!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Prepared by J. Laurie Snell, Bill Peterson and Charles Grinstead, with help from Fuxing Hou, Ma.Katrina Munoz Dy,Pamela J. Lombardi, Meghana Reddy and Joan Snell.
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 site:
===========================================================
They (statistics) are the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the science of man.
===========================================================Francis Galton
Contents of Chance News 7.03
<<<========<<
>>>>>==============>
In December we invited experts on topics that often occur in Chance
News to give lectures that would help a teacher or student better
understand chance news. We had ten lectures. The topics covered
were:
Risks in everyday life
Arnold Barnett, MIT
Local Weather Forecasting
Mark Breen, Fairbanks Museum
National Weather Forecasting
Daniel Wilks, Cornell University
Stock Market Valuation
Sheri Aggarwal, Dartmouth College
The census 2000
Tommy Wright, Census Bureau
Probability and Statistics in Gaming
Olaf Vancura, Harvard University
Polls
David Moore, Gallup Organization
DNA fingerprinting in the Court
Bruce Weir, North Carolina State University
Statistical issues in ESP research
Ray Hyman, University of Oregon
Statistics in Sports
Hal Stern, Iowa State University
The lectures were all excellent. We videotaped them and have made
them available on the web (all but the Census 2000 lecture which
will be there soon.) Now we need some help in seeing what kind of
computers are necessary to view them. We would very much
appreciate your trying to view one or more of these videos and
letting us know the outcome. You can access them from the Chance
homepage by selecting "Chance Lectures".
Don't be dissapointed if
they don't work right for you. It just means that you may need a
faster machine (at least 160 Hz we think) or more bandwidth (at
least 56Kbs). They may work o.k. with less. Please let us know of
anything you find wrong such as missing slides etc., and also tell us
what kind of a computer and network connection you were using.
Of course, we would love to hear that nothing went wrong and they
were great! Thanks.
Each homework assignment is based on reading one or more
newspaper or magazine articles. Students are asked to answer a
series of questions suggested by the articles. Michael's answers to the
questions are also provided. The newspaper articles are all from the
New York Times, Wall Street Journal, or Scientific American and so
should be easily available from your library.
<<<========<<
>>>>>==============>
Daughters Give Birth on Same Day.
The Los Angeles Times, 14 March, 1998, A15
Times Wire Reports
The number of grandchildren of Ernie and Lynn Carey, who live in Utah, increased by three, one day last week. This may not seem like a very surprising event, until it is pointed out that in this case, no two of the grandchildren have the same parents. Yes, that's right, folks, three of the Careys' daughters gave birth on the same day! What are the odds of this happening?
DISCUSSION QUESTIONS:
(1) The article said that the probability of this event happening is 1 in 50 million. How do you suppose that they arrived at this figure?
(2) Do you think that the events of two sisters giving birth on the
same day are independent?
<<<========<<
>>>>>==============>
Passive smoking doesn't cause cancer - official.
Sunday Telegraph, 8 March, 1998
Victoria MacDonald
This article begins with the statement:
The world's leading health organization (WHO) has withheld from publication a study which shows that not only might there be no link between passive smoking and lung cancer but that it could even have a protective effect.The article states that WHO had presented only a summary of its results in an internal report. The writer says that WHO would not comment further on the article accept to say that the full study had been submitted for publication to a science journal. She says that the report would certainly be an embarrassment to WHO because of its anti-smoking stance.
She states that it was the one of the biggest study ever carried out and involved 650 cases of lung cancer patients and 1,542 healthy people.
DISCUSSION QUESTIONS:
(1) Do you think that the Telegraph did not know that making the results of a study public before publication may jeopardizes its publication in many major medical journals?
(2) Do you think that the Telegraph really believes that one study,
with about 2000 subjects, can settle an issue like second-hand smoke
where the effect, if it exists, is likely to small? In other words does
the Telegraph believe its own headline?
<<<========<<
>>>>>==============>
Scientists back our passive smoking report
The Sunday Telegraph, 15 March, 1998, 23
Victoria MacDonald and Robert Matthews
We learn in this article that WHO denounced the Sunday Telegraph report as "false and misleading". MacDonald and Matthews give more details about the results of the WHO study to defend the claims made in the previous Telegraph article. They state that the WHO report indicated that those exposed to environmental tobacco smoke from their spouse have a relative risk of 1.16 with a 95% confidence interval ranging from .93 to 1.44. They say that this indicates a non- significant result and the possibility of even a negative result.
MacDonald and Matthews go on to say that other scientists support the claims made in the original Telegraph article. They report that Professor Sir Richard Doll, who first demonstrated the link between direct smoking and lung cancer, said, "On its own, the study is not definitive, but it contributes to the weight of evidence."
DISCUSSION QUESTIONS:
(1) Explain why it an important fact that the above confidence interval contains the number 1.
(2) What do you think that the corresponding 90% confidence interval would look like?
(3) This article states that, just prior to this report, a review of the
evidence, combining the results of previous studies over the past
four years, indicated that passive smoking increases the risk of lung
cancer by 26 percent. Do you think the addition of the present study
will change this estimate significantly?
<<<========<<
>>>>>==============>
The Universe and the Teacup: The Mathematics of Truth and Beauty.
Hartcourt Brace 1998
K. C. Cole
ISBN: 0151003238
Hard cover. $13.20 from Amazon
Unlike scientists, science writers cannot concentrate on one narrow aspect of their field. Their profession requires that they follow current results in mathematics and science across disciplines. In the process they can see relations between sciences that many of us miss.
The recent book "The Jungles of Randomness" by science writer Ivars Peterson (See Chance News 6.12) wonderfully described how probability and statistics are applied to modern research in a wide range of sciences. Now K. C. Cole, science writer for the Los Angeles Times, has written a lively book that shows the many beautiful connections between mathematics and science.
While Cole travels throughout mathematics and science, she also has a lot to say about probability and statistics and we will limit our remarks to these areas.
Chapter 3 is entitled "Calculated Risks." The message of this chapter is that we don't make very intelligent use of what mathematics tells us about the risks we face. We are more influenced by many psychological considerations. We exaggerate risks that are beyond our control and underestimate the risks that we do control. This leads us to worry more about the dangers of flying than of driving and more about the danger of asbestos than skiing or skydiving.
As Tversky and his colleagues showed us, "The threat of a loss has a greater effect on a decision than the possibility of an equivalent gain." We are less likely to accept an operation when we are told there is a 5% chance of failure than when we are told it is successful 95% of the time.
Cole gives numerous other examples of our irrationality in making decisions based on known risks. She uses analogies to good effect. She learned the following frog story from mathematician Sam Saunders: "A frog placed in hot water will struggle to escape, but the same frog, placed in cool water that's slowly warmed up, will sit peaceful until its cooked. Analogously, we run from a burning house but ignore the accumulated effect of poor life styles such as: fatty foods, lack of exercise, and smoking.
Another analogy that we liked occurred in Chapter 12 where Cole talks about chance. She remarks that Ivar Ekland in "The Broken Dice" observed that the difference between determinism and chance is often a matter of scale. To illustrate this Cole observes that the game of billiards is thought of as a deterministic process. You aim the ball correctly and it will go in the pocket. The outcome of the role of a die is thought of as a chance process. If the die is made much larger the outcome is more deterministic while if you make the billiard balls smaller and confront them with barriers, you have a pin-ball machine which is more of a chance process.
Explaining coincidences to the general reader is always a challenge. Cole does not try to convince the reader that a single unusual event, for example three sisters giving birth on the same day, is not really a coincidence.
Rather, she shows how repeated experiments can establish that something that might be a coincidence is, in fact, not a coincidence. She does this in terms of experiments which led to the conclusion that quarks exist. She quotes Physicist William Carrithers, who help in the search for quarks, as saying: all we could do was to measure the probability that certain attributes were produced by quarks and not by more prosaic processes. Initial evidence estimated this to be about 1 in 400. After repeated experiments this probability became more like 1 in 10,000 and the result was announced.
To see the effect of repeated experiments, the reader is again offered an analogy. You are trying to show a particular coin has heads. You cannot see the coin but can learn the outcome of a toss of the coin. One toss coming up heads is not very convincing but 100 tosses all of which turn up heads, becomes pretty convincing.
Fellow science writer Robert Matthews, in his review (New Scientist, 28 February 1998) of Cole's book, refers to the current argument about curriculum reform and suggests that we "abandon the stultifying and alienating emphasis on ever more pointless technique and stress instead ways of 'seeing' the science and mathematics in the world around us". He suggests that a good start would be to use Cole's book as a text for a first course in mathematics appreciation. We agree!
DISCUSSION QUESTION:
Cole describes the 1 in 400 chance as about the same as the chance of
pulling two aces from the top of a deck of cards? How good is this
analogy? Can you think of a better one?
<<<========<<
>>>>>==============>
Joan Garfield sent us the following tidbit.
Sunday's (March 22, 1998) Star Tribune, has a column "C.J." that mentions that Dave Kapell, the developer/creator of Magnetic poetry kits, thinks Madonna used his sequel kit to inspire her new song "Candy Perfume Girl." She denies it. He claims that only 3.2 percent of the song's 128 words are not found in his kit, and he even had a mathematician (Bill Kamp) check it. Kamp said the odds that the song and the sequel kit would have that many words in common are 1 in 4.4 trillion!
DISCUSSION QUESTION:
How do you think Kamp computed his odds?
<<<========<<
>>>>>==============>
U.S. 12th-graders miss the mark.
Christian Science Monitor, 25 Feb. 1998 p.1
Gail Russell Chaddock
The results of the Third International Mathematics and Science Study(TIMSS) were announced Feb. 24 and American 12th-graders scored well below the world average. No country ranked lower than the U.S. in advanced mathematics and physics.
This continued the downward trend in U.S. students as they reach higher grades. In a previous test for 4th-grade students, those in the U.S. were above average in math and were second only to South Korea in science. Then on the TIMSS test of 8th-grade students, U.S. students had already dropped below international averages. Now the 12th graders have fallen even further behind.
Poor performances in the past have been blamed on the fact that U.S. high schools accept all students and other countries are selective. This is no longer the case. All 21 countries participating in the TIMSS study enroll more than 90 percent of the secondary-school- age students. In addition, the explanation that U.S. students watch too much television no longer applies since the TIMSS report showed that U.S. students are just about average in hours watched per week.
A report of this test in the New York Times brought a barrage of explanations in op-editorials and letters to the editor. Some simply said that the results are unacceptable and something should be done about it. Others attempted to explain the poor showing. An education expert pointed out that European schools have more centralized systems. Because of this they are able to make a concerted effort to improve education much faster than can the U.S. The same writer remarked that the average age of the students in some countries is two to four years older than most of the U.S. students.
Others blamed the poor performance on the fact that U.S. teachers are poorly paid and often do not have the qualifications to teach science. Others think we should not worry and encourage the new active learning without worrying so much about what facts are learned. This leads another writer to comment: "just think, by 'designing and carrying out their own experiments,' students might reinvent the Pythagorean theorem!" A nobel laureate says "Maybe we have let student wander all over hell in high school, but that reserves some energy for later when it is better spent."
Many writers wanted to know how the U.S. could be doing so well in technology if the students were not learning math and science. But then one writer points out that, since 1976, workers with wages in the 50th percentile have lost about 15% of earning power, while the lowest tenth has lost 25 percent in real wages, suggesting things would only get even worse for students poorly prepared in high- school.
After reading a report of this study in the Boston Globe, reader Marcello Pagano wrote us:
What I found amusing was that, in the same issue that the Boston Globe (Feb 25, 1998) reported the poor showing of American students, almost to prove the point, the editors published a letter that contained the following (referring to a previous issue of special education); "Instead of bemoaning that the 17 percent of students in special ed get 20 percent of education funds, he (John Silber) should applaud teachers and therapists for doing so much with that 3 percent diff- erence."Marcello points out that, if a group comprising 17% of the students get 20 percent of the funds, then they are getting 22 percent more than the students not in the group, not 3 percent.
You can find the full TIMSS report at here.
DISCUSSION QUESTIONS:
(1) How did Marcello get the 22%?
(2) The hardest question on the TIMSS test proved to be the following:
A string is wrapped symmetrically around a circular rod. The string goes exactly 4 times around the rod. The circumference of the rod is 4cm and the length is 12 cm. (A picture is provided showing the string wrapping around the rod going from the top to the bottom of the rod.)Only 10% solved this problem correctly with only 4% of the U.S. students getting a correct solution. How would you have done on this question? Does this question suggest that more drill is needed in math classes or that more time be spent on how to think about a math problem?
(3) Comment on the various explanations given for the poor
performance of U.S. high school students?
<<<========<<
>>>>>==============>
Here is some more cheerful news.
I.Q. scores are up, and psychologists wonder why.
New York Times, 24 Feb. 1998, F1
Trish Hall
While SAT scores go up and down (they declined from the mid 1960's to the early 1980's), the IQ scores just keep going up. This was not noticed for a long time because, unlike SAT scores, IQ tests are normed each year to keep the median 100. Political scientist James Flynn discovered this increase in the 1980's when he looked at scores on tests for the military that have not been normed. This increase in I.Q. has been called the "Flynn effect."
Flynn's observations have been verified by others but no-one has a satisfactory explanation for the increase. The largest increases have occurred on the Ravens Progressive Matrices, designed to measure abstract reasoning and thought to be free of the influence of culture and education. It is estimated that someone who would have scored in the 90th percentile in the early 1990's would score only in the 5th percentile today.
DISCUSSION QUESTIONS:
(1) One explanation for the increasing I.Q scores is that children have much more experience working with mazes and puzzles that are almost identical with those used on I.Q. tests. They even see these mazes on advertisements. Doesn't this suggest that you can study to improve your IQ scores?
(2) An IQ tester once came to the office of the famed geometer John Conway to give him an I.Q. test. One look at Conway's office would suggest that this is a man who has solved puzzles all his life. How do you think Conway would do on an I.Q test?
(3) We seem to have an increasing number of computer wizards and
not Mozarts. What, if anything, does that suggest about IQ as a
measure of intelligence.
<<<========<<
>>>>>==============>
Sometimes good discussion questions come from real life rather than
from a newspaper article. Here is one such question.
DISCUSSION QUESTION:
One of my (Bill Peterson's) colleagues at Middlebury wrote to the Math Dept. to report the bad luck his daughter had experienced in her room-draw-lottery at Carleton College. In three years her numbers were 460, 460 and 399 out of 470. Her question "How rare was it?" led to an interesting discussion. Simply using (1- 460/470)(1-460/470)(1-399/470) overstates the case, as it ignores other triples that would likely be viewed as equally bad. One proposal was an additive measure: what is the chance of a sum of draws greater than or equal to 460 + 460 + 399? Another was multiplicative: what is the chance of a triple (i,j,k) with
We ultimately preferred the sum. Sorting out the pros and cons
could make an interesting discussion question for class.
<<<========<<
>>>>>==============>
We find another interesting probability problem on the
ABC web site
Reporter Robert Kulwich says that he is working on a story for Nightline about the tendency of women around the world to have fewer children.
He says that there are now about 57 million Italians and families who have on average 1.2 children. Assuming that this rate continues and there is no migration or immigration, Kulwich wants to estimate when the population of Italy will become 0.
He reports that demographer Karl Haub of the Population Reference Bureau estimated that "after 2000, the population of Italy would drop by 50 percent every 42 years."
Kulwich then gives some additional assumptions: average lifetime is 78, all women have children and can have children only until age 46. He then asks when the population can be expected to be 0. He reports that Haub estimated this would happen in the year 3880 by one method and the low 4000's A.D.s. by another method. Kulwich asks readers to help him with this estimate.
We persuaded Charles Grinstead to give his thoughts on the problem. He writes:
The assumptions in this problem are that, on the average, each woman gives birth to 1.2 children. We will also assume that the average age of the mothers at the time they give birth is about 30. We further assume that about half of the babies are female.
There are several points to be made before we attempt to describe a way to calculate the demise of Italy. First, it makes little difference what the life expectancy of the population is. To see this, imagine two different populations; in one, the women stop having children at age 46 and live to an average age of 78, and in the other, the women stop having children at age 46 and all of the women die at age 46. It is easy to see that the only difference in the number of women in the two populations is the number of women in the first population who are over 46 years old. When the population dwindles, as it does in the case at hand, this difference becomes very small.
Second, if we look at the ancestry of any woman at any time in the future, we see that there is a unique path from her back to a woman who is now alive, if we restrict the path to only go through women. This means that if we look forward into the future, we can simply consider how many women are alive at any particular time who have, as an ancestor, a given woman who is alive today, and then multiply this number by the number of women of child-bearing age who are alive today.
We now proceed to the calculation. Given a particular woman who is alive today, in 30 years she is replaced by .6 women. Thus, if F(x) is the number of female offspring of this woman at time x, then F(30n) = (.6)^n. Thus,
If there are W women in the current population (remember, we may assume they are all under 46), then at time x, there are W*F(x) women.
To find the approximate halving time, we simply set W*F(x) = .5*W, and solve for x. We obtain 40.7 years. Next, if we guess that W = 20,000,000 (a rough estimate of the number of women now alive in Italy under the age of 46), and we set W*F(x) = 1, we obtain a value of x of 987 years. Thus, in 987 years from the present, there will be only one woman of childbearing age (and presumably, approximately one man) in Italy.
Thus the estimate for halving the population is quite close to that of demographer Haub. However, Charles gives Italy only about 1000 more years to live while Haub gives them about 2000.
DISCUSSION QUESTION:
What can you contribute to this problem?
<<<========<<
>>>>>==============>
Alcohol tie to breast cancer affirmed.
The Boston Globe, 18 February 1998, pA3.
Richard A. Knox
Described as one of the largest investigations of the link between alcohol consumption and breast cancer, a Harvard School of Public Health study gathered data from a half dozen studies involving 330,000 women worldwide. Consuming two to five alcoholic beverages a day was associated with a lifetime breast cancer risk of about 17%, compared to the overall rate of 12.5%. This represents a 40% increase in risk, nearly as large a risk factor as having a family history of breast cancer.
Because the study was so large, it was possible to control for other risk factors and focus on the contribution of alcohol. But according to investigator Stephanie Smith-Warner, factors such as menopausal status, family history of breast cancer and smoking did not seem to affect the alcohol-related risk.
A 1996 Harvard study gave an indication of the mechanism for the link. It was found that women who drink regularly and take post- menopausal estrogen supplements had three times the blood estrogen level of non-drinkers who took estrogen.
The article notes that these findings further complicate the trade-off between heart disease and breast cancer risks. Other studies have reported that moderate alcohol consumption lowers the incidence of heart disease and mortality in both men and women.
DISCUSSION QUESTIONS:
(1) What do you think is meant by "moderate" alcohol use? Do you think that two to five drinks per day covers a lot of ground here?
(2) Can you think of any reasons why it would be difficult to break
out the dose-response effect more carefully?
<<<========<<
>>>>>==============>
Missile destruction raises fears of space debris.
The Christian Science Monitor, 19 February 1998, pA4.
David L. Chandler
During a test flight last month, a Minuteman missile disappeared from radar 240 miles above the Pacific. While there is no hard evidence, analysts are speculating that the missile was destroyed in a collision with a piece of space debris. A tracking telescope in the Marshall Islands reportedly saw an object headed toward the missile before it disappeared
Some 7000 pieces of orbital debris are catalogued and tracked by the U.S. Space Command, whose detection equipment can only pick up pieces about the size of a baseball or larger. But the high speeds at which the debris is moving--typically 17,000 mph--means that much smaller pieces are very dangerous. In a December 1997 report, the National Research Council stated that about 95% of the debris big enough to inflict "critical damage" on the space shuttle was too small to be tracked. An object 0.1 inches across could potentially penetrate the crew cabin, resulting in an air leak.
The article reports that a Pegasus rocket exploded in June of 1996, leaving 700 pieces of debris large enough to be tracked, and "far more that were too small for detection."
DISCUSSION QUESTIONS:
(1) How do you suppose the 95% figure (for the portion of debris that is undetectable) was estimated?
(2) What do you make of the fact that the Pegasus represents a tenth of all debris currently being tracked?
(3) What information would you need to estimate the chance that a
shuttle flight would suffer a collision with space debris?
<<<========<<
>>>>>==============>
Elderly gamblers and health risks.
The New York Times, 22 February 1998, Sec. 14 (Long Island), p. 27.
Letter to the editor, by Mark L. Taff, M.D. and Lauren R. Boglioli, M.D.
Prompted by February 8 article "For the Gambling Elderly, Help in Fighting Addiction", the doctors write to share results of their research on 398 casino-related deaths in Atlantic City from 1982 to 1986. They report that 83% of these were due to sudden cardiac arrest, and most of the victims were elderly, white, retired males with previously diagnosed medical conditions.
The doctors identify a number of stresses related to gambling activity, including travel, excitement, exposure to cigarette smoke, overeating and drinking. They urge the elderly to properly medicate themselves and recommend that casinos provide emergency health care services.
DISCUSSION QUESTIONS:
(1) Can you think of other causes of casino-related deaths? Should we be surprised that most of the victims are elderly?
(2) In terms of mortality risk, how do you think a trip to the casino
compares with other kinds of travel?
<<<========<<
>>>>>==============>
Personal health: Left-handed approach to survival?
Newsday, 24 February 1998, C11
Ridgely Ochs
Ochs reports running across a British study in the February issue of the Journal of Epidemiology and Community Health with the imposing title "Is forced dextrality an explanation for the fall in the prevalence of sinistrality with age?"
Ochs cites a famous 1981 study by Stanley Coren, popularized in Coren's 1992 book "The Left-hander Syndrome" (Free Press). Coren's data showed that, among 5147 people ranging in age from 8 to 100, 15% at age 10 were left-handed, compared to only 5% at age 50 and less than 1% at age 80. The apparent conclusion is that left-handers have lower survival rates. Coren felt that the lefty effect translated into about a two to five year decrease in life expectancy.
Simon Ellis, author of the recent British study, points out that, if Coren is right, then the effect is about of the same magnitude as the difference in genders. He worries that the insurance companies might be led to quote different rates to left-handers. Ellis' own study looked at 6097 people aged 15-70. He found that the prevalence of left-handers decreased from 11.2% at age 15 to 4.4% at age 70.
Ellis doesn't blame a decreased ability of left-handers to survive. Rather, he explains that social prejudices against left-handers have forced them, over the course of their lives, to switch hands. In order to adjust for this, Ellis removed questions about writing and drawing, skills deemed particularly susceptible to being changed by outside influences. Still, this didn't completely remove the lefty effect among older people, and Ellis reports that the case is still not closed.
Ochs did a phone interview with Coren, who said that President Clinton is a lefty and he suffers bad allergies. Former President Bush is a lefty and suffers from a thyroid condition known as Graves disease. Ochs observed that Barbara Bush also has Graves disease, but Coren didn't know if she was left-handed!
DISCUSSION QUESTIONS:
(1) It appears that people cannot accurately remember if they were born lefty and forced to switch. Does this surprise you?
(2) What kind of study would be required to settle the issue?
<<<========<<
>>>>>==============>
John Perazzo sent us the following contribution with discussion
questions.
Valentines day sees record marriages.
Las Vegas Sun, 19 Feb. 1998
Ed Koch
This article reported that 2,590 marriage licenses were issued over the Valentine's Day weekend (Friday - Monday), up from the 2,126 issued last year.
To further demonstrate this increase, the years 1988 (59,423), 1990 (76,060), 1993 (86,608), 1995 (101,755), 1996 (104,920), and 1997 (110,696), were listed with the total licenses issued that year. Comments were made that other years did not show an increase.
DISCUSSION QUESTIONS:
(1) Do the totals shown confirm the idea that the number of licenses issued is going up and up and up?
(2) What interpretation can be made of the leveling off of the
licenses issued?
<<<========<<
>>>>>==============>
'Soothing': A happy-marriage predictor.
Washington Post, 13 March, 1998
Barbara Mathias Regel
This article reports on a study led by marriage researcher John Gottman. The results of the study were published in the current issue of the Journal of Marriage and Family.
The study followed 130 newlyweds for six years to try to understand what kind of behaviors are good predictors for a happy marriage.
The researchers found that most discussions about marital problems are initiated by women, especially in volatile marriages. Men initiate the discussions only 13% of the time. They found that husbands who accept the influence of their wives tended to have happy, stable marriages, while a pattern of negative startup by the wife and a refusal to accept her influence predicted divorce.
For the past 25 years, therapists in marriage counseling have tended to use "active listening". For example, the therapist asks the wife to state her complaint directly to the husband. The husband is then asked to paraphrase what his wife said and how she feels. Then he is asked to validate her feelings.
According to Gottman this method of counseling is based upon Psychological theory and not on empirical studies and is just not working.
He reports that divorce and stability were predicted with 83% accuracy by evaluating the softening/accepting behavior of the couples, and so feels that counseling should take this into account. He says: "I'm quite confident that active listening is not working very well. But I don't want to offend or alienate clinicians....I'm not saying that active listening will never work. We all have the same enemy, ignorance. We have the same goal -- to treat."
DISCUSSION QUESTION:
Most writers reporting on this study said something like "We did not
have to be told this!" Do you believe them?
<<<========<<
>>>>>==============>
Several readers suggested interesting web sites related to chance.
Anne Webster Grant suggested the web site "Public Awareness and
Schools Support for Mathematics" (PASS Maths).
Your students will enjoy the article on
Coincidences by Geoffrey Grimmett. Here they will learn, among
other things, why the length of the longest run of heads in n tosses of
a coin is about log(n) to the base 2 and can check this out from the
site by simulation.
<<<========<<
>>>>>==============>
GÈ Groenewegen wrote that he had a good experience using a
software package PQRS in his introductory statistics class for
business students.
PQRS (Probabilities, Quantiles and Random Samples) was developed by S. Knypstra from the State University of Groningen, the Netherlands. It is freeware and it can be obtained at PQRS.
PQRS allows the user to choose any of the standard probability distributions with appropriate parameters and a number z. The program graphs the distribution (density) with z in the appropriate place on the x axis. It then shows the probability that the outcome is < z and > z. For a discrete distribution it also gives the probability the outcome is = z. The user can also choose a sample of size n using the distribution and save the results in a file.
Knypstra has used PQRS as part of a larger package: Sila (Statistical
Inference Laboratory) designed to assist in teaching statistical
inference. This package is available at
http://www.eco.rug.nl/medewerk/knypstra/sila.html
Both PQRS and Sila run only on Windows.
<<<========<<
>>>>>==============>
Jane Milar mentioned an interesting web site maintained by the
Annenberg/CPB Project that produced the well known statistical
video series "Against all Odds".
Here you will find a series of
exhibits associated with the Annenberg projects. One of these is on
polling and provides a nice introduction to polls.
<<<========<<
>>>>>==============>
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!