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Solution Manual for Fundamentals of Statistics, 5th Edition

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    S OLUTIONS M ANUAL GEX P UBLISHING S ERVICES F UNDAMENTALS OF S TATISTICS : I NFORMED D ECISIONS U SING D ATA F IFTH E DITION Michael Sullivan, III Joliet Junior College

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    Table of Contents Preface Chapter 1 Data Collection 1.1 Introduction to the Practice of Statistics .............................................................................................................1 1.2 Observational Studies versus Designed Experiments .........................................................................................3 1.3 Simple Random Sampling ..................................................................................................................................6 1.4 Other Effective Sampling Methods.....................................................................................................................8 1.5 Bias in Sampling ...............................................................................................................................................10 1.6 The Design of Experiments ..............................................................................................................................13 Chapter 1 Review Exercises ........................................................................................................................................19 Chapter 1 Test..............................................................................................................................................................22 Chapter 2 Summarizing Data in Tables and Graphs 2.1 Organizing Qualitative Data .............................................................................................................................25 2.2 Organizing Quantitative Data ...........................................................................................................................33 2.3 Graphical Misrepresentations of Data...............................................................................................................47 Chapter 2 Review Exercises ........................................................................................................................................49 Chapter 2 Test..............................................................................................................................................................54 Chapter 3 Numerically Summarizing Data 3.1 Measures of Central Tendency .........................................................................................................................57 3.2 Measures of Dispersion.....................................................................................................................................64 3.3 Measures of Central Tendency and Dispersion from Grouped Data ................................................................80 3.4 Measures of Position and Outliers ....................................................................................................................89 3.5 The Five-Number Summary and Boxplots .......................................................................................................97 Chapter 3 Review Exercises ......................................................................................................................................106 Chapter 3 Test............................................................................................................................................................111 Chapter 4 Describing the Relation between Two Variables 4.1 Scatter Diagrams and Correlation ...................................................................................................................116 4.2 Least-Squares Regression ...............................................................................................................................133 4.3 The Coefficient of Determination ...................................................................................................................144 4.4 Contingency Tables and Association ..............................................................................................................148 Chapter 4 Review Exercises ......................................................................................................................................155 Chapter 4 Test............................................................................................................................................................160 Chapter 5 Probability 5.1 Probability Rules.............................................................................................................................................164 5.2 The Addition Rule and Complements.............................................................................................................170 5.3 Independence and the Multiplication Rule .....................................................................................................177 5.4 Conditional Probability and the General Multiplication Rule.........................................................................181 5.5 Counting Techniques ......................................................................................................................................187 5.6 Putting It Together: Which Method Do I Use? ...............................................................................................192 Chapter 5 Review Exercises ......................................................................................................................................194 Chapter 5 Test............................................................................................................................................................197

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    Chapter 6 Discrete Probability Distributions 6.1 Discrete Random Variables.............................................................................................................................200 6.2 The Binomial Probability Distribution............................................................................................................208 Chapter 6 Review Exercises ......................................................................................................................................222 Chapter 6 Test............................................................................................................................................................225 Chapter 7 The Normal Probability Distribution 7.1 Properties of the Normal Distribution .............................................................................................................228 7.2 Applications of the Normal Distribution.........................................................................................................231 7.3 Assessing Normality .......................................................................................................................................250 7.4 The Normal Approximation to the Binomial Probability Distribution ...........................................................253 Chapter 7 Review Exercises ......................................................................................................................................257 Chapter 7 Test............................................................................................................................................................262 Chapter 8 Sampling Distributions 8.1 Distribution of the Sample Mean ....................................................................................................................266 8.2 Distribution of the Sample Proportion ............................................................................................................278 Chapter 8 Review Exercises ......................................................................................................................................284 Chapter 8 Test............................................................................................................................................................287 Chapter 9 Estimating the Value of a Parameter 9.1 Estimating a Population Proportion ................................................................................................................289 9.2 Estimating a Population Mean ........................................................................................................................295 9.3 Putting It Together: Which Procedure Do I Use?............................................................................................305 Chapter 9 Review Exercises ......................................................................................................................................310 Chapter 9 Test............................................................................................................................................................314 Chapter 10 Hypothesis Tests Regarding a Parameter 10.1 The Language of Hypothesis Testing..............................................................................................................317 10.2 Hypothesis Tests for a Population Proportion.................................................................................................320 10.3 Hypothesis Tests for a Population Mean.........................................................................................................329 10.4 Putting It Together: Which Method Do I Use? ...............................................................................................338 Chapter 10 Review Exercises ....................................................................................................................................342 Chapter 10 Test..........................................................................................................................................................346 Chapter 11 Inferences on Two Samples 11.1 Inference about Two Population Proportions..................................................................................................348 11.2 Inference about Two Means: Dependent Samples ..........................................................................................361 11.3 Inference about Two Means: Independent Samples........................................................................................371 11.4 Putting It Together: Which Method Do I Use? ...............................................................................................385 Chapter 11 Review Exercises (Online) ......................................................................................................................398 Chapter 11 Test..........................................................................................................................................................404 Chapter 12 Additional Inferential Techniques 12.1 Goodness-of-Fit Test.......................................................................................................................................410 12.2 Tests for Independence and the Homogeneity of Proportions.........................................................................421 12.3 Testing the Significance of the Least-Squares Regression Model ..................................................................439 12.4 Confidence and Prediction Intervals ...............................................................................................................445 Chapter 12 Review Exercises ....................................................................................................................................451 Chapter 12 Test..........................................................................................................................................................458

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    Chapter 1 Data Collection Section 1.1 1. Statistics is the science of collecting, organizing, summarizing, and analyzing information in order to draw conclusions and answer questions. In addition, statistics is about providing a measure of confidence in any conclusions. 2. The population is the group to be studied as defined by the research objective. A sample is any subset of the population. 3. Individual 4. Descriptive; Inferential 5. Statistic; Parameter 6. Variables 7. 18% is a parameter because it describes a population (all of the governors). 8. 72% is a parameter because it describes a population (the entire class). 9. 32% is a statistic because it describes a sample (the high school students surveyed). 10. 9.6% is a statistic because it describes a sample (the youths surveyed). 11. 0.366 is a parameter because it describes a population (all of Ty Cobb’s at-bats). 12. 43.92 hours is a parameter because it describes a population (all the men who have walked on the moon). 13. 23% is a statistic because it describes a sample (the 6076 adults studied). 14. 44% is a statistic because it describes a sample (the 100 adults interviewed). 15. Qualitative 16. Quantitative 17. Quantitative 18. Qualitative 19. Quantitative 20. Quantitative 21. Qualitative 22. Qualitative 23. Discrete 24. Continuous 25. Continuous 26. Discrete 27. Continuous 28. Continuous 29. Discrete 30. Continuous 31. Nominal 32. Ordinal 33. Ratio 34. Interval 35. Ordinal 36. Nominal 37. Ratio 38. Interval 39. 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. 40. The population consists of all bottles of Coca- Cola 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. 41. 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. 42. 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. 43. 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. 44. The population consists of all full-time students enrolled at this large community college. The sample consists of the 128 full- time students who were surveyed by the administration.

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    2 Chapter 1: Data Collection 45. 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.) 46. 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, Roadster 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. 47. (a) The research objective is to determine if adolescents aged 18–21 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. 48. (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. 49. (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 31% to 39% of Americans believe the federal government wastes much of the money collected in taxes. 50. (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 1018 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. 51. 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.

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    Section 1.2: Observational Studies vs. Designed Experiments 3 52. (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. 53. (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. 54. 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. 55. The values of a discrete random variable result from counting. The values of a continuous random variable result from a measurement. 56. 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.) 57. 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. 58. 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. 59. 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. 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. 3. Confounding exists in a study when the effects of two or more explanatory variables are not separated. So any relation that appears to exist between a certain explanatory variable and the response variable may be due to some other variable or variables not accounted for in the study. A lurking variable is a variable not accounted for in a study, but one that affects the value of the response variable. 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.

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    4 Chapter 1: Data Collection 4. 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. 5. Cross-sectional studies collect information at a specific point in time (or over a very short period of time). Case-control studies are retrospective (they look back in time). Also, individuals that have a certain characteristic (such as cancer) in a case-control study are matched with those that do not have the characteristic. Case-control studies are typically superior to cross-sectional studies. They are relatively inexpensive, provide individual level data, and give longitudinal information not available in a cross-sectional study. 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.

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    Section 1.2: Observational Studies vs. Designed Experiments 5 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. 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 quantitative. 22. (a) The explanatory variable is web page design; qualitative (b) The response variables are time on site and amount spent. Both are quantitative. (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) 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.

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    6 Chapter 1: Data Collection (c) The response variable is whether the individual has lung cancer or not. This is a qualitative variable. (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. Section 1.3 1. The frame is a list of all the individuals in the population. 2. Simple random sampling occurs when every possible sample of size n has an equally likely chance of occurring. 3. Sampling without replacement means that no individual may be selected more than once as a member of the sample. 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. 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} (b) There is a 1 in 21 chance that the pair The United Nations and Amnesty International will be selected. 9. (a) Starting at row 5, column 22, using two- digit 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 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.

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    Section 1.3: Simple Random Sampling 7 (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 four- digit 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. 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, 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, Carter, Madison, Obama, Pierce, Buchanan, Ford, Clinton. (b) Repeating part (a) with a seed of 99, our sample would be L. B. Johnson, Truman, Pierce, Garfield, Obama, Grant, George H. Bush, T. Roosevelt. 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.

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    8 Chapter 1: Data Collection 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. 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 self- selected, which will frequently result in small portions of the population being overrepresented. 4. Cluster sample 5. Stratified sample 6. False. In a systematic random sample, every k th 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 8 th chip, starting with the 3 rd 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 5 th person attending the lecture, starting with the 3 rd person. 21. Stratified sampling. Shawn takes a sample of measurements during each of the four time intervals (strata). 22. Simple random sampling. Each club member has the same chance of being selected for the survey. 23. The numbers corresponding to the 20 clients selected are 16 , 16 25 41 + = , 41 25 66 + = , 66 25 91 + = , 91 25 116 + = , 141, 166, 191, 216, 241, 266, 291, 316, 341, 366, 391, 416, 441, 466, 491. 24. Since the number of clusters is more than 100, but less than 1000, we assign each cluster a three-digit label between 001 and 795. Starting at row 8, column 38 in Table I of Appendix A, and proceeding downward, the 10 clusters selected are numbered 763, 185, 377, 304, 626, 392, 315, 084, 565, and 508. Note that we discard 822 and 955 in reading the table because we have no clusters with these labels. We also discard the second occurrence of 377 because we cannot select the same cluster twice.

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    Section 1.4: Other Effective Sampling Methods 9 25. Answers will vary. To obtain the sample, number the Democrats 1 to 16 and obtain a simple random sample of size 2. Then number the Republicans 1 to 16 and obtain a simple random sample of size 2. Be sure to use a different starting point in Table I or a different seed for each stratum. For example, using a TI-84 Plus graphing calculator with a seed of 38 for the Democrats and 40 for the Republicans, the numbers selected would be 6, 9 for the Democrats and 14, 4 for the Republicans. If we had numbered the individuals down each column, the sample would consist of Haydra, Motola, Thompson, and Engler. 26. Answers will vary. To obtain the sample, number the managers 1 to 8 and obtain a simple random sample of size 2. Then number the employees 1 to 21 and obtain a simple random sample of size 4. Be sure to use a different starting point in Table I or a different seed for each stratum. For example, using a TI-84 Plus graphing calculator with a seed of 18 for the managers and 20 for the employees, the numbers selected would be 4, 1 for the managers and 20, 3, 11, 9 for the employees. If we had numbered the individuals down each column, the sample would consist of Lindsey, Carlisle, Weber, Bryant, Hall, and Gow. 27. (a) 4502 90.04 90 50 N n = = → ; Thus, 90 k = . (b) Randomly select a number between 1 and 90. Suppose that we select 15. Then the individuals to be surveyed will be the 15th, 105th, 195th, 285th, and so on up to the 4425th employee on the company list. 28. (a) 945035 7269.5 7269 130 N n = = → ; Thus, 7269 k = . (b) Randomly select a number between 1 and 7269. Suppose that we randomly select 2000. Then we will survey the individuals numbered 2000, 9269, 16,538, and so on up to the individual numbered 939,701. 29. Simple Random Sample: Number the students from 1 to 1280. Use a table of random digits or a random- number generator to randomly select 128 students to survey. Stratified Sample: Since class sizes are similar, we would want to randomly select 128 4 32 = students from each class to be included in the sample. Cluster Sample: Since classes are similar in size and makeup, we would want to randomly select 128 4 32 = classes and include all the students from those classes in the sample. 30. No. The clusters were not randomly selected. This would be considered convenience 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.

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    10 Chapter 1: Data Collection 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 over- or 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. Bias means that the results of the sample are not representative of the population. There are three types of bias: sampling bias, response bias, and nonresponse bias. Sampling bias is due to the use of a sample to describe a population. This includes bias due to convenience sampling. Response bias involves intentional or unintentional misinformation. This would include lying to a surveyor or entering responses incorrectly. Nonresponse bias results when individuals choose not to respond to questions or are unable to be reached. A census can suffer from response bias and nonresponse bias, but would not suffer from sampling bias. 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. 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.

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    Section 1.5: Bias in Sampling 11 (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 multi- lingual. 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. 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.

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    12 Chapter 1: Data Collection 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. For random digit dialing, the frame is anyone with a phone (whose number is not on a do- not-call registry). Even those with unlisted numbers can still be reached through this method. Any household without a phone, households on the do-not-call registry, and homeless individuals are excluded. This could result in sampling bias due to undercoverage if the excluded individuals differ in some way than those included in the frame. 24. Answers will vary. 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. Some nonsampling errors presented in the article as leading to incorrect exit polls were poorly trained interviewers, interviewer bias, and over representation of female voters. 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. (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 elicit 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.

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