Gnarly Gnews

(The free, humorous bi-monthly math newsletter.)
The Gnarly Gnews ™
Published by the SMP Company, PO Box 1563, Santa Fe, NM 87504
Copyright 2008 by Montgomery Phister, Jr.
www.gnarlymath.com
Vol. 9, No. 4, March/April, 2008
Gnarly Gnews World HQ
Page 1

**Please Don't Believe Everything You Read or Hear**
This Issue will Help You Size up What Newspapers Tell Us And What we Hear and See on TV and Radio Every day we make important decisions on how to live --what we should eat, what medicines we should take, where we should go to school, what we should worry about, whom we should vote for, and so on -- and what we should buy -- food, clothes, toys, cars, and so on -- from the "media". That is, from newspapers, magazines, TV shows, Radio broadcasts, and Internet sources. And often the media is wrong. In this issue of the Gnews we'll suggest how to cope with: 1. Math Errors and mathematical confusion. 2. Error or bias in TV and radio. 3. 'Facts' created out of nothing. 4. Statistical errors. 5. Different interpretations of the same data. So here goes: 1. Math errors and confusion. Reporters and editors are often math haters, or at least are unable to do the simplest of math calculations: finding percentages, understanding the average and the median, even multiplying and dividing numbers. Here are some example headlines: William Safire Loves Words. 1152-Word Novel is Proof A 1152-word novel? A page contains maybe 300 to 500 words. So that would be maybe a 3-page book. The reporter meant a 1152-page novel. 35 Million Greeting cards are sold each year, worth a total of $8 Billion. $8 billion divided by 35 million gives $228 per card! Christmas Tree cut for Rockefeller Center is 45 feet in diameter. A tree 45 feet in diameter? The accompanying photo showed a tree about 15 feet in diameter. Since the circumference of a tree is pi times its diameter, a tree with a circumference of 45 feet would have a 14-foot diameter. So the reporter meant Circumference, not diameter. Perhaps he didn't know the difference? To Improve your swimming, find a pool heated to 28 degrees F. But 28 degrees F is below freezing! Should have been 28 degrees C, which is about 82 degrees F. "The $322 represents a 0.06 percent raise (that's six-tenths of one percent)" (From a letter to the editor from a teacher, who objected to the size of her raise.) Six-tenths of a percent is 0.6 percent, not .06 percent. The $322 is some percent times the teacher's salary, so her salary is $322 divided by percent. A .06% raise means her salary was $322/0.0006, or $536,667 per year. For .6%, the salary would be $322/0.006, or $53,667. The teacher (and the newspaper editor who didn't save the teacher from embarrassment by questioning the figure before printing it) was confused about percentages. And probably she didn't realize that we could compute her salary from the data she sent. Intel seeks $18 billion bond deal This was the headline. The text of the article said "Intel seeks $18 million in bonds." The editor or proofreader at the newspaper was careless. I'm reminded of a story about Senator Moynihan who some years ago said, "This program will cost only $100 billion," though in fact it would cost $100 million. A Study of the Math Error Problem Newspaper people are aware of this problem, though it's not clear that they're trying to educate their reporters and editors. However, in 2003 Scott Maier wrote a report in the Newspaper Research Journal titled "How Sources, Reporters View Math Errors in News". (If you're interested, you'll find it here.) The report left no doubt that errors are very common, and that newspapers are quite aware of the problem. For example, Columnist James J. Kilpatrick of the New York Times wrote, "As a class, writers are arithmetical morons." But the general conclusion of this report was that mathematical incompetence is not as bad as people say. Only 13% of 573 errors were mathematical; many of the errors were minor; some were open to interpretation. That conclusion doesn't jibe with Kilpatrick's quote, nor with the examples we've presented above. I suspect that many errors were never detected: after all, the 'detectors' are themselves likely to be math illiterates. 2. Media error and bias. The non-written media is especially subject to error. When you watch the TV news, and TV or radio discussions, it's wise to keep that in mind. a) TV especially, but also radio, is particularly undependable because we're strongly influenced by the voice and attitude of the person giving us the information. It seems that about 40% of our opinion is based on the tone of voice of the presenter, 55% on his or her body language, and only 5% on the actual words. If we were reading what was presented, these things of course could have no effect on us. As an example suppose a TV newsperson says, "And now we'll have a report on police brutality from our local chief of police", but the face he makes and the way he lifts his shoulders make it clear he has no faith in the police. If you read what he said, you'd have no trouble trusting the chief's report. b) What you hear first sticks with you. You tend to remember it later, even if it's wrong. And in the case if TV and radio, you may never learn that it's wrong. Terrorist Threat made to Dallas Cowboy Field The announced Dallas Cowboy terrorist report was a hoax put out by a TV station to get viewers to watch the 10 o'clock news. Newspapers corrected it the next day, but many viewers must remember it with horror. 3. It may seem hard to believe, but sometimes people with axes to grind lie, and create 'facts' out of nothing. If a fact seems important to the reporter, he or she may publish it without checking whether it's true. Three Million Homeless in the United States. The 3 million figure was made up by Mitch Snyder, a gentleman who devotes himself to the problems of the homeless. He later admitted to reporters that he had invented the number. The real number of homeless (in 1990) was about 600,000, according to the Urban Institute. 4. Statistical errors are common in many news items. Such items come from surveys, medical research, government reports, and other studies. Sometimes the data can be interpreted different ways. Sometimes the sample is too small to justify the conclusions. Sometimes a later study will give a different result Flu Vaccines for people over 65 don't affect hospital stay, time off work, or death from influenza. This headline referred to a 2006 study published in the British Medical Journal. A 2007 study published in The New England Journal of Medicine reviewed 10 yrs of data from health maintenance organizations & found that Vaccinated older people are 7% less likely to need hospitalization for influenza or pneumonia, and their death rate is halved. Which study should we believe? Many Women at Elite Colleges Set Career Path to Motherhood. This story, published in the New York Times, was based on a survey of 90 women at Yale University. The reporter did not say what % is 'many'. And of course the sample is ridiculously small. Poor Diet and Physical Inactivity were Responsible for 400,000 deaths in 2000, a 33% jump from 1990. This was from the Centers for Disease Control and Prevention, published in the Journal of the American Medical Association and of course reported in many papers. However, the Journal later reported that the deaths were not 400,000, but 65,000. Most of us will remember the first number we heard, and perhaps won't even see the correction. Mississippi River was 100 million miles long One common statistical error can happen if we learn that something is changing, and suppose that change took place in the past and will continue forever. Mark Twain, in his book Life on the Mississippi made this point by writing: "In the space of one hundred and seventy-six years the Lower Mississippi has shortened itself two hundred and forty-two miles. That is an average of a trifle over one mile and a third per year. Therefore, any calm person, who is not blind or idiotic, can see that in the Old Oolitic Silurian Period, just a million years ago next November, the Lower Mississippi River was upwards of one million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-rod. And by the same token any person can see that seven hundred and forty-two years from now the Lower Mississippi will be only a mile and three-quarters long, and Cairo and New Orleans will have joined their streets together, and be plodding comfortably along under a single mayor and a mutual board of aldermen. There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact." I committed a similar blunder the other day when an elderly friend told me he had shrunk 4 inches in the past two years. I asked how tall he was now, and he said five foot four. So I explained that since he's shrinking 2 inches per year and is now 64 inches tall, in 31 years his height will be two inches. Arkansas Governor Warns of 600,000 obesity deaths. Over the past few years much has been written and said about fat people in America. We've been shown tables that give 'ideal' weight based on height and age, and have been told how our lives will be shortened if we are overweight. But the other day we saw this headline: Overweight? Study Finds in your Favor This new report explains that a little extra weight -- perhaps 25 pounds over the 'ideal' -- may in fact help people survive some illnesses. It doesn't say we should all gain weight. It just says the new information is 'puzzling', and that earlier reports exaggerated the dangers of obesity. 5. People often interpret the same data in different ways. If the situation is complicated, that's easy to do. Consider these headlines: Global Warming will cause Sea Level to rise 20 feet. WHAT SOCIAL SECURITY CRISIS? If it ain't broke, no need to fix it. The stories we keep reading about Global Warming and Social Security are examples of this type of problem. On alternative days, we read that there is no Social Security problem, and then that Social Security is going down the tubes. We learn that Global Warming will be a catastrophe, and then that the threat of Global Warming is greatly exaggerated. We'll be discussing these two subjects in more detail elsewhere in this issue. What Should we Conclude from all this? Let's begin by remembering that editors believe that we readers and listeners are mostly interested in bad things: accidents, deaths, recalls of food or other products, storms, earthquakes, social problems, and so on. And that would be tolerable, except that such news is presented without showing the whole picture. For example, are you likely to read or hear, "This tragic accident was caused by a drunk driver. Luckily drunk driving accidents in this city have fallen by 20% over the past five years."? But in addition you've seen, from the above examples, that we must be very careful when we read or hear the news, for the media makes all sorts of mistakes; and even when they report a fact -- for example, the result of a study -- the study itself may be wrong or misleading. (Continued at "Here's How to be Careful" in the right column)	Social Security is in Fine Shape Social Security is Facing Disaster Which of these predictions should we believe? Since our young friend Chloe looks at things from a young person's point of view, we asked her to explain. "It seems, like, complicated," she began, "but an example can make it easy? Suppose your school has a deal for you. Starting in the 6th grade you pay them $5 every month? $60 per year? And they promise that starting in the 12th grade they'll, you know, shell out $100 to you every year. For the rest of your life? Sounds awesome, no? So what would happen? "The school's going to grow, right? Every year there'll be more students. Let's say there are 100 for starters, but every year there are 20 more? So 120 in the second year? 140 in the third? And so on? And that's grody, because every year more students are kicking in their $60. "So let's look at the ol' bookkeeping? Every year the school puts the dough it gets into a bank? The red line shows what's in that bank account. The black line flashes, like, what comes in each year? And the blue line what goes out. "So that bank account gets, you know, fatter and fatter those first six years. When the school doesn't have to shell anything out? But in the seventh it has to start taking out dough to pay the $100 it promised. Up to year 11, it deposits more than it takes out? So the surplus in the bank keeps growing. BUT. From year 11 on its expenses (blue line) are bigger than its income (black line). And finally in year 23 it, you know, runs out of money. New 6th graders are still paying their $60/year? But the payouts are more and more humungous. The system is bankrupt. The school can't keep its promise to pay $100/year for the rest of your life. "And that's a pretty good picture of what's goin' down with Social Security (SS). The graph above shows how Social Security ticks. (The $T means trillions of dollars!) The red curve is what, in the school example, was the dough in the bank account? For Social Security, it's what's called the 'Social Security Trust Fund'. We'll talk about that in, like, a minute. And again the blue line is the income? The money paid by folks not yet retired? The black line is what's paid to retirees. And, as you can see, after the year 2015 the money goin' out is less than the money comin' in. The same as year 12 in our school example? And in the year 2042 or thereabouts the 'Trust fund' money is all gone. The same as year 23 in the school example. "The folks who say there's no Social Security crisis agree there'll be a problem in 2042, but say 'no problem' for 2016 when expenses are greater than income. 'All we have to do', they claim, 'is take money out of the Trust Fund. "But that's, like, a hoax. "What is this trust fund? It's money invested in US Bonds. That sounds groovy, right? But, hey, let's get down to the nitty-gritty. "The pic on the right shows what was happening in the beginning (the 1930's) when no one was old enough to be paid anything. Those young folks are shown on the bottom. They sent in all that cash, and the guvm'nt said it bought bonds for the trust fund. But that's just paper. In fact, the cash was still there. And our Congress and President used that cash to pay for good stuff like buildings and subways and doctor services. "Years later lots of people are, like, gramps and grannies? Retired? They begin to get SS benefits? So as you see on the left, some or most of the cash the younger people pay in is handed over to these oldies. Hey! Not as much cash available now for the good stuff! So what to do? "How about taking some money out of the Trust Fund to make up the difference? Well, sure the guvm'nt can say it sold some Trust Fund bonds. But that doesn't give 'em any cash. The only way to make up the difference is to cook up some new taxes -- or to cut back on spending (and Congress sure doesn't like to do that!) 2015 will be crisis time: that's when payments start being bigger than income? Look down below. Starting 2015, new taxes gotta, you know, help pay benefits to the old folks? as well as paying for the good stuff? That's when something's gonna have to happen: even bigger new taxes, or spending cuts, or ... or how about some big tweak in the SS system itself! And is 2042 important? When the Trust 'runs out of money'? Makes no diff. The trust bonds may be gone, but the problems after 2042 will be the same kind as those after 2015. Worse because the difference between payments and income keeps, you know, snowballing. "How does this compare with our School example? Well, the school really had mazuma in the bank. So their problem didn't come up until they took the last dollar out of the bank account. Why is the bank account different from the 'Trust Fund'? When the bank takes your deposit, it doesn't put it in a vault. It loans it out to people who are going to build a house? Or start a business? Or pay college tuition for their kids? And each of these investments by the bank is backed by something real: a house or a business or a parent who's put up his house as a guarantee he'll pay the tuition back to the bank. The US bonds in the 'Trust Fund' aren't backed by anything tangible. It's like, if the school had written down an IOU each year instead of shoving the dough in the bank. Then they could have spent your $60 per year on books and salaries? And educational trips for the teachers? But the school's 'cashing' those IOU's is just like the US Gov'ment 'cashing' the US Bonds. "How could this problem (the 2015 problem) have been prevented? Suppose the guvm'nt had bought gold every year. Instead of issuing bonds? Of course, they couldn't have shelled out the income on good stuff because the cash went to the people who owned the gold. So there'd have to be new taxes to, you know, pay for the good stuff? Then starting in 2015 they'd sell some of the gold every year. To pay the old folks? Of course, there'd still have to be new taxes to pay for the good stuff. But the Trust Fund would be based on something real -- gold. And between 2015 and 2042 (when the Fund sold the last of its gold) there wouldn't be a problem. "So what're we gonna do to stave off the problem in 2015, when payouts are bigger than income? There are a bunch of proposals. Some possibilities: "1. Increase the payroll taxes that pay for SS. "2. Reduce SS payments to retired people. "3. Increase the retirement age, so old folks can't retire until later. "4. Shift over to a system where there is a real Trust Fund holding real assets -- like Xerox and IBM and Microsoft and a bunch of other stocks. "5. Do all or some combo of these fixes. "Nobody knows how it's gonna be solved. But the later we put off solving it, the tougher it's gonna be. Here's How to be Careful and Sensible (Continued from the left column) So here are some questions you might ask yourself before you take some action, or worry, after seeing or hearing something in the media. 1. Check the arithmetic. If it's wrong, the conclusion may be wrong, as well. 2. Be especially careful of the 'news' you get from TV or radio. Remember that it's more influenced by the announcer than by the facts, and that in their hurry to tell the bad news, reporters may get their facts wrong. 3. If a number seems extraordinary, it may have been invented to prove a point. 4. Be aware that statistical errors are common. Find out the size of the sample -- how many subjects were included in the survey or study from which a conclusion was drawn. Also check on the source of the report. You may find the organization providing the data has a reason for coming to the conclusion reached. And keep in mind that a study reported today may be contradicted by a new study conducted later. 5. Remember that there may be other points of view that should be examined before you come to a conclusion. If you're proposing to do something based on what you read or hear, take the time to look for other opinions or reports on the subject. The Web is a fine place to search for such alternatives. Sam'll Answer DEAR SAM: What should I worry about? The TV and newspapers give me so many choices! WORRY, WORRY, WORRY DEAR WORRY, Your best bet is not to worry at all. In the next few years, all the worry choices will have changed.

The Gnarly Gnews ™

Vol. 9, No. 4, March/April, 2008
Gnarly Gnews World HQ
Page 2

New York City to be Underwater!
Polar Bears Drowning as Ice Melts       We keep hearing that all climate scientists agree that Global Warming is upon us and that we must immediately take steps to reduce greenhouse gases. Anyone who tries to express a different opinion is ridiculed. For example, in May of 2007 a U.N. official said, "This discussion is behind us. It's over ... it's completely immoral now to question the issue [of Global Warming]"
      What we will do in this article (at the risk of being immoral) is present a point of view you won't read in your newspaper or see on your TV. In particular, we'll show that:
      1. The temperature of the earth goes through cycles of cold and warm periods. The current situation is the beginning of a warming period that won't be any worse than those in the past.
      2. Carbon dioxide, which is supposed to be a prime cause of global warming, has often been much higher in the past, when man wasn't around. It comes from many places, and what we humans add is a very small part of the total.
      3. Mathematical models, which are used to predict future weather, give widely different predictions, and should therefore be treated with suspicion.
     NOTE. We're saying, and we believe, that the popular view is wrong. But of course our immoral view may be wrong, too. You make up your own mind.

Earth Temperature

1. Temperature of the Earth How is earth temperature measured? There are hundreds of weather stations all over the world. Today's earth temperature is found by averaging these figures.
However, the oldest of these records date back to the early 1700's, and the great majority start in 1850 or later. Earlier temperatures are deduced from many different sources.

The insides of glaciers and ice sheets. Scientists find pollen, which tells something about the plant growth in the past. The thickness of layers of ice tells the amount of rain or snow. Sometimes they find volcanic ash. Air trapped in the ice lets them measure how much carbon dioxide was present in the past.
Tree rings tell the amount of rain in the past. Some very old trees give data hundreds of years back. Petrified tree rings take us even further back.
Sediment layers in rock tell us something of climate billions of years ago.

From these sources climate scientists can deduce temperatures and plot graphs. Here are three graphs showing what's been happening:

This one shows that temperatures are close to the average of what they have been over millions of years.

This second graph shows air temperatures in the Antarctic, as figured out from ice samples. It goes back 400,000 years. It seems that every 100,000 years or so we cycle between ice ages and warm periods.

The above curve is similar to the previous one, but covers a shorter period of time. Notice that between about 80,000 and 20,000 years ago the world was very cold. Most of the earth was frozen. But in the past 20,000 years a warm cycle took over.

The above curve comes from a different source and shows temperatures since the birth of Christ.

      Finally, this last curve shows U.S. temperatures in the past century.
      The conclusion we might reasonably draw after looking at these curves is that world temperature, over the past 3 millions year, has been alternatively hot and cold. And that in the past it has been much hotter than it is now.
      A US Government summary says that from 1895 to 1999 the average temperature was 52.8 degrees Fahrenheit. In the last century US temps have risen 1.6 deg F. per 25 years, or 6.4 degrees per century. In the same time global surface temperatures have increased 1.1 deg F per century. But in the last 25 years the trend is 3.0 deg per century.
      But if we look at the next-to-last curve above, we see about a 1 degree rise from 500 AD to 1100 AD, a perhaps 1.5 degree drop from 1100 to 1600, and a 1.5 degree rise since 1600. Is it reasonable to believe we'll get a three degree rise by year 2100?

2. Carbon Dioxide the Villain? The "Greenhouse Effect" is the action of various gases in the atmosphere, which allow heat to enter the atmosphere but prevent its leaving. Vice-President Gore and other tell us that carbon dioxide, produced by coal plants, automobiles, and other sources, is the most important of the Greenhouse gases, and that we must do whatever it takes to cut back on the creation of this gas.

As the above graph shows, carbon dioxide, like global temperature, as cycled up an down over the ages. Recent figures (shown at the right of the curve) seem to be above the largest in the past. But current measurements are made with modern instruments, while past data is what we assume in looking at pollen, ice, and so forth. So there may well have been more carbon dioxide in the past 400,000 years than there is now. We know that 20 million years ago the concentration was over 5000 ppm.
But carbon dioxide is only one of the greenhouse gases. 0.15% Carbon Dioxide a problem?

The most important is water vapor, a fact which is strangely not mentioned in the usual papers on the "Greenhouse effect". As the chart on the right shows, water accounts for about 95% of the greenhouse gases, which keep heat from escaping and thus keep the earth warm. (In fact, keep it from freezing!) Carbon dioxide accounts for most of the rest (not counting methane and other gases whose concentrations are very small.) And of the carbon dioxide, only 3% is the result of human activity -- driving cars, operating coal-burning electrical plants, and so on. The other 97% comes from trees and vegetation, oceans, and land surfaces.
Three percent of five percent is 0.15%. That is the percentage of man's contribution to global warming. So if we were to completely stop driving and burning fuels, there'd be only a very slight effect on the world's temperature.

3. Predictions and Mathematical Models Predictions generally have two sources: Data from the past, assumed to continue into the future (which we've discussed above); and mathematical models. The models can come either from

scientific principles, or from data. But what in the world is a Mathematical Model? Let me explain.
Earlier in this edition of the Gnews I described two predictions. One on the length of the Mississippi River, by Mark Twain, and the other on the height of a friend of mine.
Let's take Mark Twain first. He told us that the Mississippi got shorter by 242 miles in the 176 years between roughly 1700 and 1875. That means it shortened by 242/176, or 1.375 miles per year. The graph on the left shows this effect, assuming that the length in 1700 was 1250 miles. This equation (Length=1250-1.375T) is a 'mathematical model' of river length,

based on data at two different times and assuming that length decreases by 1.375 mile per year. Probably it's a pretty good model, and could be used to estimate river length in, say 1750.
But what Mark Twain did, was assume that this model was good for all time. Notice in the graph on the right the slope of the line is still 1.375 miles per year. So in 1 million BC the river was 1.38 million miles long, hanging out over the Gulf of Mexico. And in 2617 it will be zero miles long, so that Cairo, Illinois, and New Orleans join their streets together.
The mathematical model is perhaps ok for the years 1700

to 1875. But it's wrong to assume it's good for all time.
Then there's my friend's height. On the left are those two points: 68 inches two years ago, and 64 today. So he's shorter by 2 inches per year, and the equation is height=68 minus 2T. No sweat.
But if we use the same mathematical model -- a reduction of two inches per year -- and extend it forward and back, we get the ridiculous result there on the right column.

(Continued in right column at Important Math Models)

Another Ice Age is Coming!
Drop in Food Production could Begin
only 10 Years from Now. The first of these headlines was in Time Magazine, June 24, 1974. The second is a quote from a Newsweek article on April 28, 1975. In 1974 the American Institute of Physics awarded Fortune magazine a "Science Writing Award" for its article on Global Cooling. Cycles of Global Warming and Cooling

In fact, the press has been going back and forth between Global Warming and Cooling for 100 years. For example, in February 1895 the New York Times warned, "Geologists Think the World May Be Frozen Up again". The above graph shows US temperatures over the last century, and you see periods of cooling alternated with periods of warming -- though the general trend was towards warming.
      Today's headlines are quite different. Many (but not all) of our weather scientists, and most of the media experts, are now in a panic about Global Warming.
      But let's be cautious. Remember the great "Year 2000 Disaster", in the late 1990's, when every newspaper warned that business and government computers, unprepared for a change from 1999 to 2000, would give us a disaster of unprecedented proportions throughout the world? The media was wrong. There was no disaster.
      It may be the media (and many climate experts) are wrong now. Let's keep an open mind.

Today's Gnarly Weather Today will be a hot day or a cold day. Who knows?

An Inconvenient Truth (?) Much of what we think about Global Warming has come from the publicity surrounding Former Vice-President Al Gore. His movie "An Inconvenient Truth" has won him an Oscar, and it and his book of the same name won a Nobel Peace prize. In it he says the earth is warming, and we must reduce automobile and industrial emissions of carbon dioxide to prevent climate disasters. A Convenient Lie?

      What disasters? Here are some of his examples:
      1.Global warming is caused by increases in carbon dioxide in the atmosphere. (But there's evidence that 11,000 years ago there was as much CO₂ in the atmosphere as there is now.)
      2. In this century the world's seas will rise up to 20 feet. Most of Southern Florida, New York City, and Shanghai will be underwater. (In fact, the rise is predicted to be between 4 and 35 inches, with a median of 19 inches. (Maybe Mr. Gore got his feet an inches mixed up?)
      3. Global warming has been melting the snows on Mount Kilimanjaro in Africa. (But the snow melt began 100 years ago because of a local change in climate near the mountain. See American Scientist July/August 2007. )
      4. Hurricane Katrina was caused by global warming. (But most climate scientists disagree. And a recent study finds that, based on data over the last 100 years, there's been little increase in the number of hurricanes that reached the land.)
      5. Polar bears are drowning by having to swim long distances to find ice that has melted away because of global warming. (But the Scottish newspaper The Scotsman recently reported that the Canadian polar bear population has increased from 12,000 to 15,000 in the past decade.)
      6. Himalayan glaciers are shrinking because of global warming. (But a 2006 study published in the American Meteorological Society's journal reported that glaciers are growing in the Himalayas)
      7. Some low-lying islands in the Pacific have had to clear out all their citizens because of rising water. (The only recorded evacuations came about on the Carteret Islands, and were caused because local fishermen did some unwise dynamiting.)
      8. The Antarctic ice sheet is melting because of global warming. (But recent reports show that the Antarctic has been cooling for decades, and the ice sheets grew between 1992 and 2004.)
      9. Global warming is drying up Lake Chad in Africa. (But it has dried because too much water was taken from the lake by farmers. It might also be mentioned that the lake was dry in 8500 BC, 5500 BC, 1000 BC, and 100 BC)
      So some of Mr. Gore's statements, in book and film, are not quite right. In response to criticism, Mr. Gore defended his work as fundamentally accurate. “Of course," he said, "there will always be questions around the edges of the science, and we have to rely upon the scientific community to continue to ask and to challenge and to answer those questions.”
      It would be nice if Mr. Gore and the others who predict disaster actually listened to the challengers.

A Clerihew
(Al Gore, interviewed on CNN March, 1999, said: "During my service in the United States Congress, I took the initiative in creating the Internet.”)

      Al Gore
      Let his imagination soar
      Probably he didn't think the real inventors would be upset
      When he insisted that he had invented the Internet.

(A clerihew has four lines. The first line is a person's name, and the second line must rhyme with the first. The third and fourth lines must rhyme with each other.)

Important Math Models
(Continued from left column) His height was 18 feet 2 inches in 1930, when he was a boy, and he'll disappear in 2039. The small red rectangle in the second graph is what's shown in the first.
This is all very humorous, but what does it have to do Mars reversing direction.

with important math models? Here's an important model from the past. Back in 150 AD or so the Greek astronomer Ptolemy was interested in the motion of the planets. The data he (and other astronomers) had collected over the years showed a peculiar fact: some planets didn't seem to move in circles. If you watched one of them, you'd notice that at certain times it reversed direction. The figure at the right shows what happens. Mars travels slower than the Earth. Its apparent motion against the starry background is shown by the green line. As Earth 'catches up' with Mars, the red planet, which has seemed to be moving more and more slowly, seems to stop moving, to move 'backwards' for a month or two, and then to continue forward again.
Ptolemy, who believe the Earth was the center of the universe, accounted for this strange motion by assuming that the planets moved in small circles attached to big circles. The drawing on the left shows his plan for the inside planets. The sun Ptolemy's model of the heavens

and moon travel in circles about the earth, but Mercury and Venus each travel in a small circle attached to a big circle.
This is a math model based on data -- data about the motion of the planets. It could predict (pretty well) what would happen in the future, and could show what happened in the past.
Then along came Newton. His Law of Gravity said that two objects are attracted by a gravitational force that decreases with the square of the distance between them (the graph of this equation is below). That's a math model based on science. It let him predict where a planet would be 10 or 100 years from now, or tell us where it Newton's law of gravity

was 100 or 1000 years ago.
      Similar models, based on science, let us predict how a TV set will work, how big a steel beam must be so that a skyscraper won't collapse, and how to design an evaporator in an oil refinery. Climate Predictions
A Model Based on Data       Weather scientists base their predictions on these same kinds of models -- some based on data, and some on weather science.
      (Incidentally, they use such models to predict what the weather will be tomorrow, or next week, in your own community. Do you find those predictions accurate?)
      Now let's turn to our first model. It's a very famous one and was based on temperatures in the past -- on data, not on the science of weather. It made headlines when it was published in 1998 because it seemed to show that carbon dioxide from human activity was causing Global Warming. The model was called the Hockey Stick, and is shown in blue in the graph shown below. It got the name 'hockey stick' because it shows a more-or-less straight line

of temperatures from 1400 to 1900 (the handle of the stick), and then a sharp turn up (the blade). The black line sort of looks like the stick. Since man-made gases became important starting about 1900, the authors argued that this curve clearly showed that those gases caused the warming.
The curve caused a lot of controversy, and the US Congress requested clarification. As a result the Wegman Report was published in 2005 which concluded that the paper describing the hockey stick graph was obscure, incomplete, and that the conclusions were false. The Report stated that a sounder analysis of the same data would result in the red curve, above, showing temperatures between 1400 and 1500 higher even than current temperatures. The straight-line handle of the hockey stick then disappears. The Report added that the reluctance of the original hockey stick authors to make their programs and analysis available had originally made it difficult for others to reproduce or analyze their report. (If you look at the 'Global Temperature' graph in the left column for the period 0 to 2000 AD, you'll see the period after 1400 does not look like a hockey stick.) Climate Predictions
Models Based on Science There are a great many different mathematical models of the climate that are based on the science of weather. It's not clear which are the basis for the Gore report. And generally speaking the various models do not agree with one another. Mathematial models

      Above you see the results of three different math models of the climate. The red and brown ones were carried back to 1900, the blue one back to about 1975. The report this came from showed the result of 5 other models, whose curves all fell in the shaded area. Note one model predicts the temperature in 2100 will be 9 degrees warmer than year 2000, and another predicts it will be only 4 degrees warmer. Which should we believe? Only the blue line agrees with the pessimistic government report we mentioned in the left column, which guessed there would be a 3 degree rise by year 2100--3 degrees per century.
      In addition, between 1980 and 1995 more than 50 scientific papers were published each predicting what air temperature would occur if the concentration of carbon dioxide doubled. One study predicted a change of 0.35 degrees Fahrenheit. Several predicted a change of 2.5 degrees. Others predicted changes from 11.3 to 15.7 degrees. Presumably pessimists prefer the higher, and optimists the lower temperature figures. Chaos       Science models are on the temperature, density, pressure, and composition of air, on wind speed and direction, and on the intensity of solar and land-emitted radiation. But the real problem with such models is the effect of what mathematicians call chaos theory.
      In 1960 a weather scientist named Edward Lorenz was trying to predict weather using a mathematical model. The model included twelve equations, and he had them programmed on a computer so he could see how successful his model was. To run the program, he had to enter data about the starting point for his prediction. One day he wanted to watch a particular prediction again. Comparing this second result with the first, he found that it started out the same, but that as the days of his prediction passed, the old and new predictions got further and further apart until they were completely different.
      Why in the world did that happen? The computer and its program had not changed.
      Investigating, he found that for the first run his starting data was 0.506127. For the second, he just used 0.506, which he figured was close enough. But that slight difference in starting point caused an enormous difference in end results.
      He concluded that it is impossible to predict weather accurately. Whatever model you use must have a starting point (say, air temperature and pressure at many points on the earth), and a slight change in any of these numbers -- for example, using 55.2 instead of 55.21 for a temperature) will make enormous changes in the model's prediction.

Conclusions It is natural for earth temperatures to rise (and fall), and today's temperatures are not at all-time highs. Carbon dioxide concentration, too, cycles; and man-made CO₂ emissions are too small to have an effect on temperatures. And the 'hockey stick' model is misleading, while conclusions based on mathematical models are not trustworthy because different models differ widely one from another, and are unreliable because of chaos.

School Days

That mathematical model just won't do, Edward. Your homework's got to be on time tomorrow, not next month.

The Gnarly Gnews ™

Vol. 9, No. 4, March/April, 2008
Gnarly Gnews World HQ
Page 3

This Month's Riddle
Q. What's the ideal number of reporters needed to compute the average salary of reporters?
(Answer below)

Math Test for Journalists       Some time ago, IRE.org (Investigative Reporters and Editors) published on the Web a test which it proposed all reporters should take to learn how good they are at math. (The test is given below, but you can also see it by clicking here.) The problems are all very simple. It's elementary school arithmetic we're talking about. Why don't you try to solve them? Answers are given below.
      1. Last year's city budget was $8,325,198. This year, you're told, the budget will be cut by 3 percent. What will this year's budget be? Answer here
      2. The city's parks and recreation budget was cut from $2.4 million to $1.2 million. What was the percent change? Answer here
      3. In a loose crowd, each person takes up about 10 square feet. What is the best estimate of how many people gathered for a protest demonstration that loosely filled a plaza that measures 100 yards by 40 yards? Answer here
      4. Today is Nov. 3, 2002. You are doing a story about a fatal shooting last night. The police report says the victim was born on April 21, 1948. How old was the victim? Answer here
      5. One out of 12 residents of your city speaks Spanish. About what percent of the population speaks Spanish? Answer here
      6. Your city of 152,000 had a total of 82 murders last year. What was the murder rate per 100,000? Answer here
      7. An oil tanker hit a rock offshore from your city and spilled about 170,000 gallons of crude oil. You want to make that volume relevant to your readers by saying how many backyard swimming pools (which hold about 12,000 gallons) it would take to hold that much oil. About how many swimming pools would the spilled oil fill? Answer here
      8. Your city has a property tax rate of $13.50 per $1,000 assessed value (but $25,000 of the home value can't be taxed because of the state's homestead exemption.) What would be the tax bill for a typical homeowner if the median assessed value of homes is $182,000? Answer here
      9. Every year your city has a large one-day street festival that draws a big crowd. The festival is held along one 40-block stretch of street that is two miles long and about 70 feet across. The festival organizers predict that 1.1 million people will attend this year's event. What is a more realistic estimate, knowing that the area will be filled with a loose crowd (10 square feet per person) all day, and assuming that on average everyone stays for half the time? Answer here
      10. Last year your small town had three murders. This year there were none. How would you describe the change? Answer here
      11. Last year's police and fire budget for your town was $4.83 million. This year it's $4.96 million. What was the percentage change? Answer here
      12. The city managers in five towns in your circulation area are paid $76,000, $68,000, $83,000, $122,000 and $65,000 a year. What is their average salary? Answer here
      13. In question 12, what is the median salary? Answer here Answers       1. Last year's city budget was $8,325,198. This year, you're told, the budget will be cut by 3 percent. What will this year's budget be?
      Three percent of $8,325,198 is .03 x $8,325,198 or $249,756. So the new budget will be $8,325,198 - $249,756 = $8,075,442,
      2. The city's parks and recreation budget was cut from $2.4 million to $1.2 million. What was the percent change? The dollar change was 2.4-1.2=$1.2 million. The original budget was $2.4 million, so the percent cut is 1.2/2.4 times 100, or 50%.
      3. In a loose crowd, each person takes up about 10 square feet. What is the best estimate of how many people gathered for a protest demonstration that loosely filled a plaza that measures 100 yards by 40 yards? 100 yards is 300 feet and 40 yards is 120 feet. The area of the plaza is therefore 300x120=36,000 square feet. Since a person takes up 10 square feet, the crowd will be 36,000/10, or 3600 demonstrators.
      4. Today is Nov. 3, 2002. You are doing a story about a fatal shooting last night. The police report says the victim was born on April 21, 1948. How old was the victim? On April 21 this year, the victim was 2002-1948=54 years old. He'll still be that age today.
      5. One out of 12 residents of your city speaks Spanish. About what percent of the population speaks Spanish? One twelfth is 1/12=.0833. Multiply by 100 to get the percentage: 8.33%. (Since the 1 in 12 is approximate, a figure of 8% would be sensible to use.)
      6. Your city of 152,000 had a total of 82 murders last year. What was the murder rate per 100,000? 152,000 is 1.52x100,000. So the murder rate is 82/1.52= about 54 per 100,000.
      7. An oil tanker hit a rock offshore from your city and spilled about 170,000 gallons of crude oil. You want to make that volume relevant to your readers by saying how many backyard swimming pools (which hold about 12,000 gallons) it would take to hold that much oil. About how many swimming pools would the spilled oil fill? Simply divide 170,000 by 12,000 and get 14.
      8. Your city has a property tax rate of $13.50 per $1,000 assessed value (but $25,000 of the home value can't be taxed because of the state's homestead exemption.) What would be the tax bill for a typical homeowner if the median assessed value of homes is $182,000? The exemption reduces the assessed value to $182,000-$25,000=$157,000, which is 157 thousands. Each $1000 is taxed $13.50, so the tax on a median house would be $13.20x157=$2072
      9. Every year your city has a large one-day street festival that draws a big crowd. The festival is held along one 40-block stretch of street that is two miles long and about 70 feet across. The festival organizers predict that 1.1 million people will attend this year's event. What is a more realistic estimate, knowing that the area will be filled with a loose crowd (10 square feet per person) all day, and assuming that on average everyone stays for half the time? Two miles is 2x5280=10,560 feet. With a 70-foot width the area is 10,560x70=739,200 square feet. If each person occupies 10 square feet, that would mean 739,200/10=73,920 people could be present at any one time. Since each person stays half the time, the total attendance is likely to be 2x73900=147,840, or about 148,000 people.
      10. Last year your small town had three murders. This year there were none. How would you describe the change? The number started at 3. The reduction was 3. 3/3=1.0, and multiplying by 100 we get a reduction of 100%
      11. Last year's police and fire budget for your town was $4.83 million. This year it's $4.96 million. What was the percentage change? The dollar change was $4.96-$4.83=$0.13 million. Since the starting budget was $4.83 million, the percent change is 100 times .13/4.83, or 2.7%.
      12. The city managers in five towns in your circulation area are paid $76,000, $68,000, $83,000, $122,000 and $65,000 a year. What is their average salary? The average of the five is their sum divided by 5, which is $82,800.
      13. In question 12, what is the median salary? To find the median, we list the salaries in order of size, and choose the one in the middle. In order of size, the salaries are $122,000, $83,000, $76,000, $68,000, and $65,000. The middle salary is $76,000 -- the same number of managers make more than that amount as make less.

Limerick      Now everyone, youthful and old,
     Knows ice hockey's played in the cold.
        So I find it alarming
        That grim global warming
     By a hockey stick figure is sold.

What is Inflation?
Who Cares, Anyway? 1897 consumer prices

      The table on the right shows what some different things cost back in 1897. You might think it'd be good to live in those days. Everything was so inexpensive.
      But you would change your mind if you learned that the average factory worker earned about $0.80 -- eighty cents -- per day! So he'd pay most of a day's wages for a pair of trousers for his son, or to buy a bar of soap, a broom, a pair of ice skates, a pocket knife, a comb, and a clock. And five days wages to buy a guitar.
      That's five days wages, not five hours.
      Generally speaking, prices (and salaries) increase every year. That increase is called 'inflation'. Sometime they increase quickly, and sometimes slowly. And sometimes they decrease when bad times come. Then it's called deflation. History of the dollar

The graph on the left shows how inflation has gone on in the United States since 1774. We're not looking at inflation here, we're looking at a graph that shows what a dollar bought in the past, compared to what it buys today. For example, in 1897 we see that the purchasing power figure was about 24. So if you paid 8 cents for a bar of soap then, Consumer price index

it's as if you were paying 8 times 24 or $1.92 in 2006 dollars.
The government keeps track of inflation these days with a number called the 'Consumer's Price Index', or CPI. Every year they look at what Americans spend, and what prices they pay. The chart on the right shows what was included in the 2004 index. We spent

40% of our money on where we lived, which includes heating and electricity for our homes,
16% on food, about 3/8 of which was eating out,
19% on our cars (including gasoline), and on such things as buses and air fares,
6% on school, books, stamps, telephones, computers, and so on,
5% on doctors, flu shots, hospitals, medical tests, and so on,
5% on recreation like movies, TV, pets, toys, newspapers, books, and so on,
4% on clothing, and
4% on 'other', which includes stuff like legal fees, haircuts, laundry, cigarettes, and so on.

      So the government takes what it thinks are average prices for all these various things, and uses those prices along with the 'weights' in the above table, to compute the CPI.
      What does this have to do with the media? Well, often newspapers and TV commentators and such like compare past prices with today's prices. If they don't adjust past prices with the CPI, they'll give a false impression. When I was a boy back in 1937, gas cost 20 cents a gallon. From the graph we see the inflation factor was about 14 in that year, so in today's dollars a gallon of gas cost 20 times 14, or $2.80. (It's remarkable to note that in 1920 gas cost 30 cents per gallon, but taking inflation into account it cost $2.90, which isn't far from what it costs today!)
      Any time you see past prices, be sure to find out whether they've been properly 'adjusted' to account for what's happened to prices over the years.

This Month's Puzzle 1. Maybe when your Dad was a boy, in 1970, he got a weekly allowance of $4 every week. What would his allowance have to be today so that he could buy just as much as he did then?
2. Your weekly allowance is $10.00. What would your allowance have been in 1775, if it then bought you just as much as it does today?

Your Horoscope
(For persons having birthdays this year.)
If you believe Mr. Gore, then when you grow up, you will settle in a city far from the beach so that your grandchildren's homes won't be submerged when, in the year 2100, the ocean has risen 20 feet.
But for heaven's sake, whatever you do, don't pay any attention to Horoscopes!

Answer to this Month's Riddle
Q. What's the ideal number of reporters needed to compute the average salary of reporters?

A. One. If there are more, one of them will have the dreadful job of adding all their salaries and dividing by the number of reporters.
(Back to top of page)

Answer to last month's puzzle

      Last month we learned about calculus, and were asked the following.
      1. What are the values of x that give low and high points for y in the equation below. What are the values of y at those points?
y = 5x + 7
y = x² - 7x + 15
y = x³ - 4x² + 5x
      These aren't so difficult. We differentiate each equation, set the result equal to zero, and solve for x.
      For the first equation y=5x+7 so dy/dx=5. There's no x in the result. But of course the equation is a straight line with a constant slope equal to 5. So there's no minimum or maximum.

For the second, y=x²-7x+15, so dy/dx=2x-7. Setting that equal to zero, we have 2x=7, x=3.5. So there's a minimum at x=3.5. How do we know it's a minimum and not a maximum? The slope is 2x-7, right? When x is 0 (or anything less than 3.5) the slope is negative. For x greater than 3.5, 2x-7 is positive. The negative slope means the curve is headed down, and the positive slope means it's headed up. So 3.5 must be a minimum -- see the plot above. What will the value of y be at the minimum? We put 3.5 into the equation for y and get y=x²-7x+15=(3.5)²-7(3.5)+15 = 2.75. That's y at the minimum x. Check it out on the graph.

      For the last equation, y=x³-4x²+5x, dy/dx=3x²-8x+5. How do we find out where that's zero? We have to factor the function -- that is, find two terms which multiply together to get 3x²-8x+5. Let's assume we have whole numbers everywhere. To get 3, we must multiply 3 by 1, and to get 5 we must multiply 5 by 1. So the factors must be something like (3x+1)(x+5). Let's check by multiplying. The 3x times x will give us 3x² and the 5 times 1 will give us five. But what will the middle term be? The one that multiplies x? It'll be 3x times 5 plus x times 1 or 16x. So that's wrong. We want the middle term to be -8x. Do you see what the right answer will be? Try (3x-5)(x-1).
      So the minimum and maximum must be where (3x-5)(x-1)=0. That happens when either 3x-5=0 (so 3x=5 and x=5/3=1 2/3) or when x-1=0 (so x=1). Check it on the graph above.
      What are the values of y at these points? Substitute x=1 and x=5/3 into y=x³-4x²+5x and we get, for x=1,
y=1³-4²+5=1-4+5=2. If we substitute x=5/3, we get x=1.85 (Continued in the next column)

     2. At what angle should a cannon be aimed so the cannon ball will go the greatest distance. (HINT: Find the time the ball takes to reach its highest position, as it depends on angle a. Then find the distance D it goes in twice that time, again as it depends on angle a. Now find dD/da, set it to zero, and solve for angle a.)
      Check out the above diagram. The cannonball starts out at a speed s (feet per second) aimed at an angle a. Its speed can be broken into two parts. The upward speed is s sin(a), and the speed to the right is s cos(a). We can see that from the definition of sine and cosine. For example, sine(a)=opposite side divided by hypotenuse. But the opposite side is the upward speed. So upward speed divided by hypotenuse=sin(a). But the hypotenuse is s. So upwards speed/s=sin(a), or upward speed=s sin(a).
      Let's first calculate the time when the cannon ball reaches the highest point. That'll be when its upward speed is zero. At time=0, when the ball leaves the cannon, its speed is s sin(a). And gravity slows it down 32.2 feet per second every second. So the equation for upward speed is s sin(a)-32.2t. That'll be zero when
s sin(a)-32.2t = 0, or when t =s sin(a)/32.2. And if the ground is level, the ball will come down at time 2t, or
2 s sin(a)/32.2=s sin(a)/16.1.
      We want to know how far from the cannon it'll travel in that time. Now, its rightward speed is s cos(a), so the distance it travels in time t is s cos(a) t. If we substitute the value for t when the ball hits the ground, we have the maximum distance D = s cos(a) s sin(a)/16.1 = s² sin(a) cos(a)/16.1
      So here we have an equation where the maximum distance D depends on angle a. We want to find the angle a for the biggest distance, which is when dD/da=0. So how do we differentiate D= s² sin(a) cos(a)/16.1?
      Here's the trick. We make up two new variables, u and v. We set
u=cos(a) and v=sin(a). Then we use the Product Rule.
      D= s² v u/16.1 = u v s²/16.1
      dD/da=(s²/16.1)(u dv/da + v du/da).
=(s²/16.1)(cos(a) cos(a) + sin(a) (-sin(a))
=(s²/16.1)(cos²(a)-sin²(a))
      Setting this equation equal to zero and solving the result will give us the value of angle a for the longest distance D. We can ignore the constant term, and have
cos²(a)-sin²(a)=0. But from trig we know that
sin²(a)=1-cos²(a). Substituting that value of sin, we get
cos²(a)-(1-cos²(a)=0, or 2 cos²(a)=1, or
cos(a)=sqrt(1/2)=0.707. The angle whose cosine is 0.707 is 45 degrees.
      So you aim your cannon at an angle of 45 degrees above horizontal to get your longest shot.

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