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CHANCE News 3.08
(21 May to 10 June 1994)
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Prepared by J. Laurie Snell, with help from Jeanne
Albert and William Peterson, as part of the CHANCE
Course Project supported by the National Science
Foundation and the New England Consortium for
Undergraduate Science Education.
Please send comments and suggestions for articles to:
jlsnell@dartmouth.edu
Previous issues of Chance News can be found on the
Chance gopher listed on the Dartmouth gopher
(Dartmouth.edu) in the classroom folder.
The Chance welcome page on Mosaic is:
http://www.geom.umn.edu/docs/snell/chance/welcome.html
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When it is not in our power to follow what is
true, we ought to follow what is most probable.
Descartes
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Contents
From our readers
Other internet sources
Current news
======================================= FROM OUR READERS: Bob Norman pointed out that Laura Ziegler, on Morning Edition 2 June 1994 in a discussion of crime rates, remarked: "Advocates of prison alternatives say if higher incarceration rates were the answer to crime, then states with the most prisoners should have the lowest crime rate." DISCUSSION QUESTIONS: 1. In 1992, California, Texas, and New York, had the largest number of inmates. Do you think that these states had the lowest crime rates in 1992? 2. Can you explain what Ziegler meant? Would it have helped to say "the lowest crime rate relative to their size"? <<<========<<
>>>>>==========>> Irene Starr comments on the article "Separate is Better" (NY Times Magazine 22 May, 1994, by Susan Estrich, mentioned in the last Chance News): When reading the original article, I thought of a confounding factor: all-women schools tend to be private and more elite, drawing from a better educated and harder working population than the average coed school. For me that explained a lot of the difference. <<<========<<
>>>>>==========>> Mark Kernighan, a student at Swarthmore College wrote the following letter to the editor of the New York Times, relating to a story reported in Chance News 3.05. The Times chose not to print it so we will: April 18, 1994 Letters Editor The New York Times 229 West 43d St., N.Y. 10036 To the Editor: "Probability Experts May Decide Pennsylvania Vote" (news article, April 11) contains a very popular statistical error. Based on Professor Orley Ashenfelter's claim that the kind of imbalance seen in Stinson vs. Marks would appear only 6 percent of the time by chance, Peter Passell writes that "there is a 94 percent chance that irregularities in the absentee ballots, not chance alone, swung the election to the Democrat, Professor Ashenfelter concludes." Even if the first conclusion is correct, the second doesn't follow. The probability of a 1025-vote margin of victory assuming fair balloting (which Mr. Ashenfelter estimates as 6%) is not the same as the probability of fair balloting assuming a 1025-vote margin of victory (which cannot be estimated at all without considering the frequency of fraud in all similar elections). An example may make the point clearer: if you correctly predicted two coin flips in a row - a 1-in-4 stroke of luck - would you conclude that your chances of possessing ESP were 3 in 4? Any student who has read one page about Bayesian probability knows enough to avoid this trap, which I was calling the Expert Witness Fallacy long before I heard of the Philadelphia case. When professional statisticians drown so deep in their methods that they forget about the underlying common sense, it often takes a skeptical amateur to see through their mistakes. Sincerely, MARK D. KERNIGHAN Swarthmore, April 18, 1994 DISCUSSION QUESTION 1. Is Mark saying that we mix up P(A|B) and P(B|A)? 2. If both of these are defined and P(A|B) < .04 is P(B|A) < .04 <<<========<<
>>>>>==========>> OTHER RELEVANT INTERNET SOURCES The Vanderbilt Television News Archive maintains a gopher (tvnews.vanderbilt.edu) with abstracts of discussions on the major TV news (ABC, CBS, and NBC). It has a good search procedure (modestly Boolean) and if you find something of interest you can get a copy of the tape sent on loan for a reasonable period. I tried the search procedure on the topic DNA fingerprinting and found the following abstract: 1994.05.09 INSIDER'S REPORT (CRIME: DNA TESTING) 5:50:00-5:53:40 Monday NBC (Studio: Tom Brokaw) Report introduced. (Washington: Bob Kur) The controversy over the legal use of DNA evidence in prosecuting criminal cases examined; cases of executed murderer Timothy Spencer & suspects Kirk Bloodsworth & Glenn Woodall, released due to DNA evidence, cited as examples. [Benjamin Cardozo law school professor Barry SCHECK - cites the implications of DNA evidence.] [BLOODSWORTH - comments.] [Council for Responsible Genetics Nachama WILKER - raises issue of the access of criminal justice organ.] This will be an interesting segment to show the next time that DNA fingerprinting comes up in our CHANCE course. I did not find as much as I thought I would browsing around, but perhaps I did not look hard enough. It's a pity they do not include CNN in their data base. The June 8 issue of the Chronicle of Higher Education has an account of this service and how the TV networks are reacting to the Vanderbilt project. <<<========<<
>>>>>==========>> EdStat-L is a statistics education discussion group. In addition to discussing more serious questions, from time to time the group (like us) becomes fascinated with how to properly formulate a conditional probability problem. Recently, they have discussed the following problem: A bridge player (south) caught a glimpse of a card dealt to her opponent on the left (west) - it was a red ace (she could not tell the suit). West opens the game by playing the ace of diamonds. South sees that neither she nor north (whose cards are revealed) has the ace of hearts so either east or west must have it. What is the probability that west has it? The simplest interpretation of this problem is that we are asked for the contional probability that west has both red aces given that she has the ace of hearts and that neither north nor south have any red aces. A routine calculation gives the answer 12/25 for this probability. As was pointed out in the discussion on EdStat-L, the problem becomes much harder if you try to include the source of the information given. For example, it would be natural to assume that north observed a random card among the cards dealt to west. This already changes your information and the answer. You might want to take into account west's knowledge of bridge and the bidding to say something about how she chose the card to lead. Also, when south raises the question he also knows what cards north has etc, etc. For an interesting discussion of the relation of the Monty Hall problem to some serious bridge problems, see Alan Truscott's bridge column in the New York Times 4 August 1991. To subscribe to EdStat-L, send an e-mail message containing the single line: subscribe edstat-l "your name" to the address: listserv@jse.stat.ncsu.edu EdStat-L is also sci.stat.edu on the UseNet News. ======================================= CURRENT NEWS The EdStat-L problem is a variation on the problem "given that at least one head turned up in two tosses of a coin, what is the probability that two heads turned up. Marilyn vos Savant also dealt with this problem this week. Ask Marilyn Parade Magazine, 12 June 1994, p18. Marilyn vos Savant
"You discover two booths at a carnival. Each is tended by an honest man with a pair of covered coin shakers. In each shaker is a single coin, and you are allowed to bet upon the chance that both coins in that booth's shakers are heads after the man in the booth shakes them, does an inspection and can tell you that at least one of the shakers contains a head. The difference is that the man in the first booth always looks inside both of his shakers, whereas the man in the second booth looks inside only one of his shakers. Where will you stand the best chance?" Daniel Hahn, Blairstown, Iowa Marilyn gives the correct answer: the probability that both are heads is 1/3 for the man in the first booth and 1/2 for the man in the second booth. <<<========<<
>>>>>==========>> Leadership by the numbers.
The Boston Globe, 29 May 1994, p67 David Shribman
David Shribman argues that Washington is too reliant on public opinion polls, which politicians allow to set national agenda, rather than exhibiting leadership skills and setting a course for the nation. Describing a House subcommittee discussion on health care, Rep. Ron Klink (D-Pennsylvania) complained that "every member who got up to discuss an issue was quoting some poll. Every member has some half-assed poll of his own district, and members use them whatever way they want." Foreign policy in particular is singled out by the article as an area in which effective leadership can make things happen that will shape public opinion. Republican pollster Frank Lutz comments on Bill Clinton's tendency to internalize the results of polls, noting that "When I contact focus groups. I quite literally hear, word-for-word, the same things Bill Clinton is saying. I know it's not because people are copying Clinton. Words from focus groups become Bill Clinton's words." As for the impact of polling on public discussion, former Reagan and Bush speech-writer Peggy Noonan comments on the ability of polls to "harden" opinions that people haven't really formed yet, because people feel a need to avoid embarrassment by responding in a seemingly knowledgeable way. Others have noted the tendency to use key words and phrases from the Washington debate, which affects the nature of responses. A side-bar to the article has famous public figures' quotes about polls. DISCUSSION QUESTION: 1. What is a focus group? Can it be considered a poll? 2. How can a poll get people to give opinions that they haven't really formed yet? Does this happen in political polls, for example, presidential polls? <<<========<<
>>>>>==========>> Stanford U. decides to make courses harder to drop but easier to fail.
New York Times, 4 June 1994, 1-7 David Margolick
Stanford has decided to reinstate the F grade in the slightly more friendly form of NP (not passed) and to no longer allow students to (a) drop a course on the eve of final examinations and (b) take a course as many times as they wanted to. Beginning in the 1995-96 academic year students who drop between the fifth and eighth weeks of a quarter will have a W on their transcript, and if they are still in the course after the eighth week they cannot drop it. Students will be able repeat a course only once and if they do repeat it their transcript will reflect this. It is estimated that currently 90 percent of the grades at Stanford are A's or B's leading one professor to remark: 90 percent of the students think they should be in the top 10 percent of the class. Of course, Stanford is not unique: Harvard is estimated to have 43 percent A's and Princeton 40 percent. Grading policies at other well-known colleges and universities are described in this article. DISCUSSION QUESTION: On the Op-ed page of the New York Times, 13 June 1994, Donald Kennedy, former president of Stanford, remarks that he taught a large undergraduate course in Biology until 1977 and then scarcely taught at all until 1993 when he resumed teaching and taught the same Biology course. From this experience Kennedy argues that the the grades have not been inflated, rather, in 1993 the students are just much brighter and working harder than the students of 1977 and deserve grades about ten percent higher. 1. Do you think that the SAT scores at Stanford are at least ten percent higher than they were in 1977? 2. If a 1994 calculus class at Stanford were given the corresponding exam of the 1977 class do you think that the average would be at least 10 percent higher <<<========<<
>>>>>==========>> Grade inflation losing air at some colleges.
Boston Globe, 4 June 1994, National/Foreign, p.1 Anthony Flint
This is a somewhat more general discussion of how grade inflation got started and what is being done to try to reverse the trend towards ever higher grades. It is also the only article to mention the modest step that Dartmouth took, about the same time as Stanford's widely publicized change. Dartmouth has possible grades A,A-,B+,B,B-,C+,C,C-,D,E. It has the standard problems of (a) very few grades below a B and (b) average grades in sciences courses significantly lower than those in the humanities. About the same time the Stanford faculty voted to make changes, the Dartmouth faculty voted to put on the students' transcript, after their grade, the number of people in the class and the median grade for the class. At the bottom of the transcript there will be a summary telling in how many courses the student did better than the median course grade, equal to the median grade, or lower than the median grade. The faculty meeting had an amusing discussion about whether it would be better to use the median or the mean. It was clear that some faculty were reluctant to enter the fray because they did not really know if there was a difference between them. Our registrar Tom Bickel is still interested in the question of whether the mean or the median is more appropriate, so if anyone has an opinion send it to me and I'll pass it on. Also, Tom is wondering what to do when there are an even number of students in the class and he has two choices for the median grade. Should he favor the student and report the lower of the two grades or should he make a small contribution in the battle against grade inflation and report the higher of the two? He could even invent new grades half way between the current possible grades. <<<========<<
>>>>>==========>> Smoking for Two? The National Review 16 May, 1994 Jacob Sullum
Jacob Sullum is managing editor of Reason magazine. In this article he explains why he does not believe the sign on the billboard he passes on the way to work that says "secondhand smoke kills". His criticisms of the evidence and the reasoning behind the Environmental Protection Agency decision that secondhand smoke was a serious health risk are similar to those presented by Alan Gross in his article in Chance Magazine. (See Chance News 2.20) Unlike representatives for the tobacco companies, Sullum emphasizes that the case for smoking being harmful to the smoker is overwhelming. He argues that the EPA is concentrating on the secondhand smoke issue just because it seems politically easier to get people to restrict smoking to protect non-smokers than to protect smokers. DISCUSSION QUESTION 1. Commenting on the EPA's decision to use a 10 percent significance level rather the conventional 5 percent level, Sullum remarks "this change essentially doubles the odds of being wrong." Is this correct? 2. Do you feel that a better case can be made for trying to protect people who do not smoke from the effects of second hand smoke than trying to protect the smokers themselves. <<<========<<
>>>>>==========>> WHO says smoking fatal half the time. Boston Globe, 31 May 1994, p11 Associated Press
According to stimates by the World Health Organization, about half of all smokers will be killed by smoking. Starting in young adulthood, long-time smokers have a death rate three times that of nonsmokers at all ages. The report concludes that "If, as is likely, much of this excess mortality is directly attributable to smoking, then this implies that the risk of a smoker eventually being killed by the habit is on the order of 50%, half of whom will die in middle age and lose over 20 years of life." DISCUSSION QUESTION: How do you think the 50% figure was inferred? <<<========<<
>>>>>==========>> French study finds benefits in different type of margarine.
The Boston Globe, 10 June 1994, p9 Richard A. Knox
A study by the French National Institute of Health and Medical Research, reported in "The Lancet", found that a "Mediterranean diet" rich in bread, fish fruit and vegetables, with a margarine substituted for butter, greatly lowered the risk of death in a large group of heart attack survivors. The margarine involved contained canola oil, in contrast to the soy bean or corn oil margarines used more commonly in the U.S. This was cited as a possible expanation for the different outcomes observed in the Harvard study on margarine reported in the last edition of CHANCE news. Interestingly, the diet did not lower subjects' cholesterol levels, nor did it increase the level of "good cholesterol." <<<========<<
>>>>>==========>> Data affirm higher cancer risk for female spouse of a smoker
New York Times, 8 June 1994, B9 Philip J. Hilts
An article in the June 8 issue of the Journal of the American Medical Association studies the effect on women of secondhand smoke. It extends findings that were reported three years ago from preliminary data. The current data shows that a women who does not smoke, but has a husband who does, has an increased risk of about 30 percent for developing lung cancer. This is a small increase compared to the 30 to 40 times higher risk for lung cancer for a person who smokes. A woman's risk was measured by the number of "pack- years" (packs the husband smoked per day multiplied by number of years' exposure). A significant increased risk of 30 percent occurred starting at 40 pack-years, going to 80 percent for 80-pack years. Being exposed to smoke at work or in social situations on a regular basis also increased a woman's risk of lung cancer. DISCUSSION QUESTIONS: This New York Times article states that the study "compared 653 women with lung cancer who had never smoked with a control group of 1,253 women who did not have lung cancer and had never smoked". Other accounts agreed with the Los Angeles Times which reported that the researchers "compared 653 nonsmoking women with lung cancer to a control group of 1,253 randomly selected women." Which description of the control group, if either, do you think is correct and why? <<<========<<
>>>>>==========>> Smokers have higher breast cancer death risk
The New York Times, 25 May, p. C12 Jane E. Brody
This article, based on findings published this month in The American Journal of Epidemiology, describes results of an ongoing study to determine if women who smoke are more likely to die from breast cancer than women who don't smoke. Ms. Brody writes that research has found evidence of such an increased risk and that the study is "the first to link cigarette smoking to the second most common cancer killer of American women." (Lung cancer is number one.) The study followed 600,000 initially cancer-free women for six years and found that women who smoked were 25 percent more likely to die of breast cancer than nonsmokers and former smokers. Those who had smoked more or for longer periods were at a higher risk. Women who smoked at least two packs of cigarettes a day "were 75 percent more likely to develop fatal breast cancer than nonsmokers." At the same time the article points out that the study does not propose that smokers are "more likely to get the disease", but according to Dr. Eugenia E. Call who directed the research, the results suggest that "current smokers may be at increased risk of fatal breast cancer either because of poorer survival or delayed diagnosis." DISCUSSION QUESTION: The article states that heavy smokers were "75 percent more likely than nonsmokers to develop fatal breast cancer", but that the study did not find that smokers were "more likely to get the disease." Can these both be true? <<<========<<
>>>>>==========>> Gambling know-how; state draws fine line between skill, chance
St. Louis Post-Dispatch, 29 May, 1994, p. 1A Phil Linsalata
"Welcome to Missouri," Mr. Linsalata writes, "the only state in America with riverboat casinos filled to the gunwales with 'games of skill.'" In January of this year, the Missouri State Supreme Court ruled that several gambling games, including bingo and keno, are "lotteries" and hence, by the state constitution, prohibited in the riverboat casinos. The ruling states that in lotteries, "skill does not affect the probability of winning," whereas "in skill games, one person can be a better player than the others." Both poker and blackjack were deemed to "involve skill", and were thus not prohibited in the casinos. But the Court did not rule on games such as craps, roulette, and video versions of poker, blackjack, etc., all considered to have some element of skill as well as of "pure chance" in some part of the game. A constitutional amendment was required to allow these on riverboat casinos. But this all changed in April when, after such an amendment was defeated in the polls, the Missouri State Legislature redefined craps and some video games of chance so that they were allowed in the riverboat casinos when the casinos opened in May. An anti- gambling group, Citizens for Life and Liberty, have called the state's actions illegal. DISCUSSION QUESTIONS: 1. How would you define "skill" as it relates to gambling games such as poker, blackjack, and craps? For example, the article states that while "each bet on the craps table is purely random," there are numerous possible bets with different payouts and "knowing how to combine bets is a matter of knowledge," and hence skill. Do you agree? 2. Part of the debate seems to be on how to evaluate the relative amounts of skill between these games. How might this be done? <<<========<<
>>>>>==========>> Betting strategies in Final Jeopardy CHANCE Magazine, Spring 1994, p. 14 Jeremy M. G. Taylor
Final Jeopardy is the last portion of the television game show "Jeopardy". There are three players, and they are given a category of a final question. They each then bet an amount of what they have earned so far on the show (between 0 and their total earnings) before they are given the question. They win or lose the amount of the bet, depending on whether they answer the question correctly or incorrectly. The player with the most money after they have all answered the final question wins the game and keeps the money he or she has accumulated during the entire game. The winner also gets to return the next day for a new game. The players in second and third place do not keep their money but win prizes. The prizes for the second place player are generally much more valuable than those for the third place player. Mr. Taylor has analyzed 218 games of Final Jeopardy from the game show and has investigated possible betting strategies for the three players. For example, since the players all know each others' totals, one might bet the minimum required to win, assuming the other players get the question wrong. Other information a player could use to determine his or her bet might include the category of the question, the probable bets of the other players, and whether he or she is trying to win or avoid finishing in third place. The relative size of the players' totals before the final question is evidently of considerable importance in determining a good betting strategy. In particular, if the leading player before the final question has more than twice the player with the next highest amount, he or she can be assured of winning by betting conservatively. (However, a riskier bet might be made in the hopes of winning a larger amount, especially if the leader is confident about answering the question correctly.) In all, six relative "situations" are examined, with betting strategies determined for each player according to the situation. In 78 percent of the games he analyzed, the leader before the final question won the game, while in only 3 percent did the third-place player win. Mr. Taylor has determined a betting strategy which he says does not have a big effect on the leader but increases the chance of winning for the two other players and also reduces the number of times a player ends up in third place. DISCUSSION QUESTION: Mr. Taylor found that while women comprise 35 percent of the players, they end up in the leading position only 27 percent of the time. He also found that they answer the final question incorrectly slightly more often. He writes: "One wonders if the questions have some sort of inherent male bias." How might you test this hypothesis? Could women's betting strategies be different? <<<========<<
>>>>>==========>> Tissue illness and implants: no tie is seen
The New York Times, 29 May, 1994, p.16 Gina Kolata
A study financed by Dow Corning Corporation, a major producer of silicone breast implants, has failed to find an association between the implants and the disease scleroderma. In a recent class action suit settlement, Dow Corning agreed to pay up to $2 million to implant recipients who developed severe cases of scleroderma, a terribly disfiguring disease which hardens the skin into a shell and irreversibly damages internal organs. The study involved over 400 women who had scleroderma along with 1,184 healthy women. Dr. David Schottenfeld, who directed the study, interviewed 83 percent of the patients, looking for differences in background and medical history between the two groups of women. The article states that he found that less than 1 percent of the women with scleroderma had breast implants, "suggesting that implants could not be a leading cause of the disease. In fact, the women with scleroderma were even slightly less likely than the healthy women to have had breast implants." Ms. Kolata quotes several doctors who call the study "perfectly valid" and that "there is no evidence of increased risk." On the other hand, she writes that Stanley Chesley, a lawyer who helped negotiate the class action settlement, said the study was " 'of little or no impact' " and that with scleroderma and implants, " ' it's a direct cause and effect.' " DISCUSSION QUESTION: The study found that less than 1 percent of the scleroderma patients had implants and healthy women were in fact more likely to have had the implants. Does this imply that there is no increased risk from implants of developing the disease? <<<========<<
>>>>>==========>> Ozone depletion makes matters worse
St. Louis Post-Dispatch, 30 May, Everyday magazine p.1E No author given
This article describes the research on atmospheric ozone depletion by two scientists, Sasha Madronich of the National Center for Atmospheric Research in Colorado, and F.R. de Gruijl of Utrecht University in the Netherlands, that was reported in the journal Nature in November. Some experts claim that for every 1 percent drop in ozone concentration there is an associated increase of up to 3 percent in the amount of UVB radiation that reaches the surface of the earth. Using satellite data on the ozone layer from 1979 to 1992, the two researchers found that the intensity of UV rays, which cause skin cancer, increased by 6.7 percent in St. Louis. Because of this increase, incidents of skin cancer are also expected to rise in St. Louis. For some types there may be a jump of 17 percent, the article states. DISCUSSION QUESTIONS: The article states that the results on the increase in UV ray intensity have "a range of statistical uncertainty of 2.2 percent." What does this mean? The article says that this statistical uncertainty means that "the change of UV intensity in St. Louis could be as low as 4.5 percent or as high as 8.9 percent." Do you agree? <<<========<<
>>>>>==========>> Healing power of placebo underestimated
USA Today, 25 May, 1994, p. 1A Tim Friend
A new report published in the Journal of the American Medical Association suggests that the placebo effect plays a larger role in the success of treatment than doctors think. Judith Turner of the University of Washington, Seattle, studied the outcome of back pain therapy by reviewing three books and 75 articles published over 15 years. She concluded that the common belief that placebos work in about 35 percent of patients was a vast underestimation. DISCUSSION QUESTION: The article states that Ms.Turner "found studies showing effectiveness of up to 70 percent in patients getting ineffective treatments for many disorders." What do you think this means? <<<========<<
>>>>>==========>> Time to can the scan? New studies say that ultrasound scanning could harm the fetus.
The Guardian, 25 May, 1994, p. T9 Sally Weale A number of recent studies on the usefulness and safety of routine ultrasound scans for pregnant women have indicated that there may be no significant benefit from the procedure, and indeed, harm to the fetus may result. Mark Tsechovski, WHO director of disease prevention and quality of care in Europe, sent a letter to the health departments of every European country stating that "at the moment the best research shows no benefit from routine ultrasound scanning and the real possibility of risk." In a large American study, 15,000 low-risk women were split into two groups--those who had two routine scans during pregnancy and those who only had a scan if a problem was suspected. The study found no difference between the two groups in either perinatal mortality or postnatal illness of the child. As a result, The New England Journal of Medicine recommended to discontinue routine scanning. Other studies have found potentially more serious results. A Norwegian survey suggested that a child scanned before birth was 30 percent more likely to be born left-handed, while an Australian study indicated a possible link between ultrasound scanning of the fetus and low birth weight. And a Canadian study implies that scanning impedes speech development. Not everyone is convinced that the ultrasound procedure is harmful to the fetus, however. Professor Stuart Campbell, of King's College Hospital, London, says that the benefits are "overwhelming". DISCUSSION QUESTION: The article also points out that, since the 1960's, "the proportion of women undergoing scans has risen from 5 percent to 95 percent, yet the incidence of low birth weight in babies has stayed the same." Can this be compatible with the claims of the Australian study? <<<========<<
>>>>>==========>> Odds of quake in bay area set higher 9-in-10 risk of major event in next 30 years, studies say
The San Francisco Chronicle, 8 June 1994, A1 David Permian New studies have led government scientists to put the chances that a major quake will strike somewhere in the Bay Area at 90 percent in the next three decades. Four years ago this estimate was 67 percent. While this is not yet an official statement of the U.S. Geological Survey, it was made at a conference on earthquake hazards by one of the leading seismologists and endorsed by the conference chairman who heads the geological surveys project assessing future Bay Area earthquakes. The article explains a number of new discoveries that have led to this change in the probability estimate. <<<========<<
>>>>>==========>> Estimating the size of a population Teaching Statistics, Summer 1994. pp. 50-52 Roger W. Johnson
For the well-known German tank problem, you are given a sample of size n from a population of objects numbered from 1 to N and asked to estimate N. This is a favorite problem to use as an activity with the students trying to figure out and test reasonable estimates. See, for example, Section lX of "Exploring Surveys and Information from Samples" by Landwehr, Swift, and Watkins in the Quantitative Literacy Series published by Dale Seymour. Here the author gives four quite intuitive estimates all of which are unbiased. He suggests using computer simulation to decide which has the minimum variance, but he also gives the exact expressions for the variance of each of the four estimates. He indicates that some combinatorial skill is necessary to derive them, a skill that I found lacking when I once tried to compute these variances. <<<========<<
>>>>>==========>> Hot dogs, cancer in children linked. The Boston Globe, 3 June 1994, p4 Los Angeles Times
Three reports in the same issue of the journal "Cancer Causes and Controls" appear to link hot dog consumption to childhood cancer. Children who eat more than 12 hot dogs a month had nine times the risk of childhood leukemia. Children born to mothers who ate at least one hot dog a week during pregnancy have double the risk of developing brain tumor, and this doubled risk was also observed in children whose fathers ate hot dogs before the children were conceived. The researchers cautioned that the studies are preliminary and based on small numbers of cases: a total of 621 cancer victims and an equal number of healthy subjects were involved. Critics attacked the studies because they were not published in peer-reviewed journals. Others dismissed the results as typical of the "carcinogen of the week syndrome." DISCUSSION QUESTION. What questions would you like to ask about the design of these studies?
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! CHANCE News 3.08 (21 May to 10 June 1994) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Please send suggestions to: jlsnell@dartmouth.edu
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