(BQ) Part 2 book Essential statistics - Exploring the world through data has contents: Survey sampling and inference, hypothesis testing for population proportions, inferring population means, analyzing categorical variables and interpreting research.
www.downloadslide.com Survey Sampling and Inference 324 M07_GOUL1228_02_GE_C07.indd 324 03/09/16 4:24 pm www.downloadslide.com THEME If survey subjects are chosen randomly, then we can use their answers to infer how the entire population would answer We can also quantify how far off our estimate is likely to be S omewhere in your town or city, possibly at this very moment, people are participating in a survey Perhaps they are filling out a customer satisfaction card at a restaurant Maybe their television is automatically transmitting information about which show is being watched so that marketers can estimate how many people are viewing their ads They may even be text messaging in response to a television survey Most of you will receive at least one phone call from a survey company that will ask whether you are satisfied with local government services or plan to vote for one candidate over another The information gathered by these surveys is used to piece together, bit by bit, a picture of the larger world You’ve reached a pivotal point in the text In this chapter, the data summary techniques you learned in Chapters and 3, the probability you learned about in Chapter 5, and the Normal distribution, which you studied in Chapter 6, are all combined to enable us to generalize what we learn about a small sample to a larger group Politicians rely on surveys of 1000 voters not because they care how those 1000 individuals will vote Surveys are important to politicians only if they help them learn about all potential voters In this and later chapters, we study ways to understand and measure just how reliable this projection from sample to the larger world is Whenever we draw a conclusion about a large group based on observations of some parts of that group, we are making an inference Inferential reasoning lies at the foundation of science but is far from foolproof As the following case study illustrates, when we make an inference, we can never be absolutely certain of our conclusions But applying the methods introduced in this chapter ensures that if we collect data carefully, we can at least measure how certain or uncertain we are CASE STUDY Spring Break Fever: Just What the Doctors Ordered? In 2006, the American Medical Association (AMA) issued a press release (“Sex and intoxication among women more common on spring break according to AMA poll”) in which it concluded, among other things, that “eighty-three percent of the [female, college-attending] respondents agreed spring break trips involve more or heavier drinking than occurs on college campuses and 74 percent said spring break trips result in increased sexual activity.” This survey made big news, particularly since the authors of the study claimed these percentages reflected the opinions not only of the 644 women who responded to the survey but of all women who participated in spring break The AMA’s website claimed the results were based on “a nationwide random sample of 644 women who currently attend college The survey has a margin of error of + > -4 percentage points at the 95 percent level of confidence.” It all sounds very scientific, doesn’t it? However, some survey specialists were suspicious After Cliff Zukin, a specialist who was president of the American Association for Public Opinion Research, corresponded with the AMA, it changed its website posting to say the results were based not on a random sample, but instead on “a nationwide sample of 644 women M07_GOUL1228_02_GE_C07.indd 325 03/09/16 4:24 pm www.downloadslide.com who are part of an online survey panel [emphasis added].” “Margin of error” is no longer mentioned Disagreements over how to interpret these results show just how difficult inference is In this chapter you’ll see why the method used to collect data is so important to inference, and how we use probability, under the correct conditions, to calculate a margin of error to quantify our uncertainty At the end of the chapter, you’ll see why the AMA changed its report SECTION 7.1 Learning about the World through Surveys Surveys are probably the most often encountered application of statistics Most news shows, newspapers, and magazines report on surveys or polls several times a week— and during a major election, several times a day We can learn quite a bit through a survey if the survey is done correctly Survey Terminology A population is a group of objects or people we wish to study Usually, this group is large—say, the group of all U.S citizens, or all U.S citizens between the ages of 13 and 18, or all senior citizens However, it might be smaller, such as all phone calls made on your cell phone in January We wish to know the value of a parameter, a numerical value that characterizes some aspect of this population For example, political pollsters want to know what percentage of people say they will vote in the next election Drunk-driving opponents want to know what percentage of all teenagers with driver’s licenses have drunk alcohol while driving Designers of passenger airplanes want to know the mean length of passengers’ legs so that they can put the rows of seats as close together as possible without causing discomfort In this text we focus on two frequently used parameters: the mean of a population and the population proportion This chapter deals with population proportions If the population is relatively small, we can find the exact value of the parameter by conducting a census A census is a survey in which every member of the population is measured For example, if you wish to know the percentage of people in your classroom who are left-handed, you can perform a census The classroom is the population, and the parameter is the percentage of left-handers We sometimes try to take a census with a large population (such as the U.S Census), but such undertakings are too expensive for nongovernmental organizations and are filled with complications caused by trying to track down and count people who may not want to be found (For example, the U.S Census tends to undercount poor, urban-dwelling residents, as well as undocumented immigrants.) In fact, most populations we find interesting are too large for a census For this reason, we instead observe a smaller sample A sample is a collection of people or objects taken from the population of interest Once a sample is collected, we measure the characteristic we’re interested in A statistic is a numerical characteristic of a sample of data We use statistics to estimate parameters For instance, we might be interested in knowing what proportion of all registered voters will vote in the next national election The proportion of all registered voters who will vote in the next election is our parameter Our method to estimate this parameter is to survey a small sample The proportion of the sample who say they will vote in the next election is a statistic Statistics are sometimes called estimators, and the numbers that result are called estimates For example, our estimator is the proportion of people in a sample who say 326 M07_GOUL1228_02_GE_C07.indd 326 03/09/16 4:24 pm www.downloadslide.com 7.1 Learning about the World through Surveys CHAPTER 327 they will vote in the next election When we conduct this survey, we find, perhaps, that 0.75 of the sample say they will vote This number, 0.75, is our estimate KEY POINT A statistic is a number that is based on data and used to estimate the value of a characteristic of the population Thus it is sometimes called an estimator Statistical inference is the art and science of drawing conclusions about a population on the basis of observing only a small subset of that population Statistical inference always involves uncertainty, so an important component of this science is measuring our uncertainty EXAMPLE 1 Pew Poll: Age and the Internet In February 2014 (about the time of Valentine’s Day), the Pew Research Center surveyed 1428 adults in the United States who were married or in a committed partnership The survey found that 25% of cell phone owners felt that their spouse or partner was distracted by her or his cell phone when they were together QUESTIONS Identify the population and the sample What is the parameter of interest? What is the statistic? SOLUTION The population that the Pew Research Center wanted to study consists of all American adults who were married or in a committed partnership and owned a cell phone The sample, which was taken from the population consists of 1428 such people The parameter of interest is the percentage of all adults in the United States who were married or in a committed partnership and felt that their spouse or partner was distracted by her or his cell phone when they were together The statistic, which is the percentage of the sample who felt this way, is 25% TRY THIS! Exercise 7.1 An important difference between statistics and parameters is that statistics are knowable Any time we collect data, we can find the value of a statistic In Example 1, we know that 25% of those surveyed felt that their partner was distracted by the cell phone In contrast, a parameter is typically unknown We not know for certain the percentage of all people who felt this way about their partners The only way to find out would be to ask everyone, and we have neither the time nor the money to this Table 7.1 compares the known and the unknown in this situation Unknown Known Population All cell phone owners in a committed relationship Sample A small number of cell phone owners in a committed relationship Parameter Percentage of all cell-phone owners in a committed relationship who felt that their partner was distracted when they were together Statistic Percentage of the sample who felt their partner was distracted when they were together b TABLE 7.1 Some examples of unknown quantities we might wish to estimate, and their knowable counterparts Statisticians have developed notation for keeping track of parameters and statistics In general, Greek characters are used to represent population parameters For example, m (mu, pronounced “mew,” like the beginning of music) represents the mean of a M07_GOUL1228_02_GE_C07.indd 327 03/09/16 4:24 pm www.downloadslide.com 328 CHAPTER SURVEY SAMPLING AND INFERENCE population Also, s (sigma) represents the standard deviation of a population Statistics (estimates based on a sample) are represented by English letters: x (pronounced “x-bar”) is the mean of a sample, and s is the standard deviation of a sample, for instance One frequently encountered exception is the use of the letter p to represent the proportion of a population and pn (pronounced “p-hat”) to indicate the proportion of a sample Table 7.2 summarizes this notation You’ve seen most of these symbols before, but this table organizes them in a new way that is important for statistical inference c TABLE 7.2 Notation for some commonly used statistics and parameters Statistics (based on data) Parameters (typically unknown) Sample mean x (x-bar) Population mean m (mu) Sample standard deviation s Population standard deviation s (sigma) Sample variance s2 Population variance s2 Sample proportion pn (p-hat) Population proportion p What Could Possibly Go Wrong? The Problem of Bias Caution Bias Statistical bias is different from the everyday use of the term bias You might perhaps say a friend is biased if she has a strong opinion that affects her judgment In statistics, bias is a way of measuring the performance of a method over many different applications Unfortunately, it is far easier to conduct a bad survey than to conduct a good survey One of the many ways in which we can reach a wrong conclusion is to use a survey method that is biased A method is biased if it has a tendency to produce an untrue value Bias can enter a survey in three ways The first is through sampling bias, which results from taking a sample that is not representative of the population A second way is measurement bias, which comes from asking questions that not produce a true answer For example, if we ask people their income, they are likely to inflate the value In this case, we will get a positive (or “upward”) bias: Our estimate will tend to be too high Measurement bias occurs when measurements tend to record values larger (or smaller) than the true value The third way occurs because some statistics are naturally biased For example, if you use the statistic 10x to estimate the mean, you’ll typically get estimates that are ten times too big Therefore, even when no measuring or sampling bias is present, you must also take care to use an estimator that is not biased Measurement Bias In February 2010, the Albany Times Union newspaper reported on two recent surveys to determine the opinions of New York State residents on taxing soda (Crowley 2010) The Quinnipiac University Polling Institute asked, “There is a proposal for an ‘obesity tax’ or a ‘fat tax’ on non-diet sugary soft drinks Do you support or oppose such a measure?” Forty percent of respondents said they supported the tax Another firm, Kiley and Company, asked, “Please tell me whether you feel the state should take that step in order to help balance the budget, should seriously consider it, should consider it only as a last resort, or should definitely not consider taking that step: ‘Imposing a new 18 percent tax on sodas and other soft drinks containing sugar, which would also reduce childhood obesity.’” Fifty-eight percent supported the tax when asked this question One or both of these surveys have measurement bias A famous example occurred in 1993, when, on the basis of the results of a Roper Organization poll, many U.S newspapers published headlines similar to this one from the New York Times: “1 in in New Survey Express Some Doubt About the Holocaust” (April 20, 1993) Almost a year later, the New York Times reported that this alarmingly high percentage of alleged Holocaust doubters could be due to measurement error The actual question respondents were asked contained a double negative: “Does it seem possible, or does it seem impossible to you, that the Nazi extermination of the Jews never happened?” When Gallup repeated the poll but did not use a double negative, only 9% expressed doubts (New York Times 1994) M07_GOUL1228_02_GE_C07.indd 328 03/09/16 4:24 pm www.downloadslide.com 7.1╇ Learning about the World through Surveys CHAPTER 329 Sampling Bias╇ Writing good survey questions to reduce measurement bias is an art and a science This text, however, is more concerned with sampling bias, which occurs when the estimation method uses a sample that is not representative of the population (By “not representative” we mean that the sample is fundamentally different from the population.) Have you ever heard of Alfred Landon? Unless you’re a political science Â�student, you probably haven’t In 1936, Landon was the Republican candidate for U.S Â�president, running against Franklin Delano Roosevelt The Literary Digest, a popular news Â�magazine, conducted a survey with over 10 million respondents and predicted that Landon would easily win the election with 57% of the vote The fact that you probably haven’t heard of Landon suggests that he didn’t win, and in fact, he lost big, setting a record at the time for the fewest electoral votes ever received by a major-party candidate What went wrong? The Literary Digest had a biased sample The journal relied largely on polling its own readers, and its readers were more well-to-do than the general public and more likely to vote for a Republican The reputation of the Literary Digest was so damaged that two years later it disappeared and was absorbed into Time magazine The U.S presidential elections of 2004 and 2008 both had candidates who claimed to have captured the youth vote, and both times, candidates claimed the polls were biased The reason given was that the surveys used to estimate candidate support relied on landline phones, and many young voters don’t own landlines, relying instead on their cell phones Reminiscent of the 1936 Literary Digest poll, these surveys were potentially biased because their sample systematically excluded an important part of the population: those who did not use landlines (Cornish 2007) In fact, the Pew Foundation conducted a study after the 2010 congressional Â�elections This study found that polls that excluded cell phones had a sampling bias in favor of Republican candidates Today, the most commonly encountered biased surveys are probably Internet polls These can be found on many news organization websites (“Tea Party Influence in Washington, D.C is (a) on the rise (b) on the decline (c) unchanged?” www.foxnews com, February 2014.) Internet polls suffer from what is sometimes called response bias People tend to respond to such surveys only if they have strong feelings about the results; otherwise, why bother? This implies that the sample of respondents is not Â�necessarily representative of the population Even if the population in this case is, for example, all readers of the Foxnews.com website, the survey may not accurately reflect their views, because the voluntary nature of the survey means the sample will probably be biased This bias might be even worse if we took the population to be all U.S Â�residents Readers of Internet websites may very well not be representative of all U.S residents, and readers of particular websites such as Fox or CNN might be even less so To warn readers of this fact, most Internet polls have a disclaimer: “This is not a scientific poll.” What does this mean? It means we should not trust the information reported to tell us anything about anyone other than the people who responded to the poll (And remember, we can’t even trust the counts on an Internet poll, because sometimes nothing prevents people from voting many times.) KEY POINT When reading about a survey, it is important to know 1.╇ what percentage of people who were asked to participate actually did so 2.╇whether the researchers chose people to participate in the survey or people themselves chose to participate If a large percentage of those chosen to participate refused to answer questions, or if people themselves chose whether to participate, the conclusions of a survey are suspect M07_GOUL1228_02_GE_C07.indd 329 03/09/16 4:24 pm www.downloadslide.com 330 CHAPTER SURVEY SAMPLING AND INFERENCE Because of response bias, you should always question what type of people were included in a survey But the other side of this coin is that you should also question what type of people were left out Was the survey conducted at a time of day that meant that working people were less likely to participate? Were only landline phones used, thereby excluding people who had only cell phones? Was the question that was asked potentially embarrassing, so that people might have refused to answer? All of these circumstances can bias survey results Simple Random Sampling Saves the Day Caution Random If a sample is not random, there’s really nothing we can learn about the population We can’t measure the survey’s precision, and we can’t know how large the bias might be Details Simple random sampling is not the only valid method for statistical inference Statisticians collect representative samples using other methods, as well (for example, sampling with replacement) What these methods all have in common is that they take samples randomly Tech M07_GOUL1228_02_GE_C07.indd 330 How we collect a sample that has as little bias as possible and is representative of the population? Only one way works: to take a random sample As we explained in Chapter 5, statisticians have a precise definition of random A random sample does not mean that we stand on a street corner and stop whomever we like to ask them to participate in our survey (Statisticians call this a convenience sample, for obvious reasons.) A random sample must be taken in such a way that every person in our population is equally likely to be chosen A true random sample is difficult to achieve (And that’s a big understatement!) Pollsters have invented many clever ways of pulling this off, often with great success One basic method that’s easy to understand but somewhat difficult to put into practice is simple random sampling (SRS) In SRS, we draw subjects from the population at random and without replacement Without replacement means that once a subject is selected for a sample, that subject cannot be selected again This is like dealing cards from a deck Once a card is dealt for a hand, no one else can get the same card A result of this method is that every sample of the same fixed size is equally likely to be chosen As a result, we can produce unbiased estimations of the population parameters of interest and can measure the precision of our estimator In theory, we can take an SRS by assigning a number to each and every member of the population We then use a random number table or other random number generator to select our sample, ignoring numbers that appear twice EXAMPLE 2 Taking a Simple Random Sample Alberto, Justin, Michael, Audrey, Brandy, and Nicole are in a class QUESTION Select an SRS of three names from these six names SOLUTION First assign each person a number, as shown: Alberto Justin Michael Audrey Brandy Nicole Next, select three of these numbers without replacement Figure 7.1 shows how this is done in StatCrunch, and almost all statistical technologies let you this quite easily 03/09/16 4:24 pm www.downloadslide.com 7.1 Learning about the World through Surveys CHAPTER 331 b FIGURE 7.1 StatCrunch will randomly select, without replacement, three numbers from the six shown in the var1 column Using technology, we got these three numbers: 1, 2, and These correspond to Alberto, Justin, and Nicole If technology is not available, a random number table, such as the one provided in Appendix A, can be used Here are two lines from such a table: 77598 29511 98149 63991 31942 04684 69369 50814 You can start at any row or column you please Here, we choose to start at the upper left (shown in bold face) Next, read off digits from left to right, skipping digits that are not in our population Because no one has the number 7, skip this number, twice The first person selected is number 5: Brandy Then skip and and select number 2: Justin Skip and (because you already selected Brandy) and select number 1: Alberto CONCLUSION Using technology, we got a sample consisting of Alberto, Justin, and Nicole Using the random number table, we got a different sample: Brandy, Justin, and Alberto TRY THIS! EXAMPLE Exercise 7.11 3 Survey on Sexual Harassment A newspaper at a large college wants to determine whether sexual harassment is a problem on campus The paper takes a simple random sample of 1000 students and asks each person whether he or she has been a victim of sexual harassment on campus About 35% of those surveyed refuse to answer Of those who answer, 2% say they have been victims of sexual harassment M07_GOUL1228_02_GE_C07.indd 331 03/09/16 4:24 pm www.downloadslide.com 332 CHAPTER SURVEY SAMPLING AND INFERENCE QUESTION Give a reason why we should be cautious about using the 2% value as an estimate for the population percentage of those who have been victims of sexual harassment CONCLUSION There is a large percentage of students who did not respond Those who did not respond might be different from those who did, and if their answers had been included, the results could have been quite different When those surveyed refuse to respond, it can create a biased sample TRY THIS! Exercise 7.15 There are always some people who refuse to participate in a survey, but a good researcher will everything possible to keep the percentage of nonresponders as small as possible, to reduce this source of bias SECTION 7.2 Measuring the Quality of a Survey A frequent complaint about surveys is that a survey based on 1000 people can’t possibly tell us what the entire country is thinking This complaint raises interesting questions: How we judge whether our estimators are working? What separates a good estimation method from a bad? It’s difficult, if not impossible, to judge whether any particular survey is good or bad Sometimes we can find obvious sources of bias, but often we don’t know whether a survey has failed unless we later learn the true parameter value (This sometimes occurs in elections, when we learn that a survey must have had bias because it severely missed predicting the actual outcome.) Instead, statisticians evaluate the method used to estimate a parameter, not the outcome of a particular survey KEY POINT Statisticians evaluate the method used for a survey, not the outcome of a single survey Before we talk about how to judge surveys, imagine the following scenario: We are not taking just one survey of 1000 randomly selected people We are sending out an army of pollsters Each pollster surveys a random sample of 1000 people, and they all use the same method for collecting the sample Each pollster asks the same question and produces an estimate of the proportion of people in the population who would answer yes to the question When the pollsters return home, we get to see not just a single estimate (as happens in real life) but a great many estimates Because each estimate is based on a separate random collection of people, each one will differ slightly We expect some of these estimates to be closer to the mark than others just because of random variation What we really want to know is how the group did as a whole For this reason, we talk about evaluating estimation methods, not estimates M07_GOUL1228_02_GE_C07.indd 332 03/09/16 4:24 pm www.downloadslide.com 7.2 Measuring the Quality of a Survey An estimation method is a lot like a golfer To be a good golfer, we need to get the golf ball in the cup A good golfer is both accurate (tends to hit the ball near the cup) and precise (even when she misses, she doesn’t miss by very much.) It is possible to be precise and yet be inaccurate, as shown in Figure 7.2b Also, it is possible to aim in the right direction (be accurate) but be imprecise, as shown in Figure 7.2c (Naturally, some of us are bad at both, as shown in Figure 7.2d.) But the best golfers can both aim in the right direction and manage to be very consistent, which Figure 7.2a shows us (a) (b) (c) (d) CHAPTER 333 Caution Estimator and Estimates We often use the word estimator to mean the same thing as “estimation method.” An estimate, on the other hand, is a number produced by our estimation method b FIGURE 7.2 (a) Shots from a golfer with good aim and precision; the balls are tightly clustered and centered around the cup (b) Shots from a golfer with good precision but poor aim; the balls are close together but centered to the right of the cup (c) Shots from a golfer with good aim—the balls are centered around the cup—but bad precision (d) The worst-case scenario: bad precision and bad aim Think of the cup as the population parameter, and think of each golf ball as an estimate, a value of pn , that results from a different survey We want an estimation method that aims in the right direction Such a method will, on average, get the correct value of the population parameter We also need a precise method so that if we repeated the survey, we would arrive at nearly the same estimate The aim of our method, which the accuracy, is measured in terms of the bias The precision is measured by a number called the standard error Discussion of simulation studies in the next sections will help clarify how accuracy and precision are measured These simulation studies show how bias and standard error are used to quantify the uncertainty in our inference Using Simulations to Understand the Behavior of Estimators The three simulations that follow will help measure how well the sample proportion works as an estimator of the population proportion In the first simulation, imagine doing a survey of people in a very small population with only people You’ll see that the estimator of the population proportion is accurate (no bias) but, because of the small sample size, not terribly precise M07_GOUL1228_02_GE_C07.indd 333 03/09/16 4:24 pm www.downloadslide.com 571 APPENDIX C: ANSWERS TO ODD-NUMBERED EXERCISES 9.87 Step 1: H0: mbefore = mafter, Ha: mbefore mafter, where m is the population mean pulse rate (before and after coffee) Step 2: Paired t-test (repeated measures): assume conditions hold, a = 0.05 Step 3: t = 2.96 or - 2.96, p-value = 0.005 Step 4: Reject H0 Heart rates increase significantly after coffee (The average rate before coffee was 82.4, and the average rate after coffee was 87.5.) 9.89 The typical number of hours was a little higher for the boys, and the variation was almost the same xgirls = 9.8, xboys = 10.3, sgirls = 5.4 and sboys = 5.5 See the histograms paired t-test) reduces the variation, making the denominator of t smaller and so t is larger 9.97 The table shows the results The average of s2 in the table is 2.8889 (or about 2.89), and if you take the square root, you get about 1.6997 (or about 1.70), which is the value for sigma (s) given in the TI-84 output shown in the exercise This demonstrates that s2 is an unbiased estimator of s2, sigma squared Sample s s2 1, 0 Frequency Histogram of Girls 1, 0.7071 0.5 1, 2.8284 8.0 2, 0.7071 0.5 2, 0 2, 2.1213 4.5 5, 2.8284 8.0 5, 2.1213 4.5 5, 0 Sum 26.0 0 10 15 TV Hours per Week 20 Histogram of Boys Frequency 26>9 = 2.8889 9.99 Answers will vary CHAPTER 10 0 10 15 TV Hours per Week 20 Step 1: H0: mgirls = mboys, Ha: mgirls ∙ mboys, where m is the population mean number of TV viewing hours Step 2: Two-sample t-test: random samples, assume the sample sizes of 32 girls and 22 boys is large enough that slight non-Normality is not a problem, a = 0.05 Step 3: t = - 0.38 or 0.38, p-value = 0.706 Step 4: You cannot reject the null hypothesis There is not enough evidence to conclude that boys and girls differ in the typical hours of TV watched 9.91 a. The mean for the day shift was 6.67 hours of sleep, and the mean for the night shift was 5.95 hours, showing that the people on the day shift tended to get more sleep for these samples. b. Step 1: H0: mday = mnight, Ha: mday ∙ mnight, where m is the mean number of hours of sleep per night Step 2: Two-sample t-test: large samples and assume random, a = 0.05 Step 3: t = - 2.71, p-value = 0.0079, or 0.008 Step 4: Reject the null hypothesis The difference in means is significant. c. It would not capture 0, showing a significant difference in means, because that is what we found in part b 9.93 Step 1: H0: mDem = mRep, Ha: mDem mRep Step 2: Two-sample t-test: random and Normal, a = 0.05 Step 3: t = 1.57 or - 1.57, p-value = 0.072 Step 4: Do not reject H0 The means are not significantly different. b. Step 1: H0: mDem = mRep, Ha: mDem mRep Step 2: Two-sample t-test: random and Normal, a = 0.05 Step 3: t = 2.28 or - 2.28, p-value = 0.016 Step 4: Reject H0 The mean for the Democrats is significantly higher than the mean for the Republicans. c. With an increase in sample size (more data), the t-value increased and the p-value decreased, allowing us to reject the null hypothesis 9.95 a. Step 1: H0: m7@Eleven = mVons, Ha: m7@Eleven mVons Step 2: Paired t-test: assume random and Normal, a = 0.05 Step 3: t = 2.17, p-value = 0.033 Step 4: Reject H0 The mean price at 7-Eleven is significantly more than the mean price at Vons. b. Step 1: H0: m7@Eleven = mVons, Ha: m7@Eleven mVons Step 2: Two-sample t-test (although not appropriate): assume random and Normal, a = 0.05 Step 3: t = 0.57 or - 0.57, p-value = 0.289 Step 4: Do not reject H0 The mean price at 7-Eleven is not significantly more than the mean price at Vons when using the twosample t-test. c. We found a larger t-value and a smaller p-value for the appropriate paired t-test This occurs because finding the differences (for the Z03_GOUL1228_02_GE_APPC.indd 571 Answers may vary slightly due to type of technology or rounding Section 10.1 10.1 a. Proportions are used for categorical data. b. Chi-square tests are used for categorical data 10.3 Boys Girls Stuffed Toys 8 12 Mechanical Toys 14 11 The table may have a different orientation 10.5 Mean Salary: numerical and continuous Department: categorical 10.7 a. Educated Parents Uneducated Parents Total Studying 42 23 65 Not studying 14 8 22 Total 56 31 87 b. 65>87 = 74.7% c. 0.747126(56) = 41.84 d. Expected counts are shown in the table Educated Parents Uneducated Parents Total Studying 41.84 23.16 65 Not studying 14.16 7.84 22 Total 56 31 87 10.9 (42 - 41.82)2 (23 - 23.16)2 (14 - 14.16)2 (8 - 7.84)2 + + + 41.84 23.16 14.16 7.84 = 0.0006 + 0.0011 + 0.0018 + 0.0033 = 0.0068 X2 = From technology to avoid rounding: Chi-square = 0.007 05/09/16 3:48 pm www.downloadslide.com 572 APPENDIX C: ANSWERS TO ODD-NUMBERED EXERCISES Section 10.2 10.11 Independence: one sample 10.13 Homogeneity: random assignment (to four groups) 10.15 There is no need to draw an inference from the data as they cover the entire population of crude oil producing countries and not a sample The data are given in the form of rates (percentages), not frequencies (counts), and there is not enough information to convert these percentages to counts 10.17 The answers follow the guided steps Step 1: Ha: The variables Relationship Status and Obesity are not independent (are associated) Step 2: Chi-square test of independence (given) The smallest expected count is 108.54, which is much more than 5, a = 0.05 Step 3: X = 30.83, p-value 0.001 Step 4: Reject H0 There is a connection between obesity and marital status; they are not independent However, we should not generalize Causality? No, it is an observational study Percentage Obese: Dating, 81>440 = 18.4%; Cohabiting, 103>429 = 24.0%; Married, 147>424 = 34.7% 10.19 Step 1: H0: For men, watching violent TV is independent of abusiveness, Ha: For men, watching violent TV is not independent of abusiveness Step 2: Chi-square test of independence: one sample, the smallest expected count is 8.1 5, sample not random, a = 0.05 Step 3: Chisquare = 5.02, p-value = 0.025 Step 4: Reject H0: High TV violence as a child is associated with abusiveness as an adult in men, but don’t generalize to all males and don’t conclude causality 10.21 a. Independence: one sample with two variables. b. Step 1: H0: Gender and higher studies are independent Ha: Gender and higher studies are associated (not independent) Step 2: Chi-square test of independence: random sample, the smallest expected count is 69.1 5, α = 0.05 Step 3: Chi-square = 2.41, p-value = 0.300 Step 4: Reject H0 Gender and higher studies have been shown to be associated. c. The level of higher studies has been found to be significantly different for males and females 10.23 a. HS Grad rate for no preschool: 29/64, or 45.3% The preschool kids had a higher graduation rate. b. Step 1: H0: Graduation and preschool are independent, Ha: Graduation and preschool are not independent (they are associated) Step 2: Chi-square test of homogeneity: random assignment, not a random sample, the smallest expected count is 25.91 5, a = 0.05 Step 3: X = 4.67, p-value = 0.031 Step 4: Reject H0 Graduation and preschool are associated; causality, yes; generalization, no 10.25 a. For preschool, 50% graduated, and for no preschool, 21>39 = 53.8% graduated It is surprising to see that the boys who did not go to preschool had a bit higher graduation rate. b. Step 1: H0: For the boys, graduation and preschool are independent, Ha: For the boys, graduation and preschool are associated Step 2: Chi-square test for homogeneity: random assignment, not a random sample, the smallest expected count is 15.32 5, a = 0.05 Step 3: X = 0.10, p-value = 0.747 Step 4: Do not reject H0 For the boys, there is no evidence that attending preschool is associated with graduating from high school. c. The results not generalize to other groups of boys and girls, but what evidence we have suggests that although preschool might be effective for girls, it may not be for boys, at least with regard to graduation from high school 10.27 a. in the control group, 0.75(20) = 15 in the gastric bypass group, and 0.95(20) = 19 in the biliopancreatic diversion group were free from diabetes after two years b. Control Gastric Bilio Free 0 15 19 Not Free 20 5 1 c. Step 1: H0: Form of treatment and whether the patient becomes free from diabetes are independent, Ha: Form of treatment and whether the patient becomes free from diabetes are not independent Step 2: Chi-square test for homogeneity: random assignment, not random sample, smallest expected Z03_GOUL1228_02_GE_APPC.indd 572 count is 8.67 5, a = 0.05 Step 3: Chi-square = 40.86, p-value 0.001 Step 4: Reject H0 Treatment and result are not independent; they are associated The treatment causes the result, but not generalize 10.29 a. With no confederate, 6>18 (33.3%) followed the directions and took the stairs With a compliant confederate, 16>18 (88.9%) followed directions With a noncompliant confederate, 5>18 (27.8%) followed directions Thus the subjects tended to the same thing as the confederate. b. A p-value can never be larger than The p-value is about 2.7 times 10 to the negative fourth power, or 0.00027, which is less than 0.001. c. Step 1: H0: Treatment and compliance are independent, Ha: Treatment and compliance are not independent Step 2: Chi-square test of homogeneity: random assignment, not a random sample, expected counts are all 5, a = 0.05 Step 3: X = 16.44, p-value 0.001 Step 4: Reject H0 There is a significant effect; causality, yes; generalization, no This shows an association between treatment and behavior at the elevator 10.31 a. NA, 2∙20 = 10% frogs with improved jumping performance; TRS, 15∙20 = 5% frogs with improved jumping performance Thus, there is an improvement in the jumping performance of frogs that were kept in temperature-regulated structures b NA TRS Frogs with improved jumping performance 2 15 Frogs with no improvement in jumping performance 18 5 c. Step 1: H0: The jumping performance is not associated with temperature conditions Ha: The jumping performance is associated with temperature conditions Step 2: Chi-square test for homogeneity: random assignment, the smallest expected count is 8.5 > α = 0.05 Step 3: X2 = 17.29, p-value = 0.000 Step 4: Reject H0 Temperature affects jumping performance in frogs Section 10.3 10.33 a. No, we cannot generalize, because this was not a random sample b Yes, we can infer causality because of random assignment 10.35 a. You can generalize to other people admitted to this hospital who would have been assigned a double room because of the random sampling from that group. b. Yes, you can infer causality because of the random assignment 10.37 a. The treatment variable is whether the patient received EL or the placebo The response variable is whether the patient avoided the need for a platelet transfusion. b. It was a controlled experiment, as you can see from the random assignment. c. Yes, 72% of those on the drug avoided platelet transfusions, and that was better than the 19% on the placebo who avoided them. d. You can reject the hypothesis that the treatments and outcomes are independent It shows that EL had a significant effect. e. Yes, you can infer causality (EL reduces the chance ) because the study was randomized and placebo-controlled, and there was a significant effect 10.39 Because of the usage of the words “tend to exhibit,” it is suspected that the conclusion is likely to be the result of an observational study, from which causality cannot be inferred 10.41 No The study was probably observational, and we cannot infer causality from one observational study 10.43 Randomly assign about half the women to an iron supplement and half to a placebo You could flip a coin for each woman: Heads she gets the iron, and tails she gets the placebo Study and compare the death rates over one or two years 10.45 a. The treatment variable is the drug: mg of drug, 10 mg of drug, or placebo The response variable is whether the patient experienced a 20% improvement of symptoms in three months. b. It was a controlled experiment, as you can see by the random assignment. c. Yes, the rate of 20% improvement was highest at the higher level of drug and lowest for those who took the placebo. d. The small p-values show that the percentage of patients who improved with tofacitinib is higher than the percentage who improved with the placebo, and the difference in percentages did not occur by chance. e. Yes, you can conclude that the use of tofacitinib increases the 05/09/16 3:48 pm www.downloadslide.com APPENDIX C: ANSWERS TO ODD-NUMBERED EXERCISES chances of a 20% improvement in symptoms, because this is a well-designed experiment with random assignment The fact that the higher dose provided a larger chance of improvement adds to the evidence that the drug is effective 10.47 a. Take a nonrandom sample of students and randomly assign some to the reception and some to attend a “control group” meeting where they something else (such as learn the history of the college). b. Take a random sample of students and offer them the choice of attending the reception or attending a “control group” meeting where they something else (such as learn the history of the college). c. Take a random sample of students Then randomly assign some of the students in this sample to the reception and some to the “control group” meeting 10.49 The answers follow the steps shown in the Guided Exercises 1: Is the new drug better than a placebo with regard to worsening of asthma? 2: Yes, the new drug is significantly better 3: This was a controlled experiment because of the random assignment 4: Yes Mentioning that dupilumab therapy, as compared with placebo, was associated with fewer asthma exacerbations is acceptable, but a stronger statement inferring causality could have been made For example, “Dupilumab therapy caused fewer asthma exacerbations than the placebo.” 5: Because there was no random sampling from the population, we cannot generalize widely, and the results apply only to these patients 6: There was no mention of other articles 10.51 Ten percent of the tests would be wrong (assuming none of the groups were different from the others) Because 0.10 * 10 = 1, you would expect test out of 10 to appear significant just by chance Chapter Review Exercises 10.53 a. 31>65, or 47.7%, of those in the control group were arrested, and 8>58, or 13.8%, of those who attended preschool were arrested Thus there was a lower rate of arrest for those who went to preschool b Preschool No Preschool Arrest 8 31 No Arrest 50 34 Z03_GOUL1228_02_GE_APPC.indd 573 573 Step 1: H0: The treatment and arrest rate are independent, Ha: The treatment and arrest rate are associated Step 2: Chi-square for homogeneity: random assignment, not a random sample, the smallest expected count is 18.39 5, a = 0.05 Step 3: Chi-square = 16.27, p-value = 0.000055 (or 0.001) Step 4: We can reject the hypothesis of no association at the 0.05 level Don’t generalize We conclude that preschool attendance affects the arrest rate. c. Two-proportion z-test Step 1: H0: ppre = pnopre, Ha: ppre pnopre (p is the rate of arrest) Step 2: Two-proportion z-test: the smallest expected count is 18.39 10, a = 0.05 Step 3: z = 4.03, p-value = 0.000028 (or 0.001) Step 4: Reject the null hypothesis Preschool lowers the rate of arrest, but we cannot generalize. d. The z-test enables us to test the alternative hypothesis that preschool attendance lowers the risk of later arrest The Chi-square test allows for testing for some sort of association, but we can’t specify whether it is a positive or a negative association Note that the p-value for the one-sided hypothesis with the z-test is half the p-value for the two-sided hypothesis with the Chi-square test 10.55 The data are percentages (not counts), and we cannot convert them to counts without knowing the total number of live births each year 10.57 a. The death rate before the vaccine was 18.1 deaths per 100,000 children After the vaccine, the death rate fell to 11.8 deaths per 100,000 children The difference is 6.3 fewer deaths per 100,000 children after the vaccine was introduced The small p-value (less than 0.001) means we can reject the null hypothesis that the death rate was unchanged and conclude that the death rate decreased. b. Although there are many indications that the vaccine is effective, this was not a randomized study We cannot rule out the possibility that a confounding variable, not the vaccine, caused the decrease in death rates (For example, because the comparison was done using different years, a difference in weather might have contributed to the difference in disease rates.) 10.59 a. 200>347 = 57.6% of the surgery group died, and 247>348 = 71.0% of the watchful waiting group died Thus the surgery group did better (with regard to death) in the sample. b. 63>347 = 18.2% of the surgery group died from prostate cancer, and 99>348 = 28.4% of the watchful waiting group died from prostate cancer So again the surgery group did better with regard to death from prostate cancer. c. It was a controlled experiment, as you can see from the random assignment 05/09/16 3:48 pm www.downloadslide.com This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 24/11/14 5:26 PM www.downloadslide.com â•›Appendix D: Credits PHOTO CREDITS Cover Oksana Shufrych/Shutterstock Page vi (top) Mansour Bethoney/Pearson Education, Inc., (middle) Colleen Ryan, (bottom) Rebecca Wong; Page xviii CarpathianPrince/Shutterstock Page 254 Gjermund/Shutterstock; Page 257 (top left) Evgeny Karandaev/Shutterstock, (top right) Siri Stafford/Digital Vision/Getty Images, (center) Eric Isselee/Shutterstock, (bottom) Shutterstock Chapter Chapter Page 26 Mark III Photonics/Shutterstock; Page 27 Darren Baker/Fotolia; Page 30 (top) Chad Bontrager/Shutterstock, (bottom) NASA; Page 31 S Kuelcue/Shutterstock; Page 33 Tiler84/Fotolia; Page 36 Laborant/Shutterstock; Page 38 Art_zzz/Fotolia; Page 42 Netsuthep/Fotolia; Page 45 ImageryMajestic/Shutterstock; Page 46 Coprid/ Shutterstock; Page 47 Kesu/Shutterstock; Page 49 (background) Ase/Shutterstock, (binoculars) Evgeny Karandaev/Shutterstock, (right) Monkey Business/Shutterstock (Note: Background and binoculars photos also appear on pages 85, 142, 201, 257, 307, 361, 411, 476, and 527.) Chapter Page 60 Denis Vrublevski/Shutterstock; Page 61 Harvard College Library; Page 66 Junial Enterprises/Shutterstock; Page 70 Pashin Georgiy/Shutterstock; Page 71 Filmfoto/Shutterstock; Page 73 (top) George Doyle & Ciaran Griffin/Stockbyte/Getty Images, (bottom) Iofoto/Shutterstock; Page 74 Luis Louro/Shutterstock; Page 75 RedGreen/Shutterstock; Page 79 Stephen VanHorn/Shutterstock; Page 81 Cla78/ Shutterstock; Page 85 Alamy Chapter Page 106 Jiri Hera/Shutterstock; Page 107 Andy Dean/Fotolia; Page 110 Saurabh13/ Shutterstock; Page 111 Rafa Irusta/Shutterstock; Page 116 El Greco/Shutterstock; Page 120 Shutterstock; Page 122 OLJ Studio/Shutterstock; Page 126 BW Folsom/ Shutterstock; Page 129 Jon Delorey; Page 131 Jessmine/Shutterstock; Page 133 George Doyle & Ciaran Griffin/Stockbyte/Getty Images; Page 134 Arka38/ Shutterstock; Page 137 Ilona Ignatova/Shutterstock; Page 142 Pearson Education, Inc Page 272 Joyfull/Shutterstock; Page 273 Jupiterimages/Photos.com/Getty Images; Page 275 Studiotouch/Shutterstock; Page 277 Shutterstock; Page 279 Alaettin Yildirim/ Shutterstock; Page 284 Frank Greenaway/Dorling Kindersley Limited; Page 289 Uros Jonic/Shutterstock; Page 293 Pearson Education, Inc.; Page 296 Micha Rosenwirth/ Shutterstock; Page 298 Nixx Photography/Shutterstock; Page 304 Alekss/Fotolia; Page 305 PaulPaladin/Shutterstock; Page 307 Karin Hildebrand Lau/Shutterstock Chapter Page 324 Chad McDermott/Shutterstock; Page 325 Elena Yakusheva/Shutterstock; Page 327 Rido/Fotolia; Page 332 Orla/Shutterstock; Page 333 (all) Pearson Education, Inc.; Page 338 Pearson Education, Inc.; Page 340 Pashin Georgiy/ Shutterstock; Page 345 Paulo Williams/Shutterstock; Page 347 Seregam/Shutterstock; Page 353 Imagewell/Shutterstock; Page 357 Whatafoto/Shutterstock; Page 360 Xuejun Ii/Fotolia; Page 361 Justasc/Shutterstock Chapter Page 378 Koi88/Fotolia; Page 379 Robert Daly/OJO Images/Getty Images; Page 380 Yellowj/Fotolia; Page 382 Mipan/Fotolia; Page 383 Helder Almeida/Shutterstock; Page 386 Raywoo/Fotolia; Page 387 Mircea Maties/Shutterstock; Page 401 Julian Rovagnati/Shutterstock; Page 399 Shotgun/Shutterstock; Page 402 Ajt/Shutterstock; Page 406 Jan Kaliciak/Shutterstock; Page 408 Gabriele Maltinti/Fotolia; Page 411 Nathan B Dappen/Shutterstock Chapter Page 166 Steve Rosset/Shutterstock; Page 167 Hempuli/Shutterstock; Page 172 Birute Vijeikiene/Shutterstock; Page 183 Joingate/Shutterstock; Page 188 Creations/ Shutterstock; Page 190 R Gino Santa Maria/Shutterstock; Page 191 Vixit/Shutterstock; Page 201 PCN Photography/Alamy Page 428 Todd Klassy/Shutterstock; Page 429 Shutterstock; Page 433 Mmaxer/ Shutterstock; Page 435 Rihardzz/Shutterstock; Page 438 Sashkin/Shutterstock; Page 444 Elnur/Shutterstock; Page 447 John Baran/Alamy; Page 448 Africa Studio/ Shutterstock; Page 451 Martin Allinger/Shutterstock; Page 462 Africa Studio/ Shutterstock; Page 467 4matic/Fotolia; Page 469 Bajinda/Fotolia; Page 474 Valentyn Volkov/Shutterstock; Page 476 AVAVA/Shutterstock Chapter Chapter 10 Chapter Page 228 Brian Jackson/Fotolia; Page 231 Bilder/Shutterstock; Page 235 Pearson Education, Inc.; Page 237 Pearson Education, Inc.; Page 239 Pearson Education, Inc.; Page 241 (left) Koncz/Shutterstock, (right) HomeStudio/Shutterstock; Page 244 Rannev/ Shutterstock; Page 250 Shutterstock; Page 251 Guzel Studio/Shutterstock; Page 500 Ljansempoi/Shutterstock; Page 501 Chris Curtis/Shutterstock; Page 505 Robertopalace/Shutterstock; Page 507 Pakhnyushcha/Shutterstock; Page 510 Oleksiy Mark/Fotolia; Page 512 Shutterstock; Page 514 Science Magazine; Page 519 Jim Barber/ Shutterstock; Page 522 Sunabesyou/Shutterstock; Page 527 Jennifer Nickert/Shutterstock TEXT CREDITS Texas Instruments graphing calculator screenshots courtesy of Texas Instruments Data and screenshots from StatCrunch used by permission of StatCrunch Screenshots from Minitab courtesy of Minitab Corporation Screenshot from SOCR used by the permission of the Statistics Online Computational Resource, UCLA XLSTAT screenshots courtesy of Addinsoft, Inc Used with permission All Rights Reserved XLSTAT is a registered trademark of Addinsoft SARL Chapter Pages 54–55 Excerpt from U.S HOSPITALIZATIONS FOR PNEUMONIA AFTER A DECADE OF PNEUMOCOCCAL VACCINATION by M.R Griffin et al The New England Journal of Medicine, July 11, 2013 Chapter Page 83 Screenshot of President Obama’s 2013 State of the Union speech Copyright ©Brad Borevitz Used with permission Chapter Page 328 Excerpt from “1 in New Survey Express Some Doubt About the Holocaust,” New York Times, April 20, 1993; Page 361 Based on an activity from Workshop Statistics, © 2004, Dr Allan Rossman and Dr Beth Chance, California Polytechnic State University Chapter Chapter 10 Page 520 Excerpt from BOCEPREVIR FOR UNTREATED CHRONIC HCV GENOTYPE INFECTION by Poordad et al Copyright © 2011 Used by permission of The New England Journal of Medicine; Page 521 Excerpt from TEN-YEAR EFFECTS OF THE ADVANCED COGNITIVE TRAINING FOR INDEPENDENT AND VITAL ELDERLY COGNITIVE TRAINING TRIAL ON COGNITION AND EVERYDAY FUNCTIONING IN OLDER ADULTS by Rebok et al Journal of American Geriatics Society Copyright © 2014 Used by the permission of Journal of American Geriatrics Society; Page 535 Excerpt from ELTROMBOPAG BEFORE PROCEDURES IN PATIENTS WITH CIRRHOSIS AND THROMBOCYTOPENIA by Nezam Afdhal et al The New England Journal of Medicine Copyright © 2012 Published by The New England Journal of Medicine; Page 535 Excerpt from EFFECT OF INHALED GLUCOCORTICOIDS IN CHILDHOOD ON ADULT HEIGHT by L H William Kelly et al The New England Journal of Medicine Copyright © 2012 Published by The New England Journal of Medicine; Page 538 Prostate Cancer Treatment from RADICAL PROSTATECTOMY OR WATCHFUL WAITING IN EARLY PROSTATE CANCER by Anna Bill Axelson Published by The New England Journal of Medicine; Page 538 Exercise 10.60: LENALIDOMIDE AFTER STEMCELL TRANSPLANTATION FOR MULTIPLE MYELOMA, by P L McCarthy et al The New England Journal of Medicine Copyright © 2012 Published by The New England Journal of Medicine Page 381 Definition of hypothesis: Used by permission of Merriam Webster 575 Z04_GOUL1228_02_GE_APPD.indd 575 09/09/16 4:13 pm www.downloadslide.com This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 24/11/14 5:26 PM www.downloadslide.com Index A C Abstracts, 521–522 Aggregate data, regressions of, 195–196 Alpha (a), 458 Alternative hypotheses, 387–388, 397–398, 409–410 one- and two-sided, 388–389, 460–463 AND combining events with, 237–238 “given that” vs., 243–246 Anecdotes, 39–40 Asimov, Isaac, 273 Associated events, 246 Associations, 40–42 in categorical variables See Categorical variables linear, limitation of linear regression to, 193–194 positive, 168 strength of, 168, 169–170 See also Correlation coefficient Average See Mean(s) Calculators See SOCR calculator; TI-84 calculator Carter, Jimmy, 84 Casino dice, 235 Categorical data coding with numbers, 31–32 organizing, 34–38 Categorical distributions, 75–81 bar charts and, 75–77 describing, 80–81 mode of, 78–79 pie charts and, 77–78 variability of, 79–80 Categorical variables, 30, 31, 242–251 chi-square test and See Chi-square statistic; Chi-square tests conditional probabilities and, 243–246 distributions of See Categorical distributions hypothesis testing with See Hypothesis testing with categorical variables independent and dependent events and, 246–248 intuition about independence and, 248–249 sequences of independent and associated events and, 249–251 Causality (cause-and-effect relationships), 404 data collection to understand, 39–47 Causation, correlation vs., 175, 194–195 Censuses, 332 Center measures of See Mean(s); Measures of center; Median; Mode(s) of numerical distributions, 67–68, 72–73 Central Limit Theorem (CLT) checking conditions for, 348–349, 363–364 lack of universality of, 444 for proportions, 440 for sample means, 440–447 for sample proportions, 346–353 using, 349–353 Chang, Jack, 39 Chi-square distribution (X2), 508 Chi-square statistic, 505–509 finding p-value for, 508–509 symbol for (X 2), 508 Chi-square tests for associations between categorical variables, 509–518 for independence and homogeneity, 510–514, 515 B Bar graphs (bar charts), 75–77 histograms vs., 76 Bell-shaped distributions, 68 Bias, 43 definition of, 342, 437 finding, 345–346 measurement, 334 nonresponse, 337–338 response, 335 sampling, 334, 335–336 simple random sampling to avoid, 336–338 in surveys, 334–336 Bimodal distributions, 69, 72, 133 Binomial probabilities, 296–300 cumulative, 300 definition of, 296 finding by hand, 301–302 Binomial probability model, 292–305 application of, 305 finding binomial probabilities and, 296–300 shape of, 302–304 visualizing, 294–296 Blinding, 43–44 Boxplots, 135–139, 465 comparing distributions using, 138 definition of, 135 five-number summary and, 139 horizontal vs vertical, 138 limitations of, 139 potential outliers and, 135–138 random samples and randomized assignment and, 514 tests of proportions related to, 515–518 Clinical significance, 524–525 Clinton, Hillary, 338 CLT See Central Limit Theorem (CLT) Coefficient of determination (r 2), 197–199 Comparing population means, 463–477 confidence levels of differences and, 466–468 dependent (paired) samples and, 464, 473–477 hypothesis testing about mean differences and, 468–473 independent samples and, 464–466 Comparing population proportions, 360–366 random assignment vs random sampling and, 365–366 two, confidence intervals for, 361–365 Comparison(s) of distributions, use of same measures of center and spread for, 134 of distributions, using boxplots, 138 Comparison group, 39 Complements, 234 Computers See also Technology; specific software downloading data into TI-84 calculator from, 102–103 Conditional probabilities, 243–244 finding, 245 flipping the condition and, 245–246 “given that” vs AND and, 243–244 Confidence intervals calculating, 452–457 checking conditions and, 448–450 comparing population proportions with See Comparing population proportions definition of, 353 estimating difference of means with, 465–468 estimating population mean with, 448–457 estimating population proportion with, 353–359 finding when p is not known, 356–357 hypothesis tests and, 408–409, 479–480 interpreting, 358–359, 450 margin of error and, 354, 355–356 for mean of a difference with dependent samples, 473–475 measuring performance with, 450–452 probabilities vs., 451 for proportions, 408 reporting and reading, 456 577 Z05_GOUL1228_02_GE_IDX.indd 577 09/09/16 4:15 pm www.downloadslide.com 578 INDEX Confidence intervals, (continued ) setting confidence level and, 354–355 structure of, 452 for two population proportions, 361–365, 366 understanding, 456–457 usefulness of, 448 Confidence levels, 448 definition of, 354 setting, 354–355 Confounding variables (factors), 41 Constants, correlation coefficient and, 178–179 Context, of data, 33–34 Continuous outcomes, 274 finding probabilities for continuousvalued outcomes and, 278–279 representation as areas under curves, 277–278 Control group, 39 Controlled experiments, 42, 365 extending results of, 45 Convenience samples, 336 Correlation, causation vs., 175, 194–195 Correlation coefficient in context, 174–175 definition of, 172 finding, 175–177 understanding, 177–180 visualizing, 173–174 D Data, 26–50 categorical, organizing, 34–38 classifying, 30–32 collecting See Data collection context of, 33–34 definition of, 28 evaluating, 45–47 stacked and unstacked, 32 storing, 32–33 usage of term, 28 Data analysis, 29 Data collection, 39–47, 514 anecdotes and, 39–40 blinding and, 43–44 controlled experiments for, 42 extending results and, 45 observational studies for, 40–42 placebos and, 44–45 random assignment and, 43 sample size and, 42 statistics in the news and, 45–47 Data dredging, 523 Data sets, 30 larger, calculating mean for, 111–113 small, calculating mean for, 110–111 Degrees of freedom (df) for chi-square distribution, 508 for t-distribution, 446, 447 Z05_GOUL1228_02_GE_IDX.indd 578 Dependent events, 246 Dependent samples, 464, 473–477 confidence intervals for mean of a difference with, 473–475 paired t-test with, 477 test of two means and, 475–477 Dependent variable (y-variable), 187 df See Degrees of freedom (df) Dice, casino, 235 Discrete outcomes, 274–277 equations as, 276–277 tables or graphs as, 275–277 Distributions bell-shaped, 68 bimodal, 69, 72 categorical See Categorical distributions comparing, use of same measures of center and spread for, 134 comparing using boxplots, 138 left-skewed, 69 multimodal, 69, 72 numerical See Numerical distributions population, 444 right-skewed, 69 of a sample, 62, 274, 443, 444–445 of sample means, visualizing, 442–443 sampling, 438–439, 445 symmetric See Symmetric distributions t-, 446–447 unimodal, 69, 279 Dotplots, 63–64 Double-blind studies, 44 E Empirical probabilities, 232–233, 252–256 coin flip simulation and, 252–254 Law of Large Numbers and, 254–256 steps for a simulation and, 252 Empirical Rule, 118–120, 290, 351 Normal model and, 290–291 Equations, discrete distributions as, 276–277 Estimates, 339 definition of, 332 of difference of means, with confidence intervals, 465–468 of population mean, with confidence intervals, 448–457 of population proportion with confidence intervals, 353–359 Estimators, 339 definition of, 332–333 sample mean as, 440–441 sample size and, 344–345 simulations for understanding behavior of, 339–345 unbiased, 438 Events associated, 246 combining with AND, 237–238 combining with OR, 238–240 definition of, 234 dependent, 246 independent, 246–249 independent and associated, sequences of, 249–251 mutually exclusive, 240–242 Excel binomial distribution using, 322 boxplots using, 163–164 confidence intervals using, 382 correlations using, 226 Data Analysis Toolpak and, 103 data entry and, 103 dotplots using, 104 finding descriptive statistics using, 163 histograms using, 104 Normal distribution using, 321–322 one-proportion z-test using, 432 one-sample t-test using, 504 paired t-test using, 504 random integer generation using, 270 regression equation coefficients using, 226 scatterplots using, 226, 227 stemplots using, 104 two-proportion z-test using, 432–433 two-sample t-test using, 504 XLSTAT and, 103 Expectations, 503 Expected counts, testing with categorical variables and, 503–505 Expected value, 303–304 Experiments See Controlled experiments Explanatory variable (x-variable), 187 Extrapolation, regression line and, 196–197 Extreme values See Outliers F First quartile (Q1), 128 Five-number summary, 139 Force, John, 189–190 Fractions, of people, 504 Frequencies, 35, 62 relative, 65 G Galton, Francis, 197 Gauss, Karl Friedrich, 279 Gaussian distribution See Normal distribution (Normal curve) “Given that,” AND vs., 243–244 Goodness of fit, coefficient of determination and, 197–199 Graphs, 60–86 See also Categorical distributions; Distributions; Numerical distributions bar charts, 75–77 discrete distributions as, 275–277 dotplots, 63–64 future of, 83–84 histograms and, 64–66 09/09/16 4:15 pm www.downloadslide.com INDEX interpreting, 81–84 misleading, 82–83 pie charts, 77–78 stemplots, 67 H Histograms, 64–66 bar graphs (bar charts) vs., 76 relative frequency, 65 Homogeneity, chi-square tests for, 509–518 Horizontal boxplots, vertical boxplots vs., 138 Hypotheses alternative, 387–388, 397–398, 409–410 caution against changing, 406–407 definition of, 387 null, 387–388, 397–398, 409–410 Hypothesis test(s) with categorical variables See Categorical variables; Hypothesis testing with categorical variables confidence intervals and, 479–480 standard errors in, 392 Hypothesis testing for population means, 457–463, 468–473 Hypothesis testing for population proportions caution against changing hypotheses and, 406–407 checking conditions and, 394–397, 412–415 comparing proportions from two populations and, 409–415 confidence intervals and, 408–409 definition of, 386 logic for, 407–408 null and alternative hypotheses and, 387–388, 397–398, 409–410 p-values and, 392–393, 402–403 significance level and See Significance level statistical significance vs practical significance and, 405–406 steps in, 393–402 test statistic for, 391–392, 410–412 Hypothesis testing with categorical variables chi-square test and See Chi-square statistic; Chi-square tests data and, 502–503 expected counts and, 503–505 I Inclusive OR, 239 Independence, chi-square tests for, 509–518 Independent events, 246–249 incorrect assumptions of independence and, 250 multiplication rule and, 249–251 Independent samples, 363, 464–466 two-sample t-test for, 471 Independent variable (x-variable), 187 Z05_GOUL1228_02_GE_IDX.indd 579 Inference for regression See Linear regression model Inferring population means, 434–484 comparing population means and See Comparing population means estimation with confidence intervals, 448–457, 479–480 hypothesis testing and, 457–463 null hypotheses for, 476 sample means and See Sample means Influential points, linear regression model and, 195 Intercept, regression line and, 191–193 Interquartile range (IQR), 126–130 calculating, 128–130 in context, 127–128 visualizing, 126–127 Inverse Normal values, 288 IQR See Interquartile range (IQR) L Landon, Alfred, 335 Law of Large Numbers, 254–256 Left-skewed distributions, 69 Line equation of, 180–181 regression See Regression line Linear regression model See also Regression line limitation to linear associations, 193–194 Linear trends, 170 Linearity, correlation coefficient and, 179 M Margin of error definition of, 354 setting, 355–356 Marginal totals, 504 Mean(s) analyzing, overview of, 476–480 of binomial probability distribution, 303–304 calculating for larger data sets, 111–113 calculating for small data sets, 110–111 comparing with other measures of center, 130–134 in context, 110 of a difference, confidence intervals for, 473–475 estimating difference of, with confidence intervals, 465–468 limitations of, 134 population See Inferring p opulation means; Population means of a probability distribution, 280–281 regression line and, 188–190 regression toward, 197 two, test of, with dependent samples, 475–477 visualizing, 108–110 579 Measurement bias, 334 Measures of center See also Mean(s); Median; Mode(s) comparing, 130–134 Median, 123–126 calculating, 125–126 comparing with other measures of center, 130–134 in context, 124–125 definition of, 123 effect on research, 524 visualizing, 124 Meta-analysis, 523 Minitab bar charts using, 103 binomial distribution using, 321 boxplots using, 162–163 confidence intervals using, 381–382 correlations using, 226 data entry and, 103 dotplots using, 103 finding descriptive statistics using, 162 graphs using, 103 histograms using, 103 Normal distribution using, 320–321 one-proportion z-test using, 431–432 one-sample t-test using, 502–503 paired t-test using, 503 random integer generation using, 271 regression equation coefficients using, 226 scatterplots using, 226 stemplots using, 103 two-proportion z-test using, 432 two-sample t-test using, 503 Misleading graphs, 82–83 Mode(s) avoiding computer use to find, 133 comparing with other measures of center, 130–134 multiple See Bimodal distributions; Multimodal distributions Moore, David, 28 Morse, Samuel, 352–353 Mu (m), 280, 436 Multimodal distributions, 69, 72 Multiplication rule, 249 Mutually exclusive events, 240–242 N Negative associations (negative trends), 168 Nonresponse bias, 337–338 Normal distribution (Normal curve), 279–292 appropriateness of, 291–292 center and spread of, 279–280 definition of, 279 Empirical rule and, 290–291 finding measurements from percentiles for normal distribution and, 287–290 09/09/16 4:15 pm www.downloadslide.com 580 INDEX Normal distribution, (continued ) finding normal probabilities and, 281–283 finding probability with technology and, 283–284 mean and standard deviation of, 280–281 notation for, 347 standard Normal model and, 285–287 Normal model, 279–292 See also Normal distribution (Normal curve) appropriateness of, 291–292 definition of, 279 standard, 285–287 Normal probabilities, 281–283 Null hypotheses, 387–388, 397–398, 409–410 “accepting,” 470, 476 for two means, 470 Numerical distributions, 67–75 describing, 74–75 shape of, 68–72 typical value (center) of, 67–68, 72–73 variability (spread) of, 67–68, 73–74 O Obama, Barack, 83–84, 338 Observational studies, 40–42 extending results of, 45 One-proportion z-test, 402 One-proportion z-test statistic, 391–392 One-sided hypotheses, 388–389, 460–463 OR, combining events with, 238–240 Outcome variables, 39 Outliers, 71–72 comparing measures of center and, 132–133 potential, 135–138 regression line and, 195 Ozeki, Ruth, 275 P Paired samples See Dependent samples Parameters definition of, 332 statistics vs., 387 Pareto, Vilfredo, 77 Pareto charts, 77 pdfs See Probability distributions (probability distribution functions [pdfs]) Peer review, 46, 519 Percentiles definition of, 288 finding measurements for normal distribution from, 287–290 Pie charts, 77–78 Placebo(s), 40, 44–45 Placebo effect, 40 Population(s) choosing, 364 definition of, 30, 332 size of, precision and, 342–344 Z05_GOUL1228_02_GE_IDX.indd 580 Population distribution, 444 Population means hypothesis testing for, 457–463 inferring See Inferring population means Population proportions (p) comparing See Comparing population proportions estimating with confidence intervals, 353–359 hypothesis testing for See Hypothesis testing for population proportions unknown, finding confidence intervals with, 356–357 Positive associations (positive trends), 168 Potential outliers, 135–138 Precision definition of, 342, 437 population size and, 342–344 Predicted variable (y-variable), 187 Predictor variable (x-variable), 187 Probability(ies), 228–256 conditional, 243–246 confidence intervals vs., 451 definition of, 232 empirical See Empirical probabilities independent and dependent events and, 246–248 intuition about independence and, 248–249 normal, 281–283 proportions vs., 389 randomness and, 230–233 sequences of independent and associated events and, 249–251 theoretical See Theoretical probabilities Probability density curves, 277–278 Probability distributions (probability distribution functions [pdfs]) continuous, 274–275, 277–279 definition of, 274 discrete, 274–277 mean of, 280–281 standard deviation of, 280–281 Probability models, 274 Profit motive, effect on research, 524 Proportions Central Limit Theorem for, 440 comparing, 513 confidence intervals for, 408 population See Comparing population proportions; Population proportions (p) probabilities vs., 389 sample See *Sample proportions (p) tests of, relation to tests for association between categorical variables, 515–518 Publication bias, 523 p-values, 392–393, 460 calculating, 395–397 small, 401, 402–403 two-tailed, 395–396 Q Q1 See First quartile (Q1) Q2 See Second quartile (Q2) Q3 See Third quartile (Q3) Qualitative variables See Categorical variables Quality of a survey, measuring, 338–346 Quantitative variables, 30, 31 Quartiles definition of, 128 software and, 128 R r2 See Coefficient of determination (r2) Random assignment, 43, 45, 513 random sampling vs., 365–366 Random samples, 514 sample means of, 436–439 sample proportions from, 439 Random sampling, random assignment vs., 365–366 Random selection, 45 Randomized assignment, 514 Randomness, 230–233 Range, 130 interquartile See Interquartile range (IQR) Reading research papers, 518–525 abstracts and, 521–522 warning signs for poor quality and, 522–525 Regression analysis, 166–202 See also Linear regression model; Regression line coefficient of determination and, 197–199 correlation and See Correlation; Correlation coefficient definition of, 168 pitfalls to avoid with, 193–197 scatterplots and See Scatterplots Regression line, 180–193 See also Linear regression model choosing x and y and, 186–188 in context, 182–183 equation of a line and, 180–181 finding, 183–186 interpreting intercept and, 191–193 interpreting slope and, 190 as line of averages, 188–190 visualizing, 181 Regression toward the mean, 197 Relative frequencies, 65 Relative frequency histograms, 65 Research papers, reading See Reading research papers Resistance to outliers, 132 Response bias, 335 Response variable (y-variable), 187 Right-skewed distributions, 69 Roosevelt, Franklin Delano, 335 r-squared, 197–199 09/09/16 4:15 pm www.downloadslide.com INDEX S Sample(s) convenience, 336 definition of, 30, 332 dependent See Dependent samples distributions of, 62, 274, 443, 444–445 independent, 363, 464–466 not randomly selected, 404 random, sample means of, 436–439 usage of term, 30 variation of statistics from one to another, 340–342 Sample means accuracy and precision of, 437–439 Central Limit Theorem for, 440–447 as estimator, 440–441 of random samples, 436–439 standard error of, 438 t-distribution and, 446–447 types of distributions and, 444–445 visualizing distributions of, 442–443 Sample proportions (p), 333, 350 Central Limit Theorem for, 346–353 from random samples, 439 Sample size, 42 estimators and, 344–345 hypothesis testing for population proportions and, 404 Sample space, 234–235 Sample standard deviation, 446 Sampling with replacement, 336 without replacement, 336 Sampling bias, 334, 335–336 Sampling distributions, 394, 438–439, 445 definition of, 341 distribution of a sample vs., 443 Scatterplots, 168–172 describing associations and, 171–172 shape of, 170–171 strength of association and, 169–170 trend and, 168–169 SE See Standard error (SE) Second quartile (Q2), 128 Shape comparing measures of center and, 131–132 of numerical distributions, 68–72 of scatterplots, 168, 170–171 Sigma (s), 280 Sigma (Σ), 110 Significance level definition of, 390 hypothesis testing for population proportions and, 390, 404–405 Simple random sampling (SRS), 336–338 Simulations coin flip, 252–254 definition of, 233 failing to give expected theoretical value, 255 Z05_GOUL1228_02_GE_IDX.indd 581 number of trials in, 255 steps for, 252 technology and, 343 understanding behavior of estimators using, 339–345 Skewed distributions, 123–130 interquartile range of, 126–130 median of, 123–126 range of, 130 Slope equation for a line and, 180–181 of regression line, 190–191 SOCR calculator, finding probabilities using, 283–284 Sports Illustrated jinx, 197 SRS See Simple random sampling (SRS) Stacked data, 32 Standard deviation, 113–117 of binomial probability distribution, 303–304 calculating, 116–117 in context, 115–116 definition of, 114 Empirical Rule and, 118–120 of a probability distribution, 280–281 sample, 446 visualizing, 113–115 Standard error (SE), 342, 438 finding, 345–346 in hypothesis tests, 392 Standard Normal model, 285–287 Standard units, 121, 285 StatCrunch bar charts using, 105 binomial distribution using, 323 boxplots using, 164–165 confidence intervals using, 382 correlations using, 227 data entry and, 104 dotplots using, 105 finding summary statistics using, 164 histograms using, 105 Normal distribution using, 322–323 one-proportion z-test using, 433 one-sample t-test using, 505 paired t-test using, 505 pasting data and, 104 random integer generation using, 271 regression equation coefficients using, 227 scatterplots using, 227 setting up, 104 simulated sampling using, 382–383 stemplots using, 105 two-proportion z-test using, 433 two-sample t-test using, 505 Statistical inference, 333 Statistical significance, 524–525 Statistics definition of, 332, 360 in the news, 45–47 581 parameters vs., 387 variation from sample to sample, 340–342 Stemplots, 67 Streaks, probability and, 256 Strength of associations, 168, 169–170 Summation symbol (Σ), 110 Surveys, 332–346 bias and, 334–336, 345–346 measuring quality of, 338–346 simple random sampling for, 336–338 simulations to understand behavior of estimators and, 339–345 standard error and, 345–346 terminology associated with, 332–334 Symbols a (alpha), 384 a (from y = a + bx), 183 b (from y = a + bx), 183 H0 (Null hypothesis), 381 Ha (Alternative hypothesis), 381 m (mu), 280, 328, 430 p, 330 pn , 330 r, 176 r2, 197 s (sigma, lower case), 328 Σ (sigma, upper case), 110 s, 116 t, 453 x (x-bar), 110 x2 (chi-square), 505 z, 122, 286 Symmetric distributions, 68, 108–117, 279 mean of, 108–113 standard deviation of, 113–117 variance of, 117 T Tables discrete distributions as, 275–277 two-way (contingency), 35, 411 A Tale for the Time Being (Ozeki), 275 t-distribution, 446–447 Technology See also specific devices and software columns and, 101 finding probability with, 283–284 simulations and, 343 Test statistics hypothesis testing for population proportions and, 410 for one-sample t-test, 459, 463 z-, one-proportion, 391–392 Theoretical probabilities, 232–242 combining events with AND and, 237–238 combining events with OR and, 238–240 with equally likely outcomes, 234–237 finding, 233–242 mutually exclusive events and, 240–242 09/09/16 4:15 pm www.downloadslide.com 582 INDEX Third quartile (Q3), 128 TI-84 calculator binomial distribution using, 320 boxplots using, 161–162 clearing memory, 101 confidence intervals using, 381 correlations using, 225 data entry and, 101 downloading data from computer into, 102–103 finding descriptive statistics using, 161 histograms using, 101 Normal distribution using, 319 one-proportion z-test using, 431 one-sample t-test using, 501 paired t-test using, 502 random integer generation using, 270 regression equation coefficients using, 225 resetting, 101 scatterplots using, 225 two-proportion z-test using, 431 two-sample t-test using, 501–502 Treatment group, 39 Treatment variables, 39 Trends linear, 170 See also Linear regression model; Regression line positive and negative, 168 scatterplots and, 168–169 Z05_GOUL1228_02_GE_IDX.indd 582 t-statistic, 446 t-test paired, with dependent samples, 477 two-sample, 471 using software to do, 471 Two-proportion z-test, 410 Two-sided hypotheses, 388–389, 460–463 Two-tailed p-values, 395–396 Two-way tables, 35, 411, 502–503 Typical value, of numerical distributions, 67–68, 72–73 Variance, definition of, 117 Variation See also Standard deviation definition of, 28 Venn diagrams, 237 Vertical boxplots, horizontal boxplots vs., 138 W Washington, George, 83 X U x-variable, 187 regression line and, 186–188 Unbiased estimator, 438 Unimodal distributions, 69, 279 Unstacked data, 32 Y V Variability interquartile range and, 126–130 of a numerical distribution, 73–74 Variables categorical (qualitative).See Categorical variables confounding (lurking), 41 definition of, 30 numerical (quantitative), 30, 31 order of, correlation coefficient and, 177–178 y intercept, equation for a line and, 180–181 y-variable, 187 regression line and, 186–188 Z z, sign of, 411 z-scores, 120–123, 175, 285, 286, 361 calculating, 122 in context, 121–122 visualizing, 121 z-test, 459 one-proportion, 402 two-proportion, 410 09/09/16 4:15 pm www.downloadslide.com This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 24/11/14 5:26 PM www.downloadslide.com This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 24/11/14 5:26 PM www.downloadslide.com Global edition Essential Statistics Exploring the World through Data For these Global Editions, the editorial team at Pearson has collaborated with educators across the world to address a wide range of subjects and requirements, equipping students with the best possible learning tools This Global Edition preserves the cutting-edge approach and pedagogy of the original, but also features alterations, customization, and adaptation from the North American version Global edition SECoND edition Gould • Ryan • Wong Exploring the World through Data SECoND edition Gould • Ryan • Wong GLOBal edition This is a special edition of an established title widely used by colleges and universities throughout the world Pearson published this exclusive edition for the benefit of students outside the United States and Canada If you purchased this book within the United States or Canada, you should be aware that it has been imported without the approval of the Publisher or Author Essential Statistics Pearson Global Edition Gould_02_1292161221_Final.indd 17/09/16 9:42 AM ... Fortunately, the child remains visible The problem is to figure out where the mother is sitting Where is the mother? On 68% of the days, the child is within yard of the mother, so at these times the mother... estimator is the proportion of people in a sample who say 326 M07_GOUL 122 8_ 02_ GE_C07.indd 326 03/09/16 4 :24 pm www.downloadslide.com 7.1 Learning about the World through Surveys CHAPTER 327 they will... instead guessed that the mother is within yards of the child Then we would be wrong on only 5% of the days In this analogy, the mother is the population proportion Like the mother, the population proportion