., 2 =
Consider the accompanying 2 3 3 table displaying the sample proportions that fell in the various combinations of categories (e.g., 13% of those in the sample were in the first category of both factors). ..................1.............2...........3 1................ .13.......... .19 .......
The article "Birth Order and Political Success" (Psych. Reports, 1971: 1239-1242) reports that among 31 randomly selected candidates for political office who came from families with four children, 12 were firstborn, 11 were middle born, and 8 were last born. Use this data to test the null
Does the phase of the moon have any bearing on birthrate? Each of 222,784 births that occurred during a period encompassing 24 full lunar cycles was classified according to lunar phase. The following data is consistent with summary quantities that appeared in the article "The Effect of the Lunar
Each individual in a sample of nursing home patients was cross-classified both with respect to cognitive state (normal or mild impairment, moderate impairment, severe impairment) and with respect to drug status (psychotropic drug change, psychotropic user without a change, no psychotropic
The article "Application of Methods for Central Statistical Monitoring in Clinical Trials" (Clinical Trials, 2013: 783-806) made a strong case for central statistical monitoring as an alternative to more expensive onsite data verification. It suggested various methods for identifying data
The authors of the article "Predicting Professional Sports Game Outcomes from Intermediate Game Scores" (Chance, 1992: 18-22) used a chi-squared test to determine whether there was any merit to the idea that basketball games are not settled until the last quarter, whereas baseball games are over by
The accompanying two-way frequency table appears in the article "Marijuana Use in College" (Youth and Society, 1979: 323-334). Each of 445 college students was classified according to both frequency of marijuana use and parental use of alcohol and psychoactive drugs. Does the data suggest that
Much attention has recently focused on the incidence of concussions among athletes. Separate samples of soccer players, non-soccer athletes, and non-athletes were selected. The accompanying table then resulted from determining the number of concussions each individual reported on a medical history
In a study to investigate the extent to which individuals are aware of industrial odors in a certain region ("Annoyance and Health Reactions to Odor from Refineries and Other Industries in Carson, California," Environmental Research, 1978: 119-132), a sample of individuals was obtained from each of
Many shoppers have expressed unhappiness because grocery stores have stopped putting prices on individual grocery items. The article "The Impact of Item Price Removal on Grocery Shopping Behavior" (J. of Marketing, 1980: 73-93) reports on a study in which each shopper in a sample was classified by
Let p1 denote the proportion of successes in a particular population. The test statistic value in Chapter 8 for testing H0: p1 = p10 was z = (1 - p10)/√p10p20/n, where p20 = 1 - p10. Show that for the case k = 2, the chi-squared test statistic value of Section 14.1 satisfies x2 = z2.
The NCAA basketball tournament begins with 64 teams that are apportioned into four regional tournaments, each involving 16 teams. The 16 teams in each region are then ranked (seeded) from 1 to 16. During the 12-year period from 1991 to 2002, the top-ranked team won its regional tournament 22 times,
Have you ever wondered whether soccer players suffer adverse effects from hitting "headers"? The authors of the article "No Evidence of Impaired Neurocognitive Performance in Collegiate Soccer Players" (Amer. J. of Sports Med., 2002: 157-162) investigated this issue from several perspectives. a.
Do the successive digits in the decimal expansion of π behave as though they were selected from a random number table (or came from a computer's random number generator)? a. Let p0 denote the long run proportion of digits in the expansion that equal 0, and define p1,..., p9 analogously. What
The Fibonacci sequence of numbers occurs in various scientific contexts. The first two numbers in the sequence are 1,1. Then every succeeding number is the sum of the two previous numbers: 1, 1, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 8, 13, 21, . . . . The first digit of any number in this sequence can
An information-retrieval system has ten storage locations. Information has been stored with the expectation that the long-run proportion of requests for location i is given by pi = (5.5 - |i - 5.5|)/30. A sample of 200 retrieval requests gave the following frequencies for locations 1-10,
The article "The Gap Between Wine Expert Ratings and Consumer Preferences" (Intl. J. of Wine BusinessRes., 2008: 335-351) studied differences between expert and consumer ratings by considering medal ratings for wines, which could be gold (G), silver (S), or bronze (B). Three categories were then
The article "Psychiatric and Alcoholic Admissions Do Not Occur Disproportionately Close to Patients' Birthdays" (Psychological Reports, 1992: 944-946) focuses on the existence of any relationship between the date of patient admission for treatment of alcoholism and the patient's birthday. Assuming
The response time of a computer system to a request for a certain type of information is hypothesized to have an exponential distribution with parameter l 5 1 sec (so if X 5 response time, the pdf of X under H0 is f0(x) = e-x for x ‰¥ 0).a. If you had observed X1, X2, ... , Xn and wanted
Give as much information as you can about the P-value for the Wilcoxon test in each of the following situations. a. n = 12, upper-tailed test, s+ = 56 b. n = 12, upper-tailed test, s+ = 62 c. n = 12, lower-tailed test, s+ = 20 d. n = 14, two-tailed test, s+ = 21 e. n = 25, two-tailed test, s+ = 300
Say as much as you can about the P-value for the rank-sum test in each of the following situations. a. m = 5, n = 6, w = 41, upper-tailed test. b. m = 5, n = 6, w = 22, lower-tailed test. c. m = 5, n = 6, w = 45, two-tailed test. d. m = n = 12, upper-tailed test, x ranks = 4, 7, 8, 11, 12, 15, 17,
In an experiment to compare the bond strength of two different adhesives, each adhesive was used in five bondings of two surfaces, and the force necessary to separate the surfaces was determined for each bonding. For adhesive 1, the resulting values were 229, 286, 245, 299, and 250, whereas the
The article "A Study of Wood Stove Particulate Emissions" (J. of the Air Pollution Control Assoc., 1979: 724-728) reports the following data on burn time (hours) for samples of oak and pine. Test at level .05 to see whether there is any difference in true average burn time for the two types of wood.
The urinary fluoride concentration (parts per million) was measured both for a sample of livestock grazing in an area previously exposed to fluoride pollution and for a similar sample grazing in an unpolluted region:Does the data indicate strongly that the true average fluoride concentration for
The article "Multimodal Versus Unimodal Instruction in a Complex Learning Environment" (J. of Experimental Educ., 2002: 215-239) described an experiment carried out to compare students' mastery of certain software learned in two different ways. The first learning method (multimodal instruction)
The article "Measuring the Exposure of Infants to Tobacco Smoke" (New England J. of Medicine, 1984: 1075-1078) reports on a study in which various measurements were taken both from a random sample of infants who had been exposed to household smoke and from a sample of unexposed infants. The
Reconsider the situation described in Exercise 81 of Chapter 9 and the accompanying Minitab output (the Greek letter eta is used to denote a median).In Exercise 81The accompanying data on response time appeared in the article "The Extinguishment of Fires Using Low-Flow Water Hose Streams-Part II"
The article "The Lead Content and Acidity of Christchurch Precipitation" (N. Zeal. J. of Science, 1980: 311-312) reports the accompanying data on lead concentration (mg/L) in samples gathered during eight different summer rainfalls: 17.0, 21.4, 30.6, 5.0, 12.2, 11.8, 17.3, and 18.8. Assuming that
Compute the 99% signed-rank interval for true average pH μ (assuming symmetry) using the data in Exercise 15.3.In Exercise 15.3
An experiment was carried out to compare the abilities of two different solvents to extract creosote impregnated in test logs. Each of eight logs was divided into two segments, and then one segment was randomly selected for application of the first solvent, with the other segment receiving the
Here again is the data on expense ratio (%) for a sample of 20 large-cap blended mutual funds introduced in Exercise 1.53:A normal probability plot shows a distinctly nonlinear pattern, primarily because of the single outlier on each end of the data. But a dotplot and boxplot exhibit a reasonable
The following observations are amounts of hydrocarbon emissions resulting from road wear of bias-belted tires under a 522 kg load inflated at 228 kPa and driven at 64 km/hr for 6 hours ("Characterization of Tire Emissions Using an Indoor Test Facility," Rubber Chemistry and Technology, 1978: 7-25):
Compute the 90% rank-sum CI for μ1 - μ2 using the data in Exercise 11. In Exercise 11 In an experiment to compare the bond strength of two different adhesives, each adhesive was used in five bondings of two surfaces, and the force necessary to separate the surfaces was determined for each
Compute a 99% CI for μ1 - μ2 using the data in Exercise 12.In Exercise 12The article "A Study of Wood Stove Particulate Emissions" (J. of the Air Pollution Control Assoc., 1979: 724-728) reports the following data on burn time (hours) for samples of oak and pine. Test at
The accompanying data refers to concentration of the radioactive isotope strontium-90 in milk samples obtained from five randomly selected dairies in each of four different regions.Test at level .10 to see whether true average strontium-90 concentration differs for at least two of the regions.
The article "Production of Gaseous Nitrogen in Human Steady-State Conditions" (J. of Applied Physiology, 1972: 155-159) reports the following observations on the amount of nitrogen expired (in liters) under four dietary regimens: (1) fasting, (2) 23% protein, (3) 32% protein, and (4) 67% protein.
The accompanying data on cortisol level was reported in the article "Cortisol, Cortisone, and 11-Deoxycortisol Levels in Human Umbilical and Maternal Plasma in Relation to the Onset of Labor" (J. of Obstetric Gynaecology of the British Commonwealth, 1974: 737-745). Experimental subjects were
In a test to determine whether soil pretreated with small amounts of Basic-H makes the soil more permeable to water, soil samples were divided into blocks, and each block received each of the four treatments under study. The treatments were (A) water with .001% Basic-H flooded on control soil, (B)
In an experiment to study the way in which different anesthetics affect plasma epinephrine concentration, ten dogs were selected and concentration was measured while they were under the influence of the anesthetics isoflurane, halothane, and cyclopropane ("Sympathoadrenal and Hemodynamic Effects of
The article "Effects of a Rice-Rich Versus Potato-Rich Diet on Glucose, Lipoprotein, and Cholesterol Metabolism in Noninsulin-Dependent Diabetics" (Amer. J. of Clinical Nutr., 1984: 598-606) gives the accompanying data on cholesterol-synthesis rate for eight diabetic subjects. Subjects were fed a
High-pressure sales tactics or door-to-door salespeople can be quite offensive. Many people succumb to such tactics, sign a purchase agreement, and later regret their actions. In the mid-1970s, the Federal Trade Commission implemented regulations clarifying and extending the rights of purchasers to
The accompanying data is a subset of the data reported in the article "Synovial Fluid pH, Lactate, Oxygen and Carbon Dioxide Partial Pressure in Various Joint Diseases" (Arthritis and Rheumatism, 1971: 476-477). The observations are pH values of synovial fluid (which lubricates joints and tendons)
The given data on phosphorus concentration in topsoil for four different soil treatments appeared in the article "Fertilisers for Lotus and Clover Establishment on a Sequence of Acid Soils on the East Otago Uplands" (N. Zeal. J. of Exptl. Ag., 1984: 119-129). Use a distribution-free procedure to
Refer to the data of Exercise 30 and compute a 95% CI for the difference between true average concentrations for treatments II and III.In Exercise 30
The study reported in "Gait Patterns During Free Choice Ladder Ascents" (Human Movement Sci., 1983: 187-195) was motivated by publicity concerning the increased accident rate for individuals climbing ladders. A number of different gait patterns were used by subjects climbing a portable straight
The sign test is a very simple procedure for testing hypotheses about a population median assuming only that the underlying distribution is continuous. To illustrate, consider the following sample of 20 observations on component lifetime (hr):We wish to test H0: μ̃ = 25.0
Refer to Exercise 33, and consider a confidence interval associated with the sign test: the sign interval. The relevant hypotheses are now H0: μ̃ = μ̃0 versus Ha: μ̃ ≠ μ̃0. a. Suppose we decide to reject H0 if either Y ≥ 15 or Y ≤ 5. What is the smallest α for which this equivalent
Suppose we wish to test.H0: the X and Y distributions are identical versusHa: the X distribution is less spread out than the Y distributionThe accompanying figure pictures X and Y distributions for which Ha is true. The Wilcoxon rank-sum test is not appropriate in this situation because when Ha is
The ranking procedure described in Exercise 35 is somewhat asymmetric, because the smallest observation receives rank 1, whereas the largest receives rank 2, and so on. Suppose both the smallest and the largest receive rank 1, the second smallest and second largest receive rank 2, and so on, and
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