\fThis is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. This book is dedicated to the memory of a great teacher, mathematician, and friend Burton W. Jones Professor Emeritus, University of Colorado Understandable Statistics: Concepts and Methods, Tenth Edition Charles Henry Brase, Corrinne Pellillo Brase Editor in Chief: Michelle Julet Publisher: Richard Stratton Senior Sponsoring Editor: Molly Taylor Senior Editorial Assistant: Shaylin Walsh Media Editor: Andrew Coppola Marketing Manager: Ashley Pickering Marketing Communications Manager: Mary Anne Payumo Content Project Manager: Jill Clark 2012, 2009, 2006 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED. 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Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com. Printed in the United States of America 1 2 3 4 5 6 7 14 13 12 11 10 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 1 MLB Photos/Getty Images Sport/Getty Images 1.1 What Is Statistics? 1.2 Random Samples 1.3 Introduction to Experimental Design Chance favors the Bettmann/Corbis prepared mind. LOUIS PASTEUR Statistical techniques are tools of thought . . . not substitutes for thought. ABRAHAM KAPLAN For online student resources, visit the Brase/Brase, Understandable Statistics, 10th edition web site at http://www.cengage.com/statistics/brase Louis Pasteur (1822-1895) is the founder of modern bacteriology. At age 57, Pasteur was studying cholera. He accidentally left some bacillus culture unattended in his laboratory during the summer. In the fall, he injected laboratory animals with this bacilli. To his surprise, the animals did not diein fact, they thrived and were resistant to cholera. When the nal results were examined, it is said that Pasteur remained silent for a minute and then exclaimed, as if he had seen a vision, \"Don't you see they have been vaccinated!\" Pasteur's work ultimately saved many human lives. Most of the important decisions in life involve incomplete information. Such decisions often involve so many complicated factors that a complete analysis is not practical or even possible. We are often forced into the position of making a guess based on limited information. As the rst quote reminds us, our chances of success are greatly improved if we have a \"prepared mind.\" The statistical methods you will learn in this book will help you achieve a prepared mind for the study of many different elds. The second quote reminds us that statistics is an important tool, but it is not a replacement for an in-depth knowledge of the eld to which it is being applied. The authors of this book want you to understand and enjoy statistics. The reading material will tell you about the subject. The examples will show you how it works. To understand, however, you must get involved. Guided exercises, calculator and computer applications, section and chapter problems, and writing exercises are all designed to get you involved in the subject. As you grow in your understanding of statistics, we believe you will enjoy learning a subject that has a world full of interesting applications. 2 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Getting Started P R EVI EW QU ESTIONS (SECTION 1.1) What is the nature of data? (SECTION 1.1) How can you draw a random sample? What are other sampling techniques? (SECTION 1.2) (SECTION 1.2) How can you design ways to collect data? (SECTION 1.3) STUDIO SATO/amanaimages/Corbis Why is statistics important? FOCUS PROBLEM Where Have All the Fireflies Gone? Courtesy of Corrinne and Charles Brase A feature article in The Wall Street Journal discusses the disappearance of reies. In the article, Professor Sara Lewis of Tufts University and other scholars express concern about the decline in the worldwide population of reies. There are a number of possible explanations for the decline, including habitat reduction of woodlands, wetlands, and open elds; pesticides; and pollution. Articial nighttime lighting might interfere with the Morse-code-like mating ritual of the reies. Some chemical companies pay a bounty for reies because the insects contain two rare chemicals used in medical research and electronic detection systems used in spacecraft. What does any of this have to do with statistics? The truth, at this time, is that no one really knows (a) how much the world rey population has declined or (b) how to explain the decline. The population of all reies is simply too large to study in its entirety. In any study of reies, we must rely on incomplete information from samples. Furthermore, from these samples we must draw realistic conclusions that have statistical integrity. This is the kind of work that makes use of statistical methods to determine ways to collect, analyze, and investigate data. Suppose you are conducting a study to compare rey Adapted from Ohio State University Firey Files logo populations exposed to normal daylight/darkness conditions with rey populations exposed to continuous light (24 hours a day). You set up two rey colonies in a laboratory environment. The two colonies are identical except that one colony is exposed to normal daylight/darkness 3 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 4 Chapter 1 GETTING STARTED conditions and the other is exposed to continuous light. Each colony is populated with the same number of mature reies. After 72 hours, you count the number of living reies in each colony. After completing this chapter, you will be able to answer the following questions. (a) Is this an experiment or an observation study? Explain. (b) Is there a control group? Is there a treatment group? (c) What is the variable in this study? (d) What is the level of measurement (nominal, interval, ordinal, or ratio) of the variable? (See Problem 11 of the Chapter 1 Review Problems.) SECTION 1.1 What Is Statistics? FOCUS POINTS Identify variables in a statistical study. Distinguish between quantitative and qualitative variables. Identify populations and samples. Distinguish between parameters and statistics. Determine the level of measurement. Compare descriptive and inferential statistics. Introduction Decision making is an important aspect of our lives. We make decisions based on the information we have, our attitudes, and our values. Statistical methods help us examine information. Moreover, statistics can be used for making decisions when we are faced with uncertainties. For instance, if we wish to estimate the proportion of people who will have a severe reaction to a u shot without giving the shot to everyone who wants it, statistics provides appropriate methods. Statistical methods enable us to look at information from a small collection of people or items and make inferences about a larger collection of people or items. Procedures for analyzing data, together with rules of inference, are central topics in the study of statistics. Statistics Statistics is the study of how to collect, organize, analyze, and interpret numerical information from data. The statistical procedures you will learn in this book should supplement your built-in system of inferencethat is, the results from statistical procedures and good sense should dovetail. Of course, statistical methods themselves have no power to work miracles. These methods can help us make some decisions, but not all conceivable decisions. Remember, even a properly applied statistical procedure is no more accurate than the data, or facts, on which it is based. Finally, statistical results should be interpreted by one who understands not only the methods, but also the subject matter to which they have been applied. The general prerequisite for statistical decision making is the gathering of data. First, we need to identify the individuals or objects to be included in the study and the characteristics or features of the individuals that are of interest. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 1.1 Individuals Variable What Is Statistics? 5 Individuals are the people or objects included in the study. A variable is a characteristic of the individual to be measured or observed. For instance, if we want to do a study about the people who have climbed Mt. Everest, then the individuals in the study are all people who have actually made it to the summit. One variable might be the height of such individuals. Other variables might be age, weight, gender, nationality, income, and so on. Regardless of the variables we use, we would not include measurements or observations from people who have not climbed the mountain. The variables in a study may be quantitative or qualitative in nature. Quantitative variable Qualitative variable A quantitative variable has a value or numerical measurement for which operations such as addition or averaging make sense. A qualitative variable describes an individual by placing the individual into a category or group, such as male or female. For the Mt. Everest climbers, variables such as height, weight, age, or income are quantitative variables. Qualitative variables involve nonnumerical observations such as gender or nationality. Sometimes qualitative variables are referred to as categorical variables. Another important issue regarding data is their source. Do the data comprise information from all individuals of interest, or from just some of the individuals? Population data Sample data In population data, the data are from every individual of interest. In sample data, the data are from only some of the individuals of interest. It is important to know whether the data are population data or sample data. Data from a specic population are xed and complete. Data from a sample may vary from sample to sample and are not complete. Population parameter Sample statistic A population parameter is a numerical measure that describes an aspect of a population. A sample statistic is a numerical measure that describes an aspect of a sample. For instance, if we have data from all the individuals who have climbed Mt. Everest, then we have population data. The proportion of males in the population of all climbers who have conquered Mt. Everest is an example of a parameter. On the other hand, if our data come from just some of the climbers, we have sample data. The proportion of male climbers in the sample is an example of a statistic. Note that different samples may have different values for the proportion of male climbers. One of the important features of sample statistics is that they can vary from sample to sample, whereas population parameters are xed for a given population. LO O K I N G F O R WA R D In later chapters we will use information based on a sample and sample statistics to estimate population parameters (Chapter 7) or make decisions about the value of population parameters (Chapter 8). Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 6 Chapter 1 GETTING STARTED EX AM P LE 1 Using basic terminology The Hawaii Department of Tropical Agriculture is conducting a study of readyto-harvest pineapples in an experimental eld. (a) The pineapples are the objects (individuals) of the study. If the researchers are interested in the individual weights of pineapples in the eld, then the variable consists of weights. At this point, it is important to specify units of measurement and degrees of accuracy of measurement. The weights could be measured to the nearest ounce or gram. Weight is a quantitative variable because it is a numerical measure. If weights of all the ready-to-harvest pineapples in the eld are included in the data, then we have a population. The average weight of all ready-to-harvest pineapples in the eld is a parameter. Joe Solem/Riser/Getty Images (b) Suppose the researchers also want data on taste. A panel of tasters rates the pineapples according to the categories \"poor,\" \"acceptable,\" and \"good.\" Only some of the pineapples are included in the taste test. In this case, the variable is taste. This is a qualitative or categorical variable. Because only some of the pineapples in the eld are included in the study, we have a sample. The proportion of pineapples in the sample with a taste rating of \"good\" is a statistic. Throughout this text, you will encounter guided exercises embedded in the reading material. These exercises are included to give you an opportunity to work immediately with new ideas. The questions guide you through appropriate analysis. Cover the answers on the right side (an index card will t this purpose). After you have thought about or written down your own response, check the answers. If there are several parts to an exercise, check each part before you continue. You should be able to answer most of these exercise questions, but don't skip them they are important. GUIDED EXERCISE 1 Using basic terminology Television station QUE wants to know the proportion of TV owners in Virginia who watch the station's new program at least once a week. The station asks a group of 1000 TV owners in Virginia if they watch the program at least once a week. (a) Identify the individuals of the study and the variable. The individuals are the 1000 TV owners surveyed. The variable is the response does, or does not, watch the new program at least once a week. (b) Do the data comprise a sample? If so, what is the underlying population? The data comprise a sample of the population of responses from all TV owners in Virginia. (c) Is the variable qualitative or quantitative? Qualitativethe categories are the two possible responses, does or does not watch the program. (d) Identify a quantitative variable that might be of interest. Age or income might be of interest. (e) Is the proportion of viewers in the sample who watch the new program at least once a week a statistic or a parameter? Statisticthe proportion is computed from sample data. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 1.1 What Is Statistics? 7 Levels of Measurement: Nominal, Ordinal, Interval, Ratio We have categorized data as either qualitative or quantitative. Another way to classify data is according to one of the four levels of measurement. These levels indicate the type of arithmetic that is appropriate for the data, such as ordering, taking differences, or taking ratios. Levels of Measurement Levels of Measurement Nominal level The nominal level of measurement applies to data that consist of names, labels, or categories. There are no implied criteria by which the data can be ordered from smallest to largest. Ordinal level The ordinal level of measurement applies to data that can be arranged in order. However, differences between data values either cannot be determined or are meaningless. Interval level The interval level of measurement applies to data that can be arranged in order. In addition, differences between data values are meaningful. Ratio level The ratio level of measurement applies to data that can be arranged in order. In addition, both differences between data values and ratios of data values are meaningful. Data at the ratio level have a true zero. Korban Schwab/iStockphoto.com Michelle Dulieu, 2009/Used under license from Shutterstock.com EX AM P LE 2 Levels of measurement Identify the type of data. (a) Taos, Acoma, Zuni, and Cochiti are the names of four Native American pueblos from the population of names of all Native American pueblos in Arizona and New Mexico. SOLUTION: These data are at the nominal level. Notice that these data values are simply names. By looking at the name alone, we cannot determine if one name is \"greater than or less than\" another. Any ordering of the names would be numerically meaningless. (b) In a high school graduating class of 319 students, Jim ranked 25th, June ranked 19th, Walter ranked 10th, and Julia ranked 4th, where 1 is the highest rank. SOLUTION: These data are at the ordinal level. Ordering the data clearly makes sense. Walter ranked higher than June. Jim had the lowest rank, and Julia the highest. However, numerical differences in ranks do not have meaning. The difference between June's and Jim's ranks is 6, and this is the same difference that exists between Walter's and Julia's ranks. However, this difference doesn't really mean anything signicant. For instance, if you looked at grade point average, Walter and Julia may have had a large gap between their grade point averages, whereas June and Jim may have had closer grade point averages. In any ranking system, it is only the relative standing that matters. Differences between ranks are meaningless. (c) Body temperatures (in degrees Celsius) of trout in the Yellowstone River. SOLUTION: These data are at the interval level. We can certainly order the data, and we can compute meaningful differences. However, for Celsius-scale temperatures, there is not an inherent starting point. The value 0C may seem to be a starting point, but this value does not indicate the state of \"no heat.\" Furthermore, it is not correct to say that 20C is twice as hot as 10C. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 8 Chapter 1 GETTING STARTED (d) Length of trout swimming in the Yellowstone River. SOLUTION: These data are at the ratio level. An 18-inch trout is three times as long as a 6-inch trout. Observe that we can divide 6 into 18 to determine a meaningful ratio of trout lengths. In summary, there are four levels of measurement. The nominal level is considered the lowest, and in ascending order we have the ordinal, interval, and ratio levels. In general, calculations based on a particular level of measurement may not be appropriate for a lower level. P ROCEDU R E HOW TO DETERMINE THE LEVEL OF MEASUREMENT The levels of measurement, listed from lowest to highest, are nominal, ordinal, interval, and ratio. To determine the level of measurement of data, state the highest level that can be justied for the entire collection of data. Consider which calculations are suitable for the data. Level of Measurement Suitable Calculation Nominal We can order the data from smallest to largest or \"worst\" to \"best.\" Each data value can be compared with another data value. Interval We can order the data and also take the differences between data values. At this level, it makes sense to compare the differences between data values. For instance, we can say that one data value is 5 more than or 12 less than another data value. Ratio GUIDED EXERCISE 2 We can put the data into categories. Ordinal We can order the data, take differences, and also nd the ratio between data values. For instance, it makes sense to say that one data value is twice as large as another. Levels of measurement The following describe different data associated with a state senator. For each data entry, indicate the corresponding level of measurement. (a) The senator's name is Sam Wilson. Nominal level (b) The senator is 58 years old. Ratio level. Notice that age has a meaningful zero. It makes sense to give age ratios. For instance, Sam is twice as old as someone who is 29. (c) The years in which the senator was elected to the Senate are 1998, 2004, and 2010. Interval level. Dates can be ordered, and the difference between dates has meaning. For instance, 2004 is six years later than 1998. However, ratios do not make sense. The year 2000 is not twice as large as the year 1000. In addition, the year 0 does not mean \"no time.\" Continued Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 1.1 GUIDED EXERCISE 2 9 What Is Statistics? continued (d) The senator's total taxable income last year was $878,314. Ratio level. It makes sense to say that the senator's income is 10 times that of someone earning $87,831.40. (e) The senator surveyed his constituents regarding his proposed water protection bill. The choices for response were strong support, support, neutral, against, or strongly against. Ordinal level. The choices can be ordered, but there is no meaningful numerical difference between two choices. (f) The senator's marital status is \"married.\" Nominal level (g) A leading news magazine claims the senator is ranked seventh for his voting record on bills regarding public education. Ordinal level. Ranks can be ordered, but differences between ranks may vary in meaning. CR ITICAL TH I N KI NG \"Data! Data! Data!\" he cried impatiently. \"I can't make bricks without clay.\" Sherlock Holmes said these words in The Adventure of the Copper Beeches by Sir Arthur Conan Doyle. Reliable statistical conclusions require reliable data. This section has provided some of the vocabulary used in discussing data. As you read a statistical study or conduct one, pay attention to the nature of the data and the ways they were collected. When you select a variable to measure, be sure to specify the process and requirements for measurement. For example, if the variable is the weight of ready-to-harvest pineapples, specify the unit of weight, the accuracy of measurement, and maybe even the particular scale to be used. If some weights are in ounces and others in grams, the data are fairly useless. Another concern is whether or not your measurement instrument truly measures the variable. Just asking people if they know the geographic location of the island nation of Fiji may not provide accurate results. The answers may reect the fact that the respondents want you to think they are knowledgeable. Asking people to locate Fiji on a map may give more reliable results. The level of measurement is also an issue. You can put numbers into a calculator or computer and do all kinds of arithmetic. However, you need to judge whether the operations are meaningful. For ordinal data such as restaurant rankings, you can't conclude that a 4-star restaurant is \"twice as good\" as a 2-star restaurant, even though the number 4 is twice 2. Are the data from a sample, or do they comprise the entire population? Sample data can vary from one sample to another! This means that if you are studying the same statistic from two different samples of the same size, the data values may be different. In fact, the ways in which sample statistics vary among different samples of the same size will be the focus of our study from Section 6.4 on. Interpretation When you work with sample data, carefully consider the population from which they are drawn. Observations and analysis of the sample are applicable to only the population from which the sample is drawn. LO O K I N G F O R WA R D The purpose of collecting and analyzing data is to obtain information. Statistical methods provide us tools to obtain information from data. These methods break into two branches. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 10 Chapter 1 GETTING STARTED Descriptive statistics Inferential statistics Descriptive statistics involves methods of organizing, picturing, and summarizing information from samples or populations. Inferential statistics involves methods of using information from a sample to draw conclusions regarding the population. We will look at methods of descriptive statistics in Chapters 2, 3, and 9. These methods may be applied to data from samples or populations. Sometimes we do not have access to an entire population. At other times, the difculties or expense of working with the entire population is prohibitive. In such cases, we will use inferential statistics together with probability. These are the topics of Chapters 4 through 11. VI EWPOI NT The First Measured Century The 20th century saw measurements of aspects of American life that had never been systematically studied before. Social conditions involving crime, sex, food, fun, religion, and work were numerically investigated. The measurements and survey responses taken over the entire century reveal unsuspected statistical trends. The First Measured Century is a book by Caplow, Hicks, and Wattenberg. It is also a PBS documentary available on video. For more information, visit the Brase/Brase statistics site at http://www.cengage.com/statistics/brase and nd the link to the PBS First Measured Century documentary. SECTION 1.1 P ROB LEM S 1. Statistical Literacy What is the difference between an individual and a variable? 2. Statistical Literacy Are data at the nominal level of measurement quantitative or qualitative? 3. Statistical Literacy What is the difference between a parameter and a statistic? 4. Statistical Literacy For a set population, does a parameter ever change? If there are three different samples of the same size from a set population, is it possible to get three different values for the same statistic? 5. Critical Thinking Numbers are often assigned to data that are categorical in nature. (a) Consider these number assignments for category items describing electronic ways of expressing personal opinions: 1 Twitter; 2 e-mail; 3 text message; 4 Facebook; 5 blog Are these numerical assignments at the ordinal data level or higher? Explain. (b) Consider these number assignments for category items describing usefulness of customer service: 1 not helpful; 2 somewhat helpful; 3 very helpful; 4 extremely helpful Are these numerical assignments at the ordinal data level? Explain. What about at the interval level or higher? Explain. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 1.1 11 What Is Statistics? 6. Interpretation Lucy conducted a survey asking some of her friends to specify their favorite type of TV entertainment from the following list of choices: sitcom; reality; documentary; drama; cartoon; other Do Lucy's observations apply to all adults? Explain. From the description of the survey group, can we draw any conclusions regarding age of participants, gender of participants, or education level of participants? 7. Marketing: Fast Food A national survey asked 1261 U.S. adult fast-food customers which meal (breakfast, lunch, dinner, snack) they ordered. (a) Identify the variable. (b) Is the variable quantitative or qualitative? (c) What is the implied population? 8. Advertising: Auto Mileage What is the average miles per gallon (mpg) for all new cars? Using Consumer Reports, a random sample of 35 new cars gave an average of 21.1 mpg. (a) Identify the variable. (b) Is the variable quantitative or qualitative? (c) What is the implied population? 9. Ecology: Wetlands Government agencies carefully monitor water quality and its effect on wetlands (Reference: Environmental Protection Agency Wetland Report EPA 832-R-93-005). Of particular concern is the concentration of nitrogen in water draining from fertilized lands. Too much nitrogen can kill sh and wildlife. Twenty-eight samples of water were taken at random from a lake. The nitrogen concentration (milligrams of nitrogen per liter of water) was determined for each sample. (a) Identify the variable. (b) Is the variable quantitative or qualitative? (c) What is the implied population? 10. Archaeology: Ireland The archaeological site of Tara is more than 4000 years old. Tradition states that Tara was the seat of the high kings of Ireland. Because of its archaeological importance, Tara has received extensive study (Reference: Tara: An Archaeological Survey by Conor Newman, Royal Irish Academy, Dublin). Suppose an archaeologist wants to estimate the density of ferromagnetic artifacts in the Tara region. For this purpose, a random sample of 55 plots, each of size 100 square meters, is used. The number of ferromagnetic artifacts for each plot is determined. (a) Identify the variable. (b) Is the variable quantitative or qualitative? (c) What is the implied population? 11. Student Life: Levels of Measurement Categorize these measurements associated with student life according to level: nominal, ordinal, interval, or ratio. (a) Length of time to complete an exam (b) Time of rst class (c) Major eld of study (d) Course evaluation scale: poor, acceptable, good (e) Score on last exam (based on 100 possible points) (f) Age of student 12. Business: Levels of Measurement Categorize these measurements associated with a robotics company according to level: nominal, ordinal, interval, or ratio. (a) Salesperson's performance: below average, average, above average (b) Price of company's stock (c) Names of new products (d) Temperature (F) in CEO's private ofce (e) Gross income for each of the past 5 years (f) Color of product packaging Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 12 Chapter 1 GETTING STARTED 13. Fishing: Levels of Measurement Categorize these measurements associated with shing according to level: nominal, ordinal, interval, or ratio. (a) Species of sh caught: perch, bass, pike, trout (b) Cost of rod and reel (c) Time of return home (d) Guidebook rating of shing area: poor, fair, good (e) Number of sh caught (f) Temperature of water 14. Education: Teacher Evaluation If you were going to apply statistical methods to analyze teacher evaluations, which question form, A or B, would be better? Form A: In your own words, tell how this teacher compares with other teachers you have had. Form B: Use the following scale to rank your teacher as compared with other teachers you have had. 1 worst 2 below average 3 average 4 above average 5 best 15. Critical Thinking You are interested in the weights of backpacks students carry to class and decide to conduct a study using the backpacks carried by 30 students. (a) Give some instructions for weighing the backpacks. Include unit of measure, accuracy of measure, and type of scale. (b) Do you think each student asked will allow you to weigh his or her backpack? (c) Do you think telling students ahead of time that you are going to weigh their backpacks will make a difference in the weights? SECTION 1.2 Random Samples FOCUS POINTS Explain the importance of random samples. Construct a simple random sample using random numbers. Simulate a random process. Describe stratied sampling, cluster sampling, systematic sampling, multistage sampling, and convenience sampling. Simple Random Samples Eat lamb20,000 coyotes can't be wrong! This slogan is sometimes found on bumper stickers in the western United States. The slogan indicates the trouble that ranchers have experienced in protecting their ocks from predators. Based on their experience with this sample of the coyote population, the ranchers concluded that all coyotes are dangerous to their ocks and should be eliminated! The ranchers used a special poison bait to get rid of the coyotes. Not only was this poison distributed on ranch land, but with government cooperation, it also was distributed widely on public lands. The ranchers found that the results of the widespread poisoning were not very benecial. The sheep-eating coyotes continued to thrive while the general population of coyotes and other predators declined. What was the problem? The sheepeating coyotes that the ranchers had observed were not a representative sample of all coyotes. Modern methods of predator control, however, target the sheep-eating Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 1.2 13 Random Samples coyotes. To a certain extent, the new methods have come about through a closer examination of the sampling techniques used. In this section, we will examine several widely used sampling techniques. One of the most important sampling techniques is a simple random sample. Simple random sample A simple random sample of n measurements from a population is a subset of the population selected in such a manner that every sample of size n from the population has an equal chance of being selected. In a simple random sample, not only does every sample of the specied size have an equal chance of being selected, but every individual of the population also has an equal chance of being selected. However, the fact that each individual has an equal chance of being selected does not necessarily imply a simple random sample. Remember, for a simple random sample, every sample of the given size must also have an equal chance of being selected. GUIDED EXERCISE 3 Simple random sample Is open space around metropolitan areas important? Players of the Colorado Lottery might think so, since some of the proceeds of the game go to fund open space and outdoor recreational space. To play the game, you pay $1 and choose any six different numbers from the group of numbers 1 through 42. If your group of six numbers matches the winning group of six numbers selected by simple random sampling, then you are a winner of a grand prize of at least $1.5 million. (a) Is the number 25 as likely to be selected in the winning group of six numbers as the number 5? Yes. Because the winning numbers constitute a simple random sample, each number from 1 through 42 has an equal chance of being selected. (b) Could all the winning numbers be even? Yes, since six even numbers is one of the possible groups of six numbers. (c) Your friend always plays the numbers Yes. In a simple random sample, the listed group of six numbers is as likely as any of the 5,245,786 possible groups of six numbers to be selected as the winner. (See Section 4.3 to learn how to compute the number of possible groups of six numbers that can be selected from 42 numbers.) 1 2 3 4 5 6 Could she ever win? Random-number table How do we get random samples? Suppose you need to know if the emission systems of the latest shipment of Toyotas satisfy pollution-control standards. You want to pick a random sample of 30 cars from this shipment of 500 cars and test them. One way to pick a random sample is to number the cars 1 through 500. Write these numbers on cards, mix up the cards, and then draw 30 numbers. The sample will consist of the cars with the chosen numbers. If you mix the cards sufciently, this procedure produces a random sample. An easier way to select the numbers is to use a random-number table. You can make one yourself by writing the digits 0 through 9 on separate cards and mixing up these cards in a hat. Then draw a card, record the digit, return the card, and mix up the cards again. Draw another card, record the digit, and so on. Table 1 in the Appendix is a ready-made random-number table (adapted from Rand Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 14 Chapter 1 GETTING STARTED Corporation, A Million Random Digits with 100,000 Normal Deviates). Let's see how to pick our random sample of 30 Toyotas by using this random-number table. EX AM P LE 3 Random-number table Vibrant Image Studio, 2009/Used under license from Shutterstock.com Use a random-number table to pick a random sample of 30 cars from a population of 500 cars. SOLUTION: Again, we assign each car a different number between 1 and 500, inclusive. Then we use the random-number table to choose the sample. Table 1 in the Appendix has 50 rows and 10 blocks of ve digits each; it can be thought of as a solid mass of digits that has been broken up into rows and blocks for user convenience. You read the digits by beginning anywhere in the table. We dropped a pin on the table, and the head of the pin landed in row 15, block 5. We'll begin there and list all the digits in that row. If we need more digits, we'll move on to row 16, and so on. The digits we begin with are 99281 59640 15221 96079 09961 05371 Since the highest number assigned to a car is 500, and this number has three digits, we regroup our digits into blocks of 3: 992 815 964 015 221 960 790 996 105 371 To construct our random sample, we use the rst 30 car numbers we encounter in the random-number table when we start at row 15, block 5. We skip the rst three groups992, 815, and 964because these numbers are all too large. The next group of three digits is 015, which corresponds to 15. Car number 15 is the rst car included in our sample, and the next is car number 221. We skip the next three groups and then include car numbers 105 and 371. To get the rest of the cars in the sample, we continue to the next line and use the random-number table in the same fashion. If we encounter a number we've used before, we skip it. When we use the term (simple) random sample, we have very specic criteria in mind for selecting the sample. One proper method for selecting a simple random sample is to use a computer- or calculator-based random-number generator or a table of random numbers as we have done in the example. The term random should not be confused with haphazard! COMMENT LO O K I N G F O R WA R D The runs test for randomness discussed in Section 11.4 shows how to determine if two symbols are randomly mixed in an ordered list of symbols. P ROCEDU R E HOW TO DRAW A RANDOM SAMPLE 1. Number all members of the population sequentially. 2. Use a table, calculator, or computer to select random numbers from the numbers assigned to the population members. 3. Create the sample by using population members with numbers corresponding to those randomly selected. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 1.2 15 Random Samples LO O K I N G F O R WA R D Simple random samples are key components in methods of inferential statistics that we will study in Chapters 7-11. In fact, in order to draw conclusions about a population, the methods we will study require that we have simple random samples from the populations of interest. Another important use of random-number tables is in simulation. We use the word simulation to refer to the process of providing numerical imitations of \"real\" phenomena. Simulation methods have been productive in studying a diverse array of subjects such as nuclear reactors, cloud formation, cardiology (and medical science in general), highway design, production control, shipbuilding, airplane design, war games, economics, and electronics. A complete list would probably include something from every aspect of modern life. In Guided Exercise 4 we'll perform a brief simulation. A simulation is a numerical facsimile or representation of a real-world phenomenon. Simulation GUIDED EXERCISE 4 Simulation Use a random-number table to simulate the outcomes of tossing a balanced (that is, fair) penny 10 times. (a) How many outcomes are possible when you toss a coin once? Twoheads or tails (b) There are several ways to assign numbers to the two outcomes. Because we assume a fair coin, we can assign an even digit to the outcome \"heads\" and an odd digit to the outcome \"tails.\" Then, starting at block 3 of row 2 of Table 1 in the Appendix, list the rst 10 single digits. 7 1 5 4 9 4 4 8 4 3 (c) What are the outcomes associated with the 10 digits? T T T H T H H H H T (d) If you start in a different block and row of Table 1 in the Appendix, will you get the same sequence of outcomes? It is possible, but not very likely. (In Section 4.3 you will learn how to determine that there are 1024 possible sequences of outcomes for 10 tosses of a coin.) T E C H N OT E S Sampling with replacement Most statistical software packages, spreadsheet programs, and statistical calculators generate random numbers. In general, these devices sample with replacement. Sampling with replacement means that although a number is selected for the sample, it is not removed from the population. Therefore, the same number may be selected for the sample more than once. If you need to sample without replacement, generate more items than you need for the sample. Then sort the sample and remove duplicate values. Specific procedures for generating random samples using the TI-84Plus/TI-83Plus/TI-nspire (with TI-84 Plus keypad) calculator, Excel 2007, Minitab, and SPSS are shown in Using Technology at the end of this chapter. More details are given in the separate Technology Guides for each of these technologies. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 16 Chapter 1 GETTING STARTED Other Sampling Techniques Stratied sampling Systematic sampling Cluster sampling Multistage samples Convenience sampling Although we will assume throughout this text that (simple) random samples are used, other methods of sampling are also widely used. Appropriate statistical techniques exist for these sampling methods, but they are beyond the scope of this text. One of these sampling methods is called stratied sampling. Groups or classes inside a population that share a common characteristic are called strata (plural of stratum). For example, in the population of all undergraduate college students, some strata might be freshmen, sophomores, juniors, or seniors. Other strata might be men or women, in-state students or out-of-state students, and so on. In the method of stratied sampling, the population is divided into at least two distinct strata. Then a (simple) random sample of a certain size is drawn from each stratum, and the information obtained is carefully adjusted or weighted in all resulting calculations. The groups or strata are often sampled in proportion to their actual percentages of occurrence in the overall population. However, other (more sophisticated) ways to determine the optimal sample size in each stratum may give the best results. In general, statistical analysis and tests based on data obtained from stratied samples are somewhat different from techniques discussed in an introductory course in statistics. Such methods for stratied sampling will not be discussed in this text. Another popular method of sampling is called systematic sampling. In this method, it is assumed that the elements of the population are arranged in some natural sequential order. Then we select a (random) starting point and select every kth element for our sample. For example, people lining up to buy rock concert tickets are \"in order.\" To generate a systematic sample of these people (and ask questions regarding topics such as age, smoking habits, income level, etc.), we could include every fth person in line. The \"starting\" person is selected at random from the rst ve. The advantage of a systematic sample is that it is easy to get. However, there are dangers in using systematic sampling. When the population is repetitive or cyclic in nature, systematic sampling should not be used. For example, consider a fabric mill that produces dress material. Suppose the loom that produces the material makes a mistake every 17th yard, but we check only every 16th yard with an automated electronic scanner. In this case, a random starting point may or may not result in detection of fabric aws before a large amount of fabric is produced. Cluster sampling is a method used extensively by government agencies and certain private research organizations. In cluster sampling, we begin by dividing the demographic area into sections. Then we randomly select sections or clusters. Every member of the cluster is included in the sample. For example, in conducting a survey of school children in a large city, we could rst randomly select ve schools and then include all the children from each selected school. Often a population is very large or geographically spread out. In such cases, samples are constructed through a multistage sample design of several stages, with the nal stage consisting of clusters. For instance, the government Current Population Survey interviews about 60,000 households across the United States each month by means of a multistage sample design. For the Current Population Survey, the rst stage consists of selecting samples of large geographic areas that do not cross state lines. These areas are further broken down into smaller blocks, which are stratied according to ethnic and other factors. Stratied samples of the blocks are then taken. Finally, housing units in each chosen block are broken into clusters of nearby housing units. A random sample of these clusters of housing units is selected, and each household in the nal cluster is interviewed. Convenience sampling simply uses results or data that are conveniently and readily obtained. In some cases, this may be all that is available, and in many cases, it is better than no information at all. However, convenience sampling does run the risk of being severely biased. For instance, consider a newsperson who wishes to get Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 1.2 Random Samples 17 the \"opinions of the people\" about a proposed seat tax to be imposed on tickets to all sporting events. The revenues from the seat tax will then be used to support the local symphony. The newsperson stands in front of a concert hall and surveys the rst ve people exiting after a symphony performance who will cooperate. This method of choosing a sample will produce some opinions, and perhaps some human interest stories, but it certainly has bias. It is hoped that the city council will not use these opinions as the sole basis for a decision about the proposed tax. It is good advice to be very cautious indeed when the data come from the method of convenience sampling. Sampling Techniques Random sampling: Use a simple random sample from the entire population. Stratied sampling: Divide the entire population into distinct subgroups called strata. The strata are based on a specic characteristic such as age, income, education level, and so on. All members of a stratum share the specic characteristic. Draw random samples from each stratum. Systematic sampling: Number all members of the population sequentially. Then, from a starting point selected at random, include every kth member of the population in the sample. Cluster sampling: Divide the entire population into pre-existing segments or clusters. The clusters are often geographic. Make a random selection of clusters. Include every member of each selected cluster in the sample. Multistage sampling: Use a variety of sampling methods to create successively smaller groups at each stage. The nal sample consists of clusters. Convenience sampling: Create a sample by using data from population members that are readily available. CR ITICAL TH I N KI NG Sampling frame Undercoverage We call the list of individuals from which a sample is actually selected the sampling frame. Ideally, the sampling frame is the entire population. However, from a practical perspective, not all members of a population may be accessible. For instance, using a telephone directory as the sample frame for residential telephone contacts would not include unlisted numbers. When the sample frame does not match the population, we have what is called undercoverage. In demographic studies, undercoverage could result if the homeless, fugitives from the law, and so forth, are not included in the study. A sampling frame is a list of individuals from which a sample is actually selected. Undercoverage results from omitting population members from the sample frame. Sampling error In general, even when the sampling frame and the population match, a sample is not a perfect representation of a population. Therefore, information drawn from a sample may not exactly match corresponding information from the population. To the extent that sample information does not match the corresponding population information, we have an error, called a sampling error. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 18 Chapter 1 GETTING STARTED A sampling error is the difference between measurements from a sample and corresponding measurements from the respective population. It is caused by the fact that the sample does not perfectly represent the population. Nonsampling error A nonsampling error is the result of poor sample design, sloppy data collection, faulty measuring instruments, bias in questionnaires, and so on. Sampling errors do not represent mistakes! They are simply the consequences of using samples instead of populations. However, be alert to nonsampling errors, which may sometimes occur inadvertently. VI EWPOI NT Extraterrestrial Life? Do you believe intelligent life exists on other planets? Using methods of random sampling, a Fox News opinion poll found that about 54% of all U.S. men do believe in intelligent life on other planets, whereas only 47% of U.S. women believe there is such life. How could you conduct a random survey of students on your campus regarding belief in extraterrestrial life? SECTION 1.2 P ROB LEM S 1. Statistical Literacy Explain the difference between a stratied sample and a cluster sample. 2. Statistical Literacy Explain the difference between a simple random sample and a systematic sample. 3. Statistical Literacy Marcie conducted a study of the cost of breakfast cereal. She recorded the costs of several boxes of cereal. However, she neglected to take into account the number of servings in each box. Someone told her not to worry because she just had some sampling error. Comment on that advice. 4. Critical Thinking Consider the students in your statistics class as the population and suppose they are seated in four rows of 10 students each. To select a sample, you toss a coin. If it comes up heads, you use the 20 students sitting in the rst two rows as your sample. If it comes up tails, you use the 20 students sitting in the last two rows as your sample. (a) Does every student have an equal chance of being selected for the sample? Explain. (b) Is it possible to include students sitting in row 3 with students sitting in row 2 in your sample? Is your sample a simple random sample? Explain. (c) Describe a process you could use to get a simple random sample of size 20 from a class of size 40. 5. Critical Thinking Suppose you are assigned the number 1, and the other students in your statistics class call out consecutive numbers until each person in the class has his or her own number. Explain how you could get a random sample of four students from your statistics class. (a) Explain why the rst four students walking into the classroom would not necessarily form a random sample. (b) Explain why four students coming in late would not necessarily form a random sample. (c) Explain why four students sitting in the back row would not necessarily form a random sample. (d) Explain why the four tallest students would not necessarily form a random sample. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 1.2 19 Random Samples 6. Critical Thinking In each of the following situations, the sampling frame does not match the population, resulting in undercoverage. Give examples of population members that might have been omitted. (a) The population consists of all 250 students in your large statistics class. You plan to obtain a simple random sample of 30 students by using the sampling frame of students present next Monday. (b) The population consists of all 15-year-olds living in the attendance district of a local high school. You plan to obtain a simple random sample of 200 such residents by using the student roster of the high school as the sampling frame. 7. Sampling: Random Use a random-number table to generate a list of 10 random numbers between 1 and 99. Explain your work. 8. Sampling: Random Use a random-number table to generate a list of eight random numbers from 1 to 976. Explain your work. 9. Sampling: Random Use a random-number table to generate a list of six random numbers from 1 to 8615. Explain your work. 10. Simulation: Coin Toss Use a random-number table to simulate the outcomes of tossing a quarter 25 times. Assume that the quarter is balanced (i.e., fair). 11. Computer Simulation: Roll of a Die A die is a cube with dots on each face. The faces have 1, 2, 3, 4, 5, or 6 dots. The table below is a computer simulation (from the software package Minitab) of the results of rolling a fair die 20 times. DATA DISPLAY ROW C1 C2 1 2 5 3 2 2 C3 C4 C5 C6 C7 C8 C9 C10 2 4 2 5 5 4 3 5 2 3 3 5 1 3 4 4 (a) Assume that each number in the table corresponds to the number of dots on the upward face of the die. Is it appropriate that the same number appears more than once? Why? What is the outcome of the fourth roll? (b) If we simulate more rolls of the die, do you expect to get the same sequence of outcomes? Why or why not? 12. Simulation: Birthday Problem Suppose there are 30 people at a party. Do you think any two share the same birthday? Let's use the random-number table to simulate the birthdays of the 30 people at the party. Ignoring leap year, let's assume that the year has 365 days. Number the days, with 1 representing January 1, 2 representing January 2, and so forth, with 365 representing December 31. Draw a random sample of 30 days (with replacement). These days represent the birthdays of the people at the party. Were any two of the birthdays the same? Compare your results with those obtained by other students in the class. Would you expect the results to be the same or different? 13. Education: Test Construction Professor Gill is designing a multiple-choice test. There are to be 10 questions. Each question is to have ve choices for answers. The choices are to be designated by the letters a, b, c, d, and e. Professor Gill wishes to use a random-number table to determine which letter choice should correspond to the correct answer for a question. Using the number correspondence 1 for a, 2 for b, 3 for c, 4 for d, and 5 for e, use a random-number table to determine the letter choice for the correct answer for each of the 10 questions. 14. Education: Test Construction Professor Gill uses true-false questions. She wishes to place 20 such questions on the next test. To decide whether to place a true statement or a false statement in each of the 20 questions, she uses a random-number table. She selects 20 digits from the table. An even digit tells her to use a true statement. An odd digit tells her to use a false statement. Use a random-number table to pick a sequence of 20 digits, and describe the corresponding sequence of 20 true-false questions. What would the test key for your sequence look like? Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Chapter 1 GETTING STARTED 15. Sampling Methods: Benets Package An important part of employee compensation is a benets package, which might include health insurance, life insurance, child care, vacation days, retirement plan, parental leave, bonuses, etc. Suppose you want to conduct a survey of benets packages available in private businesses in Hawaii. You want a sample size of 100. Some sampling techniques are described below. Categorize each technique as simple random sample, stratied sample, systematic sample, cluster sample, or convenience sample. (a) Assign each business in the Island Business Directory a number, and then use a random-number table to select the businesses to be included in the sample. (b) Use postal ZIP Codes to divide the state into regions. Pick a random sample of 10 ZIP Code areas and then include all the businesses in each selected ZIP Code area. (c) Send a team of ve research assistants to Bishop Street in downtown Honolulu. Let each assistant select a block or building and interview an employee from each business found. Each researcher can have the rest of the day off after getting responses from 20 different businesses. (d) Use the Island Business Directory. Number all the businesses. Select a starti