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CHANCE News 4.03
(2 February to 15 February 1995)
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Prepared by J. Laurie Snell, with help from
William Peterson, Fuxing Hou and Ma.Katrina
Munoz Dy, as part of the CHANCE Course Project
supported by the National Science Foundation.
Please send comments and suggestions for articles to
jlsnell@dartmouth.edu
Back issues of Chance News and other materials for
teaching a CHANCE course are available from the
Chance Web Data Base
http://www.geom.umn.edu/docs/snell/chance/welcome.html
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Chance is lumpy.
Robert P. Abeleson
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FROM OUR READERS
Professor Abelson sent us a copy of his new book
"Statistics as Principled Argument" just published by
Lawrence Erlbaum Associates, Hillsdale NJ. I will
review this book next time -- a welcome relief after
"The Bell Curve". As a preview, the main argument of
the book appears to be that data analysis should go
beyond a formal analysis. "It should make an
interesting claim; it should tell a story that an
informed audience will care about, and it should do so
by intelligent interpretation of appropriate evidence
from empirical measurements or observations."
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Pete Hayslett sent us the following article:
Sex education programs that work.
Parade Magazine, 12 Feb. 1995, pp 18-20
Earl Ubell
In this article the author writes:
Games also teach interaction. In the old
sex-ed courses, a teacher might say:'If you
have intercourse once a month, your chances
are one in six of becoming pregnant.' Such
a dry approach soon loses teenagers' attention
Instead, today, the teacher starts a game:
TEACHER: Class, write down any number from
1 to 6. Now I'll draw a number from this box.
(Draws and reads) No. 6. All those who picked
No. 6, please stand up. You or your partner
just got pregnant. Remain standing. OK. Now
I've drawn No 3. All those with a 3, please
stand. You just got pregnant. Soon, all the
students are standing. The lesson: Frequent,
unprotected sex raises your chances of pregnancy."
DISCUSSION QUESTION:
(1) What do you think the teacher means by "If you have
intercourse once a month, your chances are one in six
of becoming pregnant"?
(2) What is wrong with this game (activity) as a way
to illustrate a probability?
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Goran Djuknic pointed out that an ad for "The Club"
(anti-theft device for cars,) claims there is a 1 in 43
chance of your car being stolen, but with The Club the
chance goes to 1 in 10,000.
DISCUSSION QUESTION
How do you think The Club promoters came up with the 1
in 10,000 figure?
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ARTICLES ABSTRACTED
>>>>>==========>> Martin Tanner remarked that we missed the following interesting article last time. Teachers' diligence finds fame, free lunch. The Morning Call (Allentown), Jan. 21, 1995, B1 Joseph P. Ferry
This article reports on what should be a very simple mathematics error, and I, at least, could not figure out what the problem was from the details of the article. The article states that Bob Swain, a Souderton High School mathematics teacher, found a mistake in an ad of the restaurant chain Boston Chicken. With much effort, he got them to change the ad. He received a free lunch with thirty of his students. He also obtained instant fame, appearing, for example, on the CBS program Good Morning America. Here is what you learn from the article about the mathematics of the problem. Boston Chicken's ad, featuring quarterback Joe Montana, claimed that there are 3,360 possible combinations of three-item meals available to customers. (I think this should have been three-item side dishes to accompany your main dish.) Swain claimed that whoever had written the ad had confused the formula for figuring permutations and combinations. By Swain's calculations, there are only 816 different combinations available. Swaim said the company would have to increase its list of side dishes from 16 to 27 to obtain the number of combinations given in the ad. Boston Chicken agreed to change their ad to say 816 possible combinations. DISCUSSION QUESTIONS. (1) (a) How did Boston Chicken arrive at the number 3360? (b) How did Swain arrive at the number 816? (2) If you are stuck on part (a) of problem (1), as I was, here is a hint: 16*15*14 = 3360. What is wrong with counting this as the number of possible three item side dishes? (2) If you are stuck on part (b) of problem (1) as I also was, show that the number of ways to choose 3 items from 16 allowing repetitions but not counting order is 816. Does this make the revised ad reasonable? (3) Show that the number of ways to choose 3 objects from 18 without repetitions is also 816. Why is this the same as the number of ways to choose 3 items from 16 with repetition? <<<========<<
>>>>>==========>> Ruma Falk suggested the next study. Study rebuts 'Acceptable' mid-life weight gain. The New York Times, 8 Feb. 1995, C11 Jane E. Brody
Current guidelines consider an increase in weight of from 10 to 40 pounds for women during mid-life acceptable and even desirable. Researchers at Harvard School of Public Health and Harvard Medical School report in the "Journal of American Medical Association" (Feb 8, 1995) on a 14 year study involving 115,818 women. The study found that women who gain 11 to 18 pounds in adult life had a 25 percent greater chance of suffering or dying of a heart attack compared with women who gained less than 11 pounds after the age of 18. Coronary risk rose to a 60 percent increase for weight gains of 18 to 25 pounds and to a 200 to 300 percent increase for weight gains above 25 pounds. Authors of the study criticized the guidelines issued in 1990 by the United States Departments of Agriculture and Health and Human Services saying "The guidelines provide false reassurance to the large fraction of the population who are not defined as overweight, but who are at substantially increased risk of coronary heart disease." They say that these guidelines were based upon studies that did not control for factors such as smoking as their study did. The 1985 guidelines, based upon the 1959 Metropolitan Life Desirable Weight Table, did not suggest that it was desirable to gain more weight in mid-life. The news director of the Public Health Service stated that the next edition of "Dietary Guidelines for Americans" to be published at the end of this year will reflect current concerns about gaining weight. The director of the Framingham Heart Study said that their studies show that significant weight increase after age 25 is also a risk factor for heart disease in men. DISCUSSION QUESTION: (1) Why is it necessary to control for smokers in a study of the effects of weight gain? (2) Do studies of this kind show that gaining weight causes heart attacks? <<<========<<
>>>>>==========>> Hard times for Britain's lost boys. New Scientist, 4 February 1995, pp 12-13 Karon Gold
In Britain, girls are overtaking boys on the GCSE exams subject by subject.(These are exams in a number of subjects taken by all 16-year-olds in Britain). In the most recent scores available, 45% of the girls received five or more GCSE grades between A to C as compared to 37% for the boys. Researchers at the University of Sheffield developed a data base of 30,000 students giving such things as social status and test scores over a wide range of ages. They found that the best predictor of a 16-year- old's GCSE result, in any subject, is the reading score between ages 10 and 12. From their data, they determined a linear predictor for an overall GCSE grade as a function of the reading score at age 12. Individual schools then made scatter plots based on these two scores for students in their school. Students above the predicted line were considered over-achievers and those under the line were considered under-achievers. Girls tended to be over-achievers and boys under- achievers. Schools then looked at records of the boys who were under-achievers to see if they could have predicted these. Standard explanations such as not being bright and not being literate were not born out. Rather, they seemed to show a pattern of poor attendance, misbehavior, and summer birthdays--a known factor for under-achievement. DISCUSSION QUESTIONS (1) The article reports that Education Secretary Gillian Sheperd warned in a speech "Although girls' academic advances are a cause for pride, there is a danger of going too far." What do you think was meant? (2) Why should students with summer birthdays be a risk factor for under-achievement? <<<========<<
>>>>>==========>> Many don't practice safe sex, survey finds. The Boston Globe, 14 Feb. 1995, p3. Associated Press
A survey of the sex lives of 1000 adults, funded by the pharmaceutical firm Burroughs-Welcome, found that 75% of Americans don't believe they can catch a sexually transmitted disease, even though doctors diagnose 12 million new (non-AIDS) cases a year. The survey turned up the familiar pattern (discussed in a previous Chance News) of men reporting more sex partners: 51% claimed six or more, whereas only 25% of women claimed that number; 28% of men claimed eleven or more, compared to 10% of women. There were also indications that people do not really know their partners' behavior. Only 18% of women under 30 years old believe their partners have had at least five partners, while 42% of the men that age and 57% of those 30-39 claim they have. DISCUSSION QUESTIONS: (1) Is there an inherent contradiction in the figures on number of partners, or is it possible to reconcile them numerically on the basis of behavior patterns? How might reporting bias contribute to the apparent discrepancy? (2) How about the figures on women's perceptions of their partners' behavior? Answer the same questions as in 1. <<<========<<
>>>>>==========>> Men, women, sex. Go figure. The Washington Post, 12 Feb. 1995, F1 Henry T. Dunbar
The author remarks that National Public Radio recently reported a survey that American men average 11 sex partners per lifetime and American women average 3. He states that the explanation usually given is that most women are virtuous but a few are very promiscuous providing all the extra sex partners for the men. He points out that no such explanation can account for the difference since every time a man adds a new partner, a women adds a new partner and the averages have to be the same under any reasonable assumptions about the population for the survey. He comments that if they meant median instead of mean it could conceivably be correct but he does not think that is the answer either. He concludes that both men and women are lying and the answer is somewhere in the middle. <<<========<<
>>>>>==========>> Note: Joe Berger sent a letter that a colleague Bill Banks sent by e-mail to National Public Radio (chastising them for quoting the statistics in the above article as if they were meaningful just because they came from scholarly survey.) He received a return message saying they did not read their e-mail. We remind you of the following two articles. M.Morris, "Telling tales explains the discrepancy in sexual partner reports", Nature 365:437-440. Einon, D. (1994) Are Men More Promiscuous than Women? Ethnology and Sociobiology 15: 131-143. Morris points out that the usual explanations center on two types of errors: sampling bias (men report encounters with women below the eligible age, prostitutes are left out, etc.) and reporting errors. He shows that sampling bias cannot reasonably account for the differences found. On the other hand, he shows that in major surveys, if the data is restricted to the approximately 90 percent of subjects who report fewer than 20 lifetime partners, the reporting ratio is reasonably close to 1. He remarks "the anomaly appears to be driven by the upper tail of the contact distribution, an example of the general principal of outlier influence in data analysis." He suggests ways to improve this part of the data to get more accurate results from these surveys. <<<========<<
>>>>>==========>> Weird dice. Math Horizons, Feb. 1995, p30 Joseph Gallian
You are in the middle of a Monopoly game and someone substitutes a pair of dice labeled 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8 for the standard dice. DISCUSSION QUESTIONS (1) Show that the probability that you move k steps when you roll the dice has not changed. (The article shows that this is the only possible pair of weird dice that would achieve this.) (2) Show that the switch does improve the value of St. James Avenue. Hint: St. James is six spaces beyond jail. And recall what happens when a player is in jail (not just visiting). <<<========<<
>>>>>==========>> Statistics Misused. Technological Review, Feb. 1995, Letters page 70 Daniel Whitney
Apparently, you cannot even write an article on the misuse of statistics without someone claiming that you misused statistics. In these letters and in the letters in the January issue, Daniel Whitney, Sydney Freidin, and Ted Tsomides criticize some of the examples used by Arnold Barnett in his article "How numbers can trick you." that appeared in the October 1994 issue. Tsomides feels that Barnett "omits or downplays some important points about the misuse of statistics". He mentions three such points: (1) public policy is often determined by statistical problems which are very little understood, for example, the heritability of I.Q., (2) scientific fraud is often cloaked in the impression of statistical significance, and (3) even honest researchers make inexcusable errors that render their statistics useless such as the Love Canal fiasco. In other words why just blame the media? One of Barnett's examples considered the Supreme Court case where odds and probability were mixed up when comparing the chance that a black defendant would receive a death sentence according to whether the victim was white or black. Tsomides suggests that Arnold himself gave a biased view of the effect of the error by choosing for his example an extreme case of a defendant who has a 99 percent chance of getting the death sentence where the victim was white. Finally, Freiden thought that "passenger hours" should replace "passenger miles" when comparing the safety of various methods of travel. Barnett does a pretty good job of defending himself against these criticisms. <<<========<<
>>>>>==========>> Note: There is an interesting discussion of comparing the safety of methods of travel by Francis Lopez-Real in the book "Teaching Statistics at its best" (pages 142-143). This is the best of Vol 6-16 of the journal "Teaching Statistics". Both are published by the Teaching Statistics Trust at the University of Sheffield, England and are wonderful resources for an elementary probability or statistics course. Lopez-Real observes that when comparing air and train travel, air appears safer when miles are used and train when hours are used. He argues that this does not show that statisticians can prove anything but rather that the choice of statistics used is important and should be considered carefully. For example, if you are convinced that miles is best, he invites you to consider the hypothetical case of three trips to Jupiter each with a crew of 4. The first two trips are successful but in the third the crew of 4 is lost. In terms of miles this would lead to the conclusion that space travel to Jupiter is safer than air or train but he wonders how many would take the next trip. <<<========<<
>>>>>==========>> Genes, Race, IQ and "The Bell Curve". OR/MS Today, Feb. 1995 Arnold Barnett
Barnett begins with the remark that there has been no review from an informed perspective and he is going to give it a try. He starts by criticizing the authors' assumption that the best way to reflect differences among a set of numerical outcomes is by standard deviation. He also feels that the authors are too enamored with the normal distribution. There is considerable discussion in the book and much in the press about the part of I.Q. that is inherited. Barnett asks what it means for intelligence to be, for example, 60 percent heritable? From the author's discussion he assumes that it means: Suppose that a large set of genetically identical newborns grew up in randomly chosen environments and their I.Q.'s were measured when they became adults. The variance of these scores would be 60% less than that for the entire population. The corresponding drop in standard deviations would be about 37%, so Barnett concludes that it would not be reasonable to assume that differences are primarily due to genetic factors. While the authors do not assume this, Barnett shows that their discussion of this issue is confusing to say the least. Barnett shows that assuming intelligence can be measured by a single number also causes real problems that would not occur if we assumed that there were ways to measure different aspects of intelligence. He uses the authors' own examples to demonstrate this. While far from a comprehensive review, Barnett identifies certain areas where he thinks the authors err and explains why he thinks they do. Finally, Barnett reflects on why the book was written. At the beginning of the article he remarks that he is a good friend of Charles Murray. He concludes by remarking that he regards Murray to be a decent and honest man but "he and Herrnstein made errors in judgment that the nation has reason to regret." <<<========<<
>>>>>==========>> Cancer study to be reviewed. The Toronto Star, 15 Feb. 1995, A4 Reuters
Last year the National Breast Cancer Screening Study determined that routine mammograms have no benefit in terms of mortality reduction for women between 40 and 49 years old. Now Dr. Daniel Kopan, director of breast imaging at Mass. General Hospital and a leading critic of the study, says he has evidence that the patient selection procedure was not random. He claims that nurses wanted to ensure that women with symptoms of breast cancer received mammograms. As a result, an independent team of researchers will investigate the random selection process and report back in two or three months. DISCUSSION QUESTION: Dr. Kopan's reasoning is not fully explained in the article. How could including symptomatic women bias the study against screening? Wouldn't it inflate the cancer detection rate and make screening look beneficial? <<<========<<
>>>>>==========>> The following article was suggests by Mary Paige. Experiments offer hope of killing leukemia cells. Chicago Tribune, 13 February 1995, p7 Wire services.
As reported in an article in the current "Science Magazine", a large biotherapy project at the University of Minnesota has shown that a substance obtained from soybeans has shown promise in killing leukemia cells resistant to traditional cancer treatments. The soybean substance was tested on mice and killed cells of B-cell precursor (BCP) leukemia. This is the most common childhood cancer and the second most common adult leukemia. The aim of this project is to find biotherapeutic treatments that will supplement, but not replace, chemotherapy or radiation. It is stated that, for the kind of leukemia studied, about 30 percent of the patients, chemotherapy and radiation do not succeed in wiping out the cancer cells. <<<========<<
>>>>>==========>> HIV transmission risk in sports minimized. The Boston Globe, 15 Feb. 1995, p8. Reuters
In an article in "The Annals of Internal Medicine," doctors from the Centers for Disease Control in Atlanta report that the risk for HIV transmission during sports events is so low that "athletes need not be excluded from participation solely because they are infected." It was recommended that prevention efforts be focused on off-the-field settings. A separate report in the same issue of the Annals studied injuries in 155 NFL (National Football League) games during 1992 estimated the risk of accidental transmission during a football game was no more than once in 58.6 seasons. DISCUSSION QUESTION: How does a risk of "once in 58.6 seasons" express the risk of transmission during a single football game? What is the one-game risk for an individual player? <<<========<<
>>>>>==========>> Correction to the last Chance News. Barry Brunson pointed out that correct reference for the letters from Gould and Murray relating to Gould's November 21 review in the "New Yorker" of the "The Bell Curve" is "The New Yorker", 26 Dec 94 - 2 Jan 95 combined issue on page 10. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! CHANCE News 4.03 (2 February to 15 February 1995) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Please send suggestions to: jlsnell@dartmouth.edu >>>==========>>|<<==========<<<
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