Journal of Statistics Education v.5, n.2 (1997)
Robert C. delMas
University of Minnesota
333 Appleby Hall
Minneapolis, MN 55455
William P. Peterson
Department of Mathematics and Computer Science
Middlebury, VT 05753-6145
This column features "bits" of information sampled from a variety of sources that may be of interest to teachers of statistics. Bob abstracts information from the literature on teaching and learning statistics, while Bill summarizes articles from the news and other media that may be used with students to provoke discussions or serve as a basis for classroom activities or student projects. We realize that due to limitations in the literature we have access to and time to review, we may overlook some potential articles for this column, and therefore encourage you to send us your reviews and suggestions for abstracts.
by Marny Frantz and Sylvia Lazarnick (1997). Mathematics Teacher, 90(4), 274-275.
The authors describe a class activity that uses real data of high interest to teach students about linear regression. The data come from contestants' times for different legs of a 100 mile "ultrarace" held in Colorado called the Hardrock 100. Students use information from the first half of the race to make predictions for each contestant's final time to complete the race and compare the estimates with the actual times. A description of the activity and an example of a dataset are included in the article.
by Charles Vonder Embse (1997). Mathematics Teacher, 90(5), 404-408.
The article describes an activity designed to help students develop an understanding of how minimizing squared residuals determines the line of best fit. Illustrations are given of how to use software such as Cabri Geometry II to create visual representations of regression lines embedded within scatterplots where the squared residuals are represented on-screen by geometric squares. As students drag the regression line to change the slope and y-intercept, the size of the geometric squares and the sum of the areas change to provide visual feedback. The line of best fit is obtained when the sum of the areas is at a minimum. A second activity is described that helps students visualize the relationship between the points of the scatterplot and the point defined by the means of the two variables. An appendix includes directions for the least-squares activity with the Geometer's Sketchpad 3, Cabri Geometry II, or TI-92 Geometry.
by James R. Murphy (1997). The American Statistician, 51(2), 155-157.
The author offers practical suggestions to help students in courses where articles on statistical methods are a required part of the reading.
by Franz Loosen (1997). The American Statistician, 51(2), 158-163.
The article describes a physical device the author has constructed to help students visualize concepts related to hypothesis testing. The device presents the state of nature, the state when the null hypothesis is true, and the state when the null hypothesis is false on three separate axes. The author argues that the device helps students because the three points of view needed to understand the conditional reasoning behind hypothesis testing are separated instead of overlapped on a single axis. The device allows the instructor to visually illustrate ideas such as the effect of varying the difference between the hypothesized and true means, and the effect of changing the sample size. Several examples are presented to illustrate how the device can be used to teach other concepts in hypothesis testing.
by Prem K. Goel, Mario Peruggia, and Baoshe An (1997). The American Statistician, 51(2), 164-169.
The article describes a software system developed by the authors in XLISP-STAT that provides computer simulations that allow students to explore probabilistic modeling while minimizing the need for a thorough understanding of advanced mathematical concepts and probability theory. The computer simulations were developed for use in an introductory course on probabilistic modeling for undergraduates in the biological and environmental sciences. The simulation environment allows students to interactively change parameters in real time and provides visual feedback on how the changes affect the sample paths and relevant summary statistics. Two simulation modules are described, one for the linear birth-death-immigration process and a second for the competition process.
by Candace Schau and Nancy Mattern (1997). The American Statistician, 51(2), 171-175.
The authors illustrate how mapping techniques such as graphic organizers and concept maps can be used to help students learn the interrelationships among statistical knowledge and concepts. The article discusses the use of mapping techniques for instructional planning, as a learning tool, and for assessment.
by Joan Garfield (1997). The Statistics Teacher Network, No. 45, 1-2.
The author describes a short summer course she has developed to provide primary grade teachers with the fundamentals of descriptive statistics. The article describes the use of news media, computer technology, group projects, and video to provide precollege teachers with a fundamental understanding of statistics and its applications.
by Murray H. Siegel (1997). The Statistics Teacher Network, No. 45, 6-7.
This article provides details on an activity in which the TI-83 graphing calculator is used to help students generate random samples and sampling distributions of sample means in order to "discover" the Central Limit Theorem. Concepts and understanding are developed through the use of guiding questions and hypotheses that are tested by creating graphs on the TI-83. Complete instructions are provided for running the simulations.
A regular component of the Teaching Bits Department is a list of articles from Teaching Statistics, an international journal based in England. Brief summaries of the articles are included. In addition to these articles, Teaching Statistics features several regular departments that may be of interest, including Computing Corner, Curriculum Matters, Data Bank, Historical Perspective, Practical Activities, Problem Page, Project Parade, Research Report, Book Reviews, and News and Notes.
The Circulation Manager of Teaching Statistics is Peter Holmes, firstname.lastname@example.org, RSS Centre for Statistical Education, University of Nottingham, Nottingham NG7 2RD, England.
"Murphy's Law of Maps" by Robert A. J. Matthews
Using probability theory, the author shows that the "urban myth" that map locations tend to lie on the edges or down the central crease is based on fact.
"Recognising Randomness" by David Green
The author presents a game called "Submarine" that is designed to help primary school children develop an understanding of two-dimensional random distributions. Readers are encourage to try out the game in their classes and assess whether it has an effect on students' understanding.
"Investigating Probability with the NBA Draft Lottery" by Robert J. Quinn
This article investigates an interesting application of probability in the world of sports. Specifically, it considers the role of permutations in the lottery used by the National Basketball Association in the United States to determine the order in which non-playoff teams select players from the college ranks.
"Modelling the Coupon Collector's Problem" by Wendy Maull and John Berry
The authors describe a dice tossing game that can be used to simulate the coupon collector's problem of trying to determine how long it will take to collect a complete set of coupons, where subsets of coupons from the complete set are sold in sealed packs. Students use a six-sided die to simulate how many tosses it takes to acquire from one to six coupons from a complete set of six. Graphs can be drawn to demonstrate the probability distributions produced by the simulation. The authors also describe how a spreadsheet can be set up to simulate the coupon collector's problem and to generate graphs that illustrate expected results that can be compared to the actual results obtained by the class.
"What Price Statistical Tables Now?" by Neville Hunt
The author illustrates how to use functions in Microsoft Excel to generate the tables required for a school-level study of statistics such as cumulative binomial probabilities, tail probabilities for various test statistics such as Student's t, chi-square, and F, and tables of random numbers.
The Economist, 15 February 1997, p. 80.
About 10% of men and 8% of women are left-handed, and archaeological evidence indicates that these percentages have been stable for thousands of years. By contrast, in species of mammals other than humans, right paw and left paw dominance seems to be evenly split. There are some observed disadvantages to left-handedness in humans. For example, left-handers tend to be shorter and older at puberty than right-handers, and, as reported by some controversial studies in recent years, they don't live as long. Thus the article asks why left-handedness has persisted in the population.
In the December edition of the Proceedings of the Royal Society, French evolutionary biologist Michel Raymond and several colleagues present their hypothesis that left-handers have persisted because they have an advantage in fighting. The researchers studied college athletes at the University of Lyons as well as world-class competitors, and found that left-handers tend to be over-represented in sports where an opponent is confronted directly, such as fencing, boxing, baseball, and tennis. The effect is not found in sprinting or swimming, where contestants compete against the clock.
The findings for world-class athletes were the most striking. For confrontational sports, the lefty effect becomes stronger as the playing distance between opponents decreases. For example, 16.7% of world's top tennis players are left-handed. But from 1979 to 1993, 33% of the men's world foils competitors were left-handed; for those who reached the quarterfinals of competitions, the percentage was 50%. By contrast, in sports that involve the hands, but where there is no tactical advantage to left-handedness, no lefty effect was found. For example, about 10.7% of discus, javelin, and shot-put champions are left-handed, which is comparable to the percentage in the general population.
The researchers suggest that, because of the predominance of right-handedness, right-handers are accustomed to fighting with right-handers. As long as left-handers are comparatively rare, they have the advantage of hitting from unexpected directions. This might confer an evolutionary advantage that balances the disadvantages of left-handedness noted earlier. According to the article, this squares with a legend about the design of medieval castle stairs, which were said to wind clockwise as a defense mechanism. Attackers mounting the stairs were forced to fight with their right arms tight to the wall.
Here are several references on the difficulties of reasoning with conditional probability, starting with a column by Marilyn vos Savant.
by Marilyn vos Savant. Parade Magazine, 30 March 1997, p. 15.
A few years back, Marilyn's column on the infamous Monty Hall problem generated protracted discussion in both the popular press and in academic journals. In a recent column she addressed the following version of the "two children" problem:
A woman and a man (unrelated) each have two children. At least one of the woman's children is a boy, and the man's older child is a boy. Do the chances that the woman has two boys equal the chances that the man has two boys?Her answers -- 1/3 for the woman and 1/2 for the man -- once again triggered some angry letters. In the present column she reproduces some of these letters. Here is one:
I am stunned at your abandonment of good sense in your response to a reader who asked [question above].After citing several more letters, Marilyn then goes through the "standard" textbook calculation to show that her answers above are correct. But is there a "standard" answer to such question? The next article argues convincingly that this and related problems are simply not well-posed.
You wrote that it's more likely the man has two boys. I can only conclude that you felt your readers were getting frustrated by your superior abilities, so you decided to raise our collective self-esteem by exhibiting the logical skills of a second-grader who has had too many turns on the teeter-totter.
--Mathew Zik, Yorktown Heights, NY
by Raymond S. Nickerson. Psychological Bulletin, 120(3), pp. 410-433.
Nickerson revisits all the famous conditional probability problems that have led people to conclude that conditional probability is just too hard to teach. These include variations on the problem just discussed in the Marilyn vos Savant column, the Monty Hall problem, the three prisoner problem, and the three drawer problem.
The author feels that arguments about solutions to these problems arise because reasonable people can interpret them in different ways and get different answers. Thus, they are not difficult, but rather poorly posed.
He builds on the discussion in the excellent article by Bar-Hillel and Falk, "Some Teasers Concerning Conditional Probabilities," Cognition 11, 109-122. For example, he discusses Bar-Hillel and Falk's version of the two child problem:
Mr. Smith is the father of two children. We meet him walking along the street with a young boy whom he proudly introduces as his son. What is the probability that Mr. Smith's other child is also a boy?A reasonable person might get the answer 1/2 to this problem assuming that Mr. Smith is equally likely to choose a daughter to go for a walk with if he has a boy and a girl. Another person might not be willing to make this assumption and would get an answer that depends on the probability Mr. Smith would pick his son for the walk if he has to choose between his son and daughter.
If Mr. Smith happened to say: "Let me introduce my son Sebastian," then another reasonable person would say that the answer also depends on the probability that a family names a son Sebastian. Come to think of it, we might demand to know if the probability that Mr. Smith would "proudly" introduce his daughter is the same as the probability that he would "proudly" introduce his son.
Once a well-posed question is agreed upon, a probabilist would use a simple tree diagram to solve the problem. It is also increasingly popular to use simple contingency table arguments for these kinds of problems, emphasizing the frequency interpretation. An excellent reference on this approach is the paper "Conditional Probability and Education Reform: Are They Compatible?," by Allan J. Rossman and Thomas H. Short, Journal of Statistics Education, 3(2).
by Karen W. Arenson. The New York Times, 19 March 1997, B12.
Popular publications on college rankings are a good source of discussion questions on measurement issues. In a new study, Hugh Davis Graham, a Vanderbilt University professor, and Nancy Diamond, a graduate student in public policy at the University of Maryland at Baltimore County, argue that ratings of universities based on reputation understate the quality of some universities while overstating others.
The study was recently published as "The Rise of American Research Universities: Elites and Challengers in the Postwar Era" (Johns Hopkins University Press). It used conventional criteria like publications and research money but avoided scoring based on reputations. Also, the scoring was based on averages per professor instead of totals for the institution, thus giving smaller schools a chance to compete with large universities. The five criteria used were federal grants for research and development; the overall number of journal articles published by faculty members; the number of articles published in a smaller number of prestigious journals in science and technology, and in the social and behavioral sciences; and awards in the arts and humanities. The data covered the 25-year period from 1965 to 1990.
The study found the State University of New York at Stony Brook to be the third-best public research university in the nation, behind the University of California at Berkeley and Santa Barbara and ahead of the University of Michigan. Among private institutions, Brandeis University tied for ninth place with Johns Hopkins while Stanford led the list.
The next article raises some related measurement issues.
by Lawrence K. Altman. The New York Times, 22 May 1997, A25.
Writing in the journal Nature ("Nepotism and Sexism in Peer-Review," 387, 22 May 1997, pp. 341-343), biologists Christine Wenneras and Agnes Wold of Sweden's Gothenburg University charge that, in the scientific peer review process, sex and connections count more than scientific merit. There has long been anecdotal evidence for this opinion, but this appears to be the first time a large set of data on the review process has become available. By appealing to Sweden's Freedom of the Press Act, Wenneras and Wold successfully obtained data from the Swedish Medical Research Council (MRC), a government body that funds medical research.
Wenneras and Wold studied the reviews of 114 applications to MRC for 20 available postdoctoral positions in 1995. While 46% of the applicants were women, only 20% of the awards went to women. (The article points out that this does not appear to have been an unusual year. In the 1990s, women have received 44% of doctorates in the biomedical sciences, but have been less than half as successful as men in getting postdoctoral positions.)
The MRC goes through a complicated procedure to get a numerical rating of the applications. First, the applications are read by five reviewers, who assign ratings on a zero-to-four scale for scientific competence, proposed methodology, and relevance of the research. The ratings are then multiplied, so each reviewer produces a total score from zero to sixty-four. Scores from the five reviewers are then averaged to give a final score. Wenneras and Wold found that women received lower scores in all categories, but were ranked especially low in competence.
To check whether the low competence ratings reflect gender bias, Wenneras and Wold constructed their own scientific competence ratings, using three basic attributes of a candidate's publication record. The first is simply the number of scientific papers the individual has published. The second, called the "impact factor," is based on the Institute for Scientific Information's report of the number of times an average paper in a journal is cited elsewhere in the last year. An individual's impact is the sum of the impact factors for all of his/her papers. The third measure is the number of times an individual's papers have been cited in the last year. Furthermore, because biomedical journals customarily list the primary contributor as the first author, each of the preceding measures -- number of publications, impact factor, and number of citations -- was computed once for total publications and once for first-authored publications.
In a key comparison, Wenneras and Wold plotted the MRC's average competence scores for men and women against the corresponding averages of their own "total publication impact" measure. Only women in the highest impact group (total impact exceeding 100 points) had MRC competence ratings higher than the men in the lowest impact group (total impact less than 20 points). To investigate this discrepancy, Wenneras and Wold carried out a multiple regression to find factors that exerted a significant influence on the competency scores given by the reviewers. They concluded that a woman needed to have about 67 more impact points to earn the same MRC competence score as a man. This could be achieved by publishing three more papers in one of the most prestigious general science journals such as Science or Nature, or 20 more papers in a top journal in a specific field.
by Richard Morin. The Washington Post, 23 March 1997, C5.
Two University of Chicago economists, John R. Lott, Jr., and David Mustard, claim that allowing citizens to carry concealed weapons would prevent thousands of rapes and murders annually while producing no significant increase in accidental deaths.
The analysis was based on county-level crime data collected by the FBI in ten states that passed "carry" laws between 1977 and 1992. Lott and Mustard compared the crime rates before and after the laws took effect, and then estimated what the impact would have been if every state in the country had similar statutes. They estimate that if the whole country had adopted right-to-carry concealed handgun provisions in 1992, at least 1,414 murders and over 4,177 rapes would have been avoided. This represents an annual savings of $5.74 billion in hospital expenses, lost earnings, and other costs. These conclusions were published in the January 1997 issue of the Journal of Legal Studies.
Other investigators are skeptical of these conclusions. Morin reports on work by Daniel Nagin and Dan Black of Carnegie-Mellon, who reviewed the numbers cited by Lott and Mustard and found some discrepancies. While the annual murder rate did go down in six of the ten states, it went up for the other four. Rape dropped in five states but increased in the other five. Nagin and Black claim that if the benefits arise from the concealed weapons laws, then the impact should not vary by such huge margins from state to state. They also point out that all of the benefits of concealed weapons disappear when Florida is taken out of the analysis.
by Marcia Angell. The New York Times Magazine, 4 May 1997, pp. 44-45.
Angell, the executive editor of The New England Journal of Medicine, describes the impact of the journal's articles on the public: "No sooner do we publish a study on diet or life style than news of its conclusions, though virtually none of its qualifying details, hits the airwaves. Within 24 hours, millions of people consider eating fewer egg yolks or more oat bran to fend off disease."
Angell observes that the public is not good at differentiating between significant and insignificant risks. She illustrates this in terms of the mammogram controversy. In January, a National Institutes of Health panel of experts concluded that regular mammograms for women in their 40's would at best save the life of 1 in every 1,000 women screened. Increased risk of cancer from radiation might occur for about 3 in every 10,000 screened. Because pre-menopausal women have denser breast tissue, mammograms are more difficult to interpret in this age group than in those over 50. Most women with suspicious-looking mammograms end up not having breast cancer. However, as a result of the mammogram test they might end up undergoing unnecessary surgery. Angell remarks: "In short, so small would be the payoff of regular mammograms at this age that the risk of driving the car to get them might well outweigh the benefits of the test. Yet the reaction of many to the conclusions of the panel was that they were callously sentencing large numbers of women to a death sentence."
To illustrate how a small risk can appear big, Angell cites a study that found post-menopausal estrogen is associated with a 30 percent increase in the risk of breast cancer. She suggests that the same risk could have been expressed in a less threatening way. Since we know that 3 or 4 percent of post-menopausal women will get breast cancer in the next 10 years, we could say that this study shows that estrogen increases that risk to 5 percent. Put another way, if you are a post-menopausal women trying to decide whether to take estrogen, this study shows that your chances of remaining free of breast cancer for 10 years would decrease from over 96 percent to about 95 percent.
Angell points out that the debate over health care costs has caused medical recommendations to be framed in terms of the entire population rather than individuals. As a result, individuals may overreact to risks that have substantial implications for the health of the overall population. For example, after being bombarded with warnings about blood cholesterol levels, people get the impression that a high cholesterol level guarantees a heart attack and low cholesterol provides total protection from heart disease. Angell notes that many people were led to make drastic changes in their diets, or undergo unpleasant drug regimens, for comparatively small individual gains.
Angell concludes by encouraging increased skepticism about what you read in the news. With a few exceptions, such as giving up smoking, most of the changes in lifestyle suggested will produce only small effects. She reminds us that "there's more to life than fretting about health risks."
The current debate swirling around estrogen replacement studies is a case in point for the preceding discussion. Here are four articles on this topic.
by Dr. Susan Love. The New York Times, 20 March 1997, A25.
by Meir Stampfer. The New York Times, 27 March 1997, A28.
by Malcolm Gladwell. The New Yorker, 9 June 1997, pp. 54-61.
The New Yorker, 14 July 1997. Letter from Susan Love with reply by Malcolm Gladwell, pp. 8-9.
Susan Love is a well-known breast surgeon and associate professor of clinical surgery at UCLA. She is a leading figure in the battle against breast cancer and has written two enormously successful books: Dr. Susan Love's Breast Book, published in 1990, and this year's Dr. Susan Love's Hormone Book.
In her March 20 article in the New York Times, Love attacks the pharmaceutical companies for trying to make menopause into a disease and encouraging women to take replacement hormones for life. She feels that short-term use of these hormones may be justified for women who have had hysterectomies or for symptoms such as hot flashes and insomnia as they approach menopause. However, she applies her "nature knows best" argument to argue against healthy women using replacement hormones to prevent diseases.
She asserts that studies showing that the use of replacement hormones prevents osteoporosis are confusing; she is not convinced that the evidence of protection against heart disease is as strong as the evidence that prolonged use of estrogen causes breast cancer.
Pharmaceutical companies defend their products by pointing out that one in three women dies of heart disease, while one in eight women gets breast cancer. Although this is true, it is important to note that in women younger than age 75 there are actually three times as many deaths from breast cancer as there are from heart disease.In his New Yorker article, Gladwell says that this statistic is central to Love's argument but claims that she has her numbers backwards. He writes:
In women younger than seventy-five, there are actually more than three times as many deaths from heart disease as from breast cancer. Even the general idea behind this argument -- that heart disease is more of a problem for older women and breast cancer is more of a problem for younger women -- is wrong. In every menopausal and postmenopausal age category, more women die of heart attacks than die of breast cancer.Gladwell then goes on to give specific statistics to back up these statements. Love's apparent mistake was pointed out in a letter to the New York Times by Harvard epidemiologist Meir Stampfer, who thought she had just mixed up the two categories in reading the government's mortality rates. However, Gladwell claims that, in a meeting with Love after this letter appeared, she defended her figures and quotes her as saying:
Most women at fifty know someone who has died of breast cancer. Most women at fifty don't know someone who has had heart disease. That's because under seventy-five there are three times as many deaths from breast cancer as from heart disease.It does seem to us that women under 50 tend to know other women who died of breast cancer, but rather exceptionally know women who died of heart disease. Therefore we decided to check to see if Gladwell was really correct with his statistics. The required data can be found from the National Center for Health Statistics (NCHS) and can be accessed from the CDC Web page (www.cdc.gov). Official mortality rates are given in deaths per 100,000 for very specific diseases. The decision of which diseases to classify as heart disease is, perhaps, rather subjective, but we followed the choice of the NCHS in a related study. (Strokes are not considered heart disease.)
Age Breast cancer Heart disease 25-35 2.7 6.0 35-44 15.2 17.1 45-54 41.6 57.2 54-64 69.8 195.7 65-74 105.6 566.2 75-84 145.9 1741.3 85-100 195.5 6252.6The results agree with those given by Gladwell. However, the story does not end here. Love replied in a letter to the New Yorker (14 July 1997) that she got her numbers from a Nurses' Health Study for which Stampfer (who wrote the letter to the editors saying she was wrong) was one of the investigators. She writes:
According to the data published by Stampfer and his colleagues (in the Nurses' Study article), there are more deaths from breast cancer than from heart disease over the eighteen years of follow-up. The exact figures vary depending on whether you are looking at smokers (most premature heart disease is in smokers) or non-smokers. In the healthy fifty-year-old nonsmoker category, there are three times as many deaths from breast cancer as from heart disease. I propose that this is the population closest to the perimenopausal woman contemplating taking estrogen.Malcolm Gladwell remarks that women in the Nurses' Health Study have a far lower incidence of heart disease than American women as a whole and remarks:
What Love is doing is a bit like arguing that we can find out whether American students know enough calculus by testing the freshman class at MIT and then touring the country pretending that your results -- that no one needs to take more math -- speak for all students.Love uses the difference between the nurses and the general population to support her argument that women are better off changing their life style than using estrogen to protect against heart disease. She writes:
The fact that breast-deaths are about the same but the heart-disease deaths are lower than those of the general population demonstrates that heart disease can be prevented by a life-style approach while breast cancer cannot.
The following story reports on a recent study in this area.
by Jane E. Brody. The New York Times, 19 June 1997, A1.
This article presents results from a study appearing in the June 19, 1997, issue of the New England Journal of Medicine, aimed at determining the benefits and risks of hormone replacement therapy. The study was based on the well-known Nurses' Health Study. This study started with 121,700 women participating in 1976 and continued until 1992. These women were questioned and examined every two years.
This was a case control study. It included those women in the Nurses' Health Study who were past menopause with no history of cardiovascular disease or cancer. The 3637 who died during the 1976-94 were the "cases." For each case, the researchers selected 10 "controls" from women who met the above criteria and matched the cases in age, age at menopause, and type of menopause.
After adjustment for confounding factors, the study found that, in the first decade of hormone use, the chance of dying was 37% lower for those on hormone therapy, primarily because of fewer deaths from heart disease among this group. This dropped to 20% after 10 or more years because of the increased risk of breast cancer. The risk of breast cancer mortality for those on estrogen therapy for more than 10 years increased by 43 percent.