Solution Manual for An Introduction to Statistical Methods and Data Analysis, 6th Edition

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’s Solutions ManualforPrepared byJackie MillerThe Ohio State UniversityJohn DraperThe Ohio State UniversityAn Introduction to Statistical Methodsand Data Analysis6THEDITIONR. Lyman OttMichael LongneckerTexas A&M University

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iiiTable of ContentsCHAPTER 1: Statistics and the Scientific Method.................................................................................1CHAPTER 2: Using Surveys and Experimental Studies to Gather Data................................................3CHAPTER 3: Data Description ............................................................................................................11CHAPTER 4: Probability and Probability Distributions ......................................................................51CHAPTER 5: Inferences about Population Central Values..................................................................75CHAPTER 6: Inferences Comparing Two Population Central Values ................................................97CHAPTER 7: Inferences about Population Variances........................................................................119CHAPTER 8: Inferences about More Than Two Population Central Values.....................................139CHAPTER 9: Multiple Comparisons..................................................................................................169CHAPTER 10:Categorical Data ..........................................................................................................179CHAPTER 11:Linear Regression and Correlation ..............................................................................223CHAPTER 12:Multiple Regression and the General Linear Model....................................................265CHAPTER 13:Further Regression Topics ...........................................................................................303CHAPTER 14:Analysis of Variance for Completely Randomized Designs .......................................365CHAPTER 15:Analysis of Variance for Blocked Designs ..................................................................391CHAPTER 16:The Analysis of Covariance.........................................................................................411CHAPTER 17:Analysis of Variance for Some Fixed-, Random-, and Mixed-Effects Models ...........433CHAPTER 18:Split-Plot, Repeated Measures, and Crossover Designs ..............................................453CHAPTER 19:Analysis of Variance for Some Unbalanced Designs ..................................................481

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1Chapter 1Statistics and the Scientific Method1.1a.The population of interest is the weight of the shrimp maintained on the specific diet for aperiod of 6 months.b.The sample is the 100 shrimp selected from the pond and maintained on the specific diet for aperiod of 6 months.c.The weight gain of the shrimp over 6 months.d.Since the sample is only a small proportion of the whole population, it is necessary to evaluatewhat the mean weight may be for any other randomly selected 100 shrimps.1.2a.The amount of radioactivity at all points in the suspect area.b.The 200 randomly selected points in the suspect area.c.The level of radioactivity in the suspect area.d.We want to relate the level of radioactivity of the 200 points in the sample to the level in thewhole suspect area. Thus we need to know how accurate a portrayal of the population isprovided by the 200 points in the sample.1.3a.All households in the city that receive welfare support.b.The 400 households selected from the city welfare rolls.c.The number of children per household for those households in the city which receive welfare.d.In order to evaluate how closely the sample of 400 households matches the number of childrenin all households in the city receiving welfare.1.4a.All football helmets produced by the five companies over a given period of time.b.The 540 helmets selected from the output of the five companies.c.The amount of shock transmitted to the neck when the helmet’s face mask is twisted.d.The neck strength of players is extremely variable for high school players. Hence, the amountof damage to the neck varies considerably from player to player for exactly the same amountof shock transmitted by the helmet.1.5a.The population of interest is the population of those who would vote in the 2004 senatorialcampaign.b.The population from which the sample was selected is registered voters in this state.c.The sample will adequately represent the population, unless there is a difference betweenregistered voters in the state and those who would vote in the 2004 senatorial campaign.d.The results from a second random sample of 5,000 registered voters will not be exactly thesame as the results from the initial sample. Results vary from sample to sample. With eithersample we hope that the results will be close to that of the views of the population of interest.

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2Chapter 1: Statistics and the Scientific Method1.6a.The professor’s population of interest is college freshmen at his university.b.The sampled population is all freshmen enrolled in HIST 101.c.Yes, there is a major difference in the two populations. Those enrolled in HIST 101 may notaccurately reflect the population of all freshmen at his university. For example, they might bemore interested in history.d.Had the professor lectured on the American Revolution, those students in HIST 101 would bemore likely to know which country controlled the original 13 states prior to the AmericanRevolution than other freshmen at the university.

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3Chapter 2Using Surveys and Experimental Studies to Gather Data2.1a.The explanatory variable is level of alcohol drinking. One possible confounding variable issmoking. Perhaps those who drink more often also tend to smoke more, which would impactincidence of lung cancer. To eliminate the effect of smoking, we could block the experimentinto groups (e.g., nonsmokers, light smokers, heavy smokers).b.The explanatory variable is obesity. Two confounding variables are hypertension and diabetes.Both hypertension and diabetes contribute to coronary problems. To eliminate the effect ofthese two confounding variables, we could block the experiment into four groups (e.g.,hypertension and diabetes, hypertension but no diabetes, diabetes but no hypertension, neitherhypertension nor diabetes).2.2a.The explanatory variable is the new blood clot medication. The confounding variable is theyear in which patients were admitted to the hospital. Because those admitted to the hospitalthe previous year were not given the new blood clot medication, we cannot be sure that themedication is working or if something else is going on. We can eliminate the effects of thisconfounding by randomly assigning stroke patients to the new blood clot medication or aplacebo.b.The explanatory variable is the software program. The confounding variable is whetherstudents choose to stay after school for an hour to use the software on the school’s computers.Those students who choose to stay after school to use the software on the school’s computersmay differ in some way from those students who do not choose to do so, and that differencemay relate to their mathematical abilities. To eliminate the effect of the confounding variable,we could randomly assign some students to use the software on the school’s computers duringclass time and the rest to stay in class and learn in a more traditional way.2.3Possibleconfoundingfactorsincludestudent-teacherratios,expendituresperpupil,previousmathematics preparation, and access to technology in the inner city schools. Adding advancedmathematics courses to inner city schools will not solve the discrepancy between minority studentsand white students, since there are other factors at work.2.4There may be a difference in student-teacher ratios, expenditures per pupil, and previous preparationbetween the schools that have a foreign language requirement and schools that do not have a foreignlanguage requirement.

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4Chapter 2: Using Surveys and Experimental Studies to Gather Data2.5The relative merits of the different types of sampling units depends on the availability of a samplingframe for individuals, the desired precision of the estimates from the sample to the population, and thebudgetary and time constraints of the project.2.6She could conduct a stratified random sample in which the plants serve as the stratum. A simplerandom sample could then be selected within each plant. This would provide information concerningthe differences between the plants along with the individual opinions of the employees.2.7The list of registered voters could be used as the sampling frame for selecting the persons to beincluded in the sample.2.8a.No. The survey in which the interviewer showed the peanut butter should be more accuratebecause it does not rely on the respondent’s memory of which brand was purchased.b.Both surveys may have survey nonresponse bias because an entire segment of the population(those not at home) cannot be contacted. Also, both surveys may have interviewer biasresulting from the way the question was posed (e.g., tone of voice). In the first survey, resultsmay be biased by the respondent’s ability to recall correctly which brand was purchased. Thesecond survey may be biased by the respondent’s unwillingness to show the interviewer thepeanut butter jar (too intrusive), or by the respondent not recognizing that the peanut butterthat was purchased waslow fat.2.9a.Alumni (men only?) who graduated from Yale in 1924.b.No.Alumniwhoseaddresseswereonfile25yearslaterwouldnotnecessarilyberepresentative of their class.c.Alumni who responded to the mail survey would not necessarily be representative of thosewhoweresentthequestionnaires.Incomefiguresmaynotbereportedaccurately(intentionally), or may be rounded off to the nearest $5,000, say, in a self-administeredquestionnaire.d.Rounding income responses would make the figure $25,111 unlikely. The fact that higherincome respondents would be more likely to respond (bragging), and the fact that incomes arelikely to be exaggerated, would tend to make the estimate too high.2.10a.Simple random sampling.b.Stratified sampling.c.Cluster sampling.

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Chapter 2: Using Surveys and Experimental Studies to Gather Data52.11a.Simple random sampling.b.Stratified sampling.c.Cluster sampling.2.13a.Stratified sampling. Stratify by job category and then take a random sample within each jobcategory. Different job categories will use software applications differently, so this samplingstrategy will allow us to investigate that.b.Systematic random sampling. Sample every tenth patient (starting from a randomly selectedpatient from the first ten patients). Provided that there is no relationship between the type ofpatient and the order that the patients come into the emergency room, this will give us arepresentative sample.2.13a.Stratified sampling. We should stratify by type of degree and then sample 5% of the alumniwithin each degree type. This method will allow us to examine the employment status for eachdegree type and compare among them.b.Simple random sampling. Once we find 100 containers we will stop. Still it will be difficult toget a completely random sample. However, since we don’t know the locations of thecontainers, it would be difficult to use either a stratified or cluster sample.2.14a.Water temperature and Type of hardenerb.Water temperature: 175F and 200F; Type of hardener:1H,2H,3Hc.Manufacturing plantsd.Plastic pipee.Location on Plastic pipef.2 pipes per treatmentg.6 treatments:(175F,1H), (175F,2H), (175F,3H), (200F,1H), (200F,2H), (200F,3H)2.15a.Factors: Location in orchard, Location on tree, Time of yearFactor levels: Location in orchard – 8 sections; Location on tree – top, middle, bottom;Time of year – October, November, December, January, February, March, April, MayBlocks: noneExperimental units: Location on tree during one of the 8 monthsMeasurement units: orangesReplications:Foreachsection,timeofyear,andlocationontree,thereisoneexperimental unit, hence 1 replication.Treatments: 192 combinations of 8 sections, 8 months, and 3 locations on tree –(iS,jM,kL), for1,...,8i;1,...,8j;1, 2,3k

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6Chapter 2: Using Surveys and Experimental Studies to Gather Datab.Factors: Type of treatmentFactor levels:1T,2TBlocks: HospitalsExperimental units: WardsMeasurement units: PatientsReplications: 2 wards per treatment in each of the 8 hospitalsTreatments:1T,2Tc.Factors: Type of treatmentFactor levels:1T,2TBlocks: Hospitals, WardsExperimental units: PatientsMeasurement units: PatientsReplications: 2 patients per treatment in each of the ward/hospital combinationsTreatments:1T,2Td.Factors: Type of schoolFactor levels: Public; Private – non-parochial; ParochialBlocks: Geographic regionExperimental units: ClassroomsMeasurement units: Students in classroomsReplications: 2 classrooms per each type of school in each of the city/region combinationsTreatments: Public; Private – non-parochial; Parochial2.16a.Factors: Temperature, Type of seafoodb.Factor levels: Temperature (0C, 5C, 10C); Type of seafood (oysters, mussels)c.Blocks: Noned.Experimental units: Package of seafoode.Measurement units: Sample from packagef.Replications: 3 packages per temperatureg.Treatments: (0C, oysters), (5C, oysters), (10C, oysters), (0C, mussels), (5C, mussels),(10C, mussels)2.17a.Randomized complete block design with blocking variable (5 farms) and 48 treatments in a 3× 4 × 4 factorial structure.b.Completely randomized design with 10 treatments (software packages) and 3 replications ofeach treatment.c.Latin square design with blocking variables (position in kiln, day), each having 8 levels. Thetreatment structure is a 2 × 4 factorial structure (type of glaze, thickness).

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Chapter 2: Using Surveys and Experimental Studies to Gather Data72.18a.Design B. The experimental units are not homogeneous since one group of consumers givesuniformly low scores and another group gives uniformly high scores, no matter what recipe isused. Using design A, it is possible to have a group of consumers that gives mostly low scoresrandomly assigned to a particular recipe. This would bias this particular recipe. Using designB, the experimental error would be reduced since each consumer would evaluate each recipe.That is, each consumer is a block and each of the treatments (recipes) is observed in eachblock. This results in having each recipe subjected to consumers who give low scores and toconsumers who give high scores.b.This would not be a problem for either design. In design A, each of the remaining 4 recipeswould still be observed by 20 consumers. In design B, each consumer would still evaluateeach of the 4 remaining recipes.2.19a.“Employee” should refer to anyone who is eligible forsick days.b.Use payroll records. Stratify by employee categories (full-time, part-time, etc.), employmentlocation (plant, city, etc.), or other relevant subgroup categories. Consider systematic selectionwithin categories.c.Sex (women more likely to be care givers), age (younger workers less likely to have elderlyrelatives), whether or not they care for elderly relatives now or anticipate doing so in the nearfuture, how many hours of care they (would) provide (to define “substantial”), etc. Thecompany might want to explore alternative work arrangements, such as flex-time, offeringemployees 4 ten-hour days, cutting back to 3/4-time to allow more time to care for relatives,etc., or other options that might be mutually beneficial and provide alternatives to taking sickdays.2.20a.Each state agency and some federal agencies have records of licensed physicians, professionalcorporations, facility licenses, etc. Professional organizations such as the American MedicalAssociation, American Hospital Administrators Association, etc., may have such lists, butthey may not be as complete as licensing records.b.What nursing specialties are available at this time at the physician’s offices or medicalfacilities? What medical specialties/facilities do they anticipate adding or expanding? Whatstaffing requirements are unfilled at this time or may become available when expansionoccurs? What is the growth/expansion time frame?c.Licensing boards may have this information. Many professional organizations have specialcategories for members who are unemployed, retired, working in fields not directly related tonursing, students who are continuing their education, etc.d.Population growth estimates may be available from the Census Bureau, university economicgrowth research, bank research studies (prevailing and anticipated load patterns), etc. Healthrisk factors and location information would be available from state health departments, theEPA, epidemiological studies, etc.e.Licensing information should be stratified by facility type, size, physician’s specialty, etc.,prior to sampling.

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8Chapter 2: Using Surveys and Experimental Studies to Gather Data2.21If phosphorous first: [P,N][10,40], [10,50], [10,60], then [20,60], [30,60]or[20,40], [20,50], [20,60], then [10,60], [30,60]or[30,40], [30,50], [30,60], then [10,60], [10,60]If nitrogen first: [N,P][40,10], [40,20], [40,30], then [50,30], [60,30]or[50,10], [50,20], [50,30], then [40,30], [60,30]or[60,10], [60,20], [60,30], then [40,30], [50,30]2.22Factor 2Factor 1IIIIIIA254565B1030502.23a.Group dogs by sex and age:GroupDogYoung female2, 7, 13, 14Young male3, 5, 6, 16Old female1, 9, 10, 11Old male4, 8, 12, 15b.Generate a random permutation of the numbers 1 to 16:15741131381121625610914Go through the list and the first two numbers that appear in each of the four groups receivetreatment1Land the other two receive treatment2L.GroupDog-TreatmentYoung female2-2L, 7-1L, 13, 14-2LYoung male3-1L, 5-2L, 6-1L, 16-2LOld female1-1L, 9-2L, 10-2L, 11-1LOld male4-1L, 8-2L, 12-2L, 15-1L2.24a.Bake one cake from each recipe in the oven at the same time. Repeat this procedurertimes.The baking period is a block with the four treatments (recipes) appearing once in each block.The four recipes should be randomly assigned to the four positions, one cake per position.Repeat this procedurertimes.

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Chapter 2: Using Surveys and Experimental Studies to Gather Data9b.If position in the oven is important, then position in the oven is a second blocking factor alongwith the baking period. Thus, we have a Latin square design. To have4r, we would need tohave each recipe appear in each position exactly once within each of four baking periods. Forexample:Period 1Period 2Period 3Period 41R2R4R1R3R4R2R3R3R4R2R3R1R2R4R1Rc.We now have an incompleteness in the blocking variable period since only four of the fiverecipes can be observed in each period. In order to achieve some level of balance in thedesign, we need to select enough periods in order that each recipe appears the same number oftimes in each period and the same total number of times in the complete experiment. Forexample, suppose we wanted to observe each recipe4rtimes in the experiment. If wouldbe necessary to have 5 periods in order to observe each recipe 4 times in each of the 4positions with exactly 4 recipes observed in each of the 5 periods.Period 1Period 2Period 3Period 4Period 51R2R5R1R4R5R3R4R2R3R3R4R2R3R1R2R5R1R4R5R2.25Discussion question; answers will vary.2.26Discussion question; answers will vary.2.27Discussion question; answers will vary.2.28Discussion question; answers will vary.2.29Discussion question; answers will vary.

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10Chapter 2: Using Surveys and Experimental Studies to Gather Data

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11Chapter 3Data Description3.1a.The following is a pie chart of the federal expenditures for the 2006 fiscal year (in billions ofdollars).National DefenseSocial SecurityMedicare & MedicaidNational Debt InterestMajor Social-A id ProgramsotherCategory475200300500500525Pie Chart of Federal Programb.The following is a bar chart of the federal expenditures for the 2006 fiscal year (in billions ofdollars).OtherMajor Social-Aid ProgramsNational Debt InterestMedicare & MedicaidSocial SecurityNational Defense6005004003002001000Federal Program2006 Expenditures (Billions of Dollars)

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12Chapter 3: Data Descriptionc.The following are a pie chart and bar chart of the federal expenditures for the 2006 fiscal year(in percentages).National DefenseSocial SecurityMedicare & MedicaidNational Debt InterestMajor Social-Aid ProgramsotherCategory19.0%8.0%12.0%20.0%20.0%21.0%OtherMajor Social-Aid ProgramsNational Debt InterestMedicare & MedicaidSocial SecurityNational Defense0.200.150.100.050.00Federal Program2006 Expenditures (Percent of whole)d.The pie chart using percentages is probably most informative to the tax-paying public. Herethe tax-paying public can compare the percentages spent by the Federal government fordomestic and defense programs as part of a whole.

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Chapter 3: Data Description133.2a.Pie charts would not be appropriate to display these data. We would not be able to see trendsover time.b.The following bar chart shows the changes across the 12 years in the public’s choice invehicle.Year20022001200019991998199719951990100806040200Percent in YearSUV/Light TruckPassenger C arVariablec.It appears that the percentage of passenger cars has decreased over the period 1990-2002. Ifthere was a substantial increase in gasoline prices, we would expect the percentage ofpassenger cars to increase.3.3a.The following bar chart shows the increase in the number of family practice physicians (inthousands of physicians) over the period 1980-2001.2001200019991998199519901980706050403020100YearFamily Practice Physicians (in thousands)

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