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               CHANCE News 4.07
        (9 April 1995 to 5 May 1995)

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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/locate/chance.

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FROM THE INTERNET

Many government documents of interest to a statistics course are available on the Internet. For example, the "Climate Assessment for 1994" was just released and is available at http://nic.fb4.noaa.gov. You will find graphs and data showing that 1994 one of the three warmest years since global record-keeping began in 1951

Another example is the report of Labor Department's Glass Ceiling Commission released March 15 that investigates difficulties women and minorities have in reaching high level positions. The report is available on the internet at http://www.ilr.cornell.edu/. It is most easily read in the Adobe's pdf format. You can download a free copy of the pdf reader from this site In the pdf format, the graphs are nicely presented and easy to read.

The pdf reader also allows you to read a daily 8 page version of the New York Times at http://nytimesfax.com/.

FROM OUR READERS

At the suggestion of Joan Garfield, to help students in her CHANCE course, we are cataloging articles from previous chance news by topic on the Chance Database in Teaching Aids.

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The local supermarket has a large supply of Wheaties and each box has a picture of one of n different baseball players. For each player, there are r boxes having his picture. George asks: How many boxes must we buy to have a 95% chance of getting a picture of each player? The expected number of boxes bought until getting a complete set would also be interesting. Finding this expected number is a standard exercise for the classical problem where r is assumed infinite.

This problem occurred in George's work in genetics and, in his application, r = (n choose 2) and n could be as large as 100.

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• 1. In America: the issue is job.

• 2. Hospital stay for childbirth cut in half.

• 3. Feeling down? How about a pollster?

• 4. Study shows exercise must be strenuous.

• 5. Trying to reconcile exercise findings.

• 6. Ask Mr. Statistics.

• 7 Latest on the darts vs. the experts.

• 8. Ask Marilyn: risk and rewards.

• 9. Is nature worth more than a shopping mall?

• 10. Ticket to trouble. (winning a lottery).

• 11. Ann Landers: breast cancer statistics.

• 12. The Bell Curve Wars.

• 13. A humanist's view of The Bell Curve.

• 14. Restricted Choice--fact or fiction.

• 15. Simpson's paradox and the hot hand.

• 16. Ask Marilyn: a routine problem.

• 17. Aks Marilyn: a question on odds.

• 18. Effectiveness of heart disease treatments.

• 19. Any time is the right time to get fit.

• 20. The SIDS case.

• 21. How many shuffles to mix a deck?

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In America; the issue is job.
The New York Times, 6 May 1995, 1-19 Editorial desk.
Bob Herbert

The article is very interesting because it discusses the effects of downsizing in corporate America and government agencies on middle America--the increasing disparity between the haves and the have nots--the disruption of family security. The article asks the reader to think about what the people they are voting for represent in this scenario.

What is most interesting is that people tend to blame the poor and their needs without thinking that all of this downsizing is creating more poor who will need the government's support that could also be "downsized".

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The Centers for Disease Control and Prevention reported that the time spent in the hospital by women who are giving birth declined almost 50% from 1970 to 1992.

The topic is current but the data is too old to be meaningful. An average hospital stay of 2 days for a vaginal birth and 4 days for Caesarean section is discussed, neither of which are true today. 1992 data for this topic is far too old to be useful-- and is actually silly since many people would be pleased with hospital stays of such length in today's world.

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President Clinton has four pollsters working for him and all major politicians have at least one. This article attempts to explain what the pollsters do for the politicians. They obviously get data on a polititions current popularity, current opinions on major issues, etc. But the author suggests that pollsters' worth increasingly lies in telling officeholders where public opinion is headed next. He remarks that the recent abrupt new interest by the Republicans in the Medicare issue was preceded by a testing in the polls and focus groups by the National Policy Forum.

DISCUSSION QUESTION

Jim Baumgartner interprets this article as saying that the polls are now in the business of taking the derivative of public opinion. What does he mean? If he is right how could pollsters do this?

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A new Harvard study that followed 17,300 middle-aged men for more than 20 years has found that only vigorous and not non vigorous exercise reduced their risk of dying during the study period. The beneficial effects of vigorous exercise on longevity have long been assumed but were not firmly established.

Men who reported doing at least 1,500 calories worth of vigorous activity each week had a 25% lower death rate during the study than those who expended less than 150 calories a week. To achieve the level of exercise associated with longevity, a person would have to do the equivalent of jogging or walking briskly for about 15 miles a week.

In general, the more active the men were, the longer they were likely to live. This effect of vigorous exercise was seen even in men who smoked or were overweight, although those with neither of these health-robbing factors did better. By contrast, no consistent beneficial effect on longevity was found among those who pursued only non vigorous activities like golf.

The enhanced longevity associated with vigorous exercise mainly resulted from a reduced number of deaths from cardiovascular disease, said Dr. I-Min Lee, an epidemiologist at the Harvard School of Public Health, who directed the study.

The study defined as vigorous any activity that raised the metabolic rate to six or more times the rate at rest. Such activities include brisk walking, jogging, singles tennis, lap swimming, fast cycling, and doing heavy chores at home.

The following activities, or a mix of them, would achieve the level of caloric expenditure associated with the lowest death rate measured in the study:

1) Walking at four to five miles an hour for 45 minutes five times a week.
2) Playing one hour of singles tennis three days a week.
3) Swimming laps for three hours a week.
4) Cycling for one hour four times a week.
5) Jogging at six to seven miles an hour for three hours a week.
6) Rollerblading for two and a half hours a week.

Participants in the study had no known heart disease, cancer, or lung disease when they entered the study in 1962 or 1966 when they completed questionnaires about their histories and health habits. Asked again in 1977 about their exercise and health habits, many of the men were found to have increased their physical activity. This increase resulted in an even greater reduction in death rates than was associated with their 1960's activity levels.

Dr. Lee said most previous studies had looked only at the total amount of activity, not its intensity, in relation to heart disease and mortality risks.

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This articles discusses the problems researches now have reconciling the new Harvard study on exercise with earlier studies showing that moderate exercise did help prevent heart attacks. These earlier studies formed the basis for recent recommendations of the Centers for Disease Control and Prevention and the American College of Sports Medicine relating to the benefits of moderate exercise. These organizations concluded that 30 minutes a day of moderate activity, even in 10-minute spurts, would confer significant health benefits.

The article includes quotations from a large number of experts including Dr. Ralph Paffenbarger who was a co- author of this most recent study as well as an earlier study showing that moderate exercise was beneficial. He says "Everyone is confused. Even the scientists are confused"

DISCUSSION QUESTIONS

(1) Should Laurie discontinue his moderate exercise?

(2) The men in the Harvard study were classified by self-reported activities. How might that be a problem in studies of this kind?

(3) Do you think it likely that there is a threshold of exercise that you must exceed to get significant benefit?

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This seems to be a regular column in "Fortune Magazine". It appears that the authors write both the questions and the answers.

The questioner first asks why Max Frankel (in an article pointing on the sloppy use of numbers in newspapers) included an example that in one paragraph reported Russian miners had demanded a "150% pay increase" and in another paragraph the miners had asked to have their salaries "increased by 2 1/2 times". The questioner asks: Was Max's example (a) utterly pointless or (b) a striking instance of the very media weakness he is writing about? Mr Statistics answers that he "makes it 1 to 5" on (b) but I bet he meant to say "5 to 1" on (b)!

The next questioner asks about the game of 26 once played in the bars in Chicago. In this game you pick a number between one and 6. You win if your number comes up at least 26 times when you roll a die 130 times (in the bar you heave a cup full of ten dice 13 times). The payoff was 4 to 1. Mr. Statistics comments that Julian Simon got an answer between 18% and 19% by simulation and he got 18.2% by exact computing giving the bar about a 9% advantage.

The column concludes with a discussion of the report of the Labor Department's Glass Ceiling Commission released March 15. The authors poke fun at the report and the press coverage of this report. Since, as remarked above, the report is available on the Internet, it would be an interesting student project to see how accurate this column's criticism is.

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The winner in the latest six month period of the Wall Street Journal's ongoing stock-picking contest was the Dow Jones Industrial Average. The Dow rose 7.3% between Oct. 12 and the end of March. The four stocks chosen by throwing darts had an average gain of 3.8% and the stocks selected by four investment professionals fell an average 14.8%.

The pros remain comfortably in the lead in this series of overlapping six-month period. In 58 contests since the current rules were adopted they have beat the darts 33 times. They are ahead of the Dow 31 to 27.

DISCUSSION QUESTION

Is the number of times the pro's have beat the darts statistically significant? How about their lead over the Dow?

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A reader wonders if the allegation of draft dodging for President Clinton is fair. She points out that the chances of being killed in Vietnam were about one in 185 while the chances of a President being killed in office are about one in 10.

Marilyn says that is an interesting way to look at it, but says that you have to look at both the risks and the rewards. She suggests that being the leader of the Free World might be worth a billion dollars and the reward for fighting in Vietnam considerably less.

DISCUSSION QUESTION

(1) What do you think about the reader's observation?

(2) Which of the following two alternatives would you choose?

• You are given a penny, and a one in 185 chance of being killed over the next three years.
  
• You are given $1,000,000,000, and a one in 10 chance of being killed over the next four years.

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The House has passed a bill (H.R. 1022) requiring, before issuing any regulation expected to result in an annual cost of $25 million or more, an analysis of the risks to human health, safety and the environment as well as of costs and benefits.(The bill can be found from http://rs9.loc.gov/home/thomas.html). (Search under risks and costs in full text of House bills)

This article discusses expert's opinions on the possibility of carrying out the requirements of this bill. The bill itself actually only asks that the risks and benefits be carefully analyzed.

A group of experts testified that, while cost-benefit analysis has a place in public policy, it is "no magic wand" and "public policy is not well served by pretending science can do what it cannot".

The bill gives the courts the authority to review the risk assessments and cost-benefits in the case of a challenge. Experts in risk-analysis feel that this could lead to paralyzing the regulatory process by interminable challenges which the courts would be ill- equipped to settle.

Some experts believe that it is impossible to estimate the value of a life, or of preserving a species. They also suggest that moral issues may play a bigger role here than economic issues.

A similar bill will be considered by the Senate.

DISCUSSION QUESTIONS:

(1) Do you think this bill will make a good law?

(2) In the article, Dr. David Ehrenfeld, a conservation biologist at Rutgers is quoted as saying that nature "is not a subject for cost-benefit analysis". What does this mean? Do you agree?

(3) The New York Times recently reported that President Clinton has proposed expanded authority for the FBI to monitor telephone conversations, as a means of preventing acts of terrorism. How could you quantify the costs and benefits of such a proposal?

(4) Here's an idea: All legislation passed by the Congress will be required to undergo a cost-benefit analysis, and no bill will become law unless Congress can show that its benefits are "likely to justify", and are "reasonably related to" its costs. What do you say?

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Lois Gould is a novelist and this account of what happens when you win the lottery is beautifully written and reads like a novel with a very unhappy ending.

Your award for a million dollar prize is an annuity over 20 years that would cost about $450,000 to buy. You get $50,000 per year. Winners typically forget all the taxes they have to pay and end up with friends and relatives who won't speak to them because the winner won't solve all their financial problems. If a winner dies before payments are completed, the estate taxes on the remaining money are due immediately with monthly penalties added after nine months.

Gould says that the $30 billion in tickets sold last year suggest that the public prefers to ignore the arithmetic. She then takes a little poetic license writing:

And who cares if a game-theory professor once calculated the odds of winning as equal to a poker player's chance of drawing four royal flushes in a row, all in spades -- then getting up from the card table and meeting four strangers, all with the same birthday?

DISCUSSION QUESTION:

How do the odds for the game-theory professor's event compare with those for winning a lottery?

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In a previous column, Ann Landers made the familiar statement "One of out of eight women will get breast cancer." A reader provides a more detailed analysis taken from the "Women's Health Letter". Among other information provided, it is stated that:"A 20 -year-old women has a one-in-2,500 chance of developing breast cancer. At age 30, it's one in 233. At age 40, it's one in 63. At age 50, it's one in 41. At age 60, it's one in 28. At age 70, it's one in 24. At age 80, it's one in 16. And at age 95, it's one in eight."

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The Bell Curve Debate.
Random House, 1995
Edited by Russell Jacoby and Naomi Glauberman

Both of these books are collections of articles published elsewhere that relate to the Bell Curve. The first book is quite short, costs only $10, and contains what I think are three of the most interesting articles written about the Bell Curve. These are: "Curveball" by Stephen Jay Gould, "Cracking Open the IQ Box" by Howard Gardner, and "Race, IQ, and Scientism" by Richard Nisbett.

In a previous Chance News we commented on the Gould article where, among other criticisms, Gould argues that the notion of general intelligence is a mathematical fiction.

Gardner argues that not only Herrnstein and Murray but psychometricians in general are intoxicated with IQ tests and ignore the large amount of research on intelligence that has been carried out in a number of fields in the last hundred years. This research has shown that there are different forms of intelligence and success in life depends in a complicated way on various aspects of intelligence which are not strongly correlated.

Nisbit examines the explanation given in The Bell Curve for the black/white I.Q. gap. On the issue of genetic contributions, he claims that the authors leave out almost all the direct evidence and discuss in detail the only study consistent with a genetic component to the gap. On the question of intervention, Nisbit says that most of the relevant evidence is presented but in a one-sided negative light that other researchers would not give for the data. On the question of convergence between black and white IQ, he states that the authors have "simply misdescribed the straightforward data that they themselves present".

The second book is more of an encyclopedia. Part I reprints most of the news articles, editorials etc. written about The Bell Curve. Part II reprints articles that relate to the history of the I.Q. debate, starting with Galton and going up to articles by Herrnstein and Jensen.

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A critique of "The Bell Curve".
Free Inquiry, Spring 1995, pp. 60-61
Norm R. Allen, Jr.

Allan remarks that history presents many examples of one group considering another intellectually inferior starting with the Romans being convinced that the inhabitants of the British Isles they had conquered were totally incapable of being educated. In this century the Chinese were considered intellectually inferior to whites and now in "The Bell Curve" blacks are considered intellectually inferior to the whites and the whites to the Asians. He remarks that the rising and falling of peoples and nations through the ages cannot be explained simply in terms of intelligence. He concludes that "Any solution to human problems must necessarily have human answers, and humanist values will be especially relevant to bring about the improvement of the human condition."

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I mentioned last time that the problem of restricted choice in bridge was another version of the Monty Hall problem. I confess I did not really understand why this was the case until I read this very clear discussion of the problem. The journal wrote to the math department at the University of Waterloo and "After further correspondence" received a solution from a student Eric Sutherland. There are three later articles challenging Sutherland's solution again showing the difficulty people have understanding conditional probability. The journal had to declare a moratorium to discussion of this problem. Since the journal is not in your typical library I will put this article in our data base.

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In a celebrated article in Chance Magazine (Vol. 2 Number 1, pp 16-21) Tversky and Gilovich argued that fans' impression of streaks in basketball shooting is not verified by statistical analysis of players performance. To show this they asked a sample of 100 fans:"When shooting free throws, does a player have a better chance of making his second shot after making his first shot than after missing his first shot?" Sixty-eight fans said "yes" and 32 said "no." Tversky and Gilovich then analyzed data from nine regulars on the Boston Celtics basketball team and found no evidence that the outcome of the second shot depended on the outcome of the first. They attributed this difference to the erroneous belief on the part of the fans in hot hands.

Wardrop argues that the difference between the perception of the fans and what actually happens in individual player's performance could be accounted for by the effects of aggregation rather than the hot hand phenomenon. He verifies that individual records do not show that a hit after a hit is significantly higher than after a miss, but shows that, if the data for the players is aggregated, you find statistically significant evidence in support of the hot hand. Thus Wardrop suggests that the fans are just victims of Simpson's paradox and see patterns in the data that do not exist.

Waldrop also points out that while the individual data does not support the hot hand hypothesis it also does not support the Bernoulli trials model, since several Celtic players showed a significant improvement in their shooting ability on the second shot.

DISCUSSION QUESTION

Armed with this result, do you think you could persuade your students that their favorite basketball player really does not have "hot hands"?

Eight of the nine players had a higher success rate on their second shots. Would this alone convince you that the Bernoulli trials model should be rejected? How might you look more closely at this?

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A reader asks:

"You have six envelopes to pick from. Two-thirds (that is, four) are empty. One-third (that is, two) contain a $100 bill. You're told to choose two envelopes at random. Which is more likely: (1) that you'll get at least one $100 bill, or (2) that you'll get no $100 bill at all?"

No controversy this time. Marilyn solves the natural interpretation (envelopes are indistinguishable, selection is made without replacement) by simply listing the 15 possible pairs of envelopes, and observing that 9 of these have at least one $100 bill.

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A reader asks:

If the odds are in favor of something happening--as a 70% chance of rain-- and it doesn't happen, does that mean the odds were wrong?

Marilyn answers:

No. In the rain example, "a 70% chance" means that out of every 10 days with those atmospheric conditions, the forecaster says it will rain on seven days and won't rain on three. The odds would be wrong only if those conditions occurred repeatedly, and we found that it rained much more or much less often than seven times out or 10.

Our favorite question is "If the forecaster says there is a 20% chance for rain" and it does not rain, is the forecaster wrong?""

DISCUSSION QUESTION

An expert claimed that the answer to our question depends on whether historically it rains less than or more than 20% of the time in Hanover on May 5. What do you think he meant by this?

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A study of 1012 randomly selected heart disease patients compared the effectiveness of angioplasty to atherectomy for clearing clogged arteries. In angioplasty, a small balloon is inserted into the arteries and inflated to open passages. In atherectomy, a device inserted in a catheter is used to scrape deposits off artery walls.

About half of the 1012 patients underwent each therapy. After one year, 11 patients in the atherectomy group had died, compared with 3 in the angioplasty group. All but one of the deaths were related to heart disease.

The director of the study said that he had expected no difference between the procedures, and noted the difference found as a cause for concern. Other physicians suggested that the atherectomy may look worse in this comparison because of the relative inexperience of doctors performing this new procedure.

DISCUSSION QUESTION: How do you assess the statistical significance of the difference? Does it matter that one of the deaths was not related to heart disease?

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Research published today in the "Journal of the American Medical Association" describes a study of 9777 men, ages 20-82, who took part in treadmill exercises between 1970 and 1989. The men were given an initial treadmill test, a second test five years after the first, and their health was monitored for another five years. Men were classified as "fit" and "unfit" based on the number of minutes that they could maintain the treadmill exercise.

For those found "unfit" in both tests, the overall death rate in the five years following the tests was 122 per 10,000. For those found "fit" in both, the rate was 40 per 10,000. For those "unfit" in the initial test, but "fit" five years later, the rate was 68 per 10,000. Benefits were found in all ages groups. For example, fit men aged 60 had 50% lower death rates than persistently unfit men of the same age.

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Waneta E. Hoyt from Newark Valley, New York was charged on March 23, 1994 with the 1960 deaths of her five children. She recently took the stand to deny the allegations that she killed her five children. Mrs. Hoyt's case was originally considered a classical case of Sudden Infant Death Syndrome (SIDS), and was used as evidence by Dr. Alfred Steinschneider in his landmark paper on the disease. In the October 1972 issue of Pediatrics, Dr. Steinschneider suggested that SIDS is linked to sever apneic (loss of breath) episodes caused by abnormaliities present at birth. Of the five children he used for his study, all exhibited a number of prolonged apneic episodes during sleep, but only Mrs. Hoyt's children died.

According to "Science "(April 8, 1994), Steinschneider's paper ("Pediatrics", Vol. 50, No. 4, October 1972;), was first brought to the attention of New York prosecutors in 1986, when William J. Fitzpatrick, then the assistant prosecutor in Onondanga County--called Linda Norton, a forensic pathologist and former medical examiner for Dallas County, Texas, for help with a different case involving the death of several children in one family. Norton, who had read Steinschneider's description of the deaths of the two Hoyt children who were in Steinschneider's study, urged Fitzpatrick to read the paper. Norton believed that the Hoyt children's deaths were homicides. An investigation was launched, and Mrs. Hoyt was charged with murder.

Despite these charges, Steinschneider stands by his results.

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Keller presents a very simple model for riffle shuffling and computes a bound on the number of shuffles required to completely mix the deck. For a deck of k cards, the model assumes (1) a cut is equally likely to take 1,2,...,k-1 off the top of the deck, (2) when the resulting two piles are riffled together, each pile has probability 1/2 of having its bottom card become the new bottom of the whole deck.

Under these assumptions, the following recurrence relation is derived for p(n), the probability that the original bottom card is on the bottom after n shuffles:

p(0) = 1, and for n = 1,2,...

p(n+1) = (1/2) * [ p(n) + (1 - p(n)) * (1/(k-1)) ]

This follows from the law of total probability: either the card is on the bottom after n shuffles and stays there at shuffle (n+1), or else it is not on the bottom after n shuffles, but gets shuffled to the bottom the next time.

A bound on the number of shuffles to thoroughly mix the deck is obtained by noting that a necessary condition is that we shuffle until the original bottom card has probability 1/k of appearing on the bottom. For k = 52 cards, 1/k = .01923, and the above recursion gives p(7) = .02590 and p(10) = .02001.

  
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               CHANCE News 4.07
           (9 April to 5 May 1995)

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Please send suggestions to: jlsnell@dartmouth.edu

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