Let us begin by emphasizing that there is no "right" way to teach with this book. We hope that Workshop Statistics will prove useful to students and instructors in a wide variety of settings. It can be used as a stand-alone text or as a supplement, with computers or graphing calculators, as in-class work or take-home assignments. Naturally, we think that the book will work best in a classroom environment that promotes the features we extol in the preface: active learning, conceptual understanding, genuine data, and use of technology.
The purpose of these notes is to supply information from our experiences teaching with Workshop Statistics that might prove useful to other instructors. We provide details about how the book has been utilized at Dickinson College, where it was initially developed. We also offer general suggestions for teaching with the book and topic-by-topic comments about the goals of each activity and common student reactions. For the benefit of those experienced in teaching with the first edition, we also point out major changes in the new edition for each topic. This Guide applies to the "generic" (red cover) version of the text, but instructions and comments specific to the Minitab, Fathom, and Graphing Calculator versions will be added soon. You can read through this document sequentially or skip directly to one of the sections listed below:
With 75-minute class meetings on a Tuesday/Thursday schedule, I use one topic for each class period. In a typical class period, I spend five minutes or so introducing the day's topic, collect data from the students (if the day's topic calls for it), and then let them work through the activities. I and a student assistant then spend most of the class time walking about the room, peering over shoulders, checking students' progress, and asking them questions about what they are doing. I try to visit each student several times during the period. Whether students have questions or not, I often ask personalized questions of them as a means of checking their understanding. If I notice a common problem, I might go to the front, ask everyone to stop working momentarily, and discuss the problem. I also put some answers on the board occasionally, allowing students to check their work. (Another approach to providing this feedback is to direct students to the web page of answers to in-class activities.) Finally, I try to spend the last five minutes discussing with the whole class what they were supposed to have learned that day.
To facilitate students' recording information in their books, I recommend that students tear the perforated pages out of their books and place them in a three-ring binder (which I also do myself) from the start of the course. I do try to insist that students write directly in the spaces left in their books and not on scratch paper, for I think it helps students to have the questions and their responses together in one place.
I give three exams during the semester. These are somewhat cumulative but focus much more on recent material; of course, intelligent application of the material in later parts of the course depends largely on understanding earlier material. The exams try to stress understanding and interpretation as well as calculation. I allow students to use their books (and, of course, their responses to its questions) on the exams. Since the classroom has only half as many computers as students, I do not ask students to use computers on the exams. Instead I present them with computer output and ask questions about the interpretation of results. I sometimes devote the class period before an exam to a review session, a rare opportunity for me to lecture about the main ideas and to present analyses of sample activities.
I assign 3-4 homework activities per topic, collecting and grading them once per week. I try to select these so that the major ideas of the topic are covered; I also try to include a mix of problems that can be done by hand vs. with technology. I sometimes ask students to hand in their in-class activities as well, although this practice quickly creates a difficult grading burden. I encourage students to work together on the homework activities, but I require that they write up their comments individually.
I spend as little time as possible showing students how to use the technology, allowing students to concentrate on more substantive concerns. I spend more time on this earlier in the course as they are getting comfortable with the software and much less time as the course progresses. I sometimes use overhead projection equipment to demonstrate something, but more often I just write instructions on the board. In general I prefer students to explore the technology for themselves rather than to watch and mimic what I do.
Every topic begins with a "Preliminaries" section that asks students questions to get them thinking about the issues of the day. These questions often ask students to make guesses about the values of variables. While such questions are not central to learning statistics, I emphasize them for several reasons. They can provide the class with a sense of fun (you might even give a prize to the closest guesser of the day), and they hopefully motivate students' interest in the day's material. More importantly, they can show students that statistics is relevant to everyday life and get them used to thinking of data as more than just detached numbers.
Many topics call for data to be collected at the beginning of class. To protect students' privacy (at least somewhat), I pass around scratch paper, ask them to record the information on them, and have the student assistant collect the strips and either write the data on the board or enter it directly into the computer.
The primary goals of this topic are to get students thinking about data, help them to appreciate different types of variables, and expose them to simple visual displays (bar graphs and dotplots) of a distribution. Substantial changes from the first edition in this topic include asking for different data to be collected in the preliminaries, paying more attention to the definition of variable (1-2a,b) postponing the use of technology until Topic 2, and emphasizing interpretations more (1-4e).
The Preliminaries aim to get students thinking about data not as naked numbers but as numerical information with a context. You might use the first question as an opportunity to explain your goals for the course and the second to emphasize the importance of students' accepting responsibility for being active learners. You also might use question 5 as a model for many questions in the book that ask for students' guesses or intuitions by mentioning that students' actual responses are less important than their taking the question seriously and putting forth sincere efforts.
Except in rare cases, we always collect data anonymously in an effort to avoid potential embarrassments. We usually have students record their data on scratch paper; then we collect it and either write it on the board or have a student assistant enter it into a computer file. Another approach is to pass around a sheet that contains a table in which students report their data, but this loses some of the anonymity. Other approaches to collecting data on students is to ask them to send their information to you before class via e-mail or to set up a form on the Web through which they can enter their data.
Be sure to bring to class rulers with centimeter markings in order to measure heights and lengths of signatures. Questions may arise (for example, does landing in an airport in a state count as having visited it?) that could lead you to discuss some of the thorny issues involved with defining variables carefully and collecting data well.
In case your class is small enough that you would prefer to combine your data with those from other classes, sample data collected on students may be found here.
Activity 1-1 introduces the simple but fundamental principle of variability. You might tell students that while this activity hardly taxes their minds, it does introduce the key idea that permeates the course and the study of statistics.
The definitions following Activity 1-1 are essential for students to understand. We have concluded that in the past we have given short shrift to making sure that students understand the terms "variable" and "observational unit." We have come to believe that these are crucial ideas for students to be comfortable with throughout the course, so more homework activities in this topic are devoted to these terms, and more activities throughout the entire book begin by asking students to identify the observational units and variables in a study. Users of the first edition should note that we use the term "quantitative variable" as opposed to "measurement variable" in this edition.
Activity 1-2 tries to get students comfortable with the definition of variable and distinctions between types of variables. While this is a fairly simple task, it sometimes becomes problematic later when students need to decide whether an inference situation concerns means or proportions. Questions (a) and (b) aim to help students to see the difference between a variable and a summary statistic based on a variable (whether or not a student has red hair vs. number of students with red hair). Question (e) tries to indicate that how one measures the variable determines its type. Emphasizing precisely what variables and observational units are is important here.
Activity 1-3 introduces the bar graph as a simple visual display of the distribution of a categorical variable. It also begins to address the necessity of writing one's conclusions when analyzing data.
Activity 1-4 asks students to tally the results of a data collection. While this is very straightforward, reading tables of tallies (or frequencies) is a crucial skill in later topics. Many students have considerable difficulty with reading tables of tallies correctly. Question (e) also emphasizes the ability to connect verbal descriptions with underlying data.
Activity 1-5 introduces the dotplot as a simple visual display of the distribution of a measurement variable. Question (b) aims to get students to identify personally with their data analysis, and questions (d) reinforces the importance of writing about one's data analysis. We suggest not giving many hints about what kinds of features to look for in a distribution; let them struggle to think of things on their own. We emphasize to students that some responses might be more insightful than others but that there are no clear-cut right/wrong answers here.
Activity 1-6 is the first to present displays based on a larger data set not collected on students themselves. It also provides a first introduction to the important idea of comparing distributions of data. Question (b) previews the concept of median. We advise students not to worry about counting the dots to find the exact median; I'm satisfied with reasonable approximations here but that vagueness frustrates some students.
The homework activities provide practice with variables and observational units (1-7 through 1-13, 1-18) and with some simple graphical analyses (1-13 through 1-17). We have found 1-11 to be surprisingly challenging for students but also helpful in getting them to understand these definitions.
The Preliminaries again consist of a mixture of questions that ask for data to be collected about students and questions that call for predictions from students. Questions 1-3 are relevant to Activity 2-1, but the data collected are analyzed only in homework activity 2-5. Questions 4 and 5 relate to Activity 2-3, questions 6 and 7 to Activity 2-2, and question 8 to homework activity 2-7. Sample data collected on students about these "Preliminaries" questions will appear HERE.
Specific uses of technology that students learn in this topic are entering data, performing arithmetical operations on data, sorting data, creating dotplots, and separating data into groups.
Activity 2-1 introduces students to the use of technology. You may need to spend some time explaining and/or demonstrating how students are to enter data into their software or calculator. The "generic" version simply uses the phrase "use technology to" throughout the book, as in parts (b), (c), and (f). Students are also supposed to begin to appreciate the role of rates as they find that the person with the most points does not have the highest points/letters ratio.
In Activity 2-2 students use technology to create a new variable (% of women) from existing ones, to produce a dotplot, and to sort the results. As this is the first activity requiring students to access data, you will probably need to show them where to access the relevant files. The sorting creates problems for many students, because they often fail to sort the specialty names along with the variables and therefore can not easily identify which specialties have the highest and lowest percentages of women. In (d) we think it's important to emphasize to students that "percentages" are proportions multiplied by 100. It is very important throughout the book that "proportion" refer to a number between 0 and 1 (inclusive), so we expect "percentage" to be between 0% and 100%. Students struggle with questions (h) and (i) as well.
Activity 2-3 aims to give students more experience using technology to manipulate and analyze variables. It also continues to address the "how do we measure this idea" theme of this topic. Questions (f)-(h) continue to emphasize the relevance of using rates rather than raw numbers for comparison purposes.
Activity 2-4 provides another example of the need to think about whether a variable is measuring what it intends to. Most students quickly realize that because many states emphasize the ACT more than the SAT, those with a small percentage of students taking the SAT tend to have high average scores because the few students who take the exam tend to be among the best.
The homework activities ask students to use technology to analyze data collected on themselves (2-5, 2-12) and from available sources (2-7 through 2-11). With the exception of 2-10, all of these activities also concern the usefulness of rates or percentages.
New uses for technology in this topic are to create histograms and to vary their bin widths.
Again there are many questions in the Preliminaries, aiming to get students thinking about some of the issues and data covered in the topic. We reiterate to students that we don't care how well they guess things like a typical weight for an Olympic rower, but we do care that they engage themselves with such questions and respond to them conscientiously. Sample data collected on students about these "Preliminaries" questions may be found here.
Activity 3-1 leads students to develop a checklist of six features to consider when describing a distribution of data. You might want to lead students through this activity with a class discussion to make sure that nobody misses the point. Emphasizing the terminology of right- vs. left-skewness is probably worthwhile, as many students find the terms counter-intuitive.
Activity 3-2 asks students to match up variables with the corresponding dotplots of their distributions. The scales have been removed from the dotplots, of course, so that students have to judge based on shape and features other than center and spread. You might advise students to see if they can construct a meaningful scale that seems to work for their selections. This activity particularly lends itself to working in groups to increase the chances that someone in the group is familiar with baseball and Monopoly. You might want to emphasize that the explanation for students' choices is more important than the choices themselves. Some other issues that have arisen include: the Monopoly sample does not include railroads, the cities for the snowfall amounts were selected from all around the U.S., and "margin of victory" means the difference in number of runs scored between the winner and loser.
An alternative to this activity is to produce dotplots or histograms based on class-generated data and have students match up the graphs with the variables. This could avoid the problem of some students not knowing about Monopoly or baseball, and it might help students to identify more personally with the data. For example, if height were included among the variables, students could look around the room to spot a tall outlier.
Activity 3-3 emphasizes the importance of context when analyzing data. Students should recognize that the dotplot has an outlier and clusters. The important point here is that they should search for causes of the outlier and clusters. Those who know about rowing will know that the outlier is the coxswain, who calls out instructions but does not help with the rowing. The lower cluster of weights all belong to rowers in "LW" (lightweight) events.
Activity 3-4 introduces the stemplot. Some students may not recognize that the easiest way to construct it is to go through the data in the order presented rather than looking for all of the single digits and then all of the teens and then all of the twenties and so on. Be ready for students to ask whether they should list repeated values (such as 13, which appears four times) multiple times; of course they should.
Activity 3-5 asks students to interpret a histogram. Some will struggle to understand how the midpoints presented in the graph correspond to endpoints of the subintervals. Question (b) should say "61 or more," and the answer to question (c) is meant to be "no" since 90 is not a subinterval endpoint. Question (e) asks students to investigate the effect of changing the number of bins (subintervals) in the histogram and to notice how substantially different the resulting graphs appear.
The homework activities ask students to create and to analyze visual displays of data. Emphasizing that students relate their verbal descriptions to the context is crucial throughout these activities, and helping students to acquire the habit of labeling the axes of their graphs is also important. Several activities (3-6, 3-7, 3-9, 3-10, 3-15, and 3-20) provide students with the displays, while many (3-8, 3-11 through 3-14, 3-16 through 3-19) ask the students to create the displays first. Activities 3-8 and 3-10 call for the use of technology, while 3-11, 3-13, and 3-17 are to be done by hand. In other activities, we recommend letting students choose for themselves whether to create graphs using technology or by hand. Activity 3-11 introduces a variation on the standard stemplot known as a split stemplot.
New uses for technology in this topic are to calculate means and medians.
The Preliminaries are briefer than in previous topics, but they again try to generate student interest and thinking about the issues and data for the topic. Sample data collected on students about these "Preliminaries" questions may be found here; the instructor in question was 60 years old at the time.
Activity 4-1 covers the basic calculations of the mean and median. We don't expect students to find the mean or median in (b); in fact, we'd prefer a more creative response. You will probably have to help many students in question (i), where they are to make the jump to the general case for identifying the location of the median. Question (l) is challenging for many students; its point is that current justices have not yet served full terms and so underestimate the mean and median years of service among prior justices who had served full terms.
Activity 4-2 ask students to make guesses about the mean and median of a set of data based on dotplots. The first set is meant to be fairly easy, as the distributions are quite symmetric, just to establish that mean and median really do measure the center of a distribution. These questions also provide your first opportunity to show students how to use technology to calculate summary statistics. In Minitab you might use the "describe" command to display descriptive statistics, in which case you might want to warn students not to concern themselves (for now) with all of the other statistics that are displayed.
Questions (e)-(i) lead students to see how the mean and median relate to each other in skewed distributions. Many students struggle with question (j), which tries to make the point that the mean is not a sensible measure with categorical variables. We believe that re-focusing students' attention on this issue of variable types is important here.
Activity 4-3 leads students to investigate the property of resistance of the mean and median. Question (b) tries to make the obvious but important point that knowing only the mean or median does not reveal anything about the spread or shape of a distribution. Questions (c) and (e) are good examples of questions where we want students to make thoughtful predictions, but we don't care much about how accurate their predictions are. We do care, of course, that they rethink their predictions in (d) and (f) if they turn out to be inaccurate. This activity works particularly well in Fathom, where students can just drag an observation around and watch the immediate effect on the mean and median.
Activity 4-4 aims to help students see an important limitation of measures of center. The moral here is that one often wants to consider the entire distribution and not just a measure of center. While the median readability level of pamphlets equals the median reading level of patients, many patients are left without a single pamphlet at or below their reading level. Question (f) tries to bring this point home in case students have missed it in question (e). Some students, not realizing the importance of the "under 3" and "above 12" designations, will need guidance with (a). You might also be prepared for many students ignoring the tallies in (b) and just treating the distributions as if they were uniform with one observation at each level; this produces median grade levels of 7.5 for patients and 11 for pamphlets. You should also notice that the dotplots are presented in a different order than the data tables (reading level first in data table but second in dotplot), which may cause some confusion.
The homework activities ask students to calculate, interpret, and analyze measures of center of distributions of data. Activity 4-5 tries to expose the common error of not sorting values before identifying the median. Activities 4-6 through 4-8 involve student-collected data; whether students should analyze them by hand or with technology is left to the instructor's discretion. Be aware that Activity 4-9 is confusing to many students who are not sure how to calculate the mean from a frequency table, so you may want to provide some guidance if you ask students to do this by hand. Activities 4-10 through 4-16 all deal with various properties of averages. Activity 4-10 illustrates that knowing a total amount allows one to calculate a mean but not a median or mode. Activity 4-11 reveals that the mean of percentages does not necessarily produce the overall percentage. We particularly like assigning homework problems such as Activities 4-12 and 4-13 that require students to construct examples to illustrate their knowledge of the properties. Activity 4-14 introduces the idea of a weighted average between two groups, and Activity 4-16 leads students to discover the effect of a linear transformation on measures of center. Activity 4-15 serves as a reminder that a single measure of center may not reveal much of interest about a dataset.
In addition to new and updated datasets, substantial changes from the first edition in this topic include the introduction of mean absolute deviation as a measure of spread and as motivation for the standard deviation. Activity 5-7 is also new, attempting to address student misconceptions about spread as related to "bumpiness" and number of distinct values.
New uses for technology in this topic are to calculate standard deviation and inter-quartile range.
Sample data collected on students about Preliminaries questions will appear HERE.
Activity 5-1 introduces the five-number summary, range, IQR, and boxplot. You may want to stress that the IQR is the difference between the quartiles; some students just stop with the quartiles themselves. Some students are confused by what question (h) intends for verification; you may want to ask students to mark the location of the quartiles on the dotplot as a visual confirmation.
You may want to tell students that different textbooks and different technologies calculate quartiles in slightly different ways. Therefore, they should not be alarmed if their hand calculations of quartiles do not match exactly the values produced by their technology.
Activity 5-2 shows students how to calculate mean absolute deviation and standard deviation as measures of spread. Even though many of the steps in the calculations are already provided in the table, many students find these calculations to be challenging and time-consuming. In particular, taking the sum of twelve entries in a column causes problems for some. This activity also provides a good opportunity to talk about rounding in calculations. We stress to students that they should retain as many digits of accuracy as possible in intermediate calculations and round only the final answer. The notation following question (h) intimidates some students, so we try to show them how the notation corresponds with the process they just completed.
We do not recommend spending much time on students' calculating standard deviations. We prefer to let technology handle those calculations so that students can focus on interpretations as illustrated in the following activities.
Activity 5-3 asks students to complete two "matching" exercises. The first is to convince them that IQR and standard deviation do in fact measure the variability of a distribution: more variable distributions produce larger values of these measures. The second is to help students see how a five-number summary relates to a boxplot and particularly to its skewness. Many students have difficulty with the vertical orientation of these boxplots, so you should probably be prepared to explain to them that it's just rotated from a horizontal boxplot and that larger values appear at the top rather than on the right.
Activity 5-4 aims to lead students to conclude that the IQR is resistant to outliers while the standard deviation and range are not. You might want to advise students that they only need to enter the values of years of service into their technology; they need not enter the Justices' names. As with Activity 4-3, this activity works particularly well in Fathom, where the numerical measures change dynamically as students drag a point in a graph.
Activity 5-5 leads students to the empirical rule, which is one answer to the "what does the value of standard deviation tell us" question. Some students may need an explanation of the "plus/minus" notation in question (b). In question (d) you might want to advise students that it's easier to count the scores that do not fall between those two values and then subtract. We try to emphasize that this empirical rule holds only roughly and even at that only for mound-shaped distributions.
Activity 5-6 introduces z-scores, an idea to which students return later (in Topic 15) when studying normal distributions. It provides another answer to the question of how to interpret the value of standard deviation. In (h) and (i) some students struggle with realizing that the higher z-score between two negatives is the one closer to zero.
Activity 5-7 is designed to confront student misperceptions about measures of spread. Many students think that F has more spread than G because its distribution is "bumpier," and many students believe that J has more spread than I since it contains more distinct values. You might want to draw their attention to these misconceptions, after they have completed the activity, in order to help students confront them head-on.
At this early point in the course some students may be tempted to enter the data into their technology by hand, but we strongly encourage you to let them use the pre-entered files that we provide.
The homework activities ask students to calculate and interpret measures of spread. Activities 5-8, 5-10, 5-16, 5-22, and 5-24 definitely require use of technology. The expectation in (a) of Activity 5-8 is that students base their answers on their knowledge of geography. Activity 5-10 tries to combat the misperception that larger values produce more variability than smaller values. Activities 5-14 and 5-21 expose limitations of boxplots as visual displays. Activity 5-16 concerns the empirical rule, while Activities 5-18 and 5-19 involve z-scores. Activity 5-22 is similar to Activity 4-12 in asking students to create example that illustrate their understanding of these measures' properties. Activity 5-24 intends to be a reminder that summaries do not tell the whole story, for the means and standard deviations are identical for all three machines. Activity 5-26 is a follow-up to Activity 4-16 in leading students to consider the effect of linear transformations on measures of spread.
Nevertheless, this is a shorter topic than the previous one. This is also a topic that could work well with in-class activities assigned as homework, if you are pressed to preserve class time. Another approach is to work some of these activities into earlier topics. For example, the side-by-side stemplot of Activity 6-1 could be done with Topic 3, and the modified boxplots of Activity 6-2 could be done with Topic 5.
In addition to new and updated datasets, substantial changes from the first edition in this topic include a new "matching" activity.
New uses for technology in this topic are to produce (modified) boxplots.
Many students find the "top 100 American films" data in the Preliminaries section appealing to work with. Collecting the data does take some time, though, so this may be a good example where collecting data more efficiently outside of class time may be desirable. Sample data collected on students about the "Preliminaries" questions may be found here.
Activity 6-1 tries to accomplish many goals, introducing the fundamental idea of a statistical tendency as well as the graphical technique of the side-by-side stemplot. In our experience students enjoy this activity because they aren't afraid to expose their geographical ignorance. We encourage them to shout out a state about which they are unsure so that we can reach a class consensus. We're careful not to tell them (until after they've completed the activity) that the "answers" concerning states' east/west status appear in Activity 6-12. We do tell students that they should end up with 26 eastern and 24 western states, although we caution them that having this 26/24 breakdown does not guarantee that they have labeled each state correctly.
Questions (e)-(g) guide students to the important concept of a statistical tendency. Since this is a crucial concept that comes up throughout the course, we try to emphasize students' understanding of this idea.
Activity 6-2 introduces both comparative boxplots and modified boxplots. You might want to point out that the data here are already sorted, so finding quartiles by hand is not as difficult as it would otherwise be. Question (a) provides a good example of an activity for which we write the numerical answers on the board and insist that students check their calculations before proceeding. (You could also point students to the answers available on the Web.) Some students need help to understand the description of the outlier test preceding question (d). Question (g) might provide a convenient time to remind students always to write in complete sentences and relate their comments to the context at hand.
Activity 6-3 is another "matching" exercise. This is another especially good one for asking students to work in groups so that they can combine their knowledge of various variables and regions of the country. We emphasize to students that a reasonable explanation is more important than reaching the "right" answer here.
Activity 6-4 follows in the spirit of leading students to more open-ended problems toward the end of topics. You probably want to explain how to use technology to produce comparative boxplots on the same scale in (b). Although the data entry in (c) can be tedious, we think it's appropriate for students to do occasional data entry by hand when it enables them to analyze data specific to themselves.
The homework activities ask students to apply these methods and concepts to analyzing more datasets. Activities 6-5, 6-6, 6-11, 6-12, 6-13, 6-15, and 6-17 are to be done by hand. Activities 6-7, 6-10, and 6-18 require technology. Technology may be used but is not essential for Activities 6-8, 6-9, 6-20, and 6-21. Activities 6-14, 6-16, and 6-19 ask students to interpret graphical information that is provided.
Since its techniques of analysis are so simple, this topic is a particularly good one for letting students work independently at their own pace. We tend to interrupt them less in this topic than in some others, but we do try to make sure that everyone grasps and can explain the finer points such as Simpson's paradox. This topic typically takes students less time to complete than many of the others.
Substantial changes from the first edition in this topic include a stronger emphasis on the concept of independence and a mention of the term relative risk, as well as new and updated datasets.
New uses for technology in this topic are to organize raw data on categorical variables into cross-tabulation tables.
While collecting data from students for the Preliminaries section, it may be worth highlighting that all of the variables being asked about are categorical and that this distinguishes this topic from earlier ones. Sample data collected on students about the "Preliminaries" questions may be found here.
Activity 7-1 is intended primarily to make sure that students understand the relationship between the two-way table and the raw data. We used to take for granted that they understood this, but we found that students had trouble constructing the table from the raw data, so now we think this is an important exercise. The important terms response and explanatory variable are also introduced here and should be highlighted.
Activity 7-2 gives students an extended example in which to learn how to analyze two-way tables. A common error occurs in (h), where many students miss the cumulative nature of the percentages and so just put marks at .379, .499, and .122 rather than "stacking" these. Also in (h) some students put "very much" on the bottom and "not much" on top, which is not the order given in the other age groups and so defeats ease of comparison. You also might want to check especially carefully that students recognize the differences among questions (j)-(l).
Activity 7-3 tests whether students can read proportions (approximately) from a segmented bar graph. Some students will need a helpful nudge to answer question (c).
Activity 7-4 leads students to discover Simpson's paradox. The hypothetical example is contrived so that most students recognize that hospital A is the better hospital despite its lower survival rate because it treats most of those in poor condition, who are naturally less likely to survive than those in fair condition. Some students take for granted the observation that those in poor condition are less likely to survive than those in fair condition, but we emphasize this aspect as well as hospital A's treating most of the "poor condition" patients. When we first used this activity we did not ask question (f), but then we realized that this is the crucial question to see if students understand the phenomenon. Some students are unsure about which hospital they would prefer, but most recognize the superiority of hospital A.
With good students we sometimes ask a follow-up question: If you call home and learn that dear Aunt Milly is in the hospital with this condition, which hospital do you hope she's in? We claim that the answer is hospital B, because that would mean that she's likely in fair condition and therefore more likely to survive. We only ask this of strong students, though, because this is a subtle and much less important point than Simpson's paradox in general.
We also mention to students that while these data are made up, they are realistic: Many small, rural hospitals have higher survival; rates than large, urban hospitals because they handle less difficult cases.
A graphic that we have found helpful for explaining the Simpson's paradox phenomenon will appear HERE.
Activity 7-5 addresses the concept of independence more directly. Question (a) asks students to create a two-way table from raw data, as in Activity 7-1, but using technology this time. If this is not important to you, you may want to save time by providing the table for students. You may also want to update the table to use the current composition of the Senate. In the summer of 2001, following Senator Jeffords' switch from Republican to Independent, there are 50 Democrats and 49 Republicans in the Senate. Ten of the Democrats are women, and three of the Republicans are women.
The homework activities ask students to apply these methods and concepts to analyzing more datasets. None requires the use of technology, although it may be helpful in Activity 7-16. Activities 7-6, 7-16, 7-17, and 7-18 deal with data collected in class. Many of these activities start by asking students to identify explanatory and response variables. Relative risk comes up in Activities 7-12, 7-13, 7-14, and 7-19. The concept of independence is emphasized in Activity 7-25, and Simpson's paradox is relevant in Activities 7-20, 7-21, and 7-22.
We particularly like Activities 7-21 and 7-22 in that they assess students' understanding of Simpson's paradox by asking them to create an example that illustrates it. Students who understand the phenomenon well have little difficulty, but those with a partial understanding find these exercises to be very challenging. Activity 7-20 refers to a very famous historical example; it returns in an in-class activity of Topic 24.
Activity 7-23 address a common misinterpretation of two-way tables. It's tempting to conclude here that the test is worthwhile based on the 63 of 100 cases in which a person predicted to stay actually does stay. That questions (a) and (b) produce the same answer, though, is supposed to convince students that the test prediction provides no valuable information. Many students struggle with understanding what the segmented bar graphs requested in (g) and (h) should look like.
Substantial changes from the first edition are primarily in updated and new datasets.
New uses for technology in this topic are to produce scatterplots and labeled scatterplots.
Be sure to bring rulers to class for question 5 of the Preliminaries section. You might want to mark off heights on the board and have students read off their heights from there. This topic provides a nice reminder that you may prefer to have students report this data to you prior to class so that you can have it compiled beforehand and not spend valuable class time on the data collection. Sample data collected on students about the "Preliminaries" questions may be found here.
Activity 8-1 starts by asking students about observational units and variables. While these ideas are fairly simple, they are crucial to reinforce in students' minds. The activity goes on to introduce the important idea of association and the scatterplot as a visual display of the association between two quantitative variables. You might want to point out that when students are asked for a scatterplot of A vs. B, the intention is for A to be on the vertical axis and B on the horizontal. You might also make students aware that the data tables in the books sometimes put the response variable first (as in this activity) and sometimes put the explanatory variable first (as in Activity 10-1). Question (f) aims to remind students about the idea of a statistical tendency, first encountered with side-by-side stemplots in Activity 6-1, and to point out that association is a tendency.
Activity 8-2 gives students practice with judging the direction and strength of an association from a scatterplot. When we encounter students having trouble with this activity, we often ask them first to distinguish the positive from the negative associations and then to concentrate on the strengths. You might want to tell students that they will revisit this example in Activity 9-1 when they encounter the correlation coefficient as a numerical measure of association. We tell students not to worry if their answers are off by one cell (say, switching moderate negative with least strong negative). Question (b) tests whether students can think more generally about direction and strength of associations. Many of the examples listed appear later.
Activity 8-3 tries to show that with paired data, one can learn much by inserting a "y=x" line on the scatterplot. Some students are confused by the fact that differing scales on the axes prevent the line from appearing at a 45 degree angle. Users of the first edition will note that we call it the "y=x line" and not a "45 degree line" in this edition.
Activity 8-4 gives students the opportunity to analyze genuine data from an important, still fairly recent, historical event. You may want to show students how to use technology to create scatterplots at this point. It may be helpful to remind them here that the convention for plotting A vs. B is to put A on the vertical axis and B on the horizontal. The moral in (c) and (d) is that one loses a great deal of information by discarding flights with no O-ring failures, for all of those flights occurred at relatively high temperatures.
Activity 8-5 shows that one can produce a labeled scatterplot to incorporate information from a categorical variable into a scatterplot.
The homework activities ask students to apply these methods and concepts to analyzing more datasets. Those that require use of technology are Activities 8-6, 8-9, 8-11, and 8-13 through 8-17. Technology would also be helpful in Activities 8-12 and 8-19. Students interpret scatterplots provided in Activities 8-7, 8-8, 8-10, and 8-18. Activity 8-7 is intended as a follow-up to Activity 8-2.
Substantial changes from the first edition include new and revised datasets and a new activity that introduces students to the effect of outliers on correlation through real as opposed to hypothetical data. Some activities have also been reordered: the calculation of correlation now precedes the activity concerning association and causation.
New uses for technology in this topic are to calculate correlation coefficients. Students also use technology for a guessing exercise involving pseudo-random bivariate normal data.
Activity 9-1 leads students to discover the basic properties of correlation. It starts with the same data and scatterplots that students analyzed in Activity 8-2, as it tries to convince students that their impressions about direction and strength of association follow what the correlation coefficient reveals. Notice that students do not work with the calculation of correlation here; they use technology to do the calculations so that they can concentrate on correlation's properties. You might want to go over questions (c)-(f) with the class as a whole to make sure that everyone has the right ideas here; we recommend doing this only after students have thought about the questions and written their own reactions to them. You might also want to make sure that everyone understands the morals of the last two questions: that correlation measures only linear and not curvilinear relationships and that clustered of data may have a strong correlation even when the data within each cluster have a weak association. Users of the first edition will notice that questions on the effect of outliers on correlation has been moved to a homework activity (9-7).
Activity 9-2 does introduce students to the effect of outliers on a correlation coefficient, in the context of variables from the Monopoly board game. Some students get anxious/lazy in (c) and want to go straight to the correlation without producing a scatterplot, but we encourage them to start with the scatterplot and then make a guess for the correlation value.
This is another activity that is particularly well-suited to the dynamic graphical capabilities of Fathom.
Notice that the formula for calculating a correlation coefficient appears at the end of this activity. The formula presented is in terms of z-scores and is not the typical formula used for computational purposes.
Activity 9-3 steps students through (finally!) the calculation of the correlation coefficient. We view this as much less important than understanding its properties, but we want students to see the formula nonetheless. Notice that the book does much of the work for the students, asking them only to fill in a few missing z-scores and cross-products. In question (c) we want students to recognize that the strong negative association causes most positive z-scores for weight to be paired with negative ones for MPG.
Activity 9-4 is one of my favorites and most successful. It guides students to the realization that a strong association between two variables does not imply a cause-and-effect relationship between them. Since the context is so ridiculous, almost no student has any difficulty in seeing that a causal explanation is not appropriate here. We argue that this is one of the very most important concepts for any statistics student to understand, so we try to make sure that every student can explain to me in their own words the moral of this activity.
You may also want to point out that since the association here is very non-linear, the correlation coefficient is not a revealing measure of association for these data. Nevertheless, the point that association does not imply causation is still valid. Students return to these data in Activity 11-3 when they study transformations to achieve linearity.
Activity 9-5 is designed to give students practice judging the value of a correlation coefficient based on a scatterplot. Students seem to have a lot of fun with it as well. Some get into contests with their partners, and others prefer to work together with their partners. We like to go around the room and guess along with the students.
For this activity students need to use slightly more sophisticated technology. We have written a Minitab macro, a TI-83 program, and a Fathom document to generate the "pseudo-random" data. The idea is to generate data from a bivariate normal distribution where the variables have equal means and standard deviations and where the correlation coefficient rho is chosen from a uniform distribution on the interval (-1,+1). A Java applet that implements this activity is available here.
In question (b), students invariably underestimate the value of the correlation coefficient between their own guesses and the actual correlation values. They are pleased to find that the actual value of this correlation exceeds their expectations, often considerably. Questions (f)-(h) lead students to question the validity of judging their guess accuracy from this correlation, though.
The homework activities for this topic ask students to explore further properties of correlation and to apply it to new datasets and situations. Activities 9-6 and 9-7 involve properties of correlation: 9-6 tries to help students realize that the slope of the line does not affect the value of the correlation, and 9-7 provides further experience wit the effects of outliers. Both of these require technology. Activity 9-8 tests whether students can anticipate the direction of an association based on data values without benefit of technology. Activities 9-9 through 9-14, 9-17 and 9-18 all require technology as students apply correlation analysis to various datasets. The distinction between association and causation is the focus of Activities 9-15 and 9-16. You might want to caution students before they work on any of these activities that it is good practice to always look at a scatterplot before computing a correlation.
Substantial changes from the first edition include approaching regression by first considering the mean value of the response variable as its predictor. This helps to lead into the interpretation of r2 as well as motivating regression in the first place. This topic also includes a new activity that cautions students against interpreting r2 without first looking at a scatterplot of the data.
New uses for technology in this topic are to calculate least squares lines, along with resulting fitted values and residuals. Drawing the least squares line directly on the scatterplot is another use of technology in this topic.
Activity 10-1 tries to get students thinking about the basic idea of using a line to summarize the relationship between two variables and the potential usefulness of that idea for making predictions. Questions (a)-(c) aim to convince students that in the absence of further information, the mean airfare is the best prediction to make, but one can use knowledge of distance to make a better prediction for airfare. You might want to indicate that there is no "right" answer in (d) for which line best summarizes the data. Depending on their algebra background and knowledge, many students might need some help with finding the slope and intercept of their line in (g) and (h. In question (i) we vehemently insist that students get used to using variable names rather than generic x and y symbols when writing the equation of a least squares line.
You might want to point out the "y-hat" notation that we use with least squares lines. The carat symbol ("hat") is just to clarify that the line produces a predicted value for the y-variable and not an actual data value. You might also make clear that we use the terms "least squares line" and "regression line" interchangeably throughout the book. Also notice that the formulas given for the least squares estimates of the slope and intercept coefficients are in terms of the means, standard deviations, and correlation between the two variables. We do not provide the computational formulas, as we expect students to use technology to calculate these estimates. We do hope that the formulas given provide some insight into least squares lines.
Students cover a lot of ground in Activity 10-2, where they apply the least squares criterion to the issue of selecting a regression line to fit the data. Some students may need considerable one-on-one help to work with the formulas in (b). You'll want to show students how to use technology to find the regression line in (c). This activity is another case where round-off errors can arise and confuse some students. In (d) and (e) we expect students to use the equation of the least squares line to calculate the predictions, not just to estimate predictions visually. Question (i) aims to warn students of the danger of extrapolation. Round-off errors committed by students can affect questions (j) and (k), which try to illustrate the interpretation of the slope coefficient.
Activity 10-3 introduces the big ideas of fit and residual and leads students to the interpretation of r2 as the proportion of variability in the y-variable explained by the least squares line with the x-variable. We intend students to answer (a) and (b) based on the regression equation but to address (c) based solely on the values given in the table; in this way students should come to better understand the relationship between fitted values and residuals.
Questions (g)-(k) lead students to look at r2 as the proportion of variability in air fares explained by knowing distance. This is a very difficult idea for students to understand. The approach here is to compare the sum of squared residuals from the regression line to the sum of squared deviations from the mean airfare; in other words: compare the squared prediction errors from the line to the squared prediction errors from the mean. You probably want to make sure that students know how to use technology to calculate sum of squared errors in (j).
Activity 10-4 tries to provide a cautionary example illustrating that it is always important to examine visual displays of data. In this instance both regression lines have very similar values of r2, but the regression model is clearly appropriate for one case and not the other.
The homework activities for this topic ask students to apply regression ideas to a variety of datasets. Activities that require use of technology are 10-6 through 10-9, 10-11, and 10-13 through 10-16. Students practice using the formulas for slope and intercept coefficients with Activities 10-5 and 10-10; they are asked to investigate some consequences of these formulas in Activity 10-18. Activity 10-17 is one of my favorites; we have used parts of it on exams as well. We also particularly like part (d) of Activity 10-16, which revisits the issue of association vs. causation.
Substantial changes to this topic from the first edition, in addition to new and updated datasets, include a new activity introducing students to residual plots.
New uses of technology in this topic are to produce residual plots and perform transformations.
Activity 11-1 starts off in (a)-(c) with a review of basic regression ideas that students learned in Topic 8. Question (d) breaks new ground by asking students to look at a scatterplot of residuals vs. longevities. This scatterplot reveals a "megaphone" pattern, indicating that the line better predicts gestation periods for animals with shorter longevities than for animals with greater longevities. Students should realize in (e) that the elephant is an outlier in both variables but does not have the biggest residual. Questions (f)-(h) guide students to find that the giraffe has the largest residual because its gestation period is much longer than expected, but removing the giraffe from the analysis has little effect. Questions (i)-(k) reveal that removing the elephant does have a substantial effect on the analysis, thus introducing students to the idea of an influential observation.
Activity 11-2 is new in this edition, another example of a "matching" activity. You might want to help students realize that the residual plots are just rotations of the original scatterplots, with the rotation serving to make the regression line into a horizontal "mean residual = 0" line. Encouraging students to draw this horizontal line on the residual plots may be helpful to them. Part (b) then asks them to do another matching, this time between the residual plots and verbal descriptions of them. The key point for students to grasp is that patterns in the residual plot indicate nonlinearity in the original relationship.
Good students occasionally ask why residual plots are necessary, since they can see the nonlinearity in the original scatterplot. My typical answer is first that the nonlinear relationship may be subtle enough that it's easier to detect in the residual plot than in the original scatterplot. Perhaps more importantly, residual plots are especially useful with multiple regression involving more than one predictor variable.
Students explore the idea of data transformations, one of the more challenging mathematical ideas to appear in the book, in Activity 11-3. Transformations are a natural answer to the question "what do we do when we spot a nonlinear relationship" that arises in Activity 11-2. Some students will need help understanding how to calculate logarithms in (b). In question (d) you might want to remind students to use "log ( people per tv )" when they write out the regression equation. Question (f) indicates how to interpret the slope coefficient with this log transformation. Students should discover a much better fit with the transformed data than with the original data.
Most of the homework activities for this topic ask students to use technology to analyze data and further investigate these regression issues of outlier, influence, residual plots, and transformations. The only homework activities not requiring technology are 11-13 and 11-15. Activity 11-13 reminds students of the danger of extrapolation, and Activity 11-15 provides a residual plot for students to interpret. Students examine transformations in Activities 11-4 and 11-5. Outliers and influence are the subjects of Activities 11-6, 11-7, 11-11, and 11-12. Residual plots are a focus of Activities 11-9, 11-10, and 11-15. Activity 11-14 helps students to see that inverting solving a regression equation for the explanatory variable does not necessarily produce the regression equation with the variable's roles switched.
Most of the material on randomness in this and following topics is presented in terms of simulations rather than probability rules. We try to help students to develop an intuitive sense for properties of randomness, particularly for the effect of sample size. We also try to relate the study of randomness immediately to concepts of statistical inference, namely confidence and significance, that come later in the course.
The primary goals of this topic are to convince students of the need for random sampling and for students to begin exploring properties of randomness. Substantial changes in this topic from the first edition, in addition to updating the U.S. Senate list, are an activity (12-3) highlighting the concept of bias and one (12-6) illustrating that population size has little effect on sampling variability.
Questions 5 and 6 of the Preliminaries collect some data that students are to analyze in a homework activity, so you might want to decide whether you want to assign those activities before you bother to collect the data. Depending on how e-mail savvy/addicted your students are, you may want to alter the questions, perhaps to cover the last day instead of the last week or the last 4 hours instead of the last 24 hours, in an effort to produce more variability in the responses. Sample data collected on students about the "Preliminaries" questions will appear HERE.
New uses of technology in this topic include generating random samples from a population.
Activity 12-1 simply introduces students to the terms population and sample. You may want to stress that they have studied data from both so far but that most of the rest of the book will focus on analyzing samples in order to infer knowledge of populations. We suggest that you quickly discuss this activity with the class, and make them aware in no uncertain terms that appreciating the distinction between population and sample is absolutely critical to understanding statistical inference, which is the subject of much of the remainder of the course.
Activity 12-2 provides some examples of biased sampling designs; you might want to cover these questions as a class discussion. For the Literary Digest example, we try to get students to identify at least two major sources of bias: that owners of cars and phones during the Depression tended to be more wealthy and Republican and that those who take the time to write in are typically less happy with the status quo incumbent than those who decline to write in.
Activity 12-3 is a new one that aims to help students understand the concept of bias by collecting some data from themselves that illustrate bias. The idea is that when students choose a sample of five Senators whose names are familiar, the samples will most likely contain more experienced Senators and therefore be biased in the direction of overestimating the mean years of service in the Senate. We ask students to answer (b) without looking back at the list, but many students complain that they do not know the names of five Senators. After they have made an honest effort, we tell them to go ahead and peek at the list but that they should only write down names they have heard of. To combine the sample means in (h), we either have students come to the board and mark their mean value on a dotplot or else have them call out their mean value while we enter them into technology. If your results are typical you should find that most students' means exceed the population mean value, often by a lot. We try to emphasize to students that getting a few samples that over- or under-estimate a population value is not evidence of bias, but rather having a systematic such tendency is evidence of bias. You might impress upon students the importance of question (k)- that increasing the sample size does not help to reduce bias.
Having encountered a biased method of sampling Senators, Activity 12-4 asks students to avoid bias by selecting a simple random sample, using a table of random digits, from the 100 members of the U.S. Senate. We don't think you can emphasize strongly enough that this is one of those rare situations in which one actually knows details of the population. We usually describe for the entire class how to read the table of random digits and then let them take their samples individually, encouraging them all to enter the table at different points. You might want to remind students that information for the population is given earlier in the activity. Since it's impossible to find 45% Democrats in a sample of 10, all students should answer "no" to the first part of question (d). In question (e), though, we hope that they recognize that this "no" response does not mean that the sampling method is biased in the sense of systematically favoring one group over another. This should be revealed by the dotplot of class results in (f), where you should find about the same number of sample means on the low as on the high side of the population value. You might want to show students this dotplot and the one from Activity 12-3 on the same scale so that the bias, and lack thereof, is more clear.
We emphasize to students that the definitions of parameter and statistic following question (g) are crucial to understanding the material in the rest of the course. While students usually find question (i) to be easy and routine, we caution them that the distinction trips up many students who do not pay proper attention to it. We think it's also worth repeating at this point that this situation is a rare one in which you know the values of population parameters.
Activity 12-5 continues with random sampling of Senators but lets technology take over the sampling and increases the sample size to 20. You'll want to explain to students how to use technology to do this repeated sampling. One option is to use a Minitab macro that we have written; another option is to use the applet available here. You might need to remind some students to record the sample proportion (not number) of Democrats. In (d) and (e) students should find that the sample means are less variable with the larger sample size, so in (f) they should say that the result of a single sample is more likely to be close to the population value with a larger sample. You may want to emphasize in your wrap-up that while sample size does affect precision, it does not affect bias.
Activity 12-6 has a very specific aim: to help students to realize the counter-intuitive principle that while the sample size directly affects the amount of sampling variability, the population size does not. This, a simple random sample of size 1500 provides just as much information about the population of the entire U.S. as it does about the population of any one state. Questions (a) and (b) try to help students to see this by asking them to take samples of size 5, first from a population of 100 and then from a population of 10,000. This is an extremely difficult point for students to accept, but we feel that it is important to emphasize because one so often encounters national surveys of 1000-1500 people.
The Gallup organization has a wonderful web site where you can find recent survey results and also a very nice description of many ideas related to sampling.
The homework activities for this topic ask students to investigate these sampling issues further. Activity 12-7 emphasizes the definitions of parameter and statistic, while Activity 12-8 asks about whether the sample of students in class is representative of various populations. Activities 12-10 through 12-14 present various studies and ask students to comment on the sampling methods. Activities 12-15 and 12-16 involve using a table of random digits. Activity 12-18 is the only one requiring technology, as it asks students to analyze sample proportions of Democrats under repeated sampling, much as they analyzed sample mean years of service in class. Activity 12-20 raises the issue of people's truthfulness, and Activity 12-21 introduces other common types of nonsampling bias. Activities 12-22 through 12-24 concern another type of bias, while Activities 12-25 through 12-27 present other more sophisticated methods of probability sampling.
The primary goals of this topic are to help students become aware of the need for well-designed experiments and to familiarize students with some of the key concepts and techniques associated with experimental design. We recommend that you consider leading a class discussion through much of this material in order to keep students roughly at the same point and so that you can interject often to clear up common questions.
Many changes from the first edition appear in this topic. First, this topic has been moved to much earlier in the course. It consists of completely new activities, designed to focus students' attention on specific concepts that many find difficult to grasp. The first activity tries to introduce students to the various types of studies and to the virtues of controlled experiments. The second activity concentrates on the difficult concept of confounding, the third on the concept of randomization, and the fourth on blindness. The fifth activity is the only holdover from the first edition, and the sixth activity asks students to study the concept of blocking.
The Preliminaries provide a convenient point at which to remind students that they should be serious and thoughtful in addressing these questions but that they are not expected to be able to answer them as knowledgeably as if they had already studied the topic. For question 5, we have in mind a very small experiment to be done using students as subjects. You should prepare strips of paper containing letter sequences with hyphens separating groups of letters. Half of the strips should say JFK-CIA-FBI-USA-SAT-GPA-GRE-IBM-NBA-CPR, and the other half should say JFKC-IAF-BIU-SASA-TGP-AGR-EIB-MN-BAC-PR. Randomly give out one version to half the students and the other type to the rest. Tell students not to look at the letters until you ask them to, and then give them twenty seconds to memorize as many as they can. Then have them write down as many letters, in order, as they can remember. Finally, have them look at the strip and determine how many letters they remembered correctly before making an error of any kind. Then collect data on the results, perhaps by asking them to write their score on their strip and turn it in. Notice, of course, that the two letter sequences are identical. The conjecture is that students who receive the sequence in convenient three-letter chunks will tend to memorize more letters correctly than those given the other version.
Sample data collected on students about the "Preliminaries" questions will appear HERE.
No new uses of technology appear in this topic.
Activity 13-1 presents four different ways of collecting data to address a question, defines terms related to the data collection strategies, and asks students to identify which studies go with which terms. Anecdotes are included to try to make clear to students that "data beat anecdotes," as David Moore puts it. You probably can not stress this point enough. Question (c) tries to establish that surveys are a step up from anecdotes but only present people's opinions and perceptions. We encourage students to get ion the habit of drawing diagrams as in (f) to describe experimental designs. Questions (g) and (h) are key for recognizing the advantages of controlled experiments over observational studies. We strongly recommend that you emphasize this point through class discussion. Question (i) establishes the principle of comparison as one of the principal ways of achieving control in an experiment.
Activity 13-2 is another example of an activity where we abandon our principle of using real data and instead use hypothetical (but realistic) data designed to make a very specific point. The goal here is to study the concept of confounding, a very difficult one for students to grasp. We suggest that you work through this activity with the class as a whole so that you can explain what's going on every step of the way, but we still recommend that you force students to be answering the questions on their own first before you discuss them.
In (c) students should report that students in the "foreign language" group did tend to do better on the SAT, but in (d) they should offer non-causal explanations for that. For instance, one could surmise that students with stronger verbal ability in the first place tend to be the ones who choose to study a foreign language, so they would do better on the SAT even if the foreign language study had no effect. Questions (e)-(g) ask students to investigate this claim by providing (hypothetical) data on these students' scores on a verbal aptitude test prior to their foreign language study. It does turn out that those with high verbal aptitude to begin with are precisely those who study a language, and so prior verbal aptitude and foreign language study are confounded. You may or may not want to mention the subtle distinction between a lurking and a confounding variable: a lurking variable is simply one that is not recorded and so may or may not be confounding, whereas a confounding variable is one whose effects can not be separated rom those of the explanatory variable. Questions (h) and (i) try to convince students that gender is not confounded with foreign language study because 5 students of each gender studied a language and five did not. (Qian is intended to be a female name.)
Activity 13-3 suggests a solution to the problem of confounding whenever one has the luxury of assigning subjects to treatments: randomization. This activity tries to convince students that randomly assigning subjects to one group or the other really does tend to balance out potential effects of lurking variables. In (c) you should let students know whether you prefer one method of performing the randomization. You might want to have students compare their results in (d) with each other so that they can see the balancing out; you might even have students produce a dotplot of the difference in the group means to see that those differences are roughly centered at zero. Question (e) is an important one, as it tries to convince students that randomization not only controls variables that you can measure but also ones that are too difficult to measure.
Note in Activity 13-4 that we first ask students to identify variables and produce a graphic of the design. We believe strongly that these are good habits to encourage in students. This activity raises the issue of blindness in the context of a very controversial study. We encourage you to lead a discussion of students' reactions to the controversial nature of this procedure.
Activity 13-5 asks students to pull together the three principles of comparison, randomization, and blindness by writing about how all three were implemented in an experiment that they have already analyzed. If you have been leading a class discussion of the earlier activities, we suggest stopping that at this point to let students write about this experiment.
Activity 13-6 addresses another very challenging concept for students to grasp: blocking. The point here is that randomly assigning cars to treatment groups is a reasonable idea, but since MPG ratings differ substantially between small and luxury cars it makes more sense to stipulate that luxury cars and small cars will each have a 50/50 split between the two groups. The sensibility of this plan is seen in that there is much less variation in the group mean differences when one first blocks on car type as opposed to assigning cars to groups completely at random.
The many homework activities for this topic lead students to consider these ideas in greater depth and to apply what they have learned. These activities present descriptions of studies to students and typically ask them to identify the variables involved, describe the design of the study, and comment on potential limitations to the conclusions that can be drawn. Activities 13-7, 13-9, 13-12, 13-13, 13-15, and 13-16 describe observational studies, while Activities 13-10, 13-11, 13-14, 13-17, 13-21, and 13-23 present experiments. Activities 13-10, 13-13, 13-17, and 13-21 ask students to analyze some data, with Activity 13-17 pertaining to the memory experiment from the Preliminaries section. Activities 13-26 and 13-27 ask students to go out ad conduct small-scale experiments. None of these activities requires use of technology, though it could be helpful in Activity 13-17.
New uses of technology in this topic are to perform probability simulations, specifically to simulate observations from a binomial distribution.
We often start by apologizing to students for the context of Activity 14-1 and saying that we hope noone finds it objectionable. Students seem to get a kick out of this version of the famous "matching problem" or probability more than they would if it were presented as, say, a group of men throwing their hats into the middle of a room and selecting one at random.
We try to pass out four index cards to each student before class begins. Most students catch on quickly to how to do the simulation in (a) and (b). The table in (c) has a lot of information, so you might want to collect data on just one of the two variables; we recommend the "all wrong?" variable since its probability is not as straightforward to assess. You can either record the data for (c) on the board as students call out their results, or have them put their results on the board themselves, or you might enter it into your technology directly as they call out their results.
For collecting the data in (f), we ask students to go to the board and put tally marks in the appropriate cell. We often make the column for "3 matches" extremely narrow and ask students to explain why we can get away with that. In (k) and (l) we begin to anticipate the issue of "unlikeliness" that drives p-values and test of significance. We hope that students will say that getting four matches is quite unlikely but getting zero, one, or two matches is not at all unlikely.
Activity 14-2 extends students' analysis of the "random babies" situation to a more theoretical analysis. Since the sample space involves only 24 outcomes, it is not too cumbersome to have students analyze the entire sample space. We do provide a listing of that sample space, though, in order not to slow students down. We let students work through (a)-(d) at their own pace but then interrupt them so that we can compare answers for (d) and make sure that everyone is in agreement there. You may want to emphasize that the analysis here depends upon assuming that the 24 possible outcomes are equally likely, which is what it means to say that the babies are returned to their mothers "at random." You may want to caution students in many situations it is not appropriate to assume equal likeliness and that, in fact, this is a common misconception that leads to serious errors. A particularly egregious example concerns an earthquake "expert" who stated on Nightline that since an earthquake either happens or doesn't, its probability is always 50%.
With luck you'll be able to highlight for students that the empirical probabilities from 14-1 closely approximate the theoretical probabilities in (d). Question (e) is meant to show students that empirical estimates generally get closer to the theoretical probabilities as the number of repetitions in the simulation increases. Questions (f) and (g) get at the idea of expected value. As students work through (f) we try to convince them that the average here is calculated just like always: summing the values and dividing by the number of values. Since there are so many repeated values, though, it's easier to take each value times its frequency and add those up for the numerator. You might show students the algebra to see that this is equivalent to taking each value times its relative frequency of occurrences and diving by the number of repetitions; this provides the analogy to working with the theoretical probabilities in (g).
One of the challenges of teaching probability ideas largely from a simulation standpoint is that students may not recognize when simulation is useful as an analysis tool to approximate probabilities and when it is intended merely as a pedagogical tool to help them to visualize what would happen in the long run. In these activities it serves both purposes.
Activity 14-3 tries to show students that probability refers to long-run outcomes, so in the short run it is very difficult to distinguish among similar probabilities. Many students are reluctant to make guesses in (a) and (c), but we try to insist that they do so just to drive home the point that it's hard and that mistakes occur in the short run. Once the coins have been flipped 50 times, it's pretty easy to tell which is which.
Activity 14-4 leads students into more efficient techniques of simulation while trying to develop more of their intuition for outcomes of chance events. Using the random digit table to simulate boy/girl births comes easily to many students, but others will need some guidance. Even though students typically work in pairs, we insist that partners use different lines of the random digit table here. Combining the results in (c) can be tricky if students have worked at varying paces and so arrive here at very different times. We again have students put tally marks of their outcomes on the board to do this compiling. We like to draw students' attention to (e), where they should realize that a 3/1 split is more likely than a 2/2 split: even though a 2/2 split is more likely than either of the two 3/1 splits, the two 3/1 splits combined are more likely than a 2/2 split.
Question (f) is another good example of our not minding if their initial guess is wrong, provided that they go back and correct it in (g) if they were wrong. We do care, though, that they make a reasonable attempt to think through (f) before doing (g). Some students tend to get lazy and not bother with the initial guess/expectation, but we want them to be thinking about the issue before they conduct the simulation.
Also in (g) you want to be sure that students know how to do this simulation using their technology. Even though it' not asked for, it's good practice to have students create a histogram of their simulation results as well as tallying the results. They should find that a four-child family is more likely to have a 50/50 split than a ten-child family. To help them understand this, you might ask if they literally expect to get exactly 5000 heads if they flip a fair coin 10,000 times.
Activity 14-5 continues students' exploration of sample size and confronts another misconception. Students should see that the hospital with the smaller sample size is much more likely to have days in which 60% or more of its births be boys, while the larger hospital is much more likely to have days with between 41% and 59% boy births. You might emphasize that this reveals that larger sample sizes tend to produce more typical results.
The homework activities for this topic aim to help to increase students' understanding of fundamental ideas of probability and to further prepare them to study statistical inference. Activity 14-6 follows upon the "random babies" activity, and Activity 14-7 asks students to consider when the equal likeliness assumption is reasonable. We have found that students have surprising difficulty with 14-7. Activity 14-10 is a favorite of mine, because more than providing a probability problem to be solved with a counting argument, it introduces the logic of significance testing by asking how likely an observed result (both officers women) would have been under a hypothesis of random selection. Students are asked to perform simulations in Activities 14-11 through 14-15. All of these simulation exercises involve principles of randomness as well, ranging from sample size effects to runs. Activities 14-14 and 14-15 also manage to include real data. Technology is needed for Activity 14-12 and 14-13, a table of random digits for Activity 14-11 and 14-14, and students may use either or a physical device in Activity 14-15.
This topic has undergone many substantial changes from the first edition. Normal distributions are motivated through both real and simulated data that follow a normal curve. The empirical rule and z-scores are emphasized much more strongly to convince students that normal calculations produce reasonable approximations. Real data on birthweights are used to provide students with practice using the standard normal table, and the idea of normal distributions as models for data is highlighted. Another "matching" activity has been added to help students see how closely sample data from populations correspond to the probability models. The algebraic notation associated with normal probability calculations has been greatly reduced from the first edition. Extensive practice using the standard normal table without a context has been eliminated.
Another change is that this study of normal distributions occurs earlier than it did in the first edition. The new "Probability" topic provided a natural segue to normal distributions. In the first edition normal distributions were not introduced until after students studied sampling distributions via simulation.
This topic lends itself to working through with the class as a whole, at least through the basics of reading the table and standardizing. We try to keep the class pretty much together as we work through this topic, as opposed to other topics where students work much more at their own pace. As always, though, we strive to have students work through a question themselves before we present the solution to them. We sometimes ask students to present their solutions for their peers in this topic.
You might draw students attention particularly to questions 3 and 4 of the Preliminaries, as they address statistical issues relevant to working with normal distributions.
Depending on how much you want students to read a table of normal probabilities, a new use for technology in this topic could be performing probability calculations involving normal distributions.
Activity 15-1 tries to motivate the normal model by reminding students that they have encountered mound-shaped distributions both with real data and with simulated sampling distributions. It also highlights the importance of z-scores and introduces use of the standard normal probability table.
In (a) we expect students to comment on the mound shapes and symmetry of the distributions. Question (c) indicates how to read the mean and standard deviation of a normal curve from its sketch. Questions (d)-(k) try to establish that comparing z-scores puts all normal distributions on a common scale, while questions (l) and (m) try to convince students that the probabilities given in the standard normal table really do approximate empirical probabilities from real data that can be modeled with a normal curve.
You should make sure that all students can read the normal table correctly in (l). You might tell them that they will need to use this skill often throughout the remainder of the course.
Activity 15-2 gives students practice with the necessary skills for reading the standard normal table. We insist that they shade in areas under the curve as well as reading the table here, and we also want them to make guesses for the probability based on the shaded area. We try to convince them that this habit helps them to keep calculations straight in their mind when they become trickier, and it also provides a way to check the reasonableness of their final answers. Question (d) involves nothing more than looking up the tabled value. In (e) students should subtract the tabled value for 2.34 from 1, and in (f) they should note that they also could have taken advantage of symmetry by looking up -2.34 directly. Question (g) is trickier, but many will figure out on their own that they need to look up tabled values for the two z-scores and subtract; some will get in the bad habit of subtracting the z-scores themselves rather than their associated tabled values. Question (h) is meant to convince students that the normal model does produce reasonable approximations. One exercise that should have been included here is one where the z-score is off the table and where the probability is therefore less than .0002 or greater than .9998; we recommend that you give students such a problem or at least tell them what kind of answer you would expect in such a circumstance.
Questions (i) and (j) ask students to read the table "in reverse," to find values that generate certain probabilities above or below. Again we insist that they first draw a sketch to get a sense of the situation and what they're looking for. Many students struggle with this, particularly with the algebra of solving for the value once the z-score has been read from the table..
Activity 15-3 tries to show students the relationship between the probability model for a population and the pattern in sample data resulting from that population. With the larger sample size of 100, students have little difficulty in matching up the sample datasets in (a) with the respective populations from which they are drawn. With a smaller sample size of 10 in part (b), this matching is much more difficult. Of course, that's the point: with small samples it's hard to tell the shape of the population distribution from the sample data. This point is an important one in doing statistical inference with small sample sizes, because the t-procedures with small samples depend on having a normally distributed population, but it is very hard to assess the normality of a population based on a small sample.
The homework activities for this topic offer students more opportunities to practice normal calculations and also to discern situations where a normal model is likely to produce reasonable approximations. Activity 15-4 provides practice with determining the mean and standard deviation from a sketch of a normal curve. Activities 15-6 and 15-7 ask students to practice normal probability calculations, with 15-7 being a more open-ended question. The last few questions of Activity 15-9 are challenging for many, as they ask about changing m and s in a production process. Activities 15-11 and 15-12 provide real data with which to compare predictions from a normal model. Activity 15-13 tries to reveal that a normal model can not be appropriate for these coin ages because the distribution must be skewed because the standard deviation is almost as large as the mean and negative values are not possible. Activity 15-14 leads students to discover the empirical rule, and Activity 15-15 introduces the idea of critical values. None of these homework activities requires the use of technology.
We encourage you to make students aware of how critically important the concepts presented in this topic are and how fundamental these ideas are to much of what comes later in the course. We also recommend that you let them know that many students find this material to be fairly difficult. You might try to allay some of their fears by pointing out that they will have multiple opportunities to deepen their understanding of these concepts and that they should focus on the "big picture" with this first pass.
This topic focuses on sampling distributions of sample proportions, and the following topic turns to sampling distributions of sample means. Again we ask students to use simulation as the tool for investigating long-term behavior of sample statistics. we question whether students "see" what we want them to in computer simulations, we favor starting with physical simulations, using candies in this case.
The primary goals of this topic are to familiarize students with the concepts of sampling variability and sampling distribution, to lead them to some important findings regarding the sampling distribution of a sample proportion, and to provide students with a first exposure to the concepts of confidence and significance.
You need to bring Reese's Pieces candies to class for this topic. These candies come in three colors: orange, brown, and yellow. Two or three one-pound bags suffice for a class of 24 or so. You could also use M&M's, but they come in more colors and therefore colors occur in smaller proportions, so the sampling distributions will not come as close to a normal distribution as it does for orange Reese's Pieces with the same sample size. One danger of using candies for this simulation is that while students certainly enjoy eating candy in statistics class, you may need to make an effort to direct their attention to the statistical principles at work here.
As you lead students through the Preliminaries questions, you might point out to students that all of the variables being asked about are categorical. This shift from dealing with quantitative variables indicates that we will work with proportions rather than means as the sample statistic of interest in this topic.
Substantial changes from the first edition include an attempt to introduce the concept of significance as well as that of confidence in this topic. This change is a result of studying sampling distributions of sample means in the following topic, where again both concepts of confidence and significance will be investigated.
There are no new uses of technology in this topic, although it is used more extensively than before for performing more simulations based on the binomial distribution.
Activity 16-1 reminds students of the crucial distinctions between population and sample, parameter and statistic. We suggest that you lead the class through this activity as a group because much of the next activity needs to be done together.
Activity 16-2 leads students through candy sampling, which should be done together since question (g) calls for the pooling of class data. We scoop out more than 25 candies for each student and then ask them to count out a sample of size 25 and then count how many they have of each color. We invite them to dispose of the excess candies in an appropriate manner. You might want to distribute the candies as they enter class in order to save some time.
Questions (b)-(e) are perhaps obvious but nonetheless critical to understanding what statistical inference is all about, so we suggest that you make those answers exceedingly clear to students. For question (g) we go around the room and ask students to report their sample proportion of oranges while we create the dotplot on the board; you could also type the values directly into technology or have students put their values on the board's dotplot themselves. We recommend leading a discussion of these questions with the entire class. In question (m) you might admit that the question is quite vague with phrases such as "most" and "reasonably close" and "some" and "way off," but students are to focus on the big idea and not the details here. You might also tell them that the next activity returns to quantify and formalize these questions.
Activity 16-3 moves to using technology to simulate the same process many more times and much more efficiently. We have written a Minitab macro called "reeses.mtb" to do this, but it amounts to little more than sampling from a binomial distribution. We have also developed a Java applet available here for simulating the sampling of Reese's Pieces that students seem to find appealing.
We don't think you can emphasize enough to students that we have to specify a certain value for the population proportion in order to make the computer run the simulation. Based on years of experience with thousands of Reese's Pieces, we believe that the population proportion of orange candies is slightly less than 50%.
In question (b) students should see a distribution that is roughly normal, that is centered around the actual population proportion of .45. Question (f) is key to the notion of confidence; many students struggle to see that the answer to (f) is the same as the middle percentage in the table of (e). We stress to them that understanding (f) is the key to understanding one of the two critical concepts of the rest of the course.
Questions (i)-(m) investigate the effect of sample size. In question (j) be prepared to warn students that the scale on the display has probably changed, making the difference hard to see, but that the distribution is indeed less spread out than before. Students should discover in (k)-(m) that the larger sample size produces more samples with proportions close to the population value.
Questions (n)-(p) use the empirical rule to introduce the idea of 95% confidence more explicitly. Questions (q) and (r) ask students to verify that the familiar expression for the standard deviation of a sample proportion is reasonable, based on its closeness to their simulated findings. Some students tend to become enamored of this standard deviation formula as if it specifies the entire sampling distribution; we try to remind them that the shape and center of the sampling distribution are just as noteworthy.
Activity 16-4 continues students' study of sampling distributions in a context where the concept of statistical significance is the relevant concept. Many students struggle mightily with the reasoning of test of significance, and this activity is meant as a gentle first pass to acquaint them with that reasoning process.
You might want to ask students to take an on-line ESP test and gather data from the class. One such test appears here, although it offers five choices of shape rather than four. You could discuss why it's probably not reasonable to conclude that the student who achieves the highest score has any particular ESP ability.
The homework activities for this topic provide experience with sampling distributions for a sample proportion. They also extend some of what students learn through in-class activities. Those that require technology for simulations are 16-9, 16-11, 16-12, and 16-15.
Activity 16-5 provides students with lots of practice distinguishing parameters and statistics; many of the examples listed appear later in the book. Activities 16-6 through 16-8 ask students to investigate various aspects of the Central Limit Theorem for a sample proportion: 16-6 and 16-8 reveal that the standard deviation of p-hat is largest when theta is to .5, and 16-7 leads students to see that the standard deviation of p-hat decreases by the reciprocal of the square root of the sample size. Activity 16-9 reinforces ideas related to sample size and sampling variability, using simulation as well as the CLT. Activities 16-10 through 16-12 and 16-15 touch on the concept of significance, while 16-13 and 16-14 concern confidence.