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Chance is perhaps the pseudonym of God when He did not want to sign.
After reading the article, I followed through with one of the discussion questions, which asked he reader to make a plot of the value of a 1996 quarter vs. year for the past 83 years. Once plotted, the reader was asked to fit a curve to the relation. I was surprised to see that a segmented straight line regression (with a break point around 1973) did an adequate job of describing the relation.
Looking on the Web, we found the equivalencies of $1 today all
the way back to 1800 using the consumer price index. You can
If you plot this data, you find that the it starts at about 10
cents and goes down to about 5 cents by 1900 with some humps along
the way at 1814 and 1864. As Jean suggests, it is quite linear
from 1973 to 1996. However, if you look at 1900 to 1973, it looks
more quadratic, like x^2, with a hump at 1920. The humps remind
us of the effects of wars. You can get data on the consumer price
index and its components from the Bureau of Labor Statistics.
Nancy Serrell sent a note saying:
Today's (July 17, 1996) Boston Globe, page A6, carries a short
AP item that made me laugh. It's typical of those social-science
Believe It Or Not stories based on "statistics."
YOUTH IS IN EYE OF THE BEHOLDER
New York - Three in four baby-boomers think they look younger
than their years, and, despite statistical impossibility, eight
in 10 say they have fewer wrinkles and similar signs of facial
aging than others their age. In a survey by Louis Harris and Associates
that was released yesterday, Americans between the ages of 30
and 50 were asked about aging and appearance.
Shades of Lake Wobegon!
The American Academy of Dermatology (ADA) recently sponsored an
Surprisingly, there's less chance your child will be hit by a speeding car than by skin cancer. One out of five Americans develops skin cancer. Don't let your child be the one.
This is voiced over a clip of cars speeding down a highway around
a red-headed girl peddling a tricycle in their midst parallel
to their flow.
(1) Jan suggests that the most people would have difficulty believing
this? Do you agree?
(2) Jan also sees problems in comparing risks like this? Do you?
Here are two new resources on the web.
This is a text (well written we must say) for a first course in probability that emphasizes the use of computing. You can read it on the web or, better, download it on your computer and read it there.
This is a scientific visualization center at Boston University
devoted to interdisciplinary research in aspects of polymeric,
random, or fractal systems. The center also develops experimental
and computational materials for
K-12 and undergraduate education.
You will find, for example, computer simulations and hands-on
activities designed to lead the student from tossing coins and
random walks to fractals. Also has you can visit the Boston Science
Museum exhibit called "The Dance of Chance".
Erasing the passed.
The Boston Globe, 14 July 1996, A1
The Stratfield school in Fairfield Connecticut is a prize-winning
prestigious school with 500 kindergartners through 5th graders.
In April the superintendent held a news conference to announce
that administrators of the Iowa Test of Basic Skills, a comprehensive
national test given annually to many 3d and 5th graders, had studied
the school's 150 answer sheets from January and observed tampering
on many exams.
This claim was based on comparisons of erasures on Stratfield's
reading comprehension tests and on those from two other high-performing
elementary schools in Fairfield. The Stratfield tests had 3.5
to 5 times the number of erasures than did the tests of the other
schools. Further, 89% of the Stratfield erasures ended in a change
to the correct answer, compared to less than 70% of the changes
at other schools. Houghton Mifflin is the parent company of the
tests and officials from this company stated that such results
"clearly and conclusively indicate tampering."
The Stratfield 3rd graders were retested. On the original test
the 3rd graders scored in the 89th percentile of the students
nationally, while on the retake they dropped to the 79th percentile.
Connecticut's education department studied tests given to the
Stratfield 4th graders in 1994 and 1995 and again found more erasures
than on tests from other schools and that 89% of the erasures
gave correct answers.
Parents and teachers remain unconvinced that there is anything
wrong with the tests. The Stratfield principal claims there is
no evidence of tampering and that the high erasure rate is evidence
of the school's emphasis on thorough, precise work.
The matter is being investigated by a former FBI agent and a forensic
scientist fresh from the O.J. Simpson trial.
"A Church Arson Epidemic? It's Smoke and Mirrors."
Wall Street Journal, 8 July, 1996, A8
The author reviews the sequence of events that led to the media
reporting a near epidemic of burning of black churches and concludes
that there is no evidence that there is an epidemic.
At the end of March, the Center for Democratic Renewal held a
press conference and released a preliminary report showing a surge
in arsons against black churches beginning in 1990. The report
claimed that there had been 90 arsons against black churches in
nine Southern states since 1990 and that the number has risen
each year, reaching 35 in 1996 as of June 18, and that each and
every culprit arrested and/or detained, has been white.
Fumento reports that, when he tried to verify the CDR claims from
government sources, he found that they did not have adequate records.
He was referred to a private group, the National Fire Protection
Association who had records of fires but not broken down by race.
Their records show a dramatic decrease in the number of church
arsons-- from 1,420 in 1980 to 520 in 1994.
Fumento contacted law officials in South Carolina, Georgia, Alabama
and Mississippi and reports his findings which are very different
from the claims of the CDR. For example, South Carolina had the
most arsons (27) on the CDR list. Officials reported that seven
of the fires were either found not to be arsons or the cause has
not been found. Eight of the 18 arrested in connection with the
fires were black. Reports in the other states indicated a wide
variety of causes for the fires and no particular pattern.
Another source of data was provided by the extensive coverage
in "U.S.A. Today". Here Fumento suggests that the authors
incorrectly interpreted their own data when they stated that:
"The numbers confirm that a sharp rise in black church arsons
started in 1994 and continues." Fumento states that the
"U.S.A Today" charts indicated that two of the states
didn't start reporting data until 1993 and a third one didn't
until 1995, and so it is not surprising that the totals increased
when they did start reporting.
Looking at just those states that have reported data since 1990,
he finds that the number (13) in 1990 was two more than in 1994
and the number in 1991 (16) was the same as in 1995. He suggests
that the large number in 1996 was in fact due to copycat crimes
caused by the undue publicity given to the CDR report.
In a letter in the July 15th "Wall Street Journal",
a writer from "U.S.A. Today" pointed out that, by looking
only at those states that have reported data since 1990, Fumento
includes information on only 6 of 11 states and in particular
leaves out Alabama, North Carolina and South Carolina where the
heart of the increase lies. It is claimed that complete data
is available since 1993 and, in 1993, there were 19 black church
arsons. In 1995 there were 27 and, in the first half of 1996,
there were 40.
A more detailed analysis of the way the press and the politicians
handled this story can be found in an article by Michael Kelly
in the July 15, 1996 issue of The "New Yorker".
(1) What information would you like to have to try to settle
the dispute over whether there is a near epidemic of black church
(2) How reliable to you feel the sources used by this author
(3) Evidently, there have been about the same number of black
churches burned as white churches. There are several times as
many white churches as black. Race has been a factor in some of
the black churches but not evidently not in the white churches.
Are their reasons other than race that could account for more
proportionally more black than white churches being burned?
Sampling wildlife populations.
Chance Magazine, Vol. 9 No.2, Spring 1996, p 9
by Bryan F. J. Manly and Lyman L. McDonald (1996).
The authors discuss several current conservation problems that
involve wildlife populations and the statistical methods used
in trying to settle these disputes.
These methods are variations of classical methods for population
estimation that go back to those used by Graunt, to estimate the
population of London in 1662, and Laplace, to estimate the population
of France in 1783. Laplace obtained a register of births for
the whole country, including a set of parishes for which he knew
the births b and population n. He then argued that the ratio
of population N to births B for the whole country should be approximately
the same as the same ratio for the parishes, i.e., N/B = n/b.
Thus he knew everything but N, so estimated N by B*n/b. This is
an example of an estimation method that has many forms and names:
capture-recapture and mark-recapture are two common names used.
The first controversy the authors discuss involves the protection
of the northern spotted owl and the forest management in the U.
S. Pacific Northwest. To help settle this controversy, studies
were carried out to see if the death rate of spotted owls is going
down as the result of current forest management policies. Here
is how one of these studies is carried out.
A number of adult spotted owls are captured, tagged and released.
A year later an attempt is made to recapture these owls. Some
will be recaptured, some will have died and some will still be
alive but missed in the recapture process. Attempts to recapture
the owls are carried at the end of each year for several years.
Let s(i) be the probability that an owl, that has survived until
the ith year, survives the i+1st year and p(i) be the probability
that an owl, captured in the ith year, is captured in the i+1st
year. The data for each owl is recorded as a sequence of 0's and
1's where a 1 means it was recaptured and 0 that it was not. For
example, for a three-year period the data for a specific owl can
be represented by one of the four sequences: 100, 101,110,111.
Here 101 means the owl was not recaptured at the end of the second
year but was at the end of the third year. The probability of
this sequence is s(1)*(1-p(2))*s(2)*p(3). The product of these
probabilities over all the owls gives the probability of obtaining
the data observed. The parameters p(i),s(i) are then chosen to
maximize this probability (the maximum likelihood method), giving
the desired estimate for the survival rates s(i).
The article shows the results of such a study for adult northern
owls for the years 1985-1992. There appeared to be a significant
negative trend in the survival probabilities for adult females
but not for males.
Another example illustrating a different type of study relates
to polar bears in Alaska. The study was designed to see how the
polar bears are surviving the harvest permitted native Alaskan
subsistence hunters. This was an aerial study aimed at estimating
the number of polar bears in the southern Beaufort Sea by a method
called the "line transect method".
This method is used when it is desired to estimate the animal
population in a region where you cannot be sure that you have
counted all the animals even in a subregion. Assume that we want
to estimate the number of animals in a strip bordered by the two
lines a distance w from a line L. A researcher travels along the
line L looking for animals. For each animal spotted he estimates
and records the perpendicular distance from the animal to the
If an animal is a distance x from L, we assume that it will be
seen with a probability g(x) with g(0) = 1. Assume further that
the animals are randomly distributed in the strip. Then the probability
that an animal in the strip is observed is the average a of g(x)
between 0 and w. Then if N is the number of animals in the strip
and n are observed, E(n) = Na. Thus n/a should be a reasonable
estimate for N.
Thus to estimate N we need only estimate a from our data consisting
of x(1), x(2), ..., x(n), the perpendicular distances to the line
for the animals seen. The x's are independent with common distribution
and, again using the fact that the perpendicular distances of
the animals are uniformly distributed, we find that the x's have
density f(x) = g(x)/a. Since g(1) = 0, w = 1/f(0). Therefore,
to estimate w we need only estimate f(0). This can again be done
by the maximum likelihood method. For example, it is often assumed
that the probability of observing an animal decreases exponentially
with the distance to L, i.e. g(x) = e^(-x/m). Then f(x) is the
exponential density f(x) = (1/m)*e^(-x/m) and so w = 1/f(0) =
m, the mean m of x(i). The obvious estimator for m is the average
of the x's which is the maximum likelihood estimate.
This is an excellent article illustrating how statistics is used
in current issues. The article is not as technical as our review
of it. Reading the article made us curious how these methods really
work, so we delved into the excellent references that the article
What problems might arise with the assumptions made in these models?
Chance problem corner.
Chance Magazine, Spring 1996, p 42
Gabor J. Szekely, editor
"Chance Magazine" has a new problem corner edited by
Gabor Szekely. The first column describes an interesting experience
Gabor had when he was visiting in Padua Italy.
One day a journalist came to the statistics department and asked
him to explain why, when she was trying to catch a bus, the buses
always seemed to be going in the wrong direction. Gabor said she
must be wrong, but she insisted that he go with her to the bus
stop to see what she means. Gabor went for several days and counted
the number of buses going the wrong direction before her bus came.
He got the numbers 2,0,0,4,5,4,3,0,1,0 with an average of about
Gabor invites you to think about the problem of finding the expected
number of buses that arrive before yours and send your solution
to him at firstname.lastname@example.org,ca. A future issue will contain your
Gabor is, apparently, too modest to tell the rest of this story
(or perhaps he does not want his readers to look up the answer).
This experience got him interested in probability paradoxes and
led to his writing a very nice book about pardoxes: "Pardoxes
in Probability Theory and Mathematical Statistics", D. Reidel,
(1) Assume that, independent of how long you have been waiting,
the probability that a bus going in your direction arrives in
the next minute is 1/60. If you arrive at the stop at 12:00 noon
each day how long on the average would you have to wait for a
(2) Buses are going both north and south at your bus stop. The
time between buses in each direction is determined as in (1).
You arrive at 12:00 and wait for a bus going North. What is the
probability that a bus from the South comes before a bus from
the North? What is the probability that you see two or more buses
from the South before your bus arrives? What is the expected
number of buses that come from the South before your bus arrives?
Seen from a rut, the lottery is essential.
The New York Times, 16 July, 1996
John P. Rash (letter to the editor)
Rash complains that an article in the 14 July 1996 "New York
Times" on lottery advertising repeats stereotypes about lottery
players' being poor and uneducated and swept into a gambling addiction.
He remarks that, before lotteries, the opportunites to improve
your life were not very expensive. Now graduate education is
expensive and so is buying stocks or starting computer companies.
Rash says he plays Lotto and has won everything except a jackpot.
He uses wheeling systems and, contrary to the claims of lottery
officials, he claims they work. He concludes by saying "Yes,
I have lost more than I've won. But in the tedious world I inhabit
along with so many other New Yorkers, I've bought a fantasy."
What is a wheeling system? Do you think it works?
Forget money; nothing can buy happiness.
The New York Times, 16 July 1996, C1
The set-point concept in weight control says that brain is wired
to turn the body's metabolism up or down to maintain a pre-set
weight. Recent research in psychology suggests that the same
is true for happiness. That is, there is a genetically determined
mood level and the vagaries of life move this up and down but
only for short periods of time and it keeps coming back to this
Studies of happiness in several countries have suggested that
the perception of happiness is not affected very much by things
things that one might expect to determine happiness, such as education,
money, and a family. A promotion or losing a lover can effect
a person's mood, but most of the effect is gone by three months
and there is not a trace of it by six months. Lottery players
are no happier a year after they win the lottery than they were
before winning it. People that are relatively happiest now will
also be the happiest 10 years from now despite the day-to-day
Support for the set-point theory has been found in large studies
that follow people over many years. It has also been supported
by research with twins. One researcher on twins was led to the
conclusion that about 50% of our sense of well-being is determined
genetically and the other half from the pleasures and sorrows
of the last hours, days, or weeks.
Scientists who study changes within the brain say that they have
found a strong relationship between the ratio of neural activity
levels on the left and right sides of the prefrontal lobes and
people's moods. Those with relatively more activity on the left
prefrontal area report more positive emotions and get more pleasure
from ordinary activities than those with more activity on the
right prefrontal area.
(1) Do you consider yourself a happy person?
(2) Is it possible to measure happiness? How would you try to
(3) Do you believe the findings of this study?
(4) The May issue of "Psychological Science" has an
article by David T. Lykken who studied 1500 pairs of twins and
compared how members of pairs of fraternal and identical twins
rate their sense of well-being. Lykken came to the conclusion
that "About half of your sense of well-being is determined
by your set- point, which is from the genetic lottery, and the
other half from the sorrows and pleasures of the last hours, days
or weeks." How do you think he arrived at this conclusion?
(5) Is it possible to reconcile the natural oscillations which
the article mentions (i.e. the standard latitude of emotions
a given person might have) with claims of a set point the study
(6) Assume you accept the hypothesis of the study that there
is such a thing as a happiness set point. Do you think it would
be possible to raise one's set point?
(7) One researcher suggests that these results indicate that
concentrating on finding small changes that keep you above your
set-point is more fruitful than pursuing big changes that might
only have a short term effect. What do you think about this suggestion?
Emotional ills tied to stunted growth in girls.
The New York Times, p.C9, 26 June 1996
A new study, published in the current issue of Pediatrics, has
found that adolescent and pre-adolescent girls who were overly
anxious grew up to be 1 to 2 inches shorter, on average, than
other girls. There was no such finding for boys, possibly because
emotional disorders are rare and more transient in boys after
puberty. The finding supports past research that showed that
children and adults of both sexes with anxiety or depression have
lower-than-normal amounts of growth hormone.
The study highlights the connection between emotional disorders
and the neuroendocrine systems. Growth hormone release is controlled
by the brain, so it's understandable that hormone secretion would
be hindered by emotional disorders.
The study focused on 716 children, about 50% girls and 50% boys,
who were randomly selected from 2 counties in upstate New York.
They were followed for 9 years, from adolescence/pre-adolescence
until adulthood. They were evaluated for depression, separation
anxiety, and overanxious disorder.
Symptoms of the three types of anxiety were found in both sexes,
but shorter adult stature was found only in girls. The strongest
link was seen in girls who were 11 to 20 when separation anxiety
was diagnosed -- they were 1.7 inches shorter than the girls in
whom no emotional problems had been found.
Scientists display substantial gains in AIDS treatment.
The New York Times, 12 July 1996, A1
Lawrence K. Altman
The international AIDS conference at Vancouver has led to a new
level of optimism for finding a cure for AIDS. This has been
the result of new small studies on the use of combinations of
drugs. The results came primarily from two studies carried out
in New York City. In each study, patients took two older drugs,
AZT and 3TC, in combination with a member of a new class of drugs
called protease inhibiters.
In the first study at the Aaron Diamond AIDS Research Center,
researchers concentrated on 9 patients who were treated within
90 days of acquiring the HIV infection. The AIDS virus became
undetectable for up to 300 days when the participants took a combination
of three drugs that included the protease inhibitor, ritonavir.
In the second study, researchers from New York University led
a four-center trial giving new combinations to people with AIDS
who had failed previous anti-HIV therapy. This study involved
97 adults with an average age of 40. Of the seven patients who
had completed 48 weeks of treatment, the new set of drugs including
the protease inhibitor, indinavir, brought HIV down to undetectable
levels in six of the patients.
These results were considered quite spectacular, but researchers
emphasized that there were a lot of problems yet to be solved.
For one thing, even though the virus could not be detected it
could still be present in other tissues and could return if the
drug therapy were discontinued.
When should the treatment begin? Starting as soon as the virus
is detected runs the risk of the body developing defenses against
the drugs making them less effective when really needed.
Also the treatment is very demanding on the patients. It can
require taking more than 15 pills a day at precise times and with
side effects that some cannot tolerate. Finally, in the United
States the cost of most of the drug combinations and monitoring
tests is estimated to be between $15,000 and $20,000.
(1) Do you think the international community will be able to find
a way to make such expensive drugs available to AIDS patients
in third world countries?
(2) Do you think it is dangerous to give so much hope based upon
such small studies?
India Suddenly Leads in H.I.V., AIDS Meeting is Told
The New York Times, A3, 8 July 1996
Lawrence K. Altman
More than 3 million of India's 950 million people are infected
with H.I.V., the AIDS virus, making it the leading country in
H.I.V. infection.. The announcement was made at the 11th international
AIDS meeting being held in Vancouver, British Columbia.
Nevertheless, the meeting also announced that impressive gains
in preventing the disease are being made in several regions of
the world. This illustrates the varying patterns in which H.I.V.
has spread, with the numbers of infection rising in some areas,
like India, while stabilizing in others, like the United States,
and even decreasing in countries like Thailand and Uganda.
There are 22 million people in the world who are infected with
the AIDS virus, with 8,500 new infections occurring daily. Two-thirds
of the infections are in sub-Saharan African countries, which
unlike the rest of the world, have more infected women than men.
Trailing India in the number of people infected are South Africa
with 1.8 million; Uganda with 1.4 million; Nigeria with 1.2 million;
and Kenya with 1.1 million. In the United States, the number
of people infected in 1992 (latest data) was between 650,000 and
A range of AIDS prevention programs is credited by Centers for
Disease Control officials for a more favorable trend in infection
rates in the United States. For instance, the number of new infections
in San Francisco dropped to 1,000/year in 1992 as compared to
8,000/year in the early 1980s. In Uganda and Thailand, public
education about safer sex practices has led to a decrease in the
number of sexual partners, increased fidelity, and abstinence.
As a result, both countries have shown a significant decrease
in the number of new infections.
Do you think the progress on developing a cure for AIDS will hurt
the attempts to get people to use safe sex?
Folic acid may cut heart disease, researchers find.
Sacramento Bee, 27 June 1996, A11
A new study, published in the "Journal of the American Medical
Association", assessed the health patterns of 5,000 Canadians
and found that those with the greatest folic acid intake had 69%
less fatal coronary heart disease than those with the lowest intake.
The finding supports similar research results reported in recent
Folic acid does nothing to cholesterol. Instead, it lowers another
substance in the blood, an amino acid called homocysteine that
may also be dangerous in higher levels. The increasing interest
in the role of homocysteine raises the issue of whether cholesterol
is the primary marker of potential heart problems--or whether
it and homocysteine are just two of many.
Existing folic acid studies, like the new one from Canada, are
not regarded as solid proof. Missing from the research is a
folic acid study with better controls -- one giving thousands
of subjects folic acid, others a pill with no active ingredients,
then tracking their health for years. Such a study, however,
would cost millions. Whether anybody is going to supply the millions
needed to assemble large-scale trials to better pin down folic
acid's medical value is questionable because folic acid is not
a chemical that a drug company can patent and then profit from.
Americans will soon be getting more folic acid in their diet.
In March, the federal Food and Drug Administration ordered that
all grain product manufacturers add the vitamin to bread, flour,
pasta, and other foods from grains by 1998. The FDA ruling was
in response to research showing that women can cut the chances
of delivering a baby with neural tube defects if they have adequate
folic acid in their bloodstream at the time of conception. To
get enough folic acid to fight heart disease, however, researchers
estimate that Americans may need to get 400 micrograms of the
vitamin every day through supplements or naturally through foods.
Pros lose out to chance, industrials.
The Wall Street Journal, 10 July, 1996, C1
The latest six months' contest in the continuing "Wall Street
Journal" battle between the darts and the pros led to another
victory for the darts. This is their fifth victory in a row.
The pros' stocks lost an average of 9.2% while those of the darts
only lost 5.3%. On the other hand the industrial average rose
This was the fifth time in a row that the darts have beaten the
pros. Unfortunately, they still have a ways to go. Looking at
the 73 contests since they began in 1990, the pros are ahead of
the darts by a score of 41 to 32. The score of the darts against
the industrial average is 38 to 35. The average gain of the pros
over the 73 contests is 10.3%, for the Dow 6% and the darts 5.8%.
A note by John R. Dorfman after this account quotes Prof. Liang
from Case Western Reserve as suggesting that professional stock
pickers should not be too quick to boast about there performance.
He suggests that about 15% of their lead comes from picking more
volatile stocks than the darts, 20% from the fact that the dart
stocks get more of their return in the form of dividends rather
than capital gains; the contest ignores dividends. Finally, he
states that about 2/3 of the pro's outperformance is due to an
announcement effect, in which stocks jump on the day they are
recommended in the Wall Street Journal. He stated that, in fact,
the darts do outperform the experts, on average, in the six months
after the contests end.
(1) Is the lead of the pros statistically significant?
(2) What do you think of Liang's analysis of the situation?
We are still pursuing Shunhui Zhu question:
The Energyguide on my new Air Conditioner states that: This model's
estimated yearly operating cost is $40. How did they arrive at
We talked to a representative of the Federal Trade Commission
who told us that this number is meant to be for comparison with
other brands only. They do not give regional averages because
there is too much variation in electricity costs and they do not
want to give the impression that they can really give an estimate
that would be meaningful for their actual costs.
A national average yearly operating cost of an air conditioner
seems to be obtained from the national average for the cost of
electricity. Thus we are led to the question of how these averages
are obtained. Here we were helped by a discussion in in a publication
called "Home Energy Magazine". In an editorial we found
the following remarks about the cost of electricity in kilowatt
The most often quoted average is from DOE's Energy Information (EIA). Their most recent average is 8.41 Cents/kWh for 1995. They get this number by dividing total residential electric bills by total residential electricity use. The EIA also projects an average rating for the upcoming year. For its Energy-Guide labels, the Federal Trade Commission uses the EIA's projected rate for the year. Current labels assume a rate of 8.67 Cents/kWh, which the EIA estimated for 1996. They projected 8.4c/kWh for 1995.
The National Association of Regulatory Utility Commissioners (NARU) calculates a different average rate. NARUC takes the rates that residential customers pay in each of about 190 "service areas" nationwide and average them, without regard for population or usage in those regions. Their average residential rate among the service territories for winter 1994-95 came to 9.1 cents/kWh.
Neither of these gives us what the average customer is paying. The EIA method is an average out of the total electricity sold, so the rates of customers who bought more electricity count more toward the average. The NARUC rate shows what the average utility is charging. We don't know of anyone who calculates what the average residential customer is paying.
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