"Take a chance on Christmas."
-- Laurie Snell=====================================================
Note: Note: There will be a Chance Workshop at Dartmouth College this summer from June 24 to June 29. The workshop is for college teachers interested in learning how we teach our quantitative literacy course called Chance and is supported by the NSF. At the end of this Chance News, you will find a description of the workshop and an application form should you be interested in applying.
Some comments on web sites: The psychology department at the University of Washington has experimented with having students in a statistics course work in groups, communicating with each other on the web using special discussion group software. The instructor poses questions about a particular statistical experiment. Students then give initial answers and justification for these answers. Then each student comments on one or more of the other students' answers. Finally, the students modify their original answers to take into account the comments of their fellow students. You can see a transcript of these discussions from previous classes
The "Institute of Statistics and Decision Sciences" at Duke University has had information about the work of the Institute. If you choose "courses" and then "basic statistics" you will find, among other things, links to a number of applets useful in teaching a beginning statistics course.
Newspapers are beginning to include links to background information on some of their major stories. Our second article, from the Washington Post web site provides a good example of this. Also, National Public Radio has archives of its programs that you can listen to on the web if you download RealAudio. Our first example is based on one of these programs.
Wild cards in poker make the game less challenging by far.
NPR: All things considered, 29 Nov. 1996
Richard Harris interviews Steve Gadbois and John Emert about their recent results relating to poker with wild cards. You can listen to the interview.
Harris, in a paper in the current "Mathematics Magazine" (Vol. 69, No.4 October 1996: 283-285) and John, in a joint paper with Dale Umbach in the Summer issue of "Chance Magazine" (Vol. 9, No. 3:17-22) report their independent discovery of a result that should concern all poker players who allow wild cards.
Suppose you are playing five card draw poker with two wild cards. Then players have to choose how to declare their hand. This is done by choosing the highest possible hand in the usual ordering of hands--an order determined by the probabilities of getting the possible hands: three of a kind, straight, full-house etc. However, these authors noticed that this very choice messes up the order. For example, with two wild cards, it becomes more probable you will be dealt a hand with three of a kind than two pair. Thus two pair should beat three of a kind. Suppose you decree this. Now players choose between two pairs and three of a kind will choose two pairs. Now it is more likely that you will be dealt two pairs than three of a kind. So, again, we have a problem. There are other pairs of hands that get out of order in this same hopeless way because of wild cards. In fact, with two wild cards you are less likely to get a bust hand than a pair so it should beat a pair!
Alas, when you have such wild cards, there is no way to re-rank the hands to make this kind of problem go away. Thus, when you play with wild cards, you will not be able to have a ranking of the hands in the order of the probability that they occur as is the case with no wild cards--a pretty basic part of poker to have to give up.
(1) If you were playing draw poker with wild cards, would you feel unhappy having your two pair beaten by three of a kind?
(2) The authors of these papers left the case where you have one-eyed jacks wild to others to an analyze. Why do you think they avoided this popular choice for wild cards?
A single number puts the economy in a new light.
Washington Post, 11 December 1996, p A1
In early winter 1995, Federal Reserve Board Chairman Alan Greenspan testified before the Congress that he thought the Consumer Price Index (CPI) substantially overstated the rate of growth in the cost of living. His testimony generated a great deal of discussion. Soon afterwards, Speaker of the House Newt Gingrich was asked about the CPI and responded by saying: "We have a handful of bureaucrats who, all professional economists agree, have made an error in their calculations. If they can't get it right in the next 30 days or so, we zero them out, we transfer the responsibility to either the Federal Reserve or the Treasury and tell them to get it right."
Well, the problem was not as simple as Gingrich thought, and a panel was set up to study the accuracy of the CPI. The panel was chaired by Stanford economist Michael J. Boskin. The panel's final report, "Toward a more accurate measure of the cost of living", was recently released. The panel concluded that, in its present form, the CPI is not a true cost of living index (a fact long recognized by the Bureau of Labor Statistics that produces it). They pointed to a number of biases that occur when it is used as such a measure, and they estimated these biases cause the CPI to overestimate the true cost of living by about 1.1%.
Pearlstein remarks that "stagnant wages, lagging productivity, lackluster economic growth, a looming Social Security crisis--suddenly, all these anxieties looked like they could be made to disappear with the wave of a statistical magic wand."
If the estimate of inflation has been 1% too high for the last 20 years, then, since the Gross Domestic Product (GDP) is measured in inflation-adjusted dollars, the apparent stagnation in the economy since 1973 gets changed to an economy that has been growing at a respectable rate of 3 to 4 percent per year on average. Projecting this into the future, the Treasury will be collecting more in tax revenue, the social security payments will be increased at a slower rate and the social security crisis will disappear. As a result of all this, the federal budget deficit over the next decade will look only about half as big as it does at the moment.
The panel suggests that this 1% error has been going on for some time and will continue if the CPI is not modified. Other economists have shown that this leads to some strange projections. For example, it would imply that, in today's dollars, the typical family in 1960 was making about $16,000, putting them just above the poverty line. And by the year 2030 a typical family will be making about $90,000 again measured in today's dollars. Neither the poverty of our parents nor the riches of our children seem quite right.
On the other hand, the conclusion, based on the current CPI, that inflation-adjusted wages are declining, does not seem right either, since surveys show Americans are spending a larger proportion of their money on luxuries, fancy medical procedures etc.
Lawrence Katz, an economist at Harvard University and recently the Labor Department's top economist, remarks that there is little doubt that the CPI, which tracks the price of a relatively fixed basket of consumer goods, gives too high an estimate for inflation. This is caused by the fact that it ignores improvements in the quality of goods, the introduction of new goods, and the substitution of one good in the basket for another not in the basket. However, he feels that it is "logically impossible" that this bias has been as high as 1.1% over a long time period.
Changing the CPI so it will become a true measure of the cost of living will require an extension to include a variety of subjective judgments about what increases the quality of life. For example, how much is access to the internet or e-mail worth to you?
The way that the CPI is carried out and the possible biases in this measure is a fascinating story. You can find a nice discussion of this, at the level you and your students can understand, in Jessica Utt's new book, "Seeing Through Statistics", Duxbury 1996. You can find more technical discussions on the Bureau of Labor Statistics web site and in the report of the Boskin committee. Links to all of these as well as to the full text of Perlstein's article can be found at the "Washington Post" web site.
Born to Rebel.
Pantheon, New York, 1996
Frank J. Sulloway
This book is the result of Sulloway's 26 years spent studying the difference between firstborn and laterborn children. It is a best seller and has been discussed in the popular press and TV programs like Nightline.
Despite sharing 50% of their DNA, Sulloway claims that siblings are almost as different as non-siblings. He attributes this difference to family dynamics. His thesis is that firstborn establish a niche in the family and, to defend this niche, they stay close to the parents and tend to end up being conservative and supportive of the status quo. Laterborn have to be more imaginative in finding ways to compete with firstborn for parental favors. This makes them more creative and innovative. They are "born to rebel".
Sulloway carried out historical studies to demonstrate that these differences between firstborn and laterborn are real. The first study he presents relates to the reception of scientists to new theories. According to his theory, scientists who are firstborn should be more likely than laterborn to develop or support "technical revolutions" such as Newtonian mechanics, while laterborn should be more likely than firstborn to support "radical ideological revolutions" such as Darwinian Evolution or Copernican astronomy"
Sulloway identified 28 such theories and, from secondary sources, developed a database of 3,890 scientists who passed judgment on these theories. Then, for a specific theory, he identified those in his database who took a position of "supporters" or "opponents". He then determined if the scientists in these two groups differ significantly as regards the proportion of firstborn and laterborn.
For example, Sulloway studied how scientists in the period 1700 to 1875 accepted or rejected Darwin-like theories of evolution. He identified 226 scientists in his database who took a position on this theory and classified them as "supporting" or "opposing". With the help of current experts, he also ranked them on a 7-point scale indicating the strength of their support. He found that, during the period prior to the publication of "The Origin of Species", 56 of the 117 laterborn, or 45%, supported the theory of evolution while, of the 109 firstborn identified, only 9, or 9%, supported this theory.
Sulloway uses an odds-ratio test to see if these differences are significant. From the above data, the odds that a laterborn will support the theory of evolution are estimated to be 56/64, and the odds for a laterborn supporting evolution are estimated to be 9/100. This gives an odds-ratio of (56/64)/(9/100) = 9.72. A standard odds-ratio test makes this observed odds ratio highly significant. Curiously, Sulloway refers to this result as indicating that "individual laterborn were 9.7 times more likely than individual firstborn to endorse evolution". This seems like a strange way to interpret an odds-ratio, but we told that this is not unusual to those who deal with odds-ratios.
Sulloway also uses correlation theory for this example. He gets correlations around .4 for the relation between birth order and support of evolution and finds that, controlled for birth order, neither sibship size nor social class is a significant predictor of support for evolution.
Sulloway gives a variety of other historical examples of the influence of birth order. For example, in the French revolution, at the trial of Louis XVI 361 deputies of the National Convention opted for the guillotine and 360 for clemency. 73% of the firstborn voted for executing the King while only 38% of the laterborn voted for this fate. Sulloway modestly remarks: "Two centuries of explanations by French historians are largely a footnote to the unrecognized power of sibling differences."
There have been numerous studies on the effect of birth order on personality traits, and the results have often been contradictory. Sulloway tries to make sense of these studies by carrying out a meta-analysis. He first makes predictions on how firstborn can be distinguished from laterborn by listing five categories of personality traits. These are: (1) Extraversion (firstborn should score higher on behaviors that tend toward "assertivess" and "dominance"), (2) Agreeableness/ Antagonism (firstborn should be more antagonistic than laterborn}, (3) Conscientiousness (firstborn should be more conscientious), (4) Neuroticism (firstborn are more anxious about their status and are more emotionally intense than laterborn. Among males, firstborn are more likely than laterborn to exhibit anger and vengefulness), and (5) Openness to Experience (laterborn should score higher than firstborn on Openness to Experience, associated with being unconventional, adventurous, and rebellious).
Sulloway chose 196 studies that he feels have been carefully controlled. He found that 72 of these studies had significant birth-order results consistent with his hypotheses, 14 yielded contrary results and the remaining 110 were not statistically significant in either direction. He remarks that: "the likelihood of obtaining 72 spurious findings by chance is less than 1 in a billion billion."
While most reviews of this book have been very favorable, Alan Wolfe, wrote a quite critical review of it. Wolfe is a history professor at Boston University. In his review in the "New Republic" (23 December, 1996, p. 29) Wolfe remarks that Sulloway pretends to be doing science but does not have the open mindedness of a good scientist. He says that the book is more dangerous as pseudo science than "The Bell Curve" because, while people were smart enough to see through the politics of "The Bell Curve", there are no obvious politics in birth order, so people are more apt to take Sulloway's claims seriously.
In considering how general Sulloway's results are, Wolfe remarks: "Statistical generalization is not quite the neutral, value-free procedure that Sulloway thinks it is." He points out that Sulloway had to make many subjective judgments such as: the decision to put singletons in the firstborn group; the decision to use functional birth order rather than biological birth order (functional birth order changes owing to sibling mortality, adoption, remarriage, and other circumstances); the decision to infer birth order when it is unknown; the decision whether a particular scientist is a radical or a conservative innovator. Wolfe also questions the generality of Sulloway's results, considering that his studies are restricted primarily to science and politics, to people who are famous enough to have biographies, to a particular period of time, and to a particular culture.
Well, we will all probably forget our statistical training and judge this theory by how well it works in the case of our own children -- it works pretty well in mine.
(1) Jim Baumgartner claims it is a well-known fact that most mathematicians are firstborn, and he himself observed once that, in a room full of mathematicians, all were firstborn. If you accept Sulloway's findings, would you think it more likely that a firstborn would become a mathematician than a laterborn? Is this the same as saying that it is more likely that a mathematician will be a firstborn than a laterborn?
(2) Sulloway was asked by Ted Koppel, on Nightline, if he felt it would be reasonable for employers to use birth order in their considerations for hiring new employees? What do you think he answered? How would you answer this question?
(3) When doing correlation analysis for binary data, can we just use standard correlation calculations as given by our friendly statistical package?
(4) The statistical package I use gives < .0001 for p values that might correspond to one chance in a billion billion. Why do you think the developers of the package chose to do this and Sulloway chose to give the more specific chance?
Margins of error.
New York Times Magazine, 15 December, 1996, p. 34
Frankel says we are "growing too dependent on virtual numbers and facts." The two major examples he discusses are the recent presidential polls and the current discussion of the consumer price index.
Frankel says that the polls in this year's presidential election were "disturbingly wide of the mark" and compares the magnitude of the error with that made in the famous 1948 Truman-Dewey election. Frankel attributes the error in this poll to the pollsters stopping interviewing early, failing to capture a "late swing". Another reason that has been given is that the pollsters allocated the undecided voters evenly, when in fact they supported Truman decisively. The more common explanation is that in this election "quota sampling" was used and it did not work very well. Under quota sampling, the interviewers are given assignments to fill quotas of people that match their representation in census data. Within their quota the interviewers were free to choose anyone they pleased. In this election it was found that they tended to choose more Republicans than they should because, for example, Republicans tended to live in nicer houses.
In this year's election only the Reuters/Zogby poll produced the correct final projection 49-41-8. Frankel remarks that the Times/CBS poll had a 16-point spread with its 50-34-9 projection. Frankel claims that, with a margin of error of 3%, a spread of more than 10% is "embarrassingly wrong". Frankel is making the common error of interpreting the 3% margin of error for a specific candidate as the margin of error for the difference. Of course, the margin of error for the difference is almost twice as big. (See Chance Magazine, Winter 1933, p. 22 and Winter 1994, p. 3, for a discussion of this problem.) For a 3% margin of error for an individual candidate the margin of error for the difference of two candidates is more like 5%. In this election, since the true difference was 8%, 10% would be well within the margin of error and 16% can hardly be called "embarrassingly wrong".
Frankel does not discuss the difference in the Zogby polling method, but other articles on this topic do. Zogby believes that party affiliation is a demographic characteristic, since most people adopt the party of their parents and peers and keep it throughout their lives. If there are more women in the poll than would be expected, the pollster makes an adjustment for this. Zogby does the same with party affiliations. Other pollsters say that, while it worked this time, it will lead him astray in future elections since party affiliations can change throughout the campaign.
Frankel turns next to the possible errors in economic indexes, such as the Gross National Product and the Consumer Price Index. He asserts that these errors are caused by the difficulty in capturing the benefits of information technology; they are with us to stay. We are going to have to quit thinking that we have precise information. He concludes with the following poetic remarks: "The greater our powers of computation, the less skilled we become at counting the things that most matter to us. The ether throbs with precise data that are often precisely wrong, and our understanding of politics and economics remains soberingly imprecise."
Milt Eisner suggested the following article and provided his comments on the article.
The "rape differential".
Washington Post, 20 Nov 1996, A19
This article appears to be statistically dishonest, although I have attempted to write to the author for clarification. The author compares "rape rates" (undefined in the article) and other-violent-crime rates for the military and the civilian populations. She notes that both rates are far lower for the military population than they are for the civilian population, but not so much lower for the former rate than for the latter rate. The author states:
"... while the U.S. Air Force rate of homicide and aggravated assault ... was 2 percent of the civilian rate for these crimes ... the Air Force rape rate was 20 percent of the civilian rate. The Marine Corps rate of "other" violent crime was 5 percent of the civilian rate, but the Marine Corps' rape rate was 27 percent of the civilian rate. [Similar figures are then given for the Navy and the Army.]"The author then, after discussing other possible explanations, speculates that
perhaps the explanation does lie in some rape-conducive aspect of military culture, as some have suggested."This quotation is highlighted in a text box by the editors.
It seems to me to be a rather remarkable inference, that if every military service has a rape rate far lower than that of the comparable civilian population, that there exists "a rape-conducive aspect of military culture." The opposite conclusion would appear to follow from the data.
Chance of a heart attack increases for those who suffer depression.
The New York Times, 17 December, 1996, C3
A report in the December 16, 1996 issue of "Circulation" reports on a study that finds that those who are subject to depression are four times more likely to get a heart attack than those who are not depressed.
The study involved 1,551 people in the Baltimore area who took part in a study in 1980 to document depression in the general public. In this study, interviewers asked questions like "In your lifetime, have you ever had two weeks or more in which you lost all interest and pleasure in things you usually cared about and enjoyed? They were also asked questions about unusual changes in their appetite and in their sleep.
Of the 444 who were classified as having been depressed, 27, or 6.1%, had a heart attack in the 13 years since the 1980 study while, of the 1,107 who had not suffered from depression, only 37, or 3.3%, had a heart attack. The four-fold advantage mentioned earlier was obtained by controlling for other known risk factors for heart attacks such as age, sex, marital status and high blood pressure.
Dr. Wayne Katon, a psychiatry professor at the University of Washington, said that he was persuaded that depression could lead to heart attacks but wondered why. He pointed out that it is possible that there is a biochemical effect of depression that elicits heart disease. There is some indication that those who are depressed have stickier platelets, which are blood cells involved in clotting. Whatever the reason, Katon said that he was amazed that cardiologists do not typically screen for depression and most internists don't look for it. He said depression is easily identified and there are very effective treatments for it.
(1) Previous studies that looked at depression as a risk factor for heart disease started with those who had had a heart attack and tried to identify those who had been depressed. What are the problems with this approach?
(2) If depression does not affect the chance of getting a heart attack, what would your estimate be for the probability that a particular person in the 1980 sample of 1,551 people would have had a heart attack? What is the probability that, by chance, there would have been 37 or less heart attacks in the group of 444 people who had been depressed?
Dan Rockmore noticed the next article and wondered how Stats Inc. would make such estimates.
The New York Times, 1 December, 1996, Section 8 Page 9
Albert Belle's agent, in preparation for salary negotiations, asked Stats Inc. to project the number of home runs Belle would have hit in 1995 and 1995 at nine other parks considered more favorable than Cleveland's Jacobs Field for right-handed hitters.
The article states that Stats Inc. found that Belle, playing for the right team, would have broken Roger Maris's home run record of 61, hitting 85 in 1995 and 68 in 1996.
We asked Stats Inc. how they calculated such things. The person we spoke to did not do the calculations but said that they would be something simple and mentioned the following two possibilities.
For each of the possible fields, use Belle's previous experience to estimate the chance that he gets a homerun each time he comes to bat at this field. Then, using these estimates, for a particular possible home team, estimate how many home runs he would get if half his at-bats were at this field and the rest were evenly divided among the other fields.
The second way would be to estimate a homerun factor for each field that would allow you to convert anyone's homeruns at field x to an equivalent number at field y.
The Stats Inc. annual publication "Baseball Scoreboard" will be out March 1 and will undoubtedly contain more information about these kind of calculations. Incidently, Belle's agent was successful in getting him a 52.5 million dollar five year contract--the highest in baseball history.
(1) How would you have estimated the number of home runs Belle would have gotten in other fields.
(2) Chass seemed to suggest that the best field was Coors Field in Colorado. Why might this be the best park for home runs?
Parade Magazine, 17 November, 1996, p. 22
Marilyn vos Savant
Marilyn is asked the following question dear to the hearts of those who teach a Chance course:
Is it possible for people who are bad at math to learn thought processes to make themselves at least mathematically functional?
--Ray Hamilton, Clackamas, Ore.Marilyn says she has never met a math book she liked very much. She finds the standard techniques tedious and circuitous. She says "I believe that effective thought processes can be learned by people with enough motivation. Math isn't as difficult as it looks--once you are comfortable with it. It looks impossible until you know how, and then it seems easy."
The next question she is asked is more directly a Chance-type question:
Say that I've just invented a random coin-flipping machine, and I operate it 50 times. The result is one of the following sequences(H = heads and T=tails). Which is more likely?1:THHHHHTHTHTHHHHHTHTTTTTTHHHHTHTHTHHHHHTTHHHHHTHHHH 2:HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
--Don, Portland Ore.Marilyn answers:
They're equally likely. It's more likely that a 50-toss result will look similar to No. 1--a mix of heads and tails--than No. 2. But for the result to be exactly like No. 1 would be rare--just as rare as No. 2.
(1) What do you think Marilyn means "It's more likely that a 50-toss result will look similar to No. 1"?
(2) A coin is tossed 5 times. Which of the 32 possible outcomes look like a random sequence? What is the probability of getting an outcome that looks like a random sequence?
Parade Magazine, 15 December, 1996, p. 22
Marilyn vos Savant
A reader writes that she has an accounting degree from a reputable college but can't grasp the difference between TV "ratings". Marilyn sympathizes and gives the following definitions:
- Rating = number of households tuned to a particular station divided by number of households that own a television set.
- Share = number of households tuned to a particular station divided by number of households with a television set on.
- Hut = number of households with a television set on divided by number of households that own a set.
A football game on ABC had the following New York ratings in New York. I assume they are percentages"
Rating 7.9 Share 15 HUT 52.7
Do these seem consistent with Marilyn's definitions?
I'm OK, you're not.
US News & World Report, 16 December, 1996, p24.
The subtitle of this article is "Why Americans think their lives are good but the nation is in peril." A collection of recent polling results is offered here to support this theme. For example:
Several important political consequences of the "I'm OK, you're not" phenomenon are noted here. Voters tend to overestimate the magnitude of national problems, and at the same time are losing faith in government's ability to address them. Moreover, to the extent they fail to perceive these problems as affecting them locally, voters cannot be mobilized to address them. The article attributes Clinton's success in the election to a savvy reading of these trends by his advisors. His campaign focused on voters' favorable views of their own lives rather than dwelling on larger societal woes such as the crime rate and the funding crisis in health care.
- About 40-50% of Americans think the nation is currently moving in the wrong direction, but 88% think their own lives and families are moving in the right direction.
- While 63% of voters say they have achieved or are close to achieving the American dream, nearly half believe it is impossible for most families in the nation today to achieve the elements of that dream.
- 65% of voters say religion is losing its influence on American life, but 62% say its influence is growing in their own lives.
- Nearly 90% of Americans think that a major problem with society is people not living up to their commitments. But over 75% say they don't find it difficult to meet their own commitments to family and job.
- 64% of Americans give the nation's public schools a grade of C or D, but 66% give the public school attended by their oldest child an A or B.
- 63% of Americans think other people's doctors are too interested in making money, but only 20% feel that way about their own doctor.
- 83% of Americans think crime is a big problem in society today, but only 17% think it is a big problem in their own community.
(1) What explanations can you think of for the "I'm OK, you're not" attitudes found in the polls.
(2) Historically, it has been viewed as nearly impossible to unseat an incumbent president during a time of peace and relative economic stability. Do we need an "I'm OK, you're not" phenomenon to explain the election, as opposed to simply "I'm OK"?
Please send comments and suggestions for articles to
For those who might be interested in our Chance Workshop this summer, we give a description of the workshop and an application to be sent by e-mail to us. Please feel free to pass this on to others who you think might be interested in this workshop.
Chance is an introductory mathematics course whose aim is to make students more informed and more critical readers of current issues in the news involving concepts of probability and statistics. Examples of such issues include: risk factors for cancer, testing for AIDS, reliability of opinion polls, DNA fingerprinting, and the use of IQ tests.
The workshop will allow college teachers to experience a shortened version of the Chance course, complete with discussions of current issues in the news, computer labs with data analysis software, handsonactivities, videos, journal writing and the use of the Internet. Previous workshop participants will describe their experiences developing a Chance course.
Participants will receive food, housing and course materials but will be asked to provide their own transportation. There will be a followup meeting at an annual meeting of the MAA or ASA. This workshop is supported by the National Science Foundation.
J. Laurie Snell, Peter Doyle, Joan Garfield, William Peterson, and Nagambal Shah.
June 24 to 29, 1997
University or College:
Where did you hear about the Workshop?
The purpose of the Chance Workshop is to assist other faculty interested in teaching a quantitative literacy course, such as our Chance course, based on current events in the news that use probability or statistics concepts. For more infromation about the Chance course see our web site: http://www.geom.umn.edu/locate/chance
Please type a short account of your own background in probability and statistics, interest in teaching a Chance course, and support you would have from your institution to do so. For the latter, a supporting letter from a chair or a dean would be very helpful.
The Chance course uses a variety of teaching techniques, including group learning, hands-on activities and journal writing, and work with Internet resources. It would be useful to know about any interest or experiences you have had with any of these. Of course, we do not expect that you have had experience with all of them, or else we would not be offering the workshop! Also, please describe the availability of computer resources for students at your institution, including statistics and simulation software packages, and Internet access tools. As far as possible, we want to tailor presentation to be applicable at your home institution.
Send this application by E-mail, FAX or ordinary mail to
J. Laurie Snell
Department of Mathematics
6188 Bradley Hall
Hanover, NH 03755-3551
The deadline for applications is March 15, 1997
Participants will be notified by April 15, 1997