INSTRUCTOR’S SOLUTIONS MANUAL DIACRITECH FUNDAMENTALS OF STATISTICS: INFORMED DECISIONS USING DATA. by digitaldownload87

Table of Contents Preface Chapter 1 Data Collection 1.1 Introduction to the Practice of Statistics ............................................................................................................. 1 1.2 Observational Studies versus Designed Experiments ......................................................................................... 4 1.3 Simple Random Sampling .................................................................................................................................. 7 1.4 Other Effective Sampling Methods .................................................................................................................... 9 1.5 Bias in Sampling ............................................................................................................................................... 11 1.6 The Design of Experiments .............................................................................................................................. 14 Chapter 1 Review Exercises ........................................................................................................................................ 21 Chapter 1 Test ............................................................................................................................................................. 24 Case Study: Chrysalises for Cash ................................................................................................................................ 26 Chapter 2 Summarizing Data in Tables and Graphs 2.1 Organizing Qualitative Data ............................................................................................................................. 28 2.2 Organizing Quantitative Data ........................................................................................................................... 38 2.3 Graphical Misrepresentations of Data .............................................................................................................. 51 Chapter 2 Review Exercises ........................................................................................................................................ 53 Chapter 2 Test ............................................................................................................................................................. 58 Case Study: The Day the Sky Roared.......................................................................................................................... 62 Chapter 3 Numerically Summarizing Data 3.1 Measures of Central Tendency ......................................................................................................................... 65 3.2 Measures of Dispersion .................................................................................................................................... 72 3.3 Measures of Central Tendency and Dispersion from Grouped Data ................................................................ 89 3.4 Measures of Position and Outliers .................................................................................................................... 99 3.5 The Five-Number Summary and Boxplots ..................................................................................................... 106 Chapter 3 Review Exercises ...................................................................................................................................... 117 Chapter 3 Test ........................................................................................................................................................... 123 Case Study: Who Was “A Mourner”? ....................................................................................................................... 126 Chapter 4 Describing the Relation between Two Variables 4.1 Scatter Diagrams and Correlation ................................................................................................................... 128 4.2 Least-Squares Regression ............................................................................................................................... 149 4.3 The Coefficient of Determination ................................................................................................................... 161 4.4 Contingency Tables and Association .............................................................................................................. 165 Chapter 4 Review Exercises ...................................................................................................................................... 178 Chapter 4 Test ........................................................................................................................................................... 184 Case Study: Thomas Malthus, Population, and Subsistence ..................................................................................... 187 Chapter 5 Probability 5.1 Probability Rules ............................................................................................................................................ 189 5.2 The Addition Rule and Complements............................................................................................................. 198 5.3 Independence and the Multiplication Rule ..................................................................................................... 206 5.4 Conditional Probability and the General Multiplication Rule ........................................................................ 210 5.5 Counting Techniques ...................................................................................................................................... 217 5.6 Simulating Probability Experiments ............................................................................................................... 221 5.7 Putting It Together: Which Method Do I Use? ............................................................................................... 223 Chapter 5 Review Exercises ...................................................................................................................................... 226 Chapter 5 Test ........................................................................................................................................................... 229 Case Study: The Case of the Body in the Bag ........................................................................................................... 230

Chapter 6 Discrete Probability Distributions 6.1 Discrete Random Variables............................................................................................................................. 233 6.2 The Binomial Probability Distribution ............................................................................................................ 243 Chapter 6 Review Exercises ...................................................................................................................................... 258 Chapter 6 Test ............................................................................................................................................................ 261 Case Study: The Voyage of the St. Andrew .............................................................................................................. 264 Chapter 7 The Normal Probability Distribution 7.1 Properties of the Normal Distribution ............................................................................................................. 266 7.2 Applications of the Normal Distribution ......................................................................................................... 269 7.3 Assessing Normality ....................................................................................................................................... 289 7.4 The Normal Approximation to the Binomial Probability Distribution ........................................................... 292 Chapter 7 Review Exercises ...................................................................................................................................... 296 Chapter 7 Test ............................................................................................................................................................ 301 Case Study: A Tale of Blood Chemistry.................................................................................................................... 304 Chapter 8 Sampling Distributions 8.1 Distribution of the Sample Mean .................................................................................................................... 306 8.2 Distribution of the Sample Proportion ............................................................................................................ 319 Chapter 8 Review Exercises ...................................................................................................................................... 325 Chapter 8 Test ............................................................................................................................................................ 328 Case Study: Sampling Distribution of the Median .................................................................................................... 330 Chapter 9 Estimating the Value of a Parameter Using Confidence Intervals 9.1 Estimating a Population Proportion ................................................................................................................ 336 9.2 Estimating a Population Mean ........................................................................................................................ 343 9.3 Putting It Together: Which Method Do I Use? ............................................................................................... 353 9.4 Estimating a Population Standard Deviation................................................................................................... 359 9.5 Estimating with Bootstrapping ........................................................................................................................ 361 Chapter 9 Review Exercises ...................................................................................................................................... 366 Chapter 9 Test ............................................................................................................................................................ 370 Case Study: Fire-Safe Cigarettes ............................................................................................................................... 372 Chapter 10 Hypothesis Tests Regarding a Parameter 10.1 The Language of Hypothesis Testing.............................................................................................................. 374 10.2 Hypothesis Tests for a Population Proportion................................................................................................. 377 10.2A Using Simulation to Perform Hypothesis Tests on a Population Proportion .................................................. 388 10.2B Hypothesis Tests for a Population Proportion Using the Normal Model ........................................................ 398 10.3 Hypothesis Tests for a Population Mean......................................................................................................... 408 10.3A Using Simulation and the Bootstrap to Perform Hypothesis Tests on a Population Mean ............................. 419 10.4 Putting It Together: Which Method Do I Use? ............................................................................................... 426 10.5 Hypothesis Tests for a Population Standard Deviation ................................................................................... 430 Chapter 10 Review Exercises .................................................................................................................................... 434 Chapter 10 Test .......................................................................................................................................................... 438 Case Study: How Old Is Stonehenge? ....................................................................................................................... 439

Chapter 11 Inference on Two Population Parameters 11.1 Inference about Two Population Proportions ................................................................................................. 442 11.1A Using Randomization Techniques to Compare Two Proportions ................................................................... 457 11.2 Inference about Two Means: Dependent Samples .......................................................................................... 465 11.2A Using Bootstrapping to Conduct Inference on Two Dependent Means .......................................................... 477 11.3 Inference about Two Means: Independent Samples ....................................................................................... 484 11.3A Using Randomization Techniques to Compare Two Independent Means ...................................................... 498 11.4 Putting It Together: Which Method Do I Use? ............................................................................................... 502 11.5 Inference about Two Populations Standard Deviations .................................................................................. 515 Chapter 11 Review Exercises .................................................................................................................................... 520 Chapter 11 Test ......................................................................................................................................................... 525 Case Study: Control in the Design of an Experiment ................................................................................................ 531 Chapter 12 Additional Inferential Methods 12.1 Goodness-of-Fit Test ...................................................................................................................................... 533 12.2 Tests for Independence and the Homogeneity of Proportions ........................................................................ 544 12.3 Testing the Significance of the Least-Squares Regression Model .................................................................. 566 12.3A Using Randomization Techniques on the Slope of the Least-Squares Regression Line ................................. 571 12.4 Confidence and Prediction Intervals ............................................................................................................... 577 Chapter 12 Review Exercises .................................................................................................................................... 583 Chapter 12 Test ......................................................................................................................................................... 590 Case Study: Feeling Lucky? Well, Are You? ............................................................................................................ 597 Appendix B B.1 Lines ............................................................................................................................................................... 600 B.2 Inference about Two Population Proportions: Dependent Samples................................................................ 608 B.3 Comparing Three or More Means (One-Way Analysis of Variance) ............................................................ 611

Preface This solutions manual accompanies Fundamentals of Statistics: Informed Decisions Using Data, 6e by Michael Sullivan, III. The Instructor’s Solutions Manual contains detailed solutions to all exercises in the text as well as the Consumer Reports® projects and the case studies. The Student’s Solutions Manual contains detailed solutions to all odd exercises in the text and all solutions to chapter reviews and tests. A concerted effort has been made to make this manual as user-friendly and error free as possible.

Chapter 1 Data Collection (f) III. A variable is the characteristics of the individuals within the population.

Section 1.1 1. (a) III. Statistics is the science of collecting, organizing, summarizing, and analyzing information in order to draw conclusions and answer questions. It is also about providing a measure of confidence in any conclusions.

3. 18% is a parameter because it describes a population (all of the governors). 4. 72% is a parameter because it describes a population (the entire class).

(b) VIII. The population is the entire group of individuals to be studied.

5. 32% is a statistic because it describes a sample (the high school students surveyed).

6. 13.3% is a statistic because it describes a sample (the 12th graders surveyed).

(d) VII. The parameter is a numerical summary of a population.

7. 0.366 is a parameter because it describes a population (all of Ty Cobb’s at-bats).

(e) I. The statistic is the numerical summary of a sample. (f) VI. The individual is a person or object that is a member of the group being studied. (g) II. Descriptive statistics involves organizing and summarizing data through tables, graphs, and numerical summaries. (h) V. Inferential statistics uses methods that take results from a sample and extends them to the population, and measures the reliability of the result. 2. (a) V. A discrete variable has either a finite number of possible values or countable number of possible values. The values of these variables typically result from counting. (b) IV. Data are information that describe characteristics of an individual. (c) VI. A continuous variable has an infinite number of possible values that are not countable. The values of these variables typically result from measurement. (d) II. A qualitative variable allows for classification of individuals based on some attribute or characteristic. (e) I. A quantitative variable provides numerical measures of individuals. The measures can be added or subtracted, and provide meaningful results.

8. 43.92 hours is a parameter because it describes a population (all the men who have walked on the moon). 9. 23% is a statistic because it describes a sample (the 6076 adults studied). 10. 44% is a statistic because it describes a sample (the 100 adults interviewed). 11. Qualitative

12. Quantitative

13. Quantitative

14. Qualitative

15. Quantitative

16. Quantitative

17. Qualitative

18. Qualitative

19. Discrete

20. Continuous

21. Continuous

22. Discrete

23. Continuous

24. Continuous

25. Discrete

26. Continuous

27. Nominal

28. Ordinal

29. Ratio

30. Interval

31. Ordinal

32. Nominal

33. Ratio

34. Interval

35. The population consists of all teenagers 13 to 17 years old who live in the United States. The sample consists of the 1028 teenagers 13 to 17 years old who were contacted by the Gallup Organization.

Chapter 1: Data Collection 36. The population consists of all bottles of CocaCola filled by that particular machine on October 15. The sample consists of the 50 bottles of Coca-Cola that were selected by the quality control manager. 37. The population consists of all of the soybean plants in this farmer’s crop. The sample consists of the 100 soybean plants that were selected by the farmer. 38. The population consists of all households within the United States. The sample consists of the 50,000 households that are surveyed by the U.S. Census Bureau. 39. The population consists of all women 27 to 44 years of age with hypertension. The sample consists of the 7373 women 27 to 44 years of age with hypertension who were included in the study. 40. The population consists of all full-time students enrolled at this large community college. The sample consists of the 128 fulltime students who were surveyed by the administration. 41. Individuals: Alabama, Colorado, Indiana, North Carolina, Wisconsin. Variables: Minimum age for driver’s license (unrestricted); mandatory belt use seating positions, maximum allowable speed limit (rural interstate) in 2011. Data for minimum age for driver’s license: 17, 17, 18, 16, 18; Data for mandatory belt use seating positions: front, front, all, all, all; Data for maximum allowable speed limit (rural interstate) 2011: 70, 75, 70, 70, 65 (mph.) The variable minimum age for driver’s license is continuous; the variable mandatory belt use seating positions is qualitative; the variable maximum allowable speed limit (rural interstate) 2011 is continuous (although only discrete values are typically chosen for speed limits.) 42. Individuals: 3 Series, 5 Series, 6 Series, 7 Series, X3, Z4 Roadster Variables: Body Style, Weight (lb), Number of Seats Data for body style: Coupe, Sedan, Convertible, Sedan, Sport utility, Coupe; Data for weight: 3362, 4056, 4277, 4564, 4012, 3505 (lb);

Data for number of seats: 4, 5, 4, 5, 5, 2. The variable body style is qualitative; the variable weight is continuous; the variable number of seats is discrete. 43. (a) The research objective is to determine if adolescents who smoke have a lower IQ than nonsmokers. (b) The population is all adolescents aged 18–21. The sample consisted of 20,211 18-year-old Israeli military recruits. (c) Descriptive statistics: The average IQ of the smokers was 94, and the average IQ of nonsmokers was 101. (d) The conclusion is that individuals with a lower IQ are more likely to choose to smoke. 44. (a) The research objective is to determine if the application of duct tape is as effective as cryotherapy in the treatment of common warts. (b) The population is all people with warts. The sample consisted of 51 patients with warts. (c) Descriptive statistics: 85% of patients in group 1 and 60% of patients in group 2 had complete resolution of their warts. (d) The conclusion is that duct tape is significantly more effective in treating warts than cryotherapy. 45. (a) The research objective is to determine the proportion of adult Americans who believe the federal government wastes 51 cents or more of every dollar. (b) The population is all adult Americans aged 18 years or older. (c) The sample is the 1017 American adults aged 18 years or older that were surveyed. (d) Descriptive statistics: Of the 1017 individuals surveyed, 35% indicated that 51 cents or more is wasted. (e) From this study, one can infer that many Americans believe the federal government wastes much of the money collected in taxes.

Section 1.1: Introduction to the Practice of Statistics 46. (a) The research objective is to determine what proportion of adults, aged 18 and over, believe it would be a bad idea to invest $1000 in the stock market. (b) The population is all adults aged 18 and over living in the United States. (c) The sample is the 1018 adults aged 18 and over living in the United States who completed the survey. (d) Descriptive statistics: Of the 1016 adults surveyed, 46% believe it would be a bad idea to invest $1000 in the stock market. (e) The conclusion is that a little fewer than half of the adults in the United States believe investing $1000 in the stock market is a bad idea. 47. (a) State is a qualitative variable because it is an individual categorization. (b) F scale is a qualitative variable because each tornado is rated according to a category. (c) Fatalities is a quantitative variable because it is a numerical measure. It is a discrete variable because it is countable. (d) Length is a quantitative variable because it is a numerical measure. It is a continuous variable because it results from measurement. 48. (a) State is a variable measured at the nominal level because values of the variable name, label, or categorize. In addition, the naming scheme does not allow for the values of the variable to be arranged in a ranked or specific order. (b) F scale is a variable measured at the ordinal level because the naming scheme allows for the values of the variable to be arranged in a ranked or specific order. (c) Fatalities is a variable measured at the ratio level because the ratio of two values makes sense and a value of zero has meaning. (d) Length is a variable measured at the ratio level because the ratio of two values makes sense and a value of zero has meaning.

49. Jersey number is nominal (the numbers generally indicate a type of position played). However, if the researcher feels that lower caliber players received higher numbers, then jersey number would be ordinal since players could be ranked by their number. 50. (a) Nominal; the ticket number is categorized as a winner or a loser. (b) Ordinal; the ticket number gives an indication as to the order of arrival of guests. (c) Ratio; the implication is that the ticket number gives an indication of the number of people attending the party. 51. (a) The research question is to determine if the season of birth affects mood later in life. (b) The sample consisted of the 400 people the researchers studied. (c) The season in which you were born (winter, spring, summer, or fall) is a qualitative variable. (d) According to the article, individuals born in the summer are characterized by rapid, frequent swings between sad and cheerful moods, while those born in the winter are less likely to be irritable. (e) The conclusion was that the season at birth plays a role in one’s temperament. 52. The population is the group to be studied as defined by the research objective. A sample is any subset of the population. 53. Quantitative variables are numerical measures such that meaningful arithmetic operations can be performed on the values of the variable. Qualitative variables describe an attribute or characteristic of the individual that allows researchers to categorize the individual. The values of a discrete random variable result from counting. The values of a continuous random variable result from a measurement. 54. The four levels of measurement of a variable are nominal, ordinal, interval, and ratio. Examples: Nominal—brand of clothing; Ordinal—size of a car (small, mid-size, large); Interval—temperature (in degrees Celsius); Ratio—number of students in a class (Examples will vary.)

Chapter 1: Data Collection 55. We say data vary, because when we draw a random sample from a population, we do not know which individuals will be included. If we were to take another random sample, we would have different individuals and therefore different data. This variability affects the results of a statistical analysis because the results would differ if a study is repeated. 56. The process of statistics is to (1) identify the research objective, which means to determine what should be studied and what we hope to learn; (2) collect the data needed to answer the research question, which is typically done by taking a random sample from a population; (3) describe the data, which is done by presenting descriptive statistics; and (4) perform inference in which the results are generalized to a larger population. 57. Age could be considered a discrete random variable. A random variable can be discrete by allowing, for example, only whole numbers to be recorded.

Section 1.2 1. The response variable is the variable of interest in a research study. An explanatory variable is a variable that affects (or explains) the value of the response variable. In research, we want to see how changes in the value of the explanatory variable affect the value of the response variable. 2. (a) III. A designed experiment is when a researcher randomly assigns the individuals in a study to groups, intentionally manipulates the value of an explanatory variable, controls other explanatory variables at fixed values, and then records the value of the response variable for each individual. (b) V. An observational study is when a researcher measures the value of the response variable without attempting to influence the value of either the response or explanatory variables. That is, the researcher observes the behavior of individuals in the study and records the values of the explanatory and response variables. (c) IV. A lurking variable is an explanatory variable that was not considered in a study, but that affects the value of the response variable in the study. In addition, this variable is typically related to other explanatory variables in the study.

(d) I. Confounding occurs when the effects of two or more explanatory variables are not separated. Therefore, any relation that may exist between an explanatory variable and the response variable may be due to some other variable not accounted for in the study. (e) II. A confounding variable is an explanatory variable that was considered in a study whose effect cannot be distinguished from a second explanatory variable in the study. 3. (a) II. A cohort study follows a group of individuals over a long period of time. Characteristics of the individuals are recorded and some individuals will be exposed to certain factors (not intentionally) and others will not. Because the data are collected over time, cohort studies are prospective. (b) III. A cross-sectional study collects information about individuals at a specific point in time, or over a short period of time. (c) I. A case-control study is retrospective, meaning it requires the researcher to look at existing records, or the subject to recall information from the past. Individuals who have certain characteristics are matched with those who don’t. 4. An observational study uses data obtained by studying individuals in a sample without trying to manipulate or influence the variable(s) of interest. In a designed experiment, a treatment is applied to the individuals in a sample in order to isolate the effects of the treatment on a response variable. Only an experiment can establish causation between an explanatory variable and a response variable. Observational studies can indicate a relationship, but cannot establish causation. 5. The choice between an observational study and an experiment depends on the circumstances involved. Sometimes there are ethical reasons why an experiment cannot be conducted. Other times the researcher may conduct an observational study first to validate a belief prior to investing a large amount of time and money into a designed experiment. A designed experiment is preferred if ethics, time, and money are not an issue.

Section 1.2: Observational Studies vs. Designed Experiments 6. A cohort study identifies the individuals to participate and then follows them over a period of time. During this period, information about the individuals is gathered, but there is no attempt to influence the individuals. Cohort studies are superior to case-control studies because cohort studies do not require recall to obtain the data. 7. There is a perceived benefit to obtaining a flu shot, so there are ethical issues in intentionally denying certain seniors access to the treatment. 8. A retrospective study looks at data from the past either through recall or existing records. A prospective study gathers data over time by following the individuals in the study and recording data as they occur. 9. This is an observational study because the researchers merely observed existing data. There was no attempt by the researchers to manipulate or influence the variable(s) of interest. 10. This is an experiment because the researchers intentionally changed the value of the explanatory variable (medication dose) to observe a potential effect on the response variable (cancer growth). 11. This is an experiment because the explanatory variable (teaching method) was intentionally varied to see how it affected the response variable (score on proficiency test). 12. This is an observational study because no attempt was made to influence the variable of interest. Voting choices were merely observed. 13. This is an observational study because the survey only observed preference of Coke or Pepsi. No attempt was made to manipulate or influence the variable of interest. 14. This is an experiment because the researcher intentionally imposed treatments on individuals in a controlled setting. 15. This is an experiment because the explanatory variable (carpal tunnel treatment regimen) was intentionally manipulated in order to observe potential effects on the response variable (level of pain). 16. This is an observational study because the conservation agents merely observed the fish to determine which were carrying parasites. No attempt was made to manipulate or influence any variable of interest.

17. (a) This is a cohort study because the researchers observed a group of people over a period of time. (b) The response variable is whether the individual has heart disease or not. The explanatory variable is whether the individual is happy or not. (c) There may be confounding due to lurking variables. For example, happy people may be more likely to exercise, which could affect whether they will have heart disease or not. 18. (a) This is a cross-sectional study because the researchers collected information about the individuals at a specific point in time. (b) The response variable is whether the woman has nonmelanoma skin cancer or not. The explanatory variable is the daily amount of caffeinated coffee consumed. (c) It was necessary to account for these variables to avoid confounding with other variables. 19. (a) This is an observational study because the researchers simply administered a questionnaire to obtain their data. No attempt was made to manipulate or influence the variable(s) of interest. This is a cross-sectional study because the researchers are observing participants at a single point in time. (b) The response variable is body mass index. The explanatory variable is whether a TV is in the bedroom or not. (c) Answers will vary. Some lurking variables might be the amount of exercise per week and eating habits. Both of these variables can affect the body mass index of an individual. (d) The researchers attempted to avoid confounding due to other variables by taking into account such variables as “socioeconomic status.” (e) No. Since this was an observational study, we can only say that a television in the bedroom is associated with a higher body mass index.

Chapter 1: Data Collection 20. (a) This is an observational study because the researchers merely observed the individuals included in the study. No attempt was made to manipulate or influence any variable of interest. This is a cohort study because the researchers identified the individuals to be included in the study, then followed them for a period of time (7 years). (b) The response variable is weight gain. The explanatory variable is whether the individual is married/cohabitating or not. (c) Answers will vary. Some potential lurking variables are eating habits, exercise routine, and whether the individual has children. (d) No. Since this is an observational study, we can only say that being married or cohabitating is associated with weight gain. 21. (a) This is a cross-sectional study because information was collected at a specific point in time (or over a very short period of time). (b) The explanatory variable is delivery scenario (caseload midwifery, standard hospital care, or private obstetric care). (c) The two response variables are (1) cost of delivery, which is quantitative, and (2) type of delivery (vaginal or not), which is qualitative. 22. (a) The explanatory variable is web page design; qualitative (b) The response variables are time on site and amount spent. Both are qualitative. (c) Answers will vary. A confounding variable might be location. Any differences in spending may be due to location rather than to web page design. 23. Answers will vary. This is a prospective, cohort observational study. The response variable is whether the worker had cancer or not, and the explanatory variable is the amount of electromagnetic field exposure. Some possible lurking variables include eating habits, exercise habits, and other health-related variables such as smoking habits. Genetics (family history) could also be a lurking

variable. This was an observational study, and not an experiment, so the study only concludes that high electromagnetic field exposure is associated with higher cancer rates. The author reminds us that this is an observational study, so there is no direct control over the variables that may affect cancer rates. He also points out that while we should not simply dismiss such reports, we should consider the results in conjunction with results from future studies. The author concludes by mentioning known ways (based on extensive study) of reducing cancer risks that can currently be done in our lives. 24. (a) This is a cohort study because a group of individuals was identified to participate in the study, and then they were observed over a period of time. (b) Because there is a link established between obesity and cell phone use, and a link between obesity and negative health outcomes, if it is determined that cellphone users are experiencing higher incidence rates of negative health outcomes, it cannot be established that the factor leading to the negative health outcome is due to the cell phone—it may be due to the lurking variable obesity. 25. Because individuals in the early 1900s were pressured to become right-handed, we would see a lower proportion of left-handers who are older in the study. This would make it seem as though left-handers die younger because the older individuals in the study are primarily right-handed. 26. (a) The research objective is to determine whether lung cancer is associated with exposure to tobacco smoke within the household. (b) This is a case-controlled study because there is a group of individuals with a certain characteristic (lung cancer but never smoked) being compared to a similar group without the characteristic (no lung cancer and never smoked). The study is retrospective because lifetime residential histories were compiled and analyzed. (c) The response variable is whether the individual has lung cancer or not. This is a qualitative variable.

Section 1.3: Simple Random Sampling (d) The explanatory variable is the number of “smoker years.” This is a quantitative variable. (e) Answers will vary. Some possible lurking variables are household income, exercise routine, and exposure to tobacco smoke outside the home. (f) The conclusion of the study is that approximately 17% of lung cancer cases among nonsmokers can be attributed to high levels of exposure to tobacco smoke during childhood and adolescence. No, we cannot say that exposure to household tobacco smoke causes lung cancer since this is only an observational study. We can, however, conclude that lung cancer is associated with exposure to tobacco smoke in the home. (g) An experiment involving human subjects is not possible for ethical reasons. Researchers would be able to conduct an experiment using laboratory animals, such as rats. 27. Web scraping can be used to extract data from tables on web pages and then upload the data to a file. Web scraping can also be used to create a data set of words from an online article (that is, fetching unstructured information and transforming it into a structured format through something called parsing and reformatting processes). In addition, web scraping can be used to dynamically call information from websites with links. 28. Answers will vary. Discussions may include the responsibility of host sites to protect private information; the ethics behind using information collected from a site to harm a competitor; the ethics behind putting stress on a host’s servers so that the website slows down; the role of social media to scrape its site for “fake news.”

Section 1.3 1. The frame is a list of all the individuals in the population.

4. Random sampling is a technique that uses chance to select individuals from a population to be in a sample. It is used because it maximizes the likelihood that the individuals in the sample are representative of the individuals in the population. In convenience sampling, the individuals in the sample are selected in the quickest and easiest way possible (e.g. the first 20 people to enter a store). Convenience samples likely do not represent the population of interest because chance was not used to select the individuals. 5. Answers will vary. We will use one-digit labels and assign the labels across each row (i.e. Pride and Prejudice – 0, The Sun Also Rises – 1, and so on). In Table I of Appendix A, starting at row 5, column 11, and proceeding downward, we obtain the following labels: 8, 4, 3 In this case, the 3 books in the sample would be As I Lay Dying, A Tale of Two Cities, and Crime and Punishment. Different labeling order, different starting points in Table I in Appendix A, or use of technology will likely yield different samples. 6. Answers will vary. We will use one-digit labels and assign the labels across each row (i.e. Mady – 0, Breanne – 1, and so on). In Table I of Appendix A, starting at row 11, column 6, and then proceeding downward, we obtain the following labels: 1, 5 In this case, the two captains would be Breanne and Payton. Different labeling order, different starting points in Table I in Appendix A, or use of technology will likely yield different results. 7. (a) {616, 630}, {616, 631}, {616, 632}, {616, 645}, {616, 649}, {616, 650}, {630, 631}, {630, 632}, {630, 645}, {630, 649}, {630, 650}, {631, 632}, {631, 645}, {631, 649}, {631, 650}, {632, 645}, {632, 649}, {632, 650}, {645, 649}, {645, 650}, {649, 650} (b) There is a 1 in 21 chance that the pair of courses will be EPR 630 and EPR 645.

2. Simple random sampling occurs when every possible sample of size n has an equally likely chance of occurring.

8. (a) {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {3, 7}, {4, 5}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}

3. Sampling without replacement means that no individual may be selected more than once as a member of the sample.

(b) There is a 1 in 21 chance that the pair The United Nations and Amnesty International will be selected.

Chapter 1: Data Collection 9. (a) Starting at row 5, column 22, using twodigit numbers, and proceeding downward, we obtain the following values: 83, 94, 67, 84, 38, 22, 96, 24, 36, 36, 58, 34,.... We must disregard 94 and 96 because there are only 87 faculty (b) Answers will vary depending on the type of technology used. If using a TI-84 Plus, the sample will be: 4, 20, 52, 5, 24, 87, 67, 86, and 39.

Note: We must disregard the second 20 because we are sampling without replacement. 10. (a) Starting at row 11, column 32, using fourdigit numbers, and proceeding downward, we obtain the following values: 2869, 5518, 6635, 2182, 8906, 0603, 2654, 2686, 0135, 7783, 4080, 6621, 3774, 7887, 0826, 0916, 3188, 0876, 5418, 0037, 3130, 2882, 0662,…. We must disregard 8906, 7783, and 7887 because there are only 7656 students in the population. Thus, the 20 students included in the sample are those numbered 2869, 5518, 6635, 2182, 0603, 2654, 2686, 0135, 4080, 6621, 3774, 0826, 0916, 3188, 0876, 5418, 0037, 3130, 2882, and 0662. (b) Answers may vary depending on the type of technology used. If using a TI-84 Plus, the sample will be: 6658, 4118, 9, 4828, 3905, 454, 2825, 2381, 495, 4445, 4455, 5759, 5397, 7066, 3404, 6667, 5074, 3777, 3206, 5216.

members in the population. We must also disregard the second 36 because we are sampling without replacement. Thus, the 9 faculty members included in the sample are those numbered 83, 67, 84, 38, 22, 24, 36, 58, and 34.

11. (a) Answers will vary depending on the technology used (including a table of random digits). Using a TI-84 Plus graphing calculator with a seed of 17 and the labels provided, our sample would be North Dakota, Nevada, Tennessee, Wisconsin, Minnesota, Maine, New Hampshire, Florida, Missouri, and Mississippi.

(b) Repeating part (a) with a seed of 18, our sample would be Michigan, Massachusetts, Arizona, Minnesota, Maine, Nebraska, Georgia, Iowa, Rhode Island, and Indiana. 12. (a) Answers will vary depending on the technology used (including a table of random digits). Using a TI-84 Plus graphing calculator with a seed of 98 and the labels provided, our sample would be Jefferson, Reagan, Madison, Trump, Pierce, Buchanan, Carter, G. W. Bush.

(b) Repeating part (a) with a seed of 99, our sample would be Nixon, Eisenhower, Pierce, Arthur, Trump, Hayes, Clinton, T. Roosevelt.

Section 1.4: Other Effective Sampling Methods 13. (a) The list provided by the administration serves as the frame. Number each student in the list of registered students, from 1 to 19,935. Generate 25 random numbers, without repetition, between 1 and 19,935 using a random number generator or table. Select the 25 students with these numbers. (b) Answers will vary. 14. (a) The list provided by the mayor serves as the frame. Number each resident in the list supplied by the mayor, from 1 to 5832. Generate 20 random numbers, without repetition, between 1 and 5832 using a random number generator or table. Select the 20 residents with these numbers. (b) Answers will vary. 15. Answers will vary. Members should be numbered 1–32, though other numbering schemes are possible (e.g. 0–31). Using a table of random digits or a random-number generator, four different numbers (labels) should be selected. The names corresponding to these numbers form the sample. 16. Answers will vary. Employees should be numbered 1–29, though other numbering schemes are possible (e.g. 0–28). Using a table of random digits or a random-number generator, four different numbers (labels) should be selected. The names corresponding to these numbers form the sample. 17. Answers will vary.

Section 1.4 1. Stratified random sampling may be appropriate if the population of interest can be divided into groups (or strata) that are homogeneous and nonoverlapping. 2. Systematic sampling does not require a frame. 3. Convenience samples are typically selected in a nonrandom manner. This means the results are not likely to represent the population. Convenience samples may also be selfselected, which will frequently result in small portions of the population being overrepresented. 4. Cluster sample

6. False. In a systematic random sample, every kth individual is selected from the population. 7. False. In many cases, other sampling techniques may provide equivalent or more information about the population with less “cost” than simple random sampling. 8. True. When the clusters are heterogeneous, the heterogeneity of each cluster likely resembles the heterogeneity of the population. In such cases, fewer clusters with more individuals from each cluster are preferred. 9. True. Because the individuals in a convenience sample are not selected using chance, it is likely that the sample is not representative of the population. 10. False. With stratified samples, the number of individuals sampled from each strata should be proportional to the size of the strata in the population. 11. Systematic sampling. The quality-control manager is sampling every 8th chip, starting with the 3rd chip. 12. Cluster sampling. The commission tests all members of the selected teams (clusters). 13. Cluster sampling. The airline surveys all passengers on selected flights (clusters). 14. Stratified sampling. The congresswoman samples some individuals from each of three different income brackets (strata). 15. Simple random sampling. Each known user of the product has the same chance of being included in the sample. 16. Convenience sampling. The radio station is relying on voluntary response to obtain the sample data. 17. Cluster sampling. The farmer samples all trees within the selected subsections (clusters). 18. Stratified sampling. The school official takes a sample of students from each of the five classes (strata). 19. Convenience sampling. The research firm is relying on voluntary response to obtain the sample data. 20. Systematic sampling. The presider is sampling every 5th person attending the lecture, starting with the 3rd person.

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Chapteer 1: Data Collection

21. Stratified d sampling. Sh hawn takes a saample of measureements during each e of the fou ur time intervalss (strata). 22. Simple random r sampling. Each club member has the same s chance off being selected for the survey. 23. The num mbers correspon nding to the 20 0 clients selected are 16 , 16 + 25 2 = 41 , 41 + 25 2 = 66 , 66 + 25 = 91 , 91 + 25 = 116 , 141, 166, 191, 216, 241 1, 266, 291, 316, 341, 366, 39 91, 416, 441, 466 6, 491. 24. Since the number of cllusters is more than 100, but less than 1000, we assign each clluster a three-dig git label betweeen 001 and 795. Starting at row 8, colum mn 38 in Tablee I of Appendiix A, and proceeeding downw ward, the 10 clusteers selected aree numbered 76 63, 185, 377, 304 4, 626, 392, 315, 084, 565, an nd 508. Note thaat we discard 822 and 955 in reading the tablee because we have h no clusterss with these lab bels. We also discard d the seco ond occurren nce of 377 becaause we cannot select the samee cluster twice.. 25. Answerss will vary. To obtain the sam mple, number the Democratss 1 to 16 and ob btain a simple random r samplee of size 2. Theen number the Repu ublicans 1 to 16 and obtain a simple random sample of sizee 2. Be sure to use u a differentt starting pointt in Table I or a different seed for each stratum. mple, using a TI-84 T Plus grap phing For exam calculato or with a seed of 38 for the Democrats D and 40 for f the Republiicans, the numb bers selected would be 6, 9 for the Democcrats and 14, 4 forr the Republicaans. If we had numbered n the indiv viduals down each e column, th he sample would consist of Hayd dra, Motola, Th hompson, and Eng gler.

26. Answerss will vary. To obtain the sam mple, number the managers 1 to 8 and obtaain a simple random r samplee of size 2. Theen number the employees 1 to 21 and obtain a siimple u a random sample of sizee 4. Be sure to use differentt starting pointt in Table I or a different seed for each stratum.

Foor example, usiing a TI-84 Pluus graphing callculator with a seed of 18 forr the managers annd 20 for the em mployees, the nnumbers sellected would bbe 4, 1 for the m managers and 200, 3, 11, 9 for thhe employees. If we had nuumbered the inddividuals downn each column, thee sample wouldd consist of Lindsey, Carlislee, W Weber, Bryant, H Hall, and Gow.

27. (a))

N 4502 = = 90.04 ® 90 ; Thus, k = 90 . n 50

(b)) Randomly sselect a numberr between 1 annd 90. Supposee that we selectt 15. Then the individuals tto be surveyedd will be the 15th, 105th,, 195th, 285th, and so on up tto the 4425th eemployee on thhe company lisst. 28. (a))

N 94503 5 = = 7269.5 ® 7269 ; Thus, n 130 k = 7269 .

(b)) Randomly sselect a numberr between 1 annd 7269. Suppoose that we ranndomly select 2000. Then we will surveyy the individuaals numbered 2000, 9269, 16,,538, and so onn up to the inddividual numbeered 939,701. 29. Sim mple Random Sample: Number thee students from m 1 to 1280. Use a table of raandom digits orr a randomnumber gennerator to randoomly select 1288 students to ssurvey. Strratified Samplee: Since class ssizes are similaar, we would 128 want to randdomly select =4 332 students from m each class too be included iin the sample. Cluuster Sample: Since classees are similar inn size and makeup, wee would want too randomly 128 select d include all thhe = 4 classes and 32 students from m those classees in the samplee. 30. Noo. The clusters were not randoomly selected. Thhis would be coonsidered convvenience sam mpling.

Section 1.5: Bias in Sampling 31. Answers will vary. One design would be a stratified random sample, with two strata being commuters and noncommuters, as these two groups each might be fairly homogeneous in their reactions to the proposal. 32. Answers will vary. One design would be a cluster sample, with classes as the clusters. Randomly select clusters and then survey all the students in the selected classes. However, care would need to be taken to make sure that no one was polled twice. Since this would negate some of the ease of cluster sampling, a simple random sample might be the more suitable design. 33. Answers will vary. One design would be a cluster sample, with the clusters being city blocks. Randomly select city blocks and survey every household in the selected blocks. 34. Answers will vary. One appropriate design would be a systematic sample, after doing a random start, clocking the speed of every tenth car, for example. 35. Answers will vary. Since the company already has a list (frame) of 6600 individuals with high cholesterol, a simple random sample would be an appropriate design. 36. Answers will vary. Since a list of all the households in the population exists, a simple random sample is possible. Number the households from 1 to N, then use a table of random digits or a random-number generator to select the sample. 37. (a) For a political poll, a good frame would be all registered voters who have voted in the past few elections since they are more likely to vote in upcoming elections. (b) Because each individual from the frame has the same chance of being selected, there is a possibility that one group may be over- or underrepresented. (c) By using a stratified sample, the strategist can obtain a simple random sample within each strata (political party) so that the number of individuals in the sample is proportionate to the number of individuals in the population. 38. Random sampling means that the individuals chosen to be in the sample are selected by chance. Random sampling minimizes the chance that one part of the population is overor underrepresented in the sample. However, it cannot guarantee that the sample will accurately represent the population.

39. Answers will vary. 40. Answers will vary.

Section 1.5 1. A closed question is one in which the respondent must choose from a list of prescribed responses. An open question is one in which the respondent is free to choose his or her own response. Closed questions are easier to analyze, but limit the responses. Open questions allow respondents to state exactly how they feel, but are harder to analyze due to the variety of answers and possible misinterpretation of answers. 2. A certain segment of the population is underrepresented if it is represented in the sample in a lower proportion than its size in the population. 3. (a) III. Bias occurs when the results of the sample are not representative of the population. (b) I. Sampling bias occurs when the techniques used to select individuals for a sample favor one part of the population over another. (c) IV. Nonresponse bias occurs when the individuals selected to be in the sample who do not respond to the survey have different opinions from those who do respond. (d) II. Response bias occurs when the answers on a survey do not reflect the true feelings of the respondent. 4. Nonsampling error is the error that results from undercoverage, nonresponse bias, response bias, or data-entry errors. Essentially, it is the error that results from the process of obtaining and recording data. Sampling error is the error that results because a sample is being used to estimate information about a population. Any error that could also occur in a census is considered a nonsampling error. 5. (a) Sampling bias. The survey suffers from undercoverage because the first 60 customers are likely not representative of the entire customer population. (b) Since a complete frame is not possible, systematic random sampling could be used to make the sample more representative of the customer population.

Chapter 1: Data Collection 6. (a) Sampling bias. The survey suffers from undercoverage because only homes in the southwest corner have a chance to be interviewed. These homes may have different demographics than those in other parts of the village. (b) Assuming that households within any given neighborhood have similar household incomes, stratified sampling might be appropriate, with neighborhoods as the strata. 7. (a) Response bias. The survey suffers from response bias because the question is poorly worded. (b) The survey should inform the respondent of the current penalty for selling a gun illegally and the question should be worded as “Do you approve or disapprove of harsher penalties for individuals who sell guns illegally?” The order of “approve” and “disapprove” should be switched from one individual to the next. 8. (a) Response bias. The survey suffers from response bias because the wording of the question is ambiguous. (b) The question might be worded more specifically as “How many hours per night do you sleep, on average?” 9. (a) Nonresponse bias. Assuming the survey is written in English, non-English speaking homes will be unable to read the survey. This is likely the reason for the very low response rate. (b) The survey can be improved by using face-to-face or phone interviews, particularly if the interviewers are multilingual.

10. (a) Nonresponse bias (b) The survey can be improved by using face-to-face or phone interviews, or possibly through the use of incentives. 11. (a) The survey suffers from sampling bias due to undercoverage and interviewer error. The readers of the magazine may not be representative of all Australian women, and advertisements and images in the magazine could affect the women’s view of themselves.

(b) A well-designed sampling plan not in a magazine, such as a cluster sample, could make the sample more representative of the population. 12. (a) The survey suffers from sampling bias due to a bad sampling plan (convenience sampling) and possible response bias due to misreported weights on driver’s licenses. (b) The teacher could use cluster sampling or stratified sampling using classes throughout the day. Each student should be weighed to get a current and accurate weight measurement. 13. (a) Response bias due to a poorly worded question (b) The question should be reworded in a more neutral manner. One possible phrasing might be “Do you believe that a marriage can be maintained after an extramarital relation?” 14. (a) Sampling bias. The frame is not necessarily representative of all college professors. (b) To remedy this problem, the publisher could use cluster sampling and obtain a list of faculty from the human resources departments at selected colleges. 15. (a) Response bias. Students are unlikely to give honest answers if their teacher is administering the survey. (b) An impartial party should administer the survey in order to increase the rate of truthful responses. 16. (a) Response bias. Residents are unlikely to give honest answers to uniformed police officers if their answer would be seen as negative by the police. (b) An impartial party should administer the survey in order to increase the rate of truthful responses. 17. No. The survey still suffers from sampling bias due to undercoverage, nonresponse bias, and potentially response bias. 18. The General Social Survey uses random sampling to obtain individuals who take the survey, so the results of their survey are more likely to be representative of the population.

Section 1.5: Bias in Sampling However, it may suffer from response bias since the survey is conducted by personal interview rather than anonymously on the Internet. The online survey, while potentially obtaining more honest answers, is basically self-selected so may not be representative of the population, particularly if most respondents are clients of the family and wellness center seeking help with health or relationship problems. 19. It is very likely that the order of these two questions will affect the survey results. To alleviate the response bias, either question B could be asked first, or the order of the two questions could be rotated randomly. 20. It is very likely that the order of these two questions will affect the survey results. To alleviate the response bias, the order of the two questions could be rotated randomly. Prohibit is a strong word. People generally do not like to be prohibited from doing things. If the word must be used, it should be offset by the word “allow.” The use of the words “prohibit” and “allow” should be rotated within the question. 21. The company is using a reward in the form of the $5.00 payment and an incentive by telling the reader that his or her input will make a difference. 22. The two choices need to be rotated so that any response bias due to the ordering of the questions is minimized. 23. Students should look up the study on the web. (a) We would expect to find sampling bias due to undercoverage. Individuals who choose not to register to vote may have some characteristics that differ from those who do register. (b) Yes. Undercoverage also would exist for RDD polls. It is likely the case that the poll is not capturing individuals in a lower socioeconomic class. RBS likely has more of this type of bias because access to cell phones is fairly prevalent today. (c) RDD had the lower response rate at 6% versus 8% in the RBS survey. This is likely due to the fact that the RBS survey had actual phone numbers to choose from, rather than random digits.

(d) The RDD survey oversampled Republicans 53% to 37%. This could be due to socioeconomic considerations. 24. Today, many individuals no longer have a land-line phone. Therefore, this segment of the population would not be included in any surveys that utilize robocalling. This would be undercoverage. Also, the use of caller ID has likely increased nonresponse bias of phone surveys since individuals may not answer calls from numbers they do not recognize. If individuals with caller ID differ in some way from individuals without caller ID, then phone surveys could also suffer from sampling bias due to undercoverage. 25. It is extremely likely, particularly if households on the do-not-call registry have a trait that is not part of those households that are not on the registry. 26. There is a higher chance that an individual at least 70 years of age will be at home when an interviewer makes contact. 27. Answers will vary. However, the research should include the fact that exit polls tended to undersample non-college-educated whites and oversampled college-educated whites. In this election, non-college educated voters broke for Trump, while college-educated voters were carried by Clinton. 28. – 32. Answers will vary. 33. The Literary Digest made an incorrect prediction due to sampling bias (an incorrect frame led to undercoverage) and nonresponse bias (due to the low response rate). 34. Answers will vary. (Gallup incorrectly predicted the outcome of the 1948 election because he quit polling weeks before the election and missed a large number of changing opinions.) 35. (a) Answers will vary. Stratified sampling by political affiliation (Democrat, Republican, etc.) could be used to ensure that all affiliations are represented. One question that could be asked is whether or not the person plans to vote in the next election. This would help determine which registered voters are likely to vote.

Chapter 1: Data Collection (b) Answers will vary. Possible explanations are that presidential election cycles get more news coverage or perhaps people are more interested in voting when they can vote for a president as well as a senator. During non-presidential cycles it is very informative to poll likely registered voters. (c) Answers will vary. A higher percentage of Democrats in polls versus turnout will lead to overstating the predicted Democrat percentage of Democratic votes.

36. It is difficult for a frame to be completely accurate since populations tend to change over time and there can be a delay in identifying individuals who have joined or left the population. 37. Nonresponse can be addressed by conducting callbacks or offering rewards. 38. Trained, skillful interviewers can illicit responses from individuals and help them give truthful responses. 39. Conducting a presurvey with open questions allows the researchers to use the most popular answers as choices on closed-question surveys. 40. Answers will vary. Phone surveys conducted in the evening may result in reaching more potential respondents; however some of these individuals could be upset by the intrusion. 41. Provided the survey was conducted properly and randomly, a high response rate will provide more representative results. When a survey has a low response rate, only those who are most willing to participate give responses. Their answers may not be representative of the whole population. 42. The order of questions on a survey should be carefully considered, so the responses are not affected by previous questions. 43. The question is ambiguous because it could be interpreted as hours per day, or hours per week, or hours for a particular class. The question could be improved by being more specific, such as, “On average, how many hours do you study each day for your statistics course?”

44. Higher response rates typically suggest that the sample represents the population well. Using rewards can help increase response rates, allowing researchers to better understand the population. There can be disadvantages to offering rewards as incentives. Some people may hurry through the survey, giving superficial answers, just to obtain the reward.

Section 1.6 1. (a) An experimental unit is a person, object, or some other well-defined item upon which a treatment is applied. (b) A treatment is a condition applied to an experimental unit. It can be any combination of the levels of the explanatory variables. (c) A response variable is a quantitative or qualitative variable that measures a response of interest to the experimenter. (d) A factor is a variable whose effect on the response variable is of interest to the experimenter. Factors are also called explanatory variables. (e) A placebo is an innocuous treatment, such as a sugar pill, administered to a subject in a manner indistinguishable from an actual treatment. (f) Confounding occurs when the effect of two explanatory variables on a response variable cannot be distinguished. (g) Blinding refers to nondisclosure of the treatment an experimental unit is receiving. There are two types of blinding: single blinding and double blinding. 2. Replication occurs when each treatment is applied to more than one experimental unit. 3. In a single-blind experiment, subjects do not know which treatment they are receiving. In a double-blind experiment, neither the subject nor the researcher(s) in contact with the subjects knows which treatment is received. 4. Completely randomized; matched-pair 5. Blocking 6. True

Section 1.6: The Design of Experiments 7. (a) The research objective of the study was to determine the association between number of times one chews food and food consumption. (b) The response variable is food consumption; quantitative. (c) The explanatory variable is chew level (100%, 150%, 200%); qualitative. (d) The experimental units are the 45 individuals aged 18 to 45 who participated in the study. (e) Control is used by determining a baseline number of chews before swallowing; same type of food is used in the baseline as in the experiment; same time of day (lunch); age (18 to 45). (f) Randomization reduces the effect of the order in which the treatments are administered. For example, perhaps the first time through the subjects are more diligent about their chewing than the last time through the study. 8. (a) The researchers used an innocuous treatment to account for effects that would result from any treatment being given (i.e. the placebo effect). The

placebo is a drug that looks and tastes like topiramate and serves as the baseline against which to compare the results when topiramate is administered. (b) Being double-blind means that neither the subject nor the researcher in contact with the subjects knows whether the placebo or topiramate is being administered. Using a double-blind procedure is necessary to avoid any intentional or unintentional bias due to knowing which treatment is being given. (c) The subjects were randomly assigned to the treatment groups (either the placebo or topiramate). (d) The population is all men and women aged 18 to 65 years diagnosed with alcohol dependence. The sample is the 371 men and women aged 18 to 65 years diagnosed with alcohol dependence who participated in the 14-week trial. (e) There are two treatments in the study: 300 mg of topiramate or a placebo daily. (f) The response variable is the percentage of heavy drinking days.

9. (a) The response variable is the achievement test scores. (b) Answers may vary. Some factors are teaching methods, grade level, intelligence, school district, and teacher. Fixed: grade level, school district, teacher Set at predetermined levels: teaching method (c) The treatments are the new teaching method and the traditional method. There are 2 levels of treatment. (d) The factors that are not controlled are dealt with by random assignment into the two treatment groups. (e) Group 2, using the traditional teaching method, serves as the control group. (f) This experiment has a completely randomized design. (g) The subjects are the 500 first-grade students from District 203 recruited for the study. (h) If students tend to perform worse in classes later in the day (due to being tired or anxious to get out of school), then time of day may be a confounding variable. Suppose group 1 is taught in the morning and group 2 is taught in the afternoon; if group 1 scores better on the achievement test, we won’t know whether this is due to the new method of teaching, or due to time of day the course is offered. One solution would be to have a rotating schedule so classes are not always taught at the same time of day. (e)

Chapter 1: Data Collection

10. (a) The response variable is the proportion of subjects with a cold. (b) Answers may vary. Some factors are gender, age, geographic location, overall health, and drug intervention. Fixed: gender, age, location Set at predetermined levels: drug intervention (c) The treatments are the experimental drug and the placebo. There are 2 levels of treatment. (d) The factors that are not controlled are dealt with by random assignment into the two groups. (e) This experiment has a completely randomized design. (f) The subjects are the 300 adult males aged 25 to 29 who have the common cold. (g)

Group 1: 150 males

Treatment 1: Experimental drug

Random assignment of males to treatments

Compare proportion of subjects with colds Group 2: 150 males

Treatment 2: Placebo

11. (a) This experiment has a matched-pairs design. (b) The response variable is the level of whiteness. (c) The explanatory variable or factor is the whitening method. The treatments are Crest Whitestrips Premium in addition to brushing and flossing, and just brushing and flossing alone. (d) Answers will vary. One other possible factor is diet. Certain foods and tobacco products are more likely to stain teeth. This could impact the level of whiteness. (e) Answers will vary. One possibility is that using twins helps control for genetic factors such as weak teeth that may affect the results of the study. 12. (a) This experiment has a matched-pairs design. (b) The response variable is the difference in test scores. (c) The treatment is the mathematics course. 13. (a) This experiment has a completely randomized design. (b) The response variable is the travel time it takes to travel 9.75 meters. It is a quantitative variable. (c) Priming is the treatment and it is set at two levels—scrambled sentence task using words associated with old age or words not associated with old age. (d) The subjects are the 30 male and female undergraduates. (e) The undergraduates did not know to which group they were assigned, and the individual assigning the students did not know to which group the student was assigned. (f) The conclusion was that the elderly priming condition subjects had a travel time significantly higher than that of the neutral priming condition. 14. (a) This experiment has a completely randomized design.

Section 1.6: The Design of Experiments (b) The population being studied is adults with insomnia. (c) The response variable is the terminal wake time after sleep onset (WASO). (d) The explanatory variable or factor is the type of intervention. The treatments are cognitive behavioral therapy (CBT), muscle relaxation training (RT), and the placebo. (e) The experimental units are the 75 adults with insomnia. (f)

Random assignment of adults to treatments

Group 1: 25 adults

Treatment 1: CBT

Group 2: 25 adults

Treatment 2: RT

Group 3: 25 adults

Treatment 3: Placebo

Compare terminal wake time after sleep onset

15. (a) This experiment has a completely randomized design. (b) The population being studied is adults over 60 years old and in good health. (c) The response variable is the standardized test of learning and memory. (d) The factor set to predetermined levels (explanatory variable) is the drug. The treatments are 40 milligrams of ginkgo 3 times per day and the matching placebo. (e) The experimental units are the 98 men and 132 women over 60 years old and in good health. (f) The control group is the placebo group. (g)

Group 1: 115 elderly adults

Treatment 1: 40 mg of Ginkgo 3 times per day

Random assignment of elderly adults to treatments

Compare performance on standardized test Group 2: 115 elderly adults

Treatment 2: Placebo

16. (a) This experiment has a completely randomized design. (b) The population being studied is obese patients. (c) The response variable is the volume of the stomach. This is a quantitative variable. (d) The treatments are the 2508 kJ diet versus the regular diet. Copyright © 2022 Pearson Education, Inc.

Chapter 1: Data Collection (e) The experimental units are the 23 obese patients.

(f)

Group 1: 14 patients

Treatment 1: 2508 kJ diet

Random assignment of patients to treatments

Compare stomach volumes Group 2: 9 patients

Treatment 2: Regular diet

17. (a) This experiment has a matched-pairs design. (b) The population being studied is females with hair loss 20 to 45 years of age. (c) The response variables are hair density and hair diameter. (d) The treatment is the injection and it is set at two levels: platelet-rich plasma (PRP) injection or saline injection. (e) The experimental units are the 30 female patients. (f) Randomization was used to choose the area of the scalp that receives the treatment. (g)

18. (a) This experiment has a matched-pairs design. (b) The response variable is the distance the yardstick falls. (c) The explanatory variable or factor is hand dominance. The treatment is dominant versus non-dominant hand. (d) The experimental units are the 15 students. (e) Professor Neil used a coin flip to eliminate bias due to starting on the dominant or non-dominant hand first on each trial. (f) Identify 15 students

Randomly assign dominant or non-dominant hand first

Administer treatment measure reaction time

For each matched pair, compute difference in reaction time

Section 1.6: The Design of Experiments 19. Answers will vary. Using a TI-84 Plus graphing calculator with a seed of 195, we would pick the volunteers numbered 8, 19, 10, 12, 13, 6, 17, 1, 4, and 7 to go into the experimental group. The rest would go into the control group. If the volunteers were numbered in the order listed, the experimental group would consist of Ann, Kevin, Christina, Eddie, Shannon, Randy, Tom, Wanda, Kim, and Colleen. 20. (a) This experiment has a completely randomized design. (b) Answers will vary. Using a TI-84 Plus graphing calculator with a seed of 223, we would pick the volunteers numbered 6, 18, 13, 3, 19, 14, 8, 1, 17, and 5 to go into group 1. 21. (a) This is an observational study because there is no intent to manipulate an explanatory variable or factor. The explanatory variable or factor is whether the individual is a green tea drinker or not, which is qualitative. (b) Some lurking variables include diet, exercise, genetics, age, gender, and socioeconomic status. (c) The experiment is a completely randomized design. (d) To make this a double-blind experiment, we would need the placebo to look, taste, and smell like green tea. Subjects would not know which treatment is being delivered. In addition, the individuals administering the treatment and measuring the changes in LDL cholesterol would not know the treatment either. (e) The treatment is the tea, which is set at three levels.

(f)

Answers will vary. Other factors you might want to control in this experiment include age, exercise, and diet of the participants.

(g)

Randomization could be used by numbering the subjects from 1 to 120. Randomly select 40 subjects and assign them to the placebo group. Then randomly select 40 from the remaining 80 subjects and assign to the one cup of green tea group. The remaining subjects will be assigned to the two cups of green tea group. By randomly assigning the subjects to the treatments, the expectation is that uncontrolled variables (such as genetic history, diet, exercise, etc.) are neutralized (even out).

(h) Exercise is a confounding variable because any change in the LDL cholesterol cannot be attributed to the tea. It may be the exercise that caused the change in LDL cholesterol. 22. (a) The research objective is to determine if alerting shoppers about the healthiness of energy-dense snack foods changes the shopping habits of overweight individuals. (b) The subjects were 42 overweight shoppers. (c) Blinding is not possible because health information is visible. (d) The explanatory variable is health information or not. (e) The number of unhealthy snacks purchased is quantitative. (f) The researchers would not be able to distinguish whether it was the priming or the weight status that played a role in purchase decisions.

Chapter 1: Data Collection

23. Answers will vary. A completely randomized design is probably best.

24. Answers will vary. A matched-pairs design matched by car model is likely the best.

25. (a) The response variable is blood pressure. (b) Three factors that have been identified are daily consumption of salt, daily consumption of fruits and vegetables, and the body’s ability to process salt. (c) The daily consumption of salt and the daily consumption of fruits and vegetables can be controlled. The body’s ability to process salt cannot be controlled. To deal with variability of the body’s ability to process salt, randomize experimental units to each treatment group. (d) Answers will vary. Three levels of treatment might be a good choice – one level below the recommended daily allowance, one equal to the recommended daily allowance, and one above the recommended daily allowance. 26. Answers will vary. 27. Answers will vary. 28. Answers will vary for the design preference. Completely Randomized Design The researcher would randomly assign each subject to either drink Coke or

Pepsi. The response variable would be whether the subject likes the soda or not. Preference rates would be compared at the end of the experiment. The subject would be blinded, but the researcher would not. Therefore, this would be a single-blind experiment. Randomly assign subjects to colas

Group 1 (half the subjects)

Coke

Group 2 (half the subjects)

Pepsi

Compare preference rates

Matched-Pairs Design The researcher would randomly determine whether each subject drinks Coke first or Pepsi first. To avoid confounding, subjects should eat something bland between drinks to remove any residual taste. The response variable would be either the proportion of subjects who prefer Coke or the proportion of subjects who prefer Pepsi. This would also be a single-blind experiment since the subject would not know which drink was first but the researcher would. The matched-pairs design is likely superior.

Chapter 1 Review Exercises

Identify Subjects

Randomly assign Coke or Pepsi first

Administer treatments measure preference

(g) Confounding occurs when the effects of two or more explanatory variables are not separated. Therefore, any relation that may exist between an explanatory variable and response variable may be due to some other variable or variables not accounted for in the study.

For each matched pair, determine which cola is preferred.

29. Answers will vary. Control groups are needed in a designed experiment to serve as a baseline against which other treatments can be compared.

(h) A lurking variable is an explanatory variable that was not considered in the study, but that affects the value of the response variable in the study.

30. (a) Answers will vary. (b) Answers will vary.

2. The three major types of observational studies are (1) cross-sectional studies, (2) case-control studies, and (3) cohort studies.

31. Answers will vary. 32. Answers will vary.

Cross-sectional studies collect data at a specific point in time or over a short period of time. Cohort studies are prospective and collect data over a period of time, sometimes over a long period of time. Case-controlled studies are retrospective, looking back in time to collect data either from historical records or from recollection by subjects in the study. Individuals possessing a certain characteristic are matched with those that do not.

33. The purpose of randomization is to minimize the effect of factors whose levels cannot be controlled. (Answers will vary.) One way to assign the experimental units to the three groups is to write the numbers 1, 2, and 3 on identical pieces of paper and to draw them out of a “hat” at random for each experimental unit. 34. Answers will vary.

Chapter 1 Review Exercises

3. The process of statistics refers to the approach used to collect, organize, analyze, and interpret data. The steps are to (1) identify the research objective, (2) collect the data needed to answer the research question, (3) describe the data, and (4) perform inference.

1. (a) The response variable is the variable of interest in the study. (b) A variable is a characteristic of an individual. (c) A qualitative variable is a variable that allows for classification of individuals based on an attribute or characteristic.

4. The three types of bias are sampling bias, nonresponse bias, and response bias. Sampling bias occurs when the techniques used to select individuals to be in the sample favor one part of the population over another. Bias in sampling is reduced when a random process is to select the sample. Nonresponse bias occurs when the individuals selected to be in the sample that do not respond to the survey have different opinions from those that do respond. This can be minimized by using callbacks and follow-up visits to increase the response rate. Response bias occurs when the answers on a survey do not reflect the true feelings of the respondent. This can be minimized by using trained interviewers, using carefully worded questions, and rotating question and answer selections.

(d) A quantitative variable is a variable that provides numerical measures of individuals. The values of quantitative variables can be added or subtracted and provide meaningful results. (e) An observational study measures the value of the response variable without attempting to influence the value of the response or explanatory variables. (f) A study is a designed experiment if a researcher randomly assigns the individuals in a study to groups, intentionally manipulates the value of an explanatory variable and controls other explanatory variables at fixed values, and then records the value of the response variable for each individual. .

Chapter 1: Data Collection 5. Nonsampling errors are errors that result from undercoverage, nonresponse bias, response bias, and data-entry errors. These errors can occur even in a census. Sampling errors are errors that result from the use of a sample to estimate information about a population. These include random error and errors due to poor sampling plans, and result because samples contain incomplete information regarding a population. 6. The following are steps in conducting an experiment: (1) Identify the problem to be solved. Give direction and indicates the variables of interest (referred to as the claim). (2) Determine the factors that affect the response variable. List all variables that may affect the response, both controllable and uncontrollable. (3) Determine the number of experimental units. Determine the sample size. Use as many as time and money allow. (4) Determine the level of each factor. Factors can be controlled by fixing their level (e.g. only using men) or setting them at predetermined levels (e.g. different dosages of a new medicine). For factors that cannot be controlled, random assignment of units to treatments helps average out the effects of the uncontrolled factor over all treatments. (5) Conduct the experiment. Carry out the experiment using an equal number of units for each treatment. Collect and organize the data produced. (6) Test the claim. Analyze the collected data and draw conclusions. 7. “Number of new automobiles sold at a dealership on a given day” is quantitative because its values are numerical measures on which addition and subtraction can be performed with meaningful results. The variable is discrete because its values result from a count.

8. “Weight in carats of an uncut diamond” is quantitative because its values are numerical measures on which addition and subtraction can be performed with meaningful results. The variable is continuous because its values result from a measurement rather than a count. 9. “Brand name of a pair of running shoes” is qualitative because its values serve only to classify individuals based on a certain characteristic. 10. 73% is a statistic because it describes a sample (the 1011 people age 50 or older who were surveyed). 11. 70% is a parameter because it describes a population (all the passes completed by Cardale Jones in the 2015 Championship Game). 12. Birth year has the interval level of measurement since differences between values have meaning, but it lacks a true zero. 13. Marital status has the nominal level of measurement since its values merely categorize individuals based on a certain characteristic. 14. Stock rating has the ordinal level of measurement because its values can be placed in rank order, but differences between values have no meaning. 15. Number of siblings has the ratio level of measurement because differences between values have meaning and there is a true zero. 16. This is an observational study because no attempt was made to influence the variable of interest. Sexual innuendos and curse words were merely observed. 17. This is an experiment because the researcher intentionally imposed treatments (experimental drug vs. placebo) on individuals in a controlled setting. 18. This was a cohort study because participants were identified to be included in the study and then followed over a period of time with data being collected at regular intervals (every 2 years).

Chapter 1 Review Exercises

19. This is convenience sampling since the pollster simply asked the first 50 individuals she encountered.

(b) This is an experimental design because the teeth were separated into groups that were assigned different treatments.

20. This is a cluster sample since the ISP included all the households in the 15 randomly selected city blocks.

21. This is a stratified sample since individuals were randomly selected from each of the three grades.

(e) Type of stain remover (gum or saliva); Qualitative

(d) Percentage of stain removed

22. This is a systematic sample since every 40 tractor trailer was tested using a random start with the 12th tractor trailer. 23. (a) Sampling bias; undercoverage or nonrepresentative sample due to a poor sampling frame. Cluster sampling or stratified sampling are better alternatives. (b) Response bias due to interviewer error. A multilingual interviewer could reduce the bias. (c) Data-entry error due to the incorrect entries. Entries should be checked by a second reader. 24. Answers will vary. Using a TI-84 Plus graphing calculator with a seed of 1990, and numbering the individuals from 1 to 21, we would select individuals numbered 14, 6, 10, 17, and 11. If we numbered the businesses down each column, the businesses selected would be Jiffy Lube, Nancy’s Flowers, Norm’s Jewelry, Risky Business Security, and Solus, Maria, DDS. 25. Answers will vary. The first step is to select a random starting point among the first 9 bolts produced. Using row 9, column 17 from Table I in Appendix A, he will sample the 3rd bolt produced, then every 9th bolt after that until a sample size of 32 is obtained. In this case, he would sample bolts 3, 12, 21, 30, and so on, until bolt 282. 26. Answers will vary. The goggles could be numbered 00 to 99, then a table of random digits could be used to select the numbers of the goggles to be inspected. Starting with row 12, column 1 of Table 1 in Appendix A and reading down, the selected labels would be 55, 96, 38, 85, 10, 67, 23, 39, 45, 57, 82, 90, and 76. 27. (a) To determine the ability of chewing gum to remove stains from teeth

(f) The 64 stained bovine incisors (g) The chewing simulator could impact the percentage of the stain removed. (h) Gum A and B remove significantly more stain. 28. (a) Matched-pairs (b) Reaction time; Quantitative (c) Alcohol consumption (d) Food consumption; caffeine intake (e) Weight, gender, etc. (f) To act as a placebo to control for the psychosomatic effects of alcohol (g) Alcohol delays the reaction time significantly in seniors for low levels of alcohol consumption; healthy seniors that are not regular drinkers. 29. Answers will vary. Since there are ten digits (0 – 9), we will let a 0 or 1 indicate that (a) is to be the correct answer, 2 or 3 indicate that (b) is to be the correct answer, and so on. Beginning with row 1, column 8 of Table 1 in Appendix A, and reading downward, we obtain the following: 2, 6, 1, 4, 1, 4, 2, 9, 4, 3, 9, 0, 6, 4, 4, 8, 6, 5, 8, 5 Therefore, the sequence of correct answers would be: b, d, a, c, a, c, b, e, c, b, e, a, d, c, c, e, d, c, e, c 30. (a) Answers will vary. One possible diagram is shown below. Randomly assign to commercial type

Humorous (25 subjects) Serious (25 subjects)

Compare percent recall

31. A matched-pairs design is an experimental design where experimental units are matched up so they are related in some way.

Chapter 1: Data Collection In a completely randomized design, the experimental units are randomly assigned to one of the treatments. The value of the response variable is compared for each treatment. In a matched-pairs design, experimental units are matched up on the basis of some common characteristic (such as husband-wife or twins). The differences between the matched units are analyzed.

32. Answers will vary. 33. Answers will vary. 34. Randomization is meant to even out the effect of those variables that are not controlled for in a designed experiment. Answers to the randomization question may vary; however, each experimental unit must be randomly assigned. For example, a researcher might randomly select 25 experimental units from the 100 units and assign them to treatment #1. Then the researcher could randomly select 25 from the remaining 75 units and assign them to treatment #2, and so on.

Chapter 1 Test 1. Collect information, organize and summarize the information, analyze the information to draw conclusions, provide a measure of confidence in the conclusions drawn from the information collected. 2. The process of statistics refers to the approach used to collect, organize, analyze, and interpret data. The steps are to (1) identify the research objective, (2) collect the data needed to answer the research question, (3) describe the data, and (4) perform inference. 3. The time to complete the 500-meter race in speed skating is quantitative because its values are numerical measurements on which addition and subtraction have meaningful results. The variable is continuous because its values result from a measurement rather than a count. The variable is at the ratio level of measurement because differences between values have meaning and there is a true zero. 4. Video game rating is qualitative because its values classify games based on certain characteristics but arithmetic operations have no meaningful results. The variable is at the ordinal level of measurement because its

values can be placed in rank order, but differences between values have no meaning. 5. The number of surface imperfections is quantitative because its values are numerical measurements on which addition and subtraction have meaningful results. The variable is discrete because its values result from a count. The variable is at the ratio level of measurement because differences between values have meaning and there is a true zero. 6. This is an experiment because the researcher intentionally imposed treatments (brandname battery versus plain-label battery) on individuals (cameras) in a controlled setting. The response variable is the battery life. 7. This is an observational study because no attempt was made to influence the variable of interest. Fan opinions about the asterisk were merely observed. The response variable is whether or not an asterisk should be placed on Barry Bonds’ 756th homerun ball. 8. A cross-sectional study collects data at a specific point in time or over a short period of time; a cohort study collects data over a period of time, sometimes over a long period of time (prospective); a case-controlled study is retrospective, looking back in time to collect data. 9. An experiment involves the researcher actively imposing treatments on experimental units in order to observe any difference between the treatments in terms of effect on the response variable. In an observational study, the researcher observes the individuals in the study without attempting to influence the response variable in any way. Only an experiment will allow a researcher to establish causality. 10. A control group is necessary for a baseline comparison. This accounts for the placebo effect that says that some individuals will respond to any treatment. Comparing other treatments to the control group allows the researcher to identify which, if any, of the other treatments are superior to the current treatment (or no treatment at all). Blinding is important to eliminate bias due to the individual or experimenter knowing which treatment is being applied.

Chapter 1 Test 11. The steps in conducting an experiment are to (1) identify the problem to be solved, (2) determine the factors that affect the response variable, (3) determine the number of experimental units, (4) determine the level of each factor, (5) conduct the experiment, and (6) test the claim. 12. Answers will vary. The franchise locations could be numbered 01 to 15 going across. Starting at row 7, column 14 of Table I in Appendix, and working downward, the selected numbers would be 08, 11, 03, and 02. The corresponding locations would be Ballwin, Chesterfield, Fenton, and O’Fallon. 13. Answers will vary. Using the available lists, obtain a simple random sample from each stratum and combine the results to form the stratified sample. Start at different points in Table I or use different seeds in a random number generator. Using a TI-84 Plus graphing calculator with a seed of 14 for Democrats, 28 for Republicans, and 42 for Independents, the selected numbers would be Democrats: 3946, 8856, 1398, 5130, 5531, 1703, 1090, and 6369 Republicans: 7271, 8014, 2575, 1150, 1888, 3138, and 2008 Independents: 945, 2855, and 1401 14. Answers will vary. Number the blocks from 1 to 2500 and obtain a simple random sample of size 10. The blocks corresponding to these numbers represent the blocks analyzed. All trees in the selected blocks are included in the sample. Using a TI-84 Plus graphing calculator with a seed of 12, the selected blocks would be numbered 2367, 678, 1761, 1577, 601, 48, 2402, 1158, 1317, and 440. 600 » 42.86 , so we let 14 k = 42 . Select a random number between 1 and 42 that represents the first slot machine inspected. Using a TI-84 Plus graphing calculator with a seed of 132, we select machine 18 as the first machine inspected. Starting with machine 18, every 42nd machine thereafter would also be inspected (60, 102, 144, 186, …, 564).

experimental units are first divided according to some common characteristic (such as gender). Then each experimental unit within each block is randomly assigned to one treatment. Within each block, the value of the response variable is compared for each treatment, but not between blocks. By blocking, we prevent the effect of the blocked variable from confounding with the treatment. 17. (a) Sampling bias due to voluntary response (b) Nonresponse bias due to the low response rate (c) Response bias due to poorly worded questions. (d) Sampling bias due to poor sampling plan (undercoverage) 18. (a) This experiment has a matched-pairs design. (b) The subjects are the 159 social drinkers who participated in the study. (c) Treatments are the types of beer glasses (straight glass or curved glass). (d) The response variable is the time to complete the drink; quantitative. (e) The type of glass used in the first week is randomly determined. This is to neutralize the effect of drinking out of a specific glass first. (f)

15. Answers will vary.

16. In a completely randomized design, the experimental units are randomly assigned to one of the treatments. The value of the response variable is compared for each treatment. In a randomized block design, the

19. (a) This experiment has a completely randomized design. (b) The factor set to predetermined levels is the topical cream concentration. The treatments are 0.5% cream, 1.0% cream, and a placebo (0% cream). (c) The study is double-blind if neither the subjects, nor the person administering the treatments, are aware of which topical cream is being applied.

Chapter 1: Data Collection associated with lower bone mineral density for women.

(d) The control group is the placebo (0% topical cream).

21. A confounding variable is an explanatory variable that cannot be separated from another explanatory variable. A lurking variable is an explanatory variable that was not considered in the study but affects the response variable in the study.

(e) The experimental units are the 225 patients with skin irritations. (f) 0.5% cream (75 patients) Randomly assign patients to creams

1.0% cream (75 patients)

Compare improvement in skin irritation

Placebo (75 patients)

20. (a) This was a cohort study because participants were identified to be included in the study and then followed over a long period of time with data being collected at regular intervals (every 4 years). (b) The response variable is bone mineral density. The explanatory variable is weekly cola consumption. (c) The response variable is quantitative because its values are numerical measures on which addition and subtraction can be performed with meaningful results. (d) The researchers observed values of variables that could potentially impact bone mineral density (besides cola consumption), so their effect could be isolated from the variable of interest.

Case Study: Chrysalises for Cash Reports will vary. The reports should include the following components: Step 1: Identify the problem to be solved. The entrepreneur wants to determine if there are differences in the quality and emergence time of broods of the black swallowtail butterfly depending on the following factors: (a) early brood season versus late brood season; (b) carrot plants versus parsley plants; and (c) liquid fertilizer versus solid fertilizer. Step 2: Determine the explanatory variables that affect the response variable. Some explanatory variables that may affect the quality and emergence time of broods are the brood season, the type of plant on which the chrysalis grows, fertilizer used for plants, soil mixture, weather, and the level of sun exposure. Step 3: Determine the number of experimental units. In this experiment, a sample of 40 caterpillars/butterflies will be used. Step 4: Determine the level of the explanatory variables: • Brood season – We wish to determine the differences in the number of deformed butterflies and in the emergence times depending on whether the brood is from the early season or the late season. We use a total of 20 caterpillars/butterflies from the early brood season and 20 caterpillars/butterflies from the late brood season.

(e) Answers will vary. Some possible lurking variables that should be accounted for are smoking status, alcohol consumption, physical activity, and calcium intake (form and quantity) (f) The study concluded that women who consumed at least one cola per day (on average) had a bone mineral density that was significantly lower at the femoral neck than those who consumed less than one cola per day. The study cannot claim that increased cola consumption causes lower bone mineral density because it is only an observational study. The researchers can only say that increased cola consumption is

• Type of plant – We wish to determine the differences in the number of deformed butterflies and in the emergence times depending on the type of plant on which the caterpillars are placed. A total of 20 caterpillars are placed on carrot plants and 20 are placed on parsley plants. .

Case Study: Chrysalises for Cash • Fertilizer – We wish to determine the differences in the number of deformed butterflies and in the emergence times depending on the type of fertilizer used on the plants. A total of 20 chrysalises grow on plants that are fed liquid fertilizer and 20 grow on plants that are fed solid fertilizer. • Soil mixture – We control the effects of soil by growing all plants in the same mixture. • Weather – We cannot control the weather, but the weather will be the same for each chrysalis grown within the same season. For chrysalises grown in different seasons, we expect the weather might be different and thus part of the reason for potential differences between seasons. Also, we can control the amount of watering that is done. • Sunlight exposure – We cannot control this variable, but the sunlight exposure will be the same for each chrysalis grown within the same season. For chrysalises grown in different seasons, we expect the sunlight exposure might be different and thus part of the reason for potential differences between seasons. Step 5: Conduct the experiment. (a) We fill eight identical pots with equal amounts of the same soil mixture. We use four of the pots for the early brood season and four of the pots for the late brood season.

Step 6: Test the claim. We determine whether any differences exist depending on season, plant type, and fertilizer type. Conclusions: Early versus late brood season: From the data presented, more deformed butterflies occur in the late season than in the early season. Five deformed butterflies occurred in the late season, while only one occurred in the early season. Also, the emergence time seems to be longer in the early season than in the late season. In the early season, all but one of the 20 emergence times were between 6 and 8 days. In the late season, all 20 of the emergence times were between 2 and 5 days. Parsley versus carrot plants: From the data presented, the plant type does not seem to affect the number of deformed butterflies that occur. Altogether, three deformed butterflies occur from parsley plants and three deformed butterflies occur from the carrot plants. Likewise, the plant type does not seem to affect the emergence times of the butterflies. Liquid versus solid fertilizer: From the data presented, the type of fertilizer seems to affect the number of deformed butterflies that occur. Five deformed butterflies occurred when the solid fertilizer was used, while only one occurred when the liquid fertilizer was used. The type of fertilizer does not seem to affect emergence times.

For the early brood season, two of the pots grow carrot plants and two grow parsley plants. One carrot plant is fertilized with a liquid fertilizer, one carrot plant is fertilized with a solid fertilizer, one parsley plant is fertilized with the liquid fertilizer, and one parsley plant is fertilized with the solid fertilizer. We place five black swallowtail caterpillars of similar age into each of the four pots. Similarly, for the late brood season, two of the pots grow carrot plants and two grow parsley plants. One carrot plant is fertilized with a liquid fertilizer, one carrot plant is fertilized with a solid fertilizer, one parsley plant is fertilized with the liquid fertilizer, and one parsley plant is fertilized with the solid fertilizer. We place five black swallowtail caterpillars of similar age into each of the four pots. (b) We determine the number of deformed butterflies and in the emergence times for the caterpillars/butterflies from each pot. Copyright © 2022 Pearson Education, Inc.

Chapter 2 Summarizing Data in Tables and Graphs Section 2.1 1. Raw data are the data as originally collected, before they have been organized or coded. 2. Number (or count); proportion (or percent) 3. The relative frequencies should add to 1, although rounding may cause the answers to vary slightly. 4. A bar graph is used to illustrate qualitative data. It is a chart in which rectangles are used to illustrate the frequency or relative frequency with which a category appears. A Pareto chart is a bar chart with bars drawn in order of decreasing frequency or relative frequency. 5. (a) The most common response was “Boss.” (b) 15% of 1053 is 158. (c) The graphic cannot be displayed as a pie chart because the percentages do not add up to 100 percent. That is, there is no “whole.” 6. (a) The largest segment in the pie chart is for “Washing your hands” so the most commonly used approach to beat the flu bug is washing your hands. 61% of respondents selected this as their primary method for beating the flu. (b) The smallest segment in the pie chart is for “Drinking Orange Juice” so the least used method is drinking orange juice. 2% of respondents selected this as their primary method for beating the flu. (c) 25% of respondents felt that flu shots were the best way to beat the flu. 7. (a) The highest bar corresponds to the position OF (outfield), so OF is the position with the most MVPs. (b) The bar for first base (1B) reaches the line for 15. Thus, there were 15 MVPs who played first base.

(c) The bar for first base (1B) is 15 on the vertical axis. The bar for third base (3B) reaches 10. 15 – 10 = 5, so there were 5 more MVPs who played first base than third base. (d) Each of the three outfield positions should be reported as MVPs, rather than treating the three positions as one position. 8. (a) 25,561,000 whites were living in poverty. (b)

11,190 ≈ 0.234 25, 561 + 9132 + 11,190 + 1985 = 23.4% In 2017, about 23.4% of the impoverished in the United States were Hispanic.

(c) This graph should use relative frequencies, rather than frequencies. The graph does not account for the different population size of each ethnic group. Without knowing the population sizes, we cannot determine whether a group is disproportionally impoverished. 9. (a) 69% of the respondents believe divorce is morally acceptable. (b) 23% believe divorce is morally wrong. So, 240 million * 0.23 = 55.2 million adult Americans believe divorce is morally wrong. (c) This statement is inferential, since it is a generalization based on the observed data. 10. (a) 5% of identity theft was loan fraud. (b) 26% of the identity fraud cases in a recent year involved credit card fraud. So, 10 million * 0.26 = 2.6 million cases of credit card fraud occurred in a recent year. 11. (a) The proportion of 18–34 year old respondents who are more likely to buy when made in America is 0.42. For 34–44 year olds, the proportion is 0.61.

Section 2.1: Organizing Qualitative Data (b) The 55+ age group has the greatest proportion of respondents who are more likely to buy when made in America.

(e)

(c) The 18–34 age group has a majority of respondents who are less likely to buy when made in America. (d) As age increases, so does the likelihood that a respondent will be more likely to buy a product that is made in America. 12. (a) The proportion of males who would like to be richer is 0.46. The proportion of females who would like to be richer is 0.41.

(f)

(b) The attribute that females desire more than males is to be thinner. (c) The attribute that males prefer over females two-to-one is to be younger. (d) Equal proportions of males and females desire to be smarter and none of these. 13. (a) Total students surveyed = 125 + 324 + 552 + 1257 + 2518 = 4776 Relative frequency of “Never” = 125 / 4776 ≈ 0.0262, and so on. Response Never

Relative Frequency 0.0262

Rarely Sometimes

0.0678 0.1156

Most of the time 0.2632 Always 0.5272

(b) 52.72% (c) 0.0262 + 0.0678 = 0.0940 or 9.40% (d)

(g) This is a descriptive statement because it is reporting a result of the sample. 14. (a) Total students surveyed = 249 + 118 + 249 + 345 + 716 + 3093 = 4770 Relative frequency of “ I do not drive” 249 ≈ 0.0522, and so on. = 4770 Response I do not drive a car Never Rarely Sometimes Most of the time Always

Relative Frequency 0.0522 0.0247 0.0522 0.0723 0.1501 0.6484

(b) 64.84% (c) 0.0247 + 0.0522 = 0.0769 or 7.69%

Chapter 2: Summarizing Data in Tables and Graphs

(d)

(e)

(f)

(g) Total students = 118 + 249 + 345 + 716 + 3093 = 4521 Relative frequency of “Never” 118 ≈ 0.0261, and so on. = 4521

Response

Relative Frequency

Never

0.0261

Rarely

0.0551

Sometimes Most of the time

0.0763 0.1584

Always

0.6841

Section 2.1: Organizing Qualitative Data The relative frequencies of all categories are very similar except that students are more likely to wear their seatbelt ‘Always’ when driving. (h) The statement is descriptive because it is describing the particular sample. 15. (a) Total adults surveyed = 377 + 192 + 132 + 81 + 243 = 1025 Relative frequency of “More than 1 hour a day” = 377 / 1025 ≈ 0.3678, and so on.

More than 1 hr a day

Relative Frequency 0.3678

Up to 1 hr a day A few times a week

0.1873 0.1288

Response

A few times a month or less 0.0790 Never

0.2371

(b) 0.2371 (about 24%) (c)

(d)

(e)

(f) The statement provides an estimate, but no level of confidence is given.

Chapter 2: Summarizing Data in Tables and Graphs

16. (a) Total adults surveyed = 103 + 204 + 130 + 79 + 5 = 521 Relative frequency of “Several times a 103 ≈ 0.197, and so on. week” = 521

Relative Response Frequency Several times a week 0.1977 Once or twice a week 0.3916 A few times a month 0.2495 Vary rarely 0.1516 Never 0.0096

(b) Total females = 1114 Platform

Females

Facebook

0.1993

Instagram

0.4803

Snapchat

0.0260

Twitter

0.0512

None

0.2433

(c)

(b) The proportion surveyed who dine out once or twice a week is 204/(103 + 204 + 130 + 79 + 5) = 0.3916 (c)

(d) Females are much more likely for Instagram to influence their online shopping, while males are more likely to have none of these platforms influence their online shopping.

(d)

18. (a) Total adults = 1936 Relative frequency for “none” is: 173/1936 = 0.09, and so on.

Number of Texts None 1 to 10 11 to 20 21 to 50 51 to 100 101+

17. (a) Total males = 1562 Platform

Males

Facebook

0.2087

Instagram

0.2433

Snapchat

0.0359

Twitter

0.0781

None

0.4341

Rel. Freq. (Adults) 0.0894 0.5052 0.1286 0.1286 0.0692 0.0790

(b) Total teens = 627 Relative frequency for “none” is: 13/627 = 0.021, and so on.

Number of Texts None 1 to 10 11 to 20 21 to 50 51 to 100 101+

Rel. Freq. (Teens) 0.0207 0.2201 0.1100 0.1802 0.1802 0.2887

Section 2.1: Organizing Qualitative Data (c)

(b)

(d) Answers will vary. Adults are much more likely to send fewer texts per day, while teens are much more likely to do more texting. 19. (a) Total males = 99; Relative frequency for “Professional Athlete” is 40/99 = 0.404, and so on.

Total number of females = 100; Relative frequency for “Professional Athlete” is 18/100 = 0.18, and so on. Dream Job Professional Athlete Actor/Actress President of the United States Rock Star Not Sure

(c) Answers will vary. Males are much more likely to want to be a professional athlete. Women are more likely to aspire to a career in acting than men. Men’s desire to become athletes may be influenced by the prominence of male sporting figures in popular culture. Women may aspire to careers in acting due to the perceived glamour of famous female actresses.

Men Women 0.4040 0.180 0.2626 0.370 0.1313 0.130 0.1313 0.0707

0.130 0.190

25 = 0.25, and so on. 100 10 = 0.10, and so on. Relative frequency for “White” sport cars = 100 Relative Frequencies Color Luxury Cars Sport Cars White 0.25 0.10 Black 0.22 0.15 Silver 0.16 0.18 Gray 0.12 0.15 Blue 0.07 0.13 Red 0.07 0.15 Gold 0.06 0.05 Green 0.03 0.02 Brown 0.02 0.07

20. (a) Relative frequency for “White” luxury cars =

Chapter 2: Summarizing Data in Tables and Graphs (b)

(c) Answers will vary. White is the most popular color for luxury cars, while silver is the most popular for sports cars. People who drive luxury cars may enjoy the clean look of a white vehicle. People who drive sports cars may prefer the flashier look of silver. 21. (a), (b) Total number of Trading Days = 30; relative frequency for Down is 15/30 = 0.5, and so on.

Winner Down No Change Up (c)

Freq. 15 2 13

(e)

Rel. Freq. 0.500 0.067 0.433 22. (a), (b) Total number of responses = 40; relative frequency for “Sunday” is 3/40 = 0.075.

Response Sunday Monday Tuesday Wednesday Thursday Friday Saturday (d)

Freq. 3 2 5 6 2 14 8

Rel. Freq. 0.075 0.050 0.125 0.150 0.050 0.350 0.200

(c) Answers will vary. If you own a restaurant, you will probably want to advertize on the days when people will be most likely to order takeout: Friday. You might consider avoiding placing an ad on Monday and Thursday, since the readers are least likely to choose to order takeout on these days.

Section 2.1: Organizing Qualitative Data (d)

(e)

(f)

23. (a) Total number of responses = 42 Day Sunday Monday Tuesday Wednesday Thursday Friday Saturday

Frequency 8 5 7 6 3 4 9

Relative Frequency 0.1905 0.1190 0.1667 0.1429 0.0714 0.0952 0.2143

(b) You would want to have the most drivers available on Saturday. (c)

Chapter 2: Summarizing Data in Tables and Graphs (d)

(i)

25. (a) Total number of tornadoes = 1473. 24. (a), (b) Total number of patients = 50 Relative frequency for “Type A” 18 = 0.36, and so on. = 50

F Scale −9 0 1 2 3 4

Blood Type Freq. Rel. Freq. A

0.36

0.08

0.12

0.44

(b)

(c) Type O is the most common. (d) Type AB is the least common. (e) We estimate that 44% of the population has type O blood. This is considered inferential statistics because a conclusion about the population is being drawn based on sample data. (f) Answers will vary; in 2008 the Red Cross reported that 45% of the population had type O blood (either + or – ). Results will differ because of sampling variability.

(c)

(g)

(h)

Frequency 64 637 607 146 16 3

Relative Frequency 0.0434 0.4325 0.4121 0.0991 0.0109 0.0020

Section 2.1: Organizing Qualitative Data (d) Answers may vary, but a bar chart is easier to read with twelve observations.

(b) It would not make sense to draw a pie chart for the highest elevation because there is no whole to which to compare the parts. 27. Answers will vary. 28. Answers will vary. 29. (a) The researcher wants to determine if online homework improves student learning over traditional pencil-and-paper homework.

(e) Texas had the most tornados in 2017 with 168. 26. (a) It would make sense to draw a pie chart for land area since the 7 continents contain all the land area on Earth. Total land area is 11,608,000 + 5,100,000 + … + 9,449,000 + 6,879,000 = 57,217,000 square miles. The relative frequency (percentage) for 11, 608, 000 = 0.2029 . Africa is 57, 217, 000 Continent Africa Antarctica

Land Area Rel. Freq. (mi 2 ) 11,608,000 0.2029 5,100,000 0.0891

Asia 17,212,000 Australia 3,132,000 Europe 3,837,000 North America 9,449,000 South America 6,879,000

0.3008 0.0547 0.0671 0.1651 0.1202

(b) This study is an experiment because the researcher is actively imposing treatments (the homework style) on subjects. (c) Answers will vary. Some examples are same teacher, same semester, and same course. (d) Assigning different homework methods to entire classes could confound the results because there may be differences between the classes. The instructor may give more instruction to one class than the other. The instructor is not blinded, so he or she may treat one group differently from the other. (e) Number of students: quantitative, discrete Average age: quantitative, continuous Average exam score: quantitative, continuous Type of homework: qualitative College experience: qualitative (f) Letter grade is a qualitative variable at the ordinal level of measurement. Answers will vary. It is possible that ordering the data from A to F is better because it might give more “weight” to the higher grade and the researcher wants to show that a higher percent of students passed using the online homework. (g) The graph being displayed is a side-byside relative frequency bar graph. (h) Yes; the “whole” is the set of students who received a grade for the course for each homework method.

Chapter 2: Summarizing Data in Tables and Graphs (i) The table shows that the two groups with no prior college experience had roughly the same average exam grade. From the bar graph, we see that the students using online homework had a lower percent for As, but had a higher percent who passed with a C or better.

30. Relative frequencies should be used when the size of two samples or populations differ. 31. Answers will vary. If the goal is to illustrate the levels of importance, then arranging the bars in a bar chart in decreasing order makes sense. Sometimes it is useful to arrange the categorical data in a bar chart in alphabetical order. A pie chart does not readily allow for arranging the data in order. 32. A bar chart is preferred when trying to compare two specific values. Pie charts are helpful for comparing parts of a whole. A pie chart cannot be drawn if the data do not include all possible values of the qualitative variable.

(e)

15 = 0.15 or 15% of the time a 7 was 100 observed.

(f) The distribution is approximately bellshaped. 10. (a) The most frequent number of cars sold in a week was 4 cars. (b) There were 9 weeks in which 2 cars sold. (c) Total frequency = 4 + 2 + 9 + 8 + 12 + 8 + 5 + 2 + 1 + 1 = 52 (as required) Percentage of time two cars are sold 9 ⋅100 = 17.3% = 52 (d) Slightly skewed to the right 11. (a) Total frequency = 2 + 3 + 13 + 42 + 58 + 40 + 31 + 8 +2 + 1 = 200 (b) 10 (e.g. 70 – 60 = 10) (c)

33. No, the percentages do not sum to 100%

Section 2.2 1. classes 2. lower; upper 3. class width 4. Skewed left means that the left tail is longer than the right tail.

IQ Score (class) 60–69 70–79 80–89 90–99 100–109 110–119 120–129 130–139 140–149 150–159

Frequency 2 3 13 42 58 40 31 8 2 1

(d) The class “100 – 109” has the highest frequency.

5. True 6. False. The class width is 10. 7. False. The distribution shape shown is skewed right. 8. False. The distribution shape is bell-shaped.

(e) The class “150 – 159” has the lowest frequency. (f)

8 + 2 +1 = 0.055 = 5.5% 200

9. (a) The value with the highest frequency is 8.

(g) No, there were no IQs above 159.

(b) The value with the lowest frequency is 2.

12. (a) The class width is 250 (e.g. 250 – 0 = 250).

(c) The value of 7 was observed 15 times. (d) The value of 5 was observed 11 times and the value of 4 was observed 7 times. Therefore, the value of 5 was observed 4 more times than the value of 4 (e.g. 11 − 7 = 4).

(b) 0–249, 250–499, 500–749, 750–999, 1000–1249, 1250–1499, 1500–1749, 1750– 1999, 2000–2249, 2250–2499, 2500– 2749, 2750–2999, 3000–3249, 3250–3499.

Section 2.2: Organizing Quantitative Data (c) The highest frequency is in class 0–249. (d) The distribution is skewed right. (e) Answers will vary. The statement is incorrect because they are comparing counts from populations of different size. Texas has a much larger population than Vermont. To make a fair comparison, the reporter should use rates of fatalities such as the number of fatalities per 1000 residents. 13. (a) Likely skewed right. Most household incomes will be to the left (perhaps in the $50,000 to $150,000 range), with fewer higher incomes to the right (in the millions). (b) Likely bell-shaped. Most scores will occur near the middle range, with scores tapering off equally in both directions. (c) Likely skewed right. Most households will have, say, 1 to 4 occupants, with fewer households having a higher number of occupants. (d) Likely skewed left. Most Alzheimer’s patients will fall in older-aged categories, with fewer patients being younger. 14. (a) Likely skewed right. More individuals would consume fewer alcoholic drinks per week, while less individuals would consume more alcoholic drinks per week. (b) Likely uniform. There will be approximately an equal number of students in each age category. (c) Likely skewed left. Most hearing-aid patients will fall in older-aged categories, with fewer patients being younger. (d) Likely bell-shaped. Most heights will occur, say, in the 66- to 70-inch range, with heights tapering off equally in both directions. 15. (a) 3 (b) 1 (c) 14 (d) 20 (e) 8.75%, 12.5% 16. (a) 2 (b) 7 (c) 6 (d) 2

17. (a) From the graph, it appears the unemployment rate in 2012 was about 8%. (b) The highest unemployment rate was about 9.8%. This occurred in 2010. (c) The highest inflation rate was about 3.9%. This occurred in 2008. (d) The unemployment rate and inflation rate were furthest in 2009. (e) The misery index for 2008 was approximately 4 + 6 = 10. The misery index for 2011 was approximately 3 + 9 = 12. According to the misery index, the year 2011 was more “miserable” than the year 2008. 18. (a) To the nearest year, the average age of a man who first married in 1980 was 25. (b) To the nearest year, the average age of a woman who first married in 1960 was 21. (c) The largest difference in the average age of men and women at which they first married occurred in 1950. The approximate age difference was 24 – 20.5 = 3.5 years. 19. (a) Total number of households = 16 + 18 + 12 + 3 + 1 = 50 Relative frequency of 0 children = 16/50 = 0.32, and so on.

(b)

(c)

Number of Children Under 5

Relative Frequency

0 1 2 3 4

0.32 0.36 0.24 0.06 0.02

12 = 0.24 or 24% of households have 50 two children under the age of 5. 18 + 12 30 = = 0.6 or 60% of 50 50 households have one or two children under the age of 5.

Chapter 2: Summarizing Data in Tables and Graphs

20. (a) Total number of free throws = 16 + 11 + 9 + 7 + 2 + 3 + 0 + 1 + 0 + 1 = 50. Relative frequency of 1 throw until a miss = 16/50 = 0.32, and so on. Number of Free Throws Until a Miss 1 2 3 4 5 6 7 8 9 10 (b)

(c)

Relative Frequency 0.32 0.22 0.18 0.14 0.04 0.06 0.00 0.02 0.00 0.02

For example, 2.0 − 1.0 = 1.0 . Therefore, the class width is 1.0. 23. (a) Total frequency = 18 + 104 + 253 + 166 + 8 = 549 Relative frequency for 22–23.9 is 18/549 = 0.0328, and so on.

Speed (ft/sec) 22–23.9 24–25.9 26–27.9 28–29.9 30–31.9

Relative Frequency 0.0328 0.1894 0.4608 0.3024 0.0146

(b)

7 = 0.14 ; 14% of the time she first 50 missed on the fourth try. 1 = 0.02 ; 2% of the time she first 50 missed on the tenth try.

(d) “At least 5” means that the basketball player misses on the 6th shot or 7th shot or 3 + 0 +1+ 0 +1 5 = = 0.10 or 8th, etc. 50 50 10% of the time.

(c)

21. (a) There are five classes. (b) Lower class limits: 22, 24, 26, 28, 30 Upper class limits: 23.9, 25.9, 27.9, 29.9, 31.9. (c) The class width can be found by subtracting consecutive lower class limits. For example, 24 − 22 = 2 . Therefore, the class width is 2. 22. (a) There are seven classes. (b) Lower class limits: 0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 Upper class limits: 0.9, 1.9, 2.9, 3.9, 4.9, 5.9, 6.9.

The percentage of players had a sprint speed between 24 and 25.9 ft/sec is 18.94%. The percentage of players that had a sprint speed less than 24 ft/sec is 3.28%.

Section 2.2: Organizing Quantitative Data 24. (a) Total frequency = 3145 + 4145 + 1264 + 241 + 770 + 130 + 14 = 9709 Relative frequency for 0–0.9 is 3145/9709 = 0.3234, and so on.

Magnitude 0–0.9 1.0–1.9 2.0–2.9 3.0–3.9 4.0–4.9 5.0–5.9 6.0–6.9

Relative Frequency 0.3239 0.4269 0.1302 0.0248 0.0793 0.0134 0.0014

(b)

Number of Televisions 0 1 2 3 4 5

Frequency

1 14 14 8 2 1

Relative Frequency 0.025 0.350 0.350 0.200 0.050 0.025

(d) The relative frequency is 0.2, so 20% of the households surveyed had 3 televisions. (e) 0.05 + 0.025 = 0.075 7.5% of the households in the survey had 4 or more televisions. (f)

(g) (c)

(h) The distribution is skewed right.

The percentage of earthquakes that registered between 4.0 and 4.9 is 7.93%. The percent of earthquakes that registered 4.9 or less is 3145 + 4145 + 1264 + 241 + 770 ≈ 0.9852 9709 = 98.52%

26. (a) The data are discrete. The possible values for the number of customers waiting for a table are countable. (b), (c) Relative frequency of 3 customers waiting = 2/40 = 0.05, and so on.

25. (a) The data are discrete. The possible values for the number of televisions in a household are countable (b), (c) The relative frequency for 0 televisions is 1/40 = 0.025, and so on.

Chapter 2: Summarizing Data in Tables and Graphs Number of Customers 3 4 5 6 7 8 9 10 11 12 13 14

Freq.

Rel. Freq.

2 3 3 5 4 8 4 4 4 0 2 1

0.050 0.075 0.075 0.125 0.100 0.200 0.100 0.100 0.100 0.000 0.050 0.025

55–59.9 60–64.9

5 5

0.037 0.037

(c)

(d)

(d) 10.0 + 10.0 + 0.0 + 5.0 + 2.5 = 27.5% of the Saturdays had 10 or more customers waiting for a table at 6 p.m. (e) 5.0 + 7.5 + 7.5 = 20.0% of the Saturdays had 5 or fewer customers waiting for a table at 6 p.m. (f)

(e) The shape of the distribution is skewed right. (f) Relative frequency of a Gini Index of 20–29.9 = 21/136 = 0.154, and so on. Gini Index Freq. Rel. Freq. 20–29.9 21 0.154 30–39.9 55 0.404 40–49.9 37 0.272 50–59.9 18 0.132 60–69.9 5 0.037

(g)

(h) The distribution is more or less symmetric. 27. (a), (b) Relative frequency of a Gini Index of 20–24.9 = 5/136 = 0.037, and so on. Gini Index

Freq.

20–24.9 25–29.9 30–34.9 35–39.9 40–44.9 45–49.9 50–54.9

5 16 28 27 20 17 13

Rel. Freq. 0.037 0.118 0.206 0.199 0.147 0.125 0.096

The shape of the distribution is skewed right. (g) Answers will vary. The graph with a class width of 5 provides more detail, so it seems to be a superior graph.

Section 2.2: Organizing Quantitative Data 28. (a), (b) Relative frequency for the median income 40,000–44,999 is 2/51 = 0.0392, and so on. Income

Freq.

Rel. Freq.

40,000–44,999

0.0392

45,000–49,999

0.0588

50,000–54,999

0.1176

55,000–59,999

0.2941

60,000–64,999

0.1961

65,000–69,999

0.0392

70,000–74,999

0.1961

75,000–79,999

0.0196

80,000–84,999

0.0392

(c)

(d)

(e) The shape of the distribution is fairly symmetric. (f) Relative frequency for the median household income 40,000–49,999 is 5/51 = 0.0980, and so on. Income 40,000–49,999 50,000–59,999 60,000–69,999 70,000–79,999 80,000–89,999

Freq. 5 21 12 11 2

Rel. Freq. 0.0980 0.4118 0.2353 0.2157 0.0392

Chapter 2: Summarizing Data in Tables and Graphs

With a class width of 10,000, the distribution looks skewed right. (g) Answers will vary, but the graph with a class width of $10,000 seems to show more details about the data so it seems better. 29. (a) There are 50 entries in the table, so the relative frequency for homeruns with an exit velocity of 90–93.9 mph is 2/50 = 0.04. Exit Velocity 90–93.9 94–97.9 98–101.9 102–105.9 106–109.9 110–113.9

Frequency

2 3 13 22 8 2

Relative Frequency 0.04 0.06 0.26 0.44 0.16 0.04

Section 2.2: Organizing Quantitative Data (b)

(c) The distribution is approximately symmetric and bell-shaped. (d) There are 50 entries in the table, so the relative frequency for homeruns with an exit velocity of 90–93.9 mph is 2/50 = 0.04. Exit Velocity 90–92.4 92.5–94.9 95–97.4 97.5–99.9 100–102.4 102.5–104.9 105–107.4 107.5–109.9 110–112.4

Frequency 1 2 1 5 10 16 10 3 2

Relative Frequency 0.02 0.04 0.02 0.10 0.20 0.32 0.20 0.06 0.04

With a class width of 2.5, the distribution looks slightly skewed to the left. (e) Answers may vary. However, the class width of 4 appears to provide a better summary.

Chapter 2: Summarizing Data in Tables and Graphs

30. (a), (b) Total number of data points = 51 Relative frequency of 0–0.499 is 7/51 = 0.1373, and so on. Cigarette Tax 0.00–0.499 0.50–0.999 1.00–1.499 1.50–1.999 2.00–2.499 2.50–2.999 3.00–3.499 3.50–3.999 4.00–4.499

Frequency

7 13 7 8 5 4 4 2 1

Relative Frequency 0.1373 0.2549 0.1373 0.1569 0.0980 0.0784 0.0784 0.0392 0.0196

(c)

The distribution appears to be right skewed. (g) Answers will vary. Both do a nice job of summarizing the data. 31. (a) We can determine a class width by subtracting the smallest value from the largest, dividing by the desired number of classes, then rounding up. For example, 27.3 − 0.0 = 3.9 → 4 7 The first lower class limit should be a number below the smallest data value. In this case, 0 is a good first lower limit since it is the smallest data value. Thus, we will have a class width of 4, and the first class will have a lower limit of 0.

(d)

(b) (e) The distribution appears to be right skewed. (f) Relative frequency of 0–0.999 is: 20/51 = 0.3922, and so on. Cigarette Tax Frequency 0.00–0.999 20 1.00–1.999 15 2.00–2.999 9 3.00–3.999 6 4.00–4.999 1

Relative Frequency 0.3922 0.2941 0.1765 0.1176 0.0196

Default Rate 0–3.9 4–7.9 8–11.9 12–15.9 16–19.9 20–23.9 24–27.9

Freq. 10 10 7 2 5 5 1

Rel. Freq. 0.25 0.25 0.175 0.05 0.125 0.125 0.025

Section 2.2: Organizing Quantitative Data 32. Answers will vary. One possibility follows. (a) We can determine a class width by subtracting the smallest value from the largest, dividing by the desired number of classes, then rounding up. For example, 23.59 − 6.37 = 2.87 → 3 6 Our first lower class limit should be a nice number below the smallest data value. In this case, 6 is a good first lower limit since it is the nearest whole number below the smallest data value. Thus, we will have a class width of 3, and the first class will have a lower limit of 6. (b), (c) Relative frequency for 6–8.99 = 15/35 = 0.4286, and so on. Volume

Freq. Rel. Freq.

6 − 8.99

0.4286

9 − 11.99

0.2571

12 − 14.99 15 − 17.99

4 4

0.1143 0.1143

18 − 20.99

0.0571

21 − 23.99

0.0286

(d)

33. Answers will vary. It is disconcerting that some schools have a negative ROI. Annual ROI <–11 −11 − −10.01 −10 − −9.01 −9 − −8.01 −8 − −7.01 −7 − −6.01 −6 − −5.01 −5 − −4.01 −4 − −3.01 −3 − 2.01 −2 − −1.01 −1 − −0.01 0–0.99 1–1.99 2–2.99 3–3.99 4–4.99 5–5.99 6–6.99 7–7.99 8–8.99 9–9.99 10–10.99 11–11.99 12–12.99 ≥13

(e)

(f) The distribution is skewed right.

Freq. 2 1 2 1 3 1 5 6 13 19 19 39 53 120 194 267 290 278 228 137 104 51 17 7 4 2

Rel. Freq. 0.001074 0.000537 0.001074 0.000537 0.00161 0.000537 0.002684 0.003221 0.006978 0.010199 0.010199 0.020934 0.028449 0.064412 0.104133 0.143317 0.155663 0.149222 0.122383 0.073537 0.055824 0.027375 0.009125 0.003757 0.002147 0.001074

Chapter 2: Summarizing Data in Tables and Graphs

34.

11.9 − 10.0 = 0.190 10.0 = 19.0% No, there have not been any years when the debt decreased since 2000.

(b) Percentage change =

38. (a)

There are several similarities in the distribution of the ideal number of children, as reported by males and females. However, females seem more likely to deem larger families as ideal. A histogram would better serve us in comparing the preferences between males and females.

(b) Percentage change 2015 to 2016 7−4 = = 0.75 = 75% 4 Percentage change 2016 to 2017 10 − 7 = = 0.429 = 42.9% 7 39.

35.

Births per woman was lowest in 1976. 40. 36.

37. (a)

Life expectancy was highest in 2014.

Section 2.2: Organizing Quantitative Data 41. (a)

Tornado Length 0–4.999 5–9.999 10–14.999 15–19.999 20–24.999 25–29.999 30–34.999 35–39.999 40–44.999 45–49.999 50–54.999 55–59.999 60–64.999 65–69.999 70–74.999 75–79.999 80–84.999

Freq.

Rel. Freq.

1126 214 72 29 14 3 6 5 1 0 1 0 0 0 1 0 1

0.7644 0.1453 0.0489 0.0197 0.0095 0.0020 0.0041 0.0034 0.0007 0 0.0007 0 0 0 0.0007 0 0.0007

The relative frequency of tornadoes between 35 and 39.999 miles in length is 0.006. (e)

Three tornadoes resulted in four or more fatalities. (f) Sixty-six tornadoes traveled through two states.

(d)

Tornado Length 0–4.999 5–9.999 10–14.999 15–19.999 20–24.999 25–29.999 30–34.999 35–39.999

Freq.

Rel. Freq.

142 17 7 0 1 0 0 1

0.8452 0.1012 0.0417 0 0.006 0 0 0.006

Time (in Seconds) Spent Viewing a Web Page 20

15 Frequency

(b) The distribution is skewed right. (c) The relative frequency of tornadoes between 5 and 9.999 miles in length is 0.145.

42. Because the data are quantitative, either a stem-and-leaf plot or a histogram would be appropriate. There were 20 people who spent less than 30 seconds, 7 people spent at least 30 seconds but less than 60 seconds, etc. One possible histogram is:

120 150 180 Time (in seconds)

210

240

270

The data appear to be skewed right with a gap and one potential outlier. It seems as if the majority of surfers spent less than one minute viewing the page, while a few surfers spent several minutes viewing the page. 43. (a) The data is quantitative and discrete. (b) This is population data because it is all recorded violations for the day.

Chapter 2: Summarizing Data in Tables and Graphs (c) There are 271 red light cameras. (d)

48. Relative frequencies should be used when comparing two data sets with different sample sizes. 49. Answers will vary. The exercise illustrates the fact that there is no such thing as the “correct” histogram. However, some histograms are better than others and class width can affect the shape of a graph. 50. Answers will vary. Sample histograms are given below.

The distribution is skewed right. (e)

Answers will vary. (f) Yes; six cameras did not record any violations. 44. Age: histogram, stem-and-leaf plot, or dot plot; Income: histogram or stem-and-leaf plot; Marital status: bar graph or pie chart; Number of vehicles: histogram, stem-and-leaf plot, or dot plot 45. Answers will vary. Reports should address the fact that the number of people going to the beach and participating in underwater activities (e.g. scuba diving, snorkeling) has also increased, so an increase in shark attacks is not unexpected. A better comparison would be the rate of attacks per 100,000 beach visitors. The number of fatalities could decrease due to better safety equipment (e.g. bite resistant suits) and better medical care. 46. Classes should not overlap to avoid any confusion as to which class an observation belongs to. 47. There is no such thing as the correct choice for a class width, however some choices are better than others. For example, if the class width is too small, the histogram will show many gaps between the bars. If the class width is too large, the histogram may not provide enough detail.

A histogram is skewed left if it has a long tail on the left side. A histogram is skewed right if it has a long tail on the right side. A histogram is symmetric if the left and right sides of the graph are roughly mirror images of each other. 51. Time-series plots are drawn with quantitative variables. They are drawn to see trends in the data.

Section 2.3: Graphical Misrepresentations of Data

Section 2.3 1. The number of shark attacks is going to be higher in the summer months due to there being more swimmers. This graphic should be changed to account for the number of swimmers in the water. 2. (a) Answers will vary. The lengths of the bars are not proportional. For example, the bar for soda is 1/3 the size of the bar for a cheeseburger, but the number of steps for a cheeseburger is just over twice that for the soda. In addition, it is unclear where the graph begins: at the base of each figure or the bottom of the platform. (b) Answers will vary. The pictures could be replaced by simple bars (of the same width) that are proportional in area. 3. (a) The vertical axis starts at $34,000 instead of $0. This tends to indicate that the median earnings for females changed at a faster rate than actually occurred. (b) This graph indicates that the median earnings for females has increased slightly over the given time period, but not as significantly as suggested by the graph in part (a).

4. (a) The vertical axis starts at 4 instead of 0. This may lead the reader to conclude, for example, the percentage of employed people aged 55–64 who are members of a union is more than double the percentage of those aged 25–34 years. (b)

5. The bar for 12p–6p covers twice as many hours as the other bars. By combining two 3hour periods, this bar looks larger compared to the others, making afternoon hours look more

dangerous. If this bar were split into two periods, the graph may give a different impression. For example, the graph may show that daylight hours are safer. 6. The article is basing its conclusion on a comparison of categories that do not cover the same number of years. A better comparison is the incidence rate (number of accidents per 100,000 licensed drivers). [Note: only about 14% of licensed drivers in 2005 were aged 24 years or younger.] 7. Answers will vary. This graph is misleading because it does not take into account the size of the population of each state. Certainly, Puerto Rico is going to pay less in total taxes than California simply because its population is so much lower. The moral of the story here is that many variables should be considered on

Chapter 2: Summarizing Data in Tables and Graphs

per capita (per person) basis. For example, this graph should be drawn to represent taxes paid per capita (per person). 8. (a) The oil reserves in 2018 were 652.6 million barrels, whereas the oil reserves in 1977 were 7.5 million barrels. The oil reserves in 2018 were about 87 times as large as in 1977 (e.g. 652.6/7.5 = 87.01). Thus, the graphic for 2018 should be roughly 87 times larger than the graphic for 1977. (b) Assuming no change in U.S. oil production, the U.S. strategic oil reserves would last approximately 64 days (e.g. 652.6/10.14 = 64.4 days). 9. (a) The graphic is misleading because the bars are not proportional. The bar for housing should be a little more than twice the length of the bar for transportation, but it is not.

(c) Answers will vary. Changing the scale on the graph will affect the message. The message is also affected by using the variable “Health Care per Capita” rather than “Health Care as a Percent of GDP.” 12. (a) A graph that is not misleading will use a vertical scale starting at $0 and bars of equal width. One example is:

(b) The graphic could be improved by adjusting the bars so that their lengths are proportional. 10.

The graph does not support the safety manager’s claim. The vertical scale starts at 0.17 instead of 0, so the difference between the bars is distorted. While there was a decrease, it appears that the decrease is roughly 10% of the 2006 rate.

11. (a) Answers will vary. Here is a time-series plot that a politician might use to support the position that health care is increasing.

(b) A graph that is misleading might use bars of unequal width or will use a vertical scale that does not start at $0. One example, as follows, is misleading because it starts at $1.25 instead of 0 without indicating a gap. This might cause the reader to conclude that cost of unleaded gasoline has risen more sharply than actually occurred.

(b) Answers will vary. Here is a time-series plot that the health care industry might use to refute the opinion of the politician; yes.

Chapter 2: Review Exercises 13. (a) A graph that is not misleading will use a vertical scale starting at 0% and bars of equal width. One example is:

16. Answers will vary. This is a histogram so the bars should touch. In addition, there are no labels and no title.

Chapter 2 Review Exercises 1. (a) There are 614 + 154 + 1448 = 2216 participants. (b) The relative frequency of the respondents indicating that it makes no difference is 1448 ≈ 0.653 2216 (b) This graphic is misleading because the vertical scale starts at 10% instead of 0% without indicating a gap. This might cause the reader to think that the proportion of overweight adults in the United States is increasing more quickly than it really is.

(d) Answers will vary. 14. (a) A bar graph (b) A reader cannot tell whether the graph ends at the top of the nipple on the baby bottle, or at the end of the milk.

2. (a) Total homicides = 844 + 149 + 69 + 162 = 1224 Relative frequency for firearms is 844/1224 = 0.6895, and so on.

Type of Weapon Firearms Knives or cutting instruments Personal weapons Other weapon

(b)

15. Answers will vary. Three-dimensional graphs are deceptive. The area for P (pitcher) looks substantially larger than the area for 3B (third base) even though both are the same percentage. Graphs should not be drawn using three dimensions. Instead, use a twodimensional graph.

Relative Frequency 0.6895 0.1217 0.0564 0.1324

Chapter 2: Summarizing Data in Tables and Graphs (c)

3. (a)

(c)

Total births (in thousands) = 2 + 194 + 765 + 1124 + 1092 + 555 + 115 + 8 + 1 = 3856 Relative frequency for 10–14 year old mothers = 2 3856 ≈ 0.0005, and so on. Cumulative frequency for 15–19 year old mothers = 2 + 194 = 196; for 20–24 year old mothers = 196 + 765 = 961, and so on. Cumulative relative frequency for 10–14 year old mothers = 2 3856 ≈ 0.0005; for 15–19 year old mothers = 196 3856 ≈ 0.0508, and so on. Age of Mother (yrs) 10–14

Freq.

Rel. Freq.

0.0005

15–19

194

0.0503

20–24

765

0.1984

25–29

1124

0.2915

30–34

1092

0.2832

35–39

555

0.1439

40–44

115

0.0298

45–49

0.0021

50–54 (b)

0.0003

(d) From the relative frequency table, the relative frequency of 20–24 is 0.1984, so the percentage is 19.84%. (e)

1092 + 555 + 115 + 8 + 1 1771 = ≈ 0.4593 3856 3856 45.93% of live births were to mothers aged 30 years or older.

4. (a), (b) Affiliation

Frequency

Democrat Independent Republican

46 16 38

(c)

The distribution is roughly symmetric and bell shaped.

Relative Frequency 0.46 0.16 0.38

Chapter 2: Review Exercises (d)

(f)

(e) Democrat appears to be the most common affiliation in Naperville.

(g) From the relative frequency table, the relative frequency of two children is 0.3000, so 30% of the couples have two children.

5. (a), (b), (c), and (d) Cumul. Family Rel. Cumul. Rel. Size Freq. Freq. Freq. Freq. 0 7 0.1167 7 0.1167 1 7 0.1167 14 0.2333 2 18 0.3000 32 0.5333 3 20 0.3333 52 0.8667 4 7 0.1167 59 0.9833 5 1 0.0167 60 1.0000

(h) From the frequency table, the relative frequency of at least two children (i.e. two or more) is 0.3000 + 0.3333 + 0.1167 + 0.0167 = 0.7667 or 76.67%. So, 76.67% of the couples have at least two children. (i)

(e) The distribution is more or less symmetric.

6. (a), (b) Homeownership Rate

Frequency

Relative Frequency

40–44.9

0.0196

45–49.9

50–54.9

0.0196

55–59.9

0.0588

60–64.9

0.2353

65–69.9

0.4118

70–74.9

0.2549

Chapter 2: Summarizing Data in Tables and Graphs (c)

(d)

(e) The distribution is skewed left. (f)

Homeownership Rate

Frequency

Relative Frequency

40–49.9

0.0196

50–59.9

0.0784

60–69.9

0.6471

70–79.9

0.2549

Chapter 2 Review Exercises

The distribution is skewed left. (g) Answers will vary. Both class widths give a good overall picture of the distribution. However, the class width of 5 appears to be the better summary. 7. (a), (b) Answers will vary. Using 2.2000 as the lower class limit of the first class and 0.0200 as the class width, we obtain the following. Class

Freq.

Rel. Freq.

2.2000 − 2.2199 2.2200 − 2.2399 2.2400 − 2.2599 2.2600 − 2.2799 2.2800 − 2.2999 2.3000 − 2.3199 2.3200 − 2.3399 2.3400 − 2.3599 2.3600 − 2.3799

2 3 5 6 4 7 5 1 1

0.0588 0.0882 0.1471 0.1765 0.1176 0.2059 0.1471 0.0294 0.0294

(c)

(d)

The distribution is slightly skewed right. 9. (a) Yes. Grade inflation seems to be happening in colleges. GPAs have increased every time period for all schools.

The distribution is roughly symmetric.

(b) GPAs increased about 5.1% for public schools. GPAs increased about 6.1% for private schools. Private schools have higher grade inflation because the GPAs are higher and they are increasing faster. (c) The graph is misleading because it starts at 2.6 on the vertical axis.

10. (a) Answers will vary. The adjusted gross income share of the top 1% of earners shows increases and decreases. The adjusted gross income share of the bottom 50% of earners shows steady decreases overall, with a few minor exceptions.

Chapter 2: Summarizing Data in Tables and Graphs (b) Answers will vary. The income tax share of the top 1% of earners shows steady increases overall, with few exceptions, including an overall decrease from 2007 to 2011 and from 2015 to 2016. The income tax share of the bottom 50% of earners has stayed relatively steady over time.

11. (a) Graphs will vary. One way to mislead would be to start the vertical scale at a value other than 0. For example, starting the vertical scale at $30,000 might make the reader believe that college graduates earn more than three times what a high school graduate earns (on average). (b) A graph that does not mislead would use equal widths for the bars and would start the vertical scale at $0. Here is an example of a graph that is not misleading:

12. (a) Flats are preferred the most (40%) and extra-high heels are preferred the least (1%). (b) The graph is misleading because the bar heights and areas for each category are not proportional.

Chapter 2 Test 1. (a) A 5 Star rating was the most popular rating with 1675 votes. (b) 35 + 67 +246 + 724 + 1675 = 2747 postings were posted on Yelp for Hot Doug’s restaurant.

(c) 1675 − 724 = 951 There were 951 more 5 Star ratings than 4 Star ratings. (d) There were 1675 5 Star ratings out of a 1675 ≈ 0.6098 total of 2747 ratings. 2747 Approximately 61% of all ratings were 5 Star ratings. (e) No, it is not appropriate to describe the shape of the distribution as skewed right. The data represented by the graph are qualitative, so the bars in the graph could be placed in any order.

Chapter 2 Test 2. (a) There were 1005 responses. The relative frequency who indicated they preferred 412 = 0.4100, and so on. new tolls was 1005 Response

Freq.

Rel. Freq.

New Tolls Inc. Gas Tax

412 181

0.4100 0.1801

No New Roads

412

0.4100

(b) The relative frequency is 0.1801, so the percentage of respondents who would like to see an increase in gas taxes is 18.01%. (c)

(e)

3. (a), (b) Rel. Education Attainment Freq. Freq. No high school 9 0.18 diploma High school graduate Some college Associate's degree Bachelor's degree

0.32

9 4 8

0.18 0.08 0.16

Advanced degree

0.08

(d)

Chapter 2: Summarizing Data in Tables and Graphs (c)

(d)

(e) The largest bar (and largest pie segment) corresponds to “High School Graduate,” so high school graduate is the most common educational level of a commuter. 4. (a), (b),

No. of Cars 1 2 3 4 5 6 7 8 9

Freq. 5 7 12 6 8 5 2 4 1

Rel. Freq. 0.10 0.14 0.24 0.12 0.16 0.10 0.04 0.08 0.02

(c)

The distribution is skewed right.