(BQ) Part 1 book Elementary statistics has contents Statistics, descriptive analysis and presentation of single variable data, descriptive analysis and presentation of bivariate data, probability, probability distributions (discrete variables), normal probability distributions, sample variability.
Index of Applications Ex: Example; AEx: Applied Example; CPT: Chapter Practice Test All others are exercises Agriculture Cherry orchard yields: 9.155 Circumferences of oranges: 9.149 Corrosive effects of soil: 10.27 Diameters of Red Delicious apples: 7.50, 7.51 Diameters of tomatoes: 9.180 Farm real estate values: 9.139 Germination trials: 5.123, 5.124 Grapefruit characteristics: 4.147, CPT 8.15 Growth of hybrid plants by soil type: CPT 11.14 Magnesium and calcium in Russian wild rye: 13.77 Maturity time of green beans: 9.138 Nitrogen fertilizer and wheat production: 13.85, CPT 13.11, CPT 13.12, CPT 13.13, CPT 13.14, CPT 13.15, CPT 13.16, CPT 13.17, CPT 13.18, CPT 13.19, CPT 13.20, CPT 13.21 Number of stamens and carpels in flowers: 13.83 Oat crop yields: 9.158, 10.162, 12.28 Peanut yield rate: 14.6, 14.36 Petal and sepal measurements of irises: 3.27 Petal width of irises: 12.53 Strontium distribution coefficient and total aluminum for soils: 13.53 Sucrose percentages in sugar cane and sugar beets: 10.121, 10.144, 13.40 Sugar cane production: 3.99 Sunflower yields: 10.49 Sweet corn yields: 10.126 Tree growth: 10.6 Tree survival: 5.61 Watermelon weights: 6.59 Weight gains for chicks: 2.157 Weight gains for pigs: 10.20 Weights of poultry flock: 9.27 Wheat production: 13.1, 13.2, 13.91 World cocoa production: 2.16 Yields of hops: 2.109 Biological Science Animals’ brain lengths and weights: 13.22 Brown trout lengths: 2.191 Cayuga Lake fish lengths: 8.35 Cricket chirp rate and temperature: 3.96 Distribution and abundance of sea otters: AEx 14.8 Lake trout lengths: CPT 7.11 Length and age of blacknose dace: 3.94 Length and weight of alligators: 3.98 Length and weight of bears: 13.6 Lengths of hatchery-raised trout: 8.173 Long-haired rabbits: 5.70 Manatee deaths in Florida: 3.37, 11.31 Mendelian theory of inheritance: Ex 11.2, 11.12, 11.13, 11.15, 11.47 Nutrient loads discharged into Biscayne Bay: AEx 13.11, 13.66 Shark attacks: AEx 1.4 Weight and girth of horses: 13.12 Weight and length of adult cicadas: 3.101, 12.54, 13.64 Weight gain of laboratory mice: 10.71, CPT 14.12 Weights of bees’ loads of pollen and nectar: 8.183 Business, Economics, and Financial Management Accuracy of tax withholding: 11.50 Ages and prices of Honda Accords: 3.72 American Express card fees: 6.111 Amounts in medical flexible spending accounts: 8.150 Amounts spent on veterinary care: 7.52 Annual fuel consumption: 8.187 Auto repair charges: 8.87, 10.10 Average home size: 8.116 Best companies to work for: 5.7 Burden of proof in tax dispute cases: 10.157 Carat weight and price of diamonds: 13.45 Checkout times: 1.12, CPT 2.11, 8.115, 8.185 Commissioned sales amounts: 8.18 Commute times: 8.188, 10.21, 12.1, 12.2, 12.62, Ex 14.2 Commuting distances: 2.160 Commuting distances and times: Ex 13.5, Ex 13.6, Ex 13.8, 13.59, Ex 13.9, Ex 13.10, 13.65 Company dress codes: 2.5 Company spending on travel: 3.4 Compensation by type of organization: 1.4 Costs of luxury cars and “feel alike” models: 3.70 County transfer taxes: 12.46 Credit card rewards and rebates: 9.77 Customer accounts at banks: 9.72 Delivery service fees: 2.25 Earnings per share for banking industry: 2.186 Earnings per share for radio industry: 2.223 Emergency funds available by gender: 3.82 Entertainment sports industry salaries: 4.65 Executive salaries: 6.46 Executives’ job search: 1.72 Expected payoff from investment: 5.125 Franchised restaurant sales: 9.179 Gas-tax money by region: 12.44 GDP and level of technology: 13.20 Gold-collar workers: 6.50 Home values in college town: 9.48 Home values in Rochester suburbs: 10.73 Hourly earnings by industry: 2.196 Hourly wages of production workers: 12.35 Hours worked by Java professionals: 1.3 House selling prices: 2.18, 10.62 Illegal tax deductions: 4.75 Jeans inventories in Levi Strauss stores: 9.49 Job growth percentage changes: 14.83 Job interview outcomes: 4.149 Leasable office space available: 6.62 Life insurance purchases: 4.113 Life insurance rates: AEx 3.6, 3.49, 3.75, Ex 4.6 Losses from online identity theft: 8.166 Magazine subscription rates: 3.35 Making business decisions: 4.119 Minimum deposit and interest rate: 3.10 Monthly car payments: 2.222 Mortgage foreclosures: 5.75 Motor-fuel taxes: 3.65 Number of client contacts and sales volume: 13.41 Numbers of automobiles by country: 4.143 Obesity and productivity: AEx 1.3 “On-hold” times for customer service: 6.61 Opinions on fringe benefits by gender: 4.156 Prices of laptop computers: 8.113 Property damage in automobile accidents: 4.152 Reducing debt: 11.21 Resale price and age of luxury autos: 3.63 Revenue per kilowatt-hour in Arkansas: 2.47 Salaries of clerk-typists: 8.186 Salaries of elementary school teachers: 2.107 Salaries of human resources clerks: 6.48 Salaries of junior executives: Ex 6.14 Salaries of labor relation managers: 7.38 Salaries of machine shop employees: CPT 2.16 Salaries of registered nurses: 7.42, 7.49 Salaries of resort club managers: 2.39, 2.51 Sales potential ranking and sales totals: CPT 14.13 Savings account amounts: 7.59 Service contractor complaints: AEx 2.17, 2.150 Spa industry profits: AEx 1.2, 1.7 Taxes per capita: 2.81, 2.85 Tax refunds: 1.11 Times to settle insurance claims: 9.103 Total personal income and value of new housing units: 13.39 Total returns in banking industry: 2.24 Turnover among nurse executives: 10.83 Unemployment rates: 3.9, 6.131 Unit pricing: 1.37 Used-car inventory: Ex 4.4 Values of funded projects: 8.15 Waiting time in post office: CPT 8.11, CPT 9.18 Yum Brands abroad: 4.53 College Life Absences at AM classes: 14.49, 14.72 ACT composite score and first-term college GPA: 13.5 Amount of trash discarded by students: 9.34 Budgeting for intramural and interscholastic sports: 9.73 Caffeine consumption: 9.50 Cars driven by students: Ex 9.8, Ex 9.13, 9.69 Cars owned by college faculty: Ex 1.5 Chemistry students by gender: 9.106 College applications: 5.21 Commute times: CPT 6.15, 9.47 Commute times and distances: 3.22 Commuting distance: 2.141, Ex 8.2, 8.124 Concrete Canoe Competition: 2.66 Cornell’s tuition and ranking: AEx 2.16, 2.149 Cost of textbooks: 1.26, Ex 1.8, 8.180, 9.25, 9.152, 9.153 Course selection criteria: 11.40 Cultural literacy of college freshmen: 12.22 Dropout rate: 6.94 Electronic study guides for accounting principles: 10.63 Final averages: 6.57 Final exam scores by teaching method: CPT 10.14 Genders and majors: Ex 3.1 GPAs and membership in fraternal organizations: Ex 10.9 Graduation rates: 4.123 Guessing on multiple-choice tests: 5.126 Hours of sleep: 2.70, 2.93, 6.39, 9.54 Hours worked per week: Ex 14.5 Introductory psychology course grades: 11.46 Living at home after graduation: 9.167 Mathematics placement exam scores: 10.50, 10.151 Monthly debt after college graduation: 2.54, 2.168 Number of colleges applied to: AEx 5.3 Number of credit hours: 2.217 Paying off college debt: 2.174 Preference for liberal arts courses by gender: Ex 11.5 Preferences for math course sections: Ex 11.1 Professor late for class: 4.163 Salaries of full professors in Colorado: 14.16 Scholarship applications: 4.118 Self-esteem scores of college students: 10.34, 10.35 Statistics final exam scores: Ex 2.6, Ex 2.12, Ex 2.13, 2.105, Ex 2.19 Statistics pass rates: 4.41 Student characteristics: 1.67 Student credit card debt: 10.1, 10.2, 10.164 Student-owned vehicle makes: 2.8 Students’ places of residence: 11.49 Summer schedule: 1.23, 4.52 Test scores: 9.55 Tuition and fees: 8.42, 10.70 Undergraduate GPA and GPA at graduation: 14.61 Weight of books and supplies: 1.24 Demographics and Population Characteristics Age and gender of licensed drivers: 4.136 Age at marriage: 7.56 Ages of auto theft offenders: 2.183 Ages of dancers: 2.36 Ages of D.C residents: 2.9, 4.134 Ages of fishermen: 2.162 Ages of heads of household: 2.220 Ages of licensed drivers: 6.49 Ages of NASCAR drivers: 2.71, 2.94 Ages of New York population: 2.169, 2.176 Ages of night-school students: 8.34 Ages of nuns: 2.42, 2.166 Ages of U.S population: 2.165, 4.155 Area (sq mi.) of U.S states: 2.193 Baby birth days: AEx 11.3, 11.22 Baby birth months: 11.52 Birth weights for babies in U.S.: 8.114 Census data: AEx 1.6, 7.1, 7.2, 7.67, 7.68 College attendance in suburban populations: 2.48 Divorce rates: 1.12 Genders of licensed drivers: 6.95 Grandparents as primary caregivers: 4.117 Gray hair by gender: 10.158 Handedness: 4.66 Height and age of children: 3.17 Height and shoe size: 3.2, 13.72 Height and weight of college women: Ex 3.7, 3.69 Heights and weights of professional soccer team: 3.26 Heights and weights of World Cup players: 3.12, 3.18 Heights of college students: Ex 10.8 Heights of high-school football players: 2.143 Heights of kindergartners: Ex 7.6, Ex 7.7, 7.32, 7.37, 7.43 Heights of male college students: 7.35 Heights of mothers and daughters: 3.11 Heights of NBA players: 2.19, 12.24 Heights of Olympic soccer players: 2.33 Heights of women in health profession: 8.199 High-poverty neighborhood populations: 2.55 Homeownership rates: 11.59 Household incomes: AEx 2.11, 4.11, 14.48 Increases in U.S population by area: 2.26, 2.142 Life expectancies by gender: 3.95 Monroe Community College demographics: 4.142 Montana’s household population: 2.6 Number of children adopted: 3.44 Number of children fathered by doctors: 2.156 Number of children per family: 5.19 Number of licensed drivers: 1.75, 2.79, 3.24 Number of people per household: 2.34 Number of push-ups and sit-ups: Ex 3.3, Ex 3.5, 3.62 Number of rooms in Texas housing units: 2.35 Number of students by grade level: 1.8 Number of telephones per household: 2.153 Number of televisions in American households: 7.22 Number of televisions in Japanese households: 5.104 Number of years of college of high-tech employees: 8.47 Ophthalmic trait and eye color: 4.154 Percentages in service and trade job categories: 14.63 Political preference by age: 11.61 Poverty and life expectancy: 3.54 Vehicle registrations and population: 3.79 Vehicles per household: 2.80, 4.115, 5.34, 5.106 Weights of adult males: 9.51, 9.52, 11.53 Weights of college students: Ex 2.5 Weights of college women: 8.82, Ex 8.21 Weights of high-school football players: 2.144 Weights of second-grade boys: Ex 8.6 Weights of 10-year-old girls: 8.184 Education and Child Development ACT exam takers: 2.195 ACT scores: 2.106, 2.126, 2.140, 2.202, 6.52, 6.109 Age at first dental exam: 2.155 AP test results: 2.32, 2.52 Attitudes of preschoolers’ parents: 1.70 Composition exam scores: CPT 9.13 Computer science aptitude test scores: 2.41, 2.53, 10.142, 13.46, 13.50 Content title and reading comprehension: 10.28 Costs of baby supplies: 2.172 Costs of day care: 6.47 Daily activities of schoolchildren: 5.11 Effects of social skills training: 12.61 Equivalence of two exams: Ex 14.6 Evaluation of teaching techniques: AEx 8.11, 8.52, 8.63, 8.64 Examination scores: Ex 2.3, Ex 2.4, 6.117, 14.71 Grade comparison for blondes and brunettes: 10.65 Hours of work per week by high-school juniors and seniors: 10.66 Hours studied for exam and grade received: 3.15, 3.19, 3.33, 3.38, 3.58, 13.48, 13.51, 13.52 Imaginary friends and coping skills: 9.76 Inherited characteristics of twins: AEx 10.3 Instructional time in social studies: 1.73 International Mathematics and Science Study results for eighth-graders: 14.17 IQ scores: 6.1, 6.2, Ex 6.10, Ex 6.11, Ex 6.13, 6.45, 6.137, 6.138, 10.133 Irrelevant answers and age: 3.20, 3.39 Kindergarten skills: 1.6 Mastery of basic math by high-school seniors: 14.21 Methods of teaching reading: 12.48 Minimum score required for grade of A: Ex 6.12 Misbehaving and smoking: 5.63 Mothers’ use of personal pronouns when talking with toddlers: AEx 9.7 National Assessment of Educational Progress in mathematics: 14.65 Number of students per computer in Canada schools: 14.45 Order of finish and scores on exams: Ex 14.14 Parental concerns in choosing a college: 10.100 Physical fitness classes: Ex 10.1 Poverty and proficiency tests: AEx 3.4, 3.16, 3.21, 3.34 Prefinal average and final exam score: 13.84 Proficiency test scores for Ohio fourth-graders: 14.32 Reading proficiency test scores for sixth-graders: 14.7, 14.14 SAT scores: 2.164, CPT 6.16 Social skills in kindergarten: AEx 1.1, AEx 1.9 Standard scores for exam grades: Ex 2.14, 2.122 Strength test scores of third-graders: 2.45, 2.75 Student computer access by grade level: AEx 12.4, 12.9 Summer jobs for high-schoolers: 5.10 TIMSS scores: 7.40, 8.37 Truancy counseling: CPT 10.21 Variability of exam scores: 1.39, 9.176 Wearing of protective clothing by teens: 14.15 Leisure and Popular Culture Ages of thoroughbred racing fans: 8.151 American Kennel Club registrations by breed: 3.83 Amounts spent on high school prom: 7.41 Aquarium inhabitants: 4.96 Art museum scheduling: 4.165 Asymmetry of euro for coin tossing: AEx 9.14 Blog creators and readers: 11.39 Carnival game probabilities: 4.77, 4.78 Casino gaming rules: AEx 14.12 Cell phone distractions while driving: 1.10 Cell phone text messaging: 9.175 Coffee break: 4.76 Contract bridge hands: 4.45 Cooling mouth after a hot taste: 11.1, 11.2, 11.69 Dimensions and base price of jet boats: 13.57 Dog-obedience training techniques: Ex 14.7 Dog ownership: 4.59, 5.20, 5.36 Downloading music and video files: 6.96 Downloading with cell phones: AEx 11.4, 11.8 Expenditures on leisure activities: 10.60 Halloween candy: 5.80 Help with household chores: 2.13 Hours of housework for men: 7.54 Hours of sleep on weekend: Ex 9.12, 10.128 Hours of television watching: 2.184, 7.23, 14.4 Hours spent housecleaning: AEx 2.7 Impact of Internet on daily life: 5.64 Instant messaging: 6.128 Internet usage: 2.1, 2.2, 2.212, 2.224, 5.79, 9.165 Length of visit to library home page: 8.50 Lengths of pop-music records: 7.62, 7.63 Lottery tickets: 5.113 Lower-leg injuries in skiing: CPT 13.22 M&M colors: 2.197, 4.1, 4.2, 4.3, 4.170, 4.171, 11.57 Misplacements of TV remote control: 2.119 Mother’s Day expenses: 8.117 Number of rolls of film dropped off for developing: 12.32 Number of TV sports reports watched per week: 2.167 Obtaining “comfort food”: 11.54, 11.55, 11.56 Personal watercraft accidents: 9.108 Powerball Lottery game: CPT 4.20 Probability of winning carnival game: Ex 4.13 Rankings of contest participants: Ex 14.13 Rebound heights of table-tennis balls: 14.80 Restaurant wait times: 8.164 Rifle-shooting competition scores: 10.136, 10.141 Shooting accuracy by method of sighting: Ex 12.6 Skittles colors: 11.23, 11.24 Spring cleaning survey: 1.9 Stamp collection value: 8.16 Swimming lessons: 4.10, 4.50, 4.97 Television ratings: 4.29, 4.49 Times for haircuts: 2.206 Time off during holidays: 1.78 Time spent in video games by children: 14.46 Tipping habits of restaurant patrons: 3.66 Use of hair coloring by blondes and brunettes: 10.86 Vacation habits of New York families: 4.141 Vacation research on Internet: 6.130 Valentine’s Day: 2.12, 2.148, 10.129 “Wired” senior citizens: 5.111 Manufacturing and Industry Absenteeism rates of employees: 11.66 Accuracy of wristwatches: 8.119 Air bag design: Ex 8.8 Amount of force needed to elicit response: 10.163 Amounts of fill: 1.38, 6.53, 6.115, 6.118, 8.194, Ex 9.17, Ex 10.10, Ex 10.15, Ex 10.17, 10.160, 12.45 Asphalt mixtures: 8.58 Asphalt sampling procedures: AEx 10.7, 10.37 Bad eggs: Ex 5.9 Breaking strength of rope: 8.182 Breaking strengths of steel bars: 7.55 Cellular phone defects: Ex 10.13 Charges for home service call by plumbers: 8.129 Comparing production methods: 10.101, 11.44 Comparing reliability of microcomputers: 10.153 Concrete shrinkage and water content: AEx 3.8 Cork characteristics: AEx 6.15, 8.79, 8.80 Cost per unit and number of units produced per manufacturing run: 3.64 Crew clean-up time: 2.133 Defective bolts: Ex 9.11 Defective parts: 3.85, 4.27, 4.48, 4.67, 4.122, Ex 4.26, 5.67, 9.168, 10.87, 11.37, 14.81 Defective products: 2.15 Defective rifle firing pins: Ex 6.19 Defective television sets: Ex 5.7 Defects in garments: 2.11 Delay times for sprinkler systems: 8.179 Delivery truck capacity: 7.61 Detonating systems for explosives: 8.51, 8.62 Diameter and shear strength of spot weld: 13.58 Diameters of ball bearings: 8.181 Diameters of wine corks: 6.55, 9.60 Drying time for paint: Ex 8.9, Ex 8.14, 8.88, 9.28 Dry weights of corks: 9.141 Effect of heat treatment of length of steel bar: 10.48 Effect of temperature on production level: Ex 12.1, 12.3 Extraction force for wine corks: 6.54, 8.41, 9.182 Failure torque of screws: 9.24 Fill weights of containers: 6.60, 7.36 Flashlight battery lifetimes: CPT6.14, 7.44 Hours worked by production workers: 12.52 Lawn mower warranties: 6.125 Lengths of commutators: 2.21, 9.58 Lengths of lunch breaks: 9.26 Lengths of machined parts: 8.33, 8.39 Lengths of nails: 9.181 Lengths of power door brackets: 2.204 Lengths of wine corks: 9.61 Lens dimensions: 2.72, 6.134, 6.135, 9.150, 10.22, 10.38, 10.74, 10.127, 12.49, 13.60, 13.61, 13.62 Lifetime mileage of tires: 2.132 Lifetimes of cigarette lighters: CPT 7.12 Lifetimes of light bulbs: 2.221, 6.116, 7.57, CPT 8.16, 9.177 Light bulb failures: 5.98 Luxury car colors: 4.56 Machine downtimes: 14.47 Maintenance expenditures for television sets: 9.154 Mileage of automobile tires: 7.58 Mileage of service truck tires: 9.147 Mislabeled shoes: 4.132 Net weights of bags of M&Ms: 6.63 Nutrition information and cost for sports drinks: 3.43, 3.59 Nutrition information for frozen dinners: 8.127 Nutrition information for hot dogs: 2.76 Nutrition information for peanut butter: 12.39 Nutrition information for soups: 2.205, 14.60 Octane ratings of gasoline: 2.182, 9.56 Online queries to PC manufacturers: 11.18 Ovality of wine corks: 8.168 Paint drying time: 2.181 Parachute inspections: 8.56 Particle size of latex paints: 2.203, Ex 8.3, 10.9 Particulate emission in generation of electricity: CPT 12.14 Performance of detergents: Ex 8.10, Ex 8.12 Product assembly times by gender: 10.159 Proofreading errors: 4.166 Quality control: 1.27, 5.72, 5.119, 7.3 Refrigerator lifetime: 6.114 Rivet strength: 8.81, 8.123 Salvaged tires: 4.151 Screw torque removal measurements: 10.149 Service provided by PC manufacturers: 10.96 Shearing strength of rivets: 6.112, CPT 7.13 Shelf life of photographic chemical: Ex 9.19 Strength and “fineness” of cotton fibers: 13.68 Strength intensity of a signal: 10.135 Sulfur dioxide emissions: Ex 8.19 Temperature of ovens during baking process: 10.119 Tensile strength of cotton fibers: 10.125 Testing rivets: 8.85, 8.128 Test-scoring machine accuracy: 6.124 Thread variance in SUV lug nuts: 10.161 Time for a component to move to next workstation: 8.48 Times for machine adjustments by two methods: 14.78 Toothpaste formulas: 8.86 Toy battery lifetimes: 9.151 Tread wear of tires: 8.190, Ex 10.4, Ex 10.6 T-shirt quality control: 5.47, 5.97, CPT 5.12 Union membership: 4.28, 6.100 Union membership and wages: 4.68 Units of work completed per day: 12.25 Variability of package weights: 9.136 Variability of weightlifting plates: 9.137 Variation in lengths of produced parts: 9.178 Wearing quality of auto tires: Ex 10.2 Weights of bread loaves: 7.34 Weights of cereal boxes: 8.163, 9.183, 9.184, 9.185 Weights of cheese wheels: 10.61 Weights of mini-laptop computers: 8.49 Weights of “1-pound” boxes: 2.161 Weights of packages shipped by air: 8.154 Weights of “10-pound” bags of potatoes: 10.64 Worker satisfaction: 4.131 Marketing and Consumer Behavior Amount spent by customers: CPT 8.17 Apple-eating preferences: 2.4 Assessing demand for new product: 1.46 Automobile brands of GM employees: 5.50 Baked potato preferences: AEx 11.7, 11.38, 11.45 Blind taste tests for cola preference: 14.19 Bottled water consumption: 6.51 Burger condiments: 9.70 Burger consumption: 9.160 Caffeine consumption: 5.1, 5.2, 5.128, 5.129 Chocolate-covered candy bar preferences: 14.74 Christmas tree sales: 2.194 Coffee drinking trends: 6.129 Coffee preferences of married men: 9.161 Consumer fraud complaints: 5.62 Customer preferences for seating arrangements in restaurants: CPT 14.19 Deodorant soap preferences: 10.88 Easter candy purchases: 4.124 Effectiveness of different types of advertisements: 12.26 Effectiveness of television commercials: 8.66, 8.67 Effect of sales display location on sales: 12.40 Features desired by home buyers: 14.64 Floor polish preferences: 11.14 Gasoline purchases by method of payment: 3.8 German restaurant ratings: 12.27 Grocery shopping habits: 1.69 Ground beef preferences: 11.17 Holiday shopping preferences: 2.151 “More space” preferences of leisure travelers: 3.3 Movie budgets, box office receipts, and Oscar nominations: 3.46 Nielsen ratings and number of viewers: 3.78 Number of commercials and sales of product: 3.40 Number of customers during noon hour: CPT 2.15 Number of customers per day: 7.53 Number of items purchased: CPT 2.12 Outdoor features desired in new homes: 5.23 Pizza crust preferences: 9.171, 9.172, 9.173, 14.20 Popcorn brand preferences: 11.63 Portuguese wine ratings and prices: 13.23 Price comparisons among grocery stores: 12.41 Product endorsements by athletes: 9.162 Radio hits: 5.112, 11.58 Radio station format preferences over time: 14.86 Ratings and street price ranks of computer monitors: 14.57 Readership of Vogue magazine: 5.108 Restaurant decor rating and cost of dinner: 13.35 Retail store customer data: 12.36, 12.37, 12.38, 12.55, 12.56, 12.57, 13.69, 13.70, 13.71, 13.86, 13.87 Satisfaction with auto service departments: 9.163 Spousal preferences for television programs: 14.84 Television brand ratings: 4.153 Turkey consumption: 7.25 Use of cleaning wipes: 2.10 Medical Science Abnormal male children and maternal age: 9.157 Acetaminophen content of cold tablets: 9.59 Acute back pain treatments: 10.147 Adverse drug reactions: 9.81, 9.82 Allergies in adults: 1.15 Amount of general anesthetic: 10.137 Amounts of water consumed daily: 8.155, 8.156 Amounts spent on prescription drugs: 9.23 Angina: 10.68 Anticoagulants and bone marrow transplantation: 14.18 Benefits of exercise: 9.1, 9.2, 9.187, 9.188 Blood pressure readings: 10.122 Blood types: 11.48 Body Mass Index (BMI) scores: 8.120, 14.70 Caffeine and dehydration: 9.71 Calculus or tartar: AEx 14.15 Cancer testing: 4.158 Carpal tunnel syndrome: 2.163 Causes of death in United States: 2.175 Cholesterol readings: 10.17, 10.30, 14.34 Clinical trials: 11.43 Clotting time and plasma heparin concentrations: AEx 13.4 Comparing methods of cataract surgery: 14.33 Crutch length and patient height: 13.63 Dental disease status by height: 4.135 Diabetes by gender: 4.25 Diastolic blood pressure readings: 10.47 Effect of calcium channel blockers on pulse rate: Ex 10.5 Effect of diet on uric acid level: 10.8 Effect of drug in lowering heart rate: 13.67 Effects of biofeedback and relaxation on blood pressure: 12.23 Exercise capacity of police recruits: 2.97 Eye-nose-throat irritations: 11.36 Fertility rate: 7.24 Flu vaccine: 1.71 Graft-versus-host disease in patients with acute myeloid leukemia: 14.13 Handedness and death rates from accident-related injuries: 10.156 Health benefits of exercise: 1.45 Health status of older Americans: 1.79 Hemoglobin test for diabetic patients: 2.44 Hip-replacement surgery: 5.105 Hospital stays after surgery: 12.21 Hypertension: 1.25 Immediate-release vs sustained-release codeine: 10.138 Infant mortality rates: 2.108, 6.133 Injury severity in younger and older children: 10.120 Length of pain relief: 12.42 Life expectancy: 1.80 Lung cancer and smoking: 4.26 Lung cancer survival rate: 9.90 Manual dexterity scores: 2.110 Marriage and health status: 1.74 Maternal death rates: 4.46 Medical assistance for accident victims: 8.57 Melanoma survival rates: 6.93 Methods for teaching anatomy: AEx 10.11, 10.67, 14.77 Mineral concentration in tissue samples: 14.59 Noise level in hospitals: CPT 8.18 Organ donation: 11.65 Percentages of nicotine in cigarettes: 14.30 “Persistent disagreements” in therapeutic recreation: 2.118 Plasma concentration of ranitidine: 13.78 Plasma protein binding of diazepam: 12.60 Prescription drug use: 1.81 Prescription drug use by seniors: 9.170 Pulse rates: 9.29, 9.30, 14.31 Response time to blood pressure medication: 8.171 Salt-free diets and diastolic blood pressure: 10.19 Self-care test scores of recently diagnosed diabetics: 10.29 Side effects of drugs: 1.22, 1.29, 5.107, 10.102 Sleep apnea: 4.60 Substance abuse: 1.82, 6.99 Surgical infections: AEx 1.7 Survival rate during surgery: 5.66 Testing effectiveness of new drugs: 8.195, 8.196 Time since last doctor visit by age: 3.84 Types of operations: Ex 2.1 Use of electrical stimulation to increase muscular strength: 13.79 Use of nuclear magnetic resonance spectroscopy for detection of malignancy: 14.77 Vertigo treatments: 12.31 Wait time for urgent care: 9.146 Weight loss on no-exercise plan: Ex 14.3 Weights before and after smoking cessation: CPT 10.15, CPT 14.11 Weights of newborns: Ex 9.4 Physical Sciences Accuracy of short-range missiles: CPT 10.17 Amount of rainfall for April: AEx 12.5, 12.10 Area and maximum depth of world lakes: 3.97 Atmospheric ammonium ions: 2.112 Atomic weight of silver: 8.40 Carbon monoxide readings in Rochester: Ex 8.13, Ex 9.5 Density of nitrogen: 2.188 Density of the earth: 2.118, 9.57 Duration and path width of solar eclipses: 3.28 Durations of eruptions of Old Faithful: 2.46 Effects of cloud seeding on rainfall amounts: 14.35 Elevations of towns in upstate New York: 2.96 Forecasting hurricanes: 5.33 Hailstone size and wind updraft speed: 13.42 Heating-degree-day data: 2.28, 2.95 High temperatures: 5.8, Ex 14.1, Ex 14.4, 14.5 Hydrogen characteristics in seasonal snow packs: 14.69 Lightning strikes: 2.38 Number of sit-ups in minute: 10.36 Old Faithful eruption data: 3.102 Parallax of the sun: 9.32 Precipitation in New York State: 6.110 Prediction of next eruption of Old Faithful: AEx 8.1, 8.17 Pressure and total aluminum content for Horn blende rims: 13.80 Roughness coefficient for quartz sand grains: 14.75 Target error of short-range rockets: 10.148 Test flow rates in dual bell rockets: 10.46 Velocity of light: 9.159, 12.34 Water content of snow: AEx 8.5, 8.43 Water pollution readings: 9.156 Weather forecasts: 4.13 Wind speeds in Honolulu: 7.39 Psychology, Sociology, and Social Issues Achievement test scores of soldiers: 10.140 Affirmative Action Program for federal contractors: AEx 5.10 African-American roles in cinema releases: 11.16 Ages of antitrust offenders: 9.22 Amount that requires consultation with spouse before spending: 8.38 Anxiety test scores: 10.139 Attitudes toward death: 10.44 Bikers and body art: 5.68 Destruction-of-property offenses among school-age boys and girls: CPT 10.16 Determining the “goodness” of a test question: 10.154 Driving while drowsy: 5.81 Drug addiction: 5.74 Earphone use on flights: 5.91 Effect of suburb position on school population: 12.30 Establishing the reliability of a test: 13.21 Family structure: Ex 4.3, 4.168, 5.6, 5.69 Fear of darkness: 11.41 Fear of dentist by age: 3.81 Fear of public speaking: 9.109 Hate crimes: Ex 2.2 Households with guns: 11.25 Ideal age: 3.5, 12.33 Ideal age to live forever: 5.12 Images of political candidates: 10.97 Job satisfaction: Ex 8.15, Ex 8.20 Job satisfaction of nurses in magnet and non-magnet hospitals: 11.19 Job satisfaction rankings by workers and boss: 14.56 Legalization of marijuana for medical purposes: 9.107 Lifetime learning activities: 5.114 Marriage proposals by women: 10.103 Memory test scores: 10.16, 10.150 Methods of disciplining children: 6.126 Personality characteristics of police academy applicants: AEx 10.20, 10.124 Proportions of Catholic and non-Catholic families in private schools: CPT 10.22 Psychology experiment: 10.7 Reporting cheating on exam: 5.109 Self-image test scores of public-assistance recipients: Ex 9.6 Sizes of communities reared in and residing in: 11.33 Teenagers’ views on contemporary issues: 14.1, 14.2, 14.89 Teen gambling: 9.74, 10.85 Television news preference and political affiliation: 3.7 Testing prospective employees: Ex 8.16 Test scores of clerk-typist applicants: 8.165 Volunteer work by children: 9.104 Worker opinions on biometric technology: 5.92 Worker-supervisor relationships: 11.35 Public Health and Safety Ages of volunteer ambulance members in upstate New York: 8.161 AIDS knowledge test scores: 10.146 Air pollution rankings of U.S cities: 14.58 Arrests for drug law violations: 12.50 Bike helmet laws: 9.86 Distance to nearest fire department: 8.152 “Five-second rule” for food safety: 2.171 Hand washing in public restrooms: 4.99 Helmet use with wheeled sports: 1.77 Medically Needy Program in Oregon: 5.22 Number of engines owned by fire departments: 8.6 Number of reported crimes by district: 11.60 Opinions on police agency organization by residence: Ex 4.24 Police officer exams: CPT 4.16 Population and violent crime rate: 13.37 Quality of service station restrooms: 11.34 Scores on Emergency Medical Services Certification Examination: 8.149 Seat belt usage: 9.89, 11.32 Seriousness weights in index of crime: AEx 13.7, 13.49 Speed limits—85th percentile rule: AEx 2.15 Tobacco settlement and population: 13.82 Traffic ticket and accident probability: 4.100 Weapons in school: 11.42 Sports Archery: 5.9 Attempted passes by NFL quarterbacks: 2.208 Baseball batting average: 5.71 Comparison of college football poll rankings: 14.87 Distance to centerfield fence in MLB stadiums: 2.116 Distribution of gold, silver, and bronze medals at Olympics: 13.76 Durability of golf balls: 12.51 Duration of MLB games: 6.113, 8.153, 8.167, 10.72 Earnings of Nationwide Tour professional golfers: 2.189 Football players’ sprint times on artificial turf and grass: 10.145 Football’s winning coaches: 4.14 Graduation rates of NCAA tournament teams: 2.113 High school basketball: 2.17 High school basketball injuries: 4.98, 11.64 Home runs in MLB: 1.76, 2.20 MLB runs scored at home and away: 2.77, 3.93 MLB team batting averages and earned run averages by league: 14.79 MLB won/loss percentages for away games by division: 12.29 Most holes played by golfers: 11.51 NBA coach records: 6.97 NBA players in Olympics: 13.38 NBA players” points scored and personal fouls committed per game: 3.1, 3.48, 3.105 NBA playoffs: 5.83 NCAA basketball: 4.44 Number of golf tournaments played by professionals: 2.170 Number of wins and earned run averages in MLB: 3.73 Odds against making professional sports team: AEx 4.7 Olympic biathlon: 5.65 Performance of Olympic gold medal winners over time: 3.25 PGA top money leaders and world rankings: 3.76 PGA tournament scores: 2.37 Points earned by NASCAR drivers: 2.207 Points per game and All-Star appearances of NBA players: 13.43 Points scored by NBA teams: 2.7 Points scored for and against NFL teams: 12.47, 13.11 Running times on cinder tracks and synthetic tracks: 14.73 “Size” measurements for MLB stadiums: 3.23 Soccer goals: 2.31 Sports championship series: 4.167 Steroid testing for athletes: 4.73, 4.116, 5.116 Stride rate and speed of serious runners: 3.68 Success rates for PGA players from various distances from the greens: 3.71 Super Bowl odds: 4.43 Surveys and Opinion Polls Attitudes of men and women on managing stress: 10.155 Cluster sampling: 1.56 Election poll: 1.68, Ex 4.8, Ex 4.9, Ex 4.14, Ex 4.15, Ex 4.20 Female president of United States: 6.98 Grid sampling: 1.49 Managers and professionals working late: 9.75 Margins of error in nationwide polls: AEx 9.9, 9.79 Opinions on budget proposal: Ex 4.11, Ex 4.12, Ex 4.19 Opinions on executive compensation: 10.99 Political election: 8.68, Ex 10.12 Poll on proposed legislation: Ex 11.4 Poll on recycling: CPT 11.18 Random sampling: 1.50, 1.51 Refusing job offer because of family considerations: 10.98 Research Randomizer: 14.52 Sample size for customer survey: 9.166 Sampling frame: 1.48, 1.57 Sampling methods: 1.41 Sampling student bodies at two schools: 7.26 “Sexiest job” poll: 6.127 Support for “get tough” policy in South America: 10.152 Systematic sample: 1.53 Telephone surveys: 1.52 Transportation Ages of urban transit rail vehicles: AEx 7.3 Airline complaints: 2.14, 2.187, 3.77, 4.12, 11.62 Airline engine reliability: 5.110 Airline on-time rates: 2.73, 2.99, 2.115, 6.132, 14.62 Airport runways: 4.51 Automobile speeds on expressway: 6.58 Baggage weights of airline passengers: 7.60 Bicycle fatalities and injuries: 4.54 Bridge conditions in North Carolina: 5.120 Car rental rates: 10.45, 14.8 Death rates on rural roads: 13.24 Distance between interchanges on interstates: 2.61 Fuel economy of SUVs: 9.33 Mass transit system data for large cities: 3.100 Maximum speed limits on interstate highways: 3.6 Miles per gallon of gasoline: 3.87, Ex 8.18, 8.172, 9.140 Motor-fuel consumption: 2.78 Number of miles of interstate highways: 2.60, 2.192, 3.45, 3.47, 3.74 On-time commutes: 4.74 Railroad riders: 4.9 Railroad violations: 4.133 Ship arrivals in harbor: 5.35, 5.103 Space shuttle reliability: 4.114 Speed of Eurostar train: 8.36 Speeds of automobiles: 2.43 Stopping distance on wet surface: 2.185, Ex 3.2, 12.43 Structurally deficient bridges: 2.98, 2.125 Taxi fares: 8.5 Thunderstorms and on-time flights: 4.150 Traffic circles: 4.148 Traffic control: 4.30, 4.137 Traffic fatalities: 2.114, 4.4 Transportation in D.C.: 4.55 Travel time index: 9.186 Wait times in airport security lines: 5.24 You determine what you need to learn …NOW! 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Community College Australia • Brazil • Canada • Mexico • Singapore • Spain United Kingdom • United States Elementary Statistics Enhanced Review Edition, Tenth Edition Robert Johnson and Patricia Kuby Senior Acquisitions Editor: Carolyn Crockett Permissions Editor: Joohee Lee Development Editor: Danielle Derbenti Production Service: Graphic World Inc Senior Assistant Editor: Ann Day Text Designer: Lisa Devenish Technology Project Manager: Fiona Chong Photo Researcher: Terri Wright Design Marketing Manager: Joseph Rogove Copy Editor: Graphic World Inc Marketing Assistant: Brian Smith Illustrator: Graphic World Inc Marketing Communications Manager: Darlene Amidon-Brent Cover Designer: Lee Friedman Project Manager, Editorial Production: Shelley Ryan Cover Image: © Getty Images Creative Director: Rob Hugel Cover Printer: Courier-Kendallville Art Director: Lee Friedman Compositor: Graphic World Inc Print Buyer: Doreen Suruki Printer: RR Donnelley/Willard ALL RIGHTS RESERVED No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, Web distribution, information storage and retrieval systems, or in any other manner—without the written permission of the publisher Thomson Higher Education 10 Davis Drive Belmont, CA 94002-3098 USA Printed in the United States of America 09 08 07 © 2008 Duxbury, an imprint of Thomson Brooks/Cole, a part of The Thomson Corporation Thomson, the Star logo, and Brooks/Cole are trademarks used herein under license Library of Congress Control Number: 2006940137 Student Edition ISBN-13: 978-0-495-38386-4 Annotated Instructor’s Edition ISBN-10: 0-495-38386-4 For more information about our products, contact us at: Thomson Learning Academic Resource Center 1-800-423-0563 For permission to use material from this text or product, submit a request online at http://www.thomsonrights.com Any additional questions about permissions can be submitted by e-mail to thomsonrights@thomson.com Brief Contents CHAPTER Statistics CHAPTER Descriptive Analysis and Presentation of Single-Variable Data 38 CHAPTER Descriptive Analysis and Presentation of Bivariate Data 144 CHAPTER Probability 204 CHAPTER Probability Distributions (Discrete Variables) 268 CHAPTER Normal Probability Distributions 312 CHAPTER Sample Variability 360 CHAPTER Introduction to Statistical Inferences 394 CHAPTER Inferences Involving One Population 472 CHAPTER 10 Inferences Involving Two Populations 544 CHAPTER 11 Applications of Chi-Square 618 CHAPTER 12 Analysis of Variance 656 CHAPTER 13 Linear Correlation and Regression Analysis 694 CHAPTER 14 Elements of Nonparametric Statistics 748 iii This page intentionally left blank SECTION 7.4 Application of the Sampling Distribution of Sample Means 379 Recall that the area (probability) under the normal curve is always exactly So as the width of the curve narrows, the height has to increase to maintain this area E X A M P L E Watch a video example at http://1pass.thomson.com or on your CD Calculating Probabilities for the Mean Height of Kindergarten Children Kindergarten children have heights that are approximately normally distributed about a mean of 39 inches and a standard deviation of inches A random sample of size 25 is taken, and the mean xෆ is calculated What is the probability that this mean value will be between 38.5 and 40.0 inches? SOLUTION We want to find P(38.5 Ͻ xෆ Ͻ 40.0) The values of xෆ, 38.5 and 40.0, must be converted to z-scores (necessary for use of Table in Appendix B) using xෆ Ϫ zϭ ᎏ : /͙n ෆ 38.5 –1.25 39.0 40.0 2.50 x z xෆ ϭ 38.5: xෆ Ϫ Ϫ0.5 38.5 Ϫ 39.0 zϭ ᎏ ϭ ᎏᎏ ϭ ᎏ ϭ Ϫ1.25 /͙n ෆ 2/͙25 ෆ 0.4 xෆ ϭ 40.0: xෆ Ϫ 40.0 Ϫ 39.0 1.0 zϭ ᎏ ϭ ᎏᎏ ϭ ᎏ ϭ 2.50 /͙n ෆ 2/͙25 ෆ 0.4 Therefore, P(38.5 Ͻ xෆ Ͻ 40.0) ϭ P(Ϫ1.25 Ͻ z Ͻ 2.50) ϭ 0.3944 ϩ 0.4938 ϭ 0.8882 E X A M P L E 7 Calculating Mean Height Limits for the Middle 90% of Kindergarten Children Use the heights of kindergarten children given in Example 7.6 Within what limits does the middle 90% of the sampling distribution of sample means for samples of size 100 fall? S O L U T I O N The two tools we have to work with are formula (7.2) and Table in Appendix B The formula relates the key values of the population to the key values of the sampling distribution, and Table relates areas to z-scores First, using Table 3, we find that the middle 0.9000 is bounded by z ϭ Ϯ1.65 90% (45%) (45%) z = –1.65 FYI Remember: If value is exactly halfway, use the larger z z 0.04 : 1.6 Ӈ 0.4495 z = 1.65 0.4500 z 0.05 0.4505 380 CHAPTER Sample Variability xෆ Ϫ Second, we use formula (7.2), z ϭ ᎏ : /͙n ෆ z ϭ Ϫ1.65: xෆ Ϫ 39.0 Ϫ1.65 ϭ ᎏᎏ 2/͙100 ෆ z ϭ 1.65: xෆ Ϫ 39 ϭ (Ϫ1.65)(0.2) xෆ ϭ 39 Ϫ 0.33 ϭ 38.67 xෆ Ϫ 39.0 1.65 ϭ ᎏᎏ 2/͙100 ෆ xෆ Ϫ 39 ϭ (1.65)(0.2) xෆ ϭ 39 ϩ 0.33 ϭ 39.33 Thus, P(38.67 Ͻ xෆ Ͻ 39.33) ϭ 0.90 Therefore, 38.67 inches and 39.33 inches are the limits that capture the middle 90% of the sample means S E C T I O N E X E R C I S E S 7.29 Consider a normal population with ϭ 43 and ϭ 5.2 Calculate the z-score for an xෆ of 46.5 from a sample of size 16 7.30 Consider a population with ϭ 43 and ϭ 5.2 a Calculate the z-score for an xෆ of 46.5 from a sample of size 35 b Could this z-score be used in calculating probabilities using Table in Appendix B? Why or why not? 7.31 In Example 7.5, explain how 0.4772 was obtained and what it is c Find the standard error of this sampling distribution d What is the probability that this sample mean will be between 45 and 55? e What is the probability that the sample mean will have a value greater than 48? f What is the probability that the sample mean will be within units of the mean? 7.34 The local bakery bakes more than a thousand 1-pound loaves of bread daily, and the weights of these loaves varies The mean weight is lb and oz., or 482 grams Assume the standard deviation of the weights is 18 grams and a sample of 40 loaves is to be randomly selected 7.32 What is the probability that the sample of kindergarten children in Example 7.6 has a mean height of less than 39.75 inches? a This sample of 40 has a mean value of xෆ, which belongs to a sampling distribution Find the shape of this sampling distribution 7.33 A random sample of size 36 is to be selected from a population that has a mean ϭ 50 and a standard deviation of 10 b Find the mean of this sampling distribution a This sample of 36 has a mean value of xෆ, which belongs to a sampling distribution Find the shape of this sampling distribution d What is the probability that this sample mean will be between 475 and 495? b Find the mean of this sampling distribution c Find the standard error of this sampling distribution e What is the probability that the sample mean will have a value less than 478? SECTION 7.4 f Application of the Sampling Distribution of Sample Means What is the probability that the sample mean will be within grams of the mean? 7.35 Consider the approximately normal population of heights of male college students with mean ϭ 69 inches and standard deviation ϭ inches A random sample of 16 heights is obtained a Describe the distribution of x, height of male college students b Find the proportion of male college students whose height is greater than 70 inches c Describe the distribution of xෆ, the mean of samples of size 16 d Find the mean and standard error of the xෆ distribution e Find P(xෆ Ͼ 70) f Find P(xෆ Ͻ 67) 7.36 The amount of fill (weight of contents) put into a glass jar of spaghetti sauce is normally distributed with mean ϭ 850 grams and standard deviation ϭ grams a Describe the distribution of x, the amount of fill per jar b Find the probability that one jar selected at random contains between 848 and 855 grams c Describe the distribution of xෆ, the mean weight for a sample of 24 such jars of sauce d Find the probability that a random sample of 24 jars has a mean weight between 848 and 855 grams 7.37 The heights of the kindergarten children mentioned in Example 7.6 (p 379) are approximately normally distributed with ϭ 39 and ϭ a If an individual kindergarten child is selected at random, what is the probability that he or she has a height between 38 and 40 inches? b A classroom of 30 of these children is used as a sample What is the probability that the class mean xෆ is between 38 and 40 inches? c If an individual kindergarten child is selected at random, what is the probability that he or she is taller than 40 inches? 381 d A classroom of 30 of these kindergarten children is used as a sample What is the probability that the class mean xෆ is greater than 40 inches? 7.38 WageWeb (http://www.wageweb.com/health1 htm) is a service of HRPDI and provides compensation information on more than 170 benchmark positions in human resources The October 2003 posting indicated that labor relation managers earn a mean annual salary of $86,700 Assume that annual salaries are normally distributed and have a standard deviation of $8850 a What is the probability that a randomly selected labor relation manager earned more than $100,000 in 2003? b A sample of 20 labor relation managers is taken, and annual salaries are reported What is the probability that the sample mean annual salary falls between $80,000 and $90,000? 7.39 Based on 53 years of data compiled by the National Climatic Data Center (http://lwf.ncdc.noaa gov/oa/climate/online/ccd/avgwind.html), the average speed of winds in Honolulu, Hawaii, equals 11.3 mph, as of June 2004 Assume that wind speeds are approximately normally distributed with a standard deviation of 3.5 mph a Find the probability that the wind speed on any one reading will exceed 13.5 mph b Find the probability that the mean of a random sample of nine readings will exceed 13.5 mph c Do you think the assumption of normality is reasonable? Explain d What effect you think the assumption of normality had on the answers to parts a and b? Explain 7.40 TIMSS 2003 (Trends in International Mathematics and Science Study) focused on the mathematics and science achievement of eighth-graders throughout the world A total of 45 countries (including the United States) participated in the 382 CHAPTER Sample Variability study The mean math exam score for U.S students was 504 with a standard deviation of 88 Source: http://nces.ed.gov/timss/TIMSS03Tables Assume the scores are normally distributed and a sample of 150 students is taken a Find the probability that the mean TIMSS score for a randomly selected group of eighthgraders would be between 495 and 510 b Find the probability that the mean TIMSS score for a randomly selected group of eighthgraders would be less than 520 c Do you think the assumption of normality is reasonable? Explain 7.41 According to the June 2004 Readers’ Digest article “Only in America,” the average amount that a 17-year-old spends on his or her high school prom is $638 Assume that the amounts spent are normally distributed with a standard deviation of $175 a Find the probability that the mean cost to attend a high school prom for 36 randomly selected high school 17-year-olds is between $550 and $700 b Find the probability that the mean cost to attend a high school prom for 36 randomly selected high school 17-year-olds is greater than $750 c Do you think the assumption of normality is reasonable? Explain 7.42 WageWeb (http://www.wageweb.com/health1 htm) provides compensation information and services on more than 160 positions As of October 1, 2003, the national average salary for a registered nurse (RN) was $47,858 Suppose the standard deviation is $7750 a Find the probability that the mean of a sample of 100 such nurses is less than $45,000 b Find the probability that the sample mean of a sample of 100 such nurses is between $46,000 and $48,000 c Find the probability that the sample mean of a sample of 100 such nurses is greater than $50,000 d Explain why the assumption of normality about the distribution of wages was not involved in the solution to parts a–c 7.43 Referring to Example 7.6 (p 379), what height would bound the lower 25% of all samples of size 25? 7.44 A popular flashlight that uses two D-size batteries was selected, and several of the same models were purchased to test the “continuous-use life” of D batteries As fresh batteries were installed, each flashlight was turned on and the time noted When the flashlight no longer produced light, the time was again noted The resulting “life” data from Rayovac batteries had a mean of 21.0 hours (source: http://www.rayovac.com) Assume these values have a normal distribution with a standard deviation of 1.38 hours a What is the probability that one randomly selected Rayovac battery will have a test life of between 20.5 and 21.5 hours? b What is the probability that a randomly selected sample of Rayovac batteries will have a mean test life of between 20.5 and 21.5 hours? c What is the probability that a randomly selected sample of 16 Rayovac batteries will have a mean test life of between 20.5 and 21.5 hours? d What is the probability that a randomly selected sample of 64 Rayovac batteries will have a mean test life of between 20.5 and 21.5 hours? e Describe the effect that the increase in sample size had on the answers for parts b–d 7.45 a Find P(4 Ͻ xෆ Ͻ 6) for a random sample of size drawn from a normal population with ϭ and ϭ b Use a computer to randomly generate 100 samples, each of size 4, from a normal probability distribution with ϭ and ϭ Calculate the mean, xෆ, for each sample SECTION 7.4 Application of the Sampling Distribution of Sample Means c How many of the sample means in part b have values between and 6? What percentage is that? d Compare the answers to parts a and c, and explain any differences that occurred 383 0, the last value with 9, the steps with 1, and the output range with H1 Use the HISTOGRAM commands on page 61 with column G as the input range, column H as the bin range, and column I as the output range MINITAB (Release 14) TI-83/84 Plus a Input the numbers and into C1 Use the CUMULATIVE NORMAL PROBABILITY DISTRIBUTION commands on page 329, replacing the mean with 5, the standard deviation with ෆ), the input column with C1, and the op(2/͙4 tional storage with C2 Find CDF(6) Ϫ CDF(4) a Use the CUMULATIVE NORMAL PROBABILITY commands on page 330, replacing the Enter with 4,6,5,1) (The standard deviation is 1; ෆ.) from 2/͙4 b Use the Normal RANDOM DATA commands on page 327, replacing generate with 100, store in with C3–C6, mean with 5, and standard deviation with Use the ROW STATISTICS commands on page 368, replacing input variables with C3–C6 and store result in with C7 c Use the HISTOGRAM commands on page 61 for the data in C7 Select Labels, Data Labels, Label Type; use y-value levels To adjust the histogram, select Binning with midpoint and midpoint positions 0:10/1 Excel a Input the numbers and into column A Activate cell B1 Use the CUMULATIVE NORMAL DISTRIBUTION commands on page 329, replacing X with A1:A2 Find CDF(6)ϪCDF(4) b Use the Normal RANDOM NUMBER GENERATION commands on page 328, replacing number of variables with 4, number of random numbers with 100, mean with 5, standard deviation with 2, and output range with C1 Activate cell G1 Use the AVERAGE INSERT FUNCTION commands in Exercise 7.13b on page 368, replacing Number1 with C1:F1 c Use the RANDOM NUMBER GENERATION Patterned Distribution commands in Exercise 6.66 on page 336, replacing the first value with b Use the Normal RANDOM DATA and STO commands on page 328, replacing the Enter with 5,2,100) Repeat these commands three more times, storing data in L2, L3, and L4, respectively Choose: STAT Ͼ EDIT Ͼ 1:Edit Highlight: L5 (column heading) Enter: (L1ϩL2ϩL3ϩL4)/4 c Use the HISTOGRAM and TRACE commands on page 62 to count Enter 0,9,1,0,45,1 for the Window 7.46 a Find P(46 Ͻ xෆ Ͻ 55) for a random sample size 16 drawn from a normal population with mean ϭ 50 and standard deviation ϭ 10 b Use a computer to randomly generate 200 samples, each of size 16, from a normal probability distribution with mean ϭ 50 and standard deviation ϭ 10 Calculate the mean, xෆ, for each sample c How many of the sample means in part b have values between 46 and 55? What percentage is that? d Compare the answers to parts a and c, and explain any differences that occurred FYI If you use a computer, see Exercise 7.45 384 CHAPTER Sample Variability CHAPTE R R EVI EW In Retrospect In Chapters and we have learned to use the standard normal probability distribution We now have two formulas for calculating a z-score: xϪ zϭ ᎏ and xෆ Ϫ zϭ ᎏ /͙n ෆ You must be careful to distinguish between these two formulas The first gives the standard score when we have individual values from a normal distribution (x values) The second formula deals with a sample mean (xෆ value) The key to distinguishing between the formulas is to decide whether the problem deals with an individual x or a sample mean xෆ If it deals with the individual values of x, we use the first formula, as presented in Chapter If the problem deals with a sample mean, xෆ, we use the second formula and proceed as illustrated in this chapter The basic purpose for considering what happens when a population is repeatedly sampled, as discussed in this chapter, is to form sampling distributions The sampling distribution is then used to describe the variability that occurs from one sample to the next Once this pattern of variability is known and understood for a specific sample statistic, we are able to make predictions about the corresponding population parameter with a measure of how accurate the prediction is The SDSM and the central limit theorem help describe the distribution for sample means We will begin to make inferences about population means in Chapter There are other reasons for repeated sampling Repeated samples are commonly used in the field of production control, in which samples are taken to determine whether a product is of the proper size or quantity When the sample statistic does not fit the standards, a mechanical adjustment of the machinery is necessary The adjustment is then followed by another sampling to be sure the production process is in control ” is the name The “standard error of the used for the standard deviation of the sampling distribution for whatever statistic is named in the blank In this chapter we have been concerned with the standard error of the mean However, we could also work with the standard error of the proportion, median, or any other statistic You should now be familiar with the concept of a sampling distribution and, in particular, with the sampling distribution of sample means In Chapter we will begin to make predictions about the values of population parameters Vocabulary and Key Concepts central limit theorem (p 370) frequency distribution (p 364) probability distribution (p 364) random sample (p 365) repeated sampling (p 366) sampling distribution (p 373) sampling distribution of sample means (pp 363, 369) standard error of the mean (p 370) z-score (p 377) Learning Outcomes ✓ Understand what a sampling distribution of a sample statistic is and that the distribution is obtained from repeated samples, all of the same size pp 363–364, EXP 7.1 ✓ Be able to form a sampling distribution for a mean, median, or range based on a small, finite population ✓ Understand that a sampling distribution is a probability distribution for a sample statistic EXP 7.1, Ex 7.6, 7.7 EXP 7.2 Chapter Exercises 385 ✓ Understand and be able to present and describe the sampling distribution of sample means and the central limit theorem pp 369–371, EXP 7.4 ✓ Understand and be able to explain the relationship between the sampling distribution of sample means and the central limit theorem pp 369–371, Ex 7.17, 7.18, 7.21 ✓ Determine and be able to explain the effect of sample size on the standard error of the mean pp 373–375, Ex 7.20, 7.26, 7.47 ✓ Understand when and how the normal distribution can be used to find probabilities corresponding to sample means EXP 7.5 ✓ Compute, describe, and interpret z-scores corresponding to known values of xෆ EXP 7.6, EXP 7.7, Ex 7.29, 7.30, 7.48 ✓ Compute z-scores and probabilities for applications of the sampling distribution of sample means Ex 7.33, 7.35 Chapter Exercises b What is the probability that the selected graduate is making between $550 and $825? Go to the StatisticsNow website http://1pass.thomson.com to • Assess your understanding of this chapter • Check your readiness for an exam by taking the Pre-Test quiz and exploring the resources in the Personalized Learning Plan 7.47 If a population has a standard deviation of 18.2 units, what is the standard error of the mean if samples of size are selected? Samples of size 25? Samples of size 49? Samples of size 100? 7.48 Consider a normal population with ϭ 24.7 and ϭ 4.5 If a random sample of 25 graduates is selected: c Describe the distribution of mean weekly salaries being earned year after graduation d What is the probability that the sample mean is between $650 and $705? e Why is the z-score used in answering parts b and d? f Why is the formula for z-score used in part d different from that used in part b? b Calculate the z-score for an xෆ of 21.5 from a sample of size 25 7.50 The diameters of Red Delicious apples in a certain orchard are normally distributed with a mean of 2.63 inches and a standard deviation of 0.25 inch c Explain how 21.5 can have such different z-scores a What percentage of the apples in this orchard have diameters less than 2.25 inches? a Calculate the z-score for an x of 21.5 7.49 The Dean of Nursing tells students being recruited for the incoming class that year after graduation, the university’s graduates can expect to be earning a mean weekly income of $675 Assume that the dean’s statement is true and that the weekly salaries year after graduation are normally distributed with a standard deviation of $85 If one graduate is randomly selected: a Describe the distribution of the weekly salaries being earned year after graduation b What percentage of the apples in this orchard are larger than 2.56 inches in diameter? A random sample of 100 apples is gathered, and the mean diameter obtained is xෆ ϭ 2.56 c If another sample of size 100 is taken, what is the probability that its sample mean will be greater than 2.56 inches? d Why is the z-score used in answering parts a–c? e Why is the formula for z-score used in part c different from that used in parts a and b? 386 CHAPTER Sample Variability 7.51 a Find a value for e such that 95% of the apples in Exercise 7.50 are within e units of the mean, 2.63 That is, find e such that P(2.63 Ϫ e Ͻ x Ͻ 2.63 ϩ e) ϭ 0.95 b Find a value for E such that 95% of the samples of 100 apples taken from the orchard in Exercise 7.50 will have mean values within E units of the mean, 2.63 That is, find E such that P (2.63 Ϫ E Ͻ xෆ Ͻ 2.63 ϩ E ) ϭ 0.95 7.52 Americans spend billions of dollars on veterinary care each year, predicted to hit $31 billion this year The health care services offered to animals rival those provided to humans, with the typical surgery costing from $1700 to $3000, or even more In 2003, on average, dog owners spent $196 on veterinary-related expenses in the prior 12 months Source: American Pet Products Manufacturers Association Assume that annual dog owner expenditure on health care is normally distributed with a mean of $196 and a standard deviation of $95 a What is the probability that a dog owner, randomly selected from the population, spent more than $300 for dog health care in 2003? b Suppose a survey of 300 dog owners is conducted, and each person is asked to report the total of their vet care bills for 2003 What is the probability that the mean annual expenditure of this sample falls between $200 and $225? c The assumption of a normal distribution in this situation is likely misguided Explain why and what effect this had on the answers 7.53 The statistics-conscious store manager at Marketview records the number of customers who walk through the door each day Years of records show the mean number of customers per day to be 586 with a standard deviation of 165 Assume the number of customers is normally distributed a What is the probability that on any given day, the number of customers exceeds 1000? b If 20 days are randomly selected, what is the probability that the mean of this sample is less than 550? c The assumption of normality allowed you to calculate the probabilities; however, this may not be a reasonable assumption Explain why and how that affects the probabilities found in parts a and b 7.54 A study from the University of Michigan, as noted in Newsweek (March 25, 2002), stated that men average 16 hours of housework each week (up from an average of 12 hours in 1965) If we assume that the number of hours in which men engage in housework each week is normally distributed with a standard deviation of 5.4 hours, what is the probability that the mean number of housework hours for a sample of 20 randomly selected men is between 15 to 18 hours? 7.55 A shipment of steel bars will be accepted if the mean breaking strength of a random sample of 10 steel bars is greater than 250 pounds per square inch In the past, the breaking strength of such bars has had a mean of 235 and a variance of 400 a Assuming that the breaking strengths are normally distributed, what is the probability that one randomly selected steel bar will have a breaking strength in the range from 245 to 255 pounds per square inch? b What is the probability that the shipment will be accepted? 7.56 An April 15, 2002, report in Time magazine stated that the average age for women to marry in the United States is now 25 years of age If the standard deviation is assumed to be 3.2 years, find the probability that a random sample of 40 U.S women would show a mean age at marriage of less than or equal to 24 years 7.57 A manufacturer of light bulbs claims that its light bulbs have a mean life of 700 hours and a standard deviation of 120 hours You purchased 144 of these bulbs and decided that you would purchase more if the mean life of your current sample exceeded 680 hours What is the probability that you will not buy again from this manufacturer? Chapter Exercises 7.58 A tire manufacturer claims (based on years of experience with its tires) that the mean mileage is 35,000 miles and the standard deviation is 5000 miles A consumer agency randomly selects 100 of these tires and finds a sample mean of 31,000 Should the consumer agency doubt the manufacturer’s claim? 7.59 For large samples, the sample sum (Αx) has an approximately normal distribution The mean of the sample sum is n ؒ and the standard deviation is ͙n ෆ ؒ The distribution of savings per account for a savings and loan institution has a mean equal to $750 and a standard deviation equal to $25 For a sample of 50 such accounts, find the probability that the sum in the 50 accounts exceeds $38,000 7.60 The baggage weights for passengers using a particular airline are normally distributed with a mean of 20 lb and a standard deviation of lb If the limit on total luggage weight is 2125 lb., what is the probability that the limit will be exceeded for 100 passengers? 7.61 A trucking firm delivers appliances for a large retail operation The packages (or crates) have a mean weight of 300 lb and a variance of 2500 a If a truck can carry 4000 lb and 25 appliances need to be picked up, what is the probability that the 25 appliances will have an aggregate weight greater than the truck’s capacity? Assume that the 25 appliances represent a random sample b If the truck has a capacity of 8000 lb., what is the probability that it will be able to carry the entire lot of 25 appliances? 7.62 A pop-music record firm wants the distribution of lengths of cuts on its records to have an average of minutes and 15 seconds (135 seconds) and a standard deviation of 10 seconds so that disc jockeys will have plenty of time for commercials within each 5-minute period The population of times for cuts is approximately normally distributed with only a negligible skew to the right You have just timed the cuts on a new release and have found that the 10 cuts average 140 seconds 387 a What percentage of the time will the average be 140 seconds or longer if the new release is randomly selected? b If the music firm wants 10 cuts to average 140 seconds less than 5% of the time, what must the population mean be, given that the standard deviation remains at 10 seconds? 7.63 Let’s simulate the sampling distribution related to the disc jockey’s concern for “length of cut” in Exercise 7.62 a Use a computer to randomly generate 50 samples, each of size 10, from a normal distribution with mean 135 and standard deviation 10 Find the “sample total” and the sample mean for each sample b Using the 50 sample means, construct a histogram and find their mean and standard deviation c Using the 50 sample “totals,” construct a histogram and find their mean and standard deviation d Compare the results obtained in parts b and c Explain any similarities and any differences observed MINITAB (Release 14) a Use the Normal RANDOM DATA commands on page 327, replacing generate with 50, store in with C1–C10, mean with 135, and standard deviation with 10 Use the ROW STATISTICS commands on page 368, selecting Sum and replacing input variables with C1–C10 and store result in with C11 Use the ROW STATISTICS commands, again selecting Mean and then replacing input variables with C1–C10 and store result in with C12 b Use the HISTOGRAM commands on page 61 for the data in C12 To adjust the histogram, select Binning with midpoint Use the MEAN and STANDARD DEVIATION commands on pages 74 and 88 for the data in C12 c Use the HISTOGRAM commands on page 61 for the data in C11 To adjust the histogram, select Binning with midpoints Use the MEAN 388 CHAPTER Sample Variability and STANDARD DEVIATION commands on pages 74 and 88 for the data in C11 d Use the DISPLAY DESCRIPTIVE STATISTICS commands on page 98 for the data in C11 and C12 Excel a Use the Normal RANDOM NUMBER GENERATION commands on page 328, replacing number of variables with 10, number of random numbers with 50, mean with 135, and standard deviation with 10 Activate cell K1 Choose: Insert function, fx Ͼ All Ͼ SUM Ͼ OK Enter: Number1: (A1:J1 or select cells) Drag: Bottom right corner of sum value box down to give other sums Activate cell L1 Use the AVERAGE INSERT FUNCTION commands in Exercise 7.13b on page 368, replacing Number1 with A1:J1 b Use the RANDOM NUMBER GENERATION Patterned Distribution commands in Exercise 6.66 on page 336, replacing the first value with 125.4, the last value with 144.6, the steps with 3.2, and the output range with M1 Use the HISTOGRAM commands on page 61 with column L as the input range and column M as the bin range Use the MEAN and STANDARD DEVIATION commands on pages 74 and 88 for the data in column L c Use the RANDOM NUMBER GENERATION Patterned Distribution commands in Exercise 6.66 on page 336, replacing the first value with 1254, the last value with 1446, the steps with 32, and the output range with M20 Use the HISTOGRAM commands on page 61 with column L as the input range and cells M20–? as the bin range Use the MEAN and STANDARD DEVIATION commands on pages 74 and 88 for the data in column K d Use the DESCRIPTIVE STATISTICS commands on page 98 for the data in columns K and L 7.64 a Find the mean and standard deviation of x for a binomial probability distribution with n ϭ 16 and p ϭ 0.5 b Use a computer to construct the probability distribution and histogram for the binomial probability experiment with n ϭ 16 and p ϭ 0.5 c Use a computer to randomly generate 200 samples of size 25 from a binomial probability distribution with n ϭ 16 and p ϭ 0.5 Calculate the mean of each sample d Construct a histogram and find the mean and standard deviation of the 200 sample means e Compare the probability distribution of x found in part b and the frequency distribution of xෆ in part d Does your information support the CLT? Explain MINITAB (Release 14) a Use the MAKE PATTERNED DATA commands in Exercise 6.66 on page 336, replacing the first value with 0, the last value with 16, and the steps with Use the BINOMIAL PROBABILITY DISTRIBUTIONS commands on page 292, replacing n with 16, p with 0.5, input column with C1, and optional storage with C2 Use the Scatterplot with Connect Line commands on page 155, replacing Y with C2 and X with C1 b Use the BINOMIAL RANDOM DATA commands on page 303, replacing generate with 200, store in with C3–C27, number of trials with 16, and probability with 0.5 Use the ROW STATISTICS commands for a mean on page 368, replacing input variables with C3–C27 and store result in with C28 Use the HISTOGRAM commands on page 61 for the data in C28 To adjust histogram, select Binning with midpoints Use the MEAN and STANDARD DEVIATION commands on pages 74 and 88 for the data in C28 Excel a Input through 16 into column A Continue with the binomial probability commands on page 292, using n ϭ 16 and p ϭ 0.5 Activate columns A and B; then continue with: Choose: Chart Wizard Ͼ Column Ͼ 1st picture Ͼ Next Ͼ Series Choose: Series Ͼ Remove Enter: Category (x)axis labels: (A1:A17 or select ‘x value’ cells) Choose: Next Ͼ Finish Chapter Project b Use the Binomial RANDOM NUMBER GENERATION commands from Exercise 5.97 on page 304, replacing number of variables with 25, number of random numbers with 200, p value with 0.5, number of trials with 16, and output range with C1 Activate cell BB1 Use the AVERAGE INSERT FUNCTION commands in Exercise 7.13b on page 368, replacing Number1 with C1:AA1 c Use the RANDOM NUMBER GENERATION Patterned Distribution commands in Exercise 6.66 on page 336, replacing the first value with 6.8, the last value with 9.2, the steps with 0.4, and the output range with CC1 Use the HISTOGRAM commands on page 61 with column BB as the input range and column CC as the bin range Use the MEAN and STANDARD DEVIATION commands on pages 74 and 88 for the data in column BB 7.65 a Find the mean and standard deviation of x for a binomial probability distribution with n ϭ 200 and p ϭ 0.3 b Use a computer to construct the probability distribution and histogram for the random variable x of the binomial probability experiment with n ϭ 200 and p ϭ 0.3 389 c Use a computer to randomly generate 200 samples of size 25 from a binomial probability distribution with n ϭ 200 and p ϭ 0.3 Calculate the mean xෆ of each sample d Construct a histogram and find the mean and standard deviation of the 200 sample means e Compare the probability distribution of x found in part b and the frequency distribution of xෆ in part d Does your information support the CLT? Explain FYI Use the commands in Exercise 7.64, making the necessary adjustments 7.66 A sample of 144 values is randomly selected from a population with mean, , equal to 45 and standard deviation, , equal to 18 a Determine the interval (smallest value to largest value) within which you would expect a sample mean to lie b What is the amount of deviation from the mean for a sample mean of 45.3? c What is the maximum deviation you have allowed for in your answer to part a? d How is this maximum deviation related to the standard error of the mean? Chapter Project 275 Million Americans As noted in the “The Sampling Issue” of Section 7.1, “275 Million Americans” (p 361), the fundamental goal of a survey is to come up with the same results that we would have obtained had we interviewed every person of the population Knowing that interviewing every person of a population is nearly impossible for most populations promotes the importance of a good representative sample In addition, we now have the sampling distribution of sample means and the central limit theorem to help us make predictions about the population by using the sample Putting Chapter to Work will help us put these new concepts together 390 CHAPTER Sample Variability Putting Chapter to Work 7.67 A second sample of 100 ages as been collected from the U.S 2000 census and is listed here [EX07-67] 14 59 64 39 12 34 27 16 18 17 33 56 60 65 73 53 43 26 42 60 87 58 42 82 21 35 64 58 53 36 66 63 66 39 62 58 49 31 27 39 35 12 28 28 20 54 41 41 63 39 37 23 79 43 28 17 12 45 52 10 11 32 32 23 86 61 50 27 19 15 51 36 83 39 35 44 59 30 31 69 40 16 40 66 15 55 32 43 41 23 46 61 30 m Describe the sampling SDSM for samples of size 1000 Be sure to include center, spread, and shape n Relate your findings in parts j, l, and m to the SDSM and the CLT Your Study a How would you describe the preceding “ages” sample data graphically? Construct the graph 7.68 Skillbuilder Applet Exercise simulates taking samples of size 50 from the population of American ages from the 2000 census, where ϭ 36.5 and ϭ 22.5 and the shape is skewed right b Using the graph that you constructed in part a, describe the shape of the distribution of sample data a Click “1” for “# Samples.” Note the 50 data values and their mean Change “slow” to “batch” and take at least 1000 samples of size 50 c How well did the sample describe the population of ages from the 2000 census shown in Section 7.1? Explain using the graphical displays b What is the mean of the sample means? How close is it to the population mean? d How would you describe the preceding “ages” sample data numerically? Calculate the statistics c What is the standard deviation of the sample means? e How well the statistics calculated in part d compare with the parameters from the 2000 census given in Section 7.1? d Based on the SDSM (as described in Section 7.3), what should you expect for the standard deviation of sample means? How close was your standard deviation from part c? f e What shape is the histogram of the 1000 means? (optional) If you completed Exercises 7.1 and 7.2, how does your graphical display and statistics compare with those constructed and calculated in Exercises 7.1 and 7.2 using a different sample of 100 ages? g Is the distribution of ages for the population of Americans in Section 7.1 normal? Is it approximately normal? h Will the SDSM apply to samples taken from this population? Explain i Will the CLT apply to samples taken from this population? Explain j Describe the SDSM for samples of size 100 Be sure to include center, spread, and shape k Compare your results in parts a and d with the theoretical answers in part j Be sure to include center, spread, and shape l Describe the SDSM for samples of size 30 Be sure to include center, spread, and shape f Relate your findings to the SDSM and the CLT Chapter Practice Test PART I: Knowing the Definitions Answer “True” if the statement is always true If the statement is not always true, replace the words shown in bold with words that make the statement always true 7.1 A sampling distribution is a distribution listing all the sample statistics that describe a particular sample 7.2 The histograms of all sampling distributions are symmetrical 7.3 The mean of the sampling distribution of xෆ’s is equal to the mean of the sample 7.4 The standard error of the mean is the standard deviation of the population from which the samples have been taken Chapter Practice Test The standard error of the mean increases as the sample size increases 7.6 The shape of the distribution of sample means is always that of a normal distribution 7.7 A probability distribution of a sample statistic is a distribution of all the values of that statistic that were obtained from all possible samples 7.8 The sampling distribution of sample means provides us with a description of the three characteristics of a sampling distribution of sample medians 7.9 A frequency sample is obtained in such a way that all possible samples of a given size have an equal chance of being selected 7.10 We not need to take repeated samples in order to use the concept of the sampling distribution 7.5 PART II: Applying the Concepts 7.11 The lengths of the lake trout in Conesus Lake are believed to have a normal distribution with a mean of 15.6 inches and a standard deviation of 3.8 inches a Kevin is going fishing at Conesus Lake tomorrow If he catches one lake trout, what is the probability that it is less than 15.0 inches long? b If Captain Brian’s fishing boat takes 10 people fishing on Conesus Lake tomorrow and they catch a random sample of 16 lake trout, what is the probability that the mean length of their total catch is less than 15 inches? 7.12 Cigarette lighters manufactured by EasyVice Company are claimed to have a mean lifetime of 20 months with a standard deviation of months The money-back guarantee allows you to return the lighter if it does not last at least 12 months from the date of purchase a If the lifetimes of these lighters are normally distributed, what percentage of the lighters will be returned to the company? b If a random sample of 25 lighters is tested, what is the probability the sample mean lifetime will be more than 18 months? 7.13 Aluminum rivets produced by Rivets Forever, Inc., are believed to have shearing 391 strengths that are distributed about a mean of 13.75 with a standard deviation of 2.4 If this information is true and a sample of 64 such rivets is tested for shear strength, what is the probability that the mean strength will be between 13.6 and 14.2? PART III: Understanding the Concepts 7.14 “Two heads are better than one.” If that’s true, then how good would several heads be? To find out, a statistics instructor drew a line across the chalkboard and asked her class to estimate its length to the nearest inch She collected their estimates, which ranged from 33 to 61 inches, and calculated the mean value She reported that the mean was 42.25 inches She then measured the line and found it to be 41.75 inches long Does this show that “several heads are better than one”? What statistical theory supports this occurrence? Explain how 7.15 The sampling distribution of sample means is more than just a distribution of the mean values that occur from many repeated samples taken from the same population Describe what other specific condition must be met in order to have a sampling distribution of sample means 7.16 Student A states, “A sampling distribution of the standard deviations tells you how the standard deviation varies from sample to sample.” Student B argues, “A population distribution tells you that.” Who is right? Justify your answer 7.17 Student A says it is the “size of each sample used” and Student B says it is the “number of samples used” that determines the spread of an empirical sampling distribution Who is right? Justify your choice Preparing for an exam? Assess your progress by taking the post-test at http://1pass.thomson.com Do you need a live tutor for homework problems? Access vMentor on the StatisticsNow website at http://1pass.thomson.com for one-on-one tutoring from a statistics expert 392 CHAPTER Sample Variability Working with Your Own Data Putting Probability to Work The sampling distribution of sample means and the central limit theorem are very important to the development of the rest of this course The proof, which requires the use of calculus, is not included in this textbook However, the truth of the SDSM and the CLT can be demonstrated both theoretically and by experimentation The following activities will help to verify both statements A The Population Construct a histogram for this sampling distribution of sample means Calculate the mean xෆ and the standard error of the mean xෆ using the probability distribution found in question Show that the results found in questions 1c, 5, and support the three claims made by the sampling distribution of sample means and the central limit theorem Cite specific values to support your conclusions Consider the theoretical population that contains the three numbers 0, 3, and in equal proportions C The Sampling Distribution, Empirically a Construct the theoretical probability distribution for the drawing of a single number, with replacement, from this population Let’s now see whether the sampling distribution of sample means and the central limit theorem can be verified empirically; that is, does it hold when the sampling distribution is formed by the sample means that result from several random samples? b Draw a histogram of this probability distribution c Calculate the mean, , and the standard deviation, , for this population B The Sampling Distribution, Theoretically Let’s study the theoretical sampling distribution formed by the means of all possible samples of size that can be drawn from the given population Construct a list showing all the possible samples of size that could be drawn from this population (There are 27 possibilities.) Find the mean for each of the 27 possible samples listed in answer to question Construct the probability distribution (the theoretical sampling distribution of sample means) for these 27 sample means Draw a random sample of size from the given population List your sample of three numbers and calculate the mean for this sample You may use a computer to generate your samples You may take three identical “tags” numbered 0, 3, and 6, put them in a “hat,” and draw your sample using replacement between each drawing Or you may use dice; let be represented by and 2; 3, by and 4; and 6, by and You may also use random numbers to simulate the drawing of your samples Or you may draw your sample from the list of random samples at the end of this section Describe the method you decide to use (Ask your instructor for guidance.) Repeat question forty-nine more times so that you have a total of 50 sample means that have resulted from samples of size Working with Your Own Data Here are 100 random samples of size that were generated by computer: 10 Construct a frequency distribution of the 50 sample means found in questions and 11 Construct a histogram of the frequency distribution of observed sample means 12 Calculate the mean xෆ and standard deviation sxෆ of the frequency distribution formed by the 50 sample means 13 Compare the observed values of xෆ and sxෆ with the values of xෆ and xෆ Do they agree? Does the empirical distribution of xෆ look like the theoretical one? 393 6 6 3 6 3 6 6 3 6 3 3 6 6 3 3 3 0 6 3 3 0 0 3 6 3 6 6 0 0 0 6 6 6 6 0 3 3 6 6 6 3 0 6 3 0 6 6 3 3 6 6 0 6 6 3 0 6 6 3 6 6 6 0 3 6 6 0 3 3 3 3 0 3 6 3 3 6 6 3 6 6 6 3 0 6 3 6 0 3 3 3 3 0 6 6 ... production: 13 .85, CPT 13 .11 , CPT 13 .12 , CPT 13 .13 , CPT 13 .14 , CPT 13 .15 , CPT 13 .16 , CPT 13 .17 , CPT 13 .18 , CPT 13 .19 , CPT 13 .20, CPT 13 . 21 Number of stamens and carpels in flowers: 13 .83 Oat crop... response: 10 .16 3 Amounts of fill: 1. 38, 6.53, 6 .11 5, 6 .11 8, 8 .19 4, Ex 9 .17 , Ex 10 .10 , Ex 10 .15 , Ex 10 .17 , 10 .16 0, 12 .45 Asphalt mixtures: 8.58 Asphalt sampling procedures: AEx 10 .7, 10 .37 Bad... cases: 10 .15 7 Carat weight and price of diamonds: 13 .45 Checkout times: 1. 12, CPT 2 .11 , 8 .11 5, 8 .18 5 Commissioned sales amounts: 8 .18 Commute times: 8 .18 8, 10 . 21, 12 .1, 12 .2, 12 .62, Ex 14 .2 Commuting