9.8 Drunk driving. A newspaper article on drunk driving cited data on traffic deaths in Rhode Island: “Forty-two percent of all fatalities occurred on Friday, Saturday, and Sunday, apparently because of increased drinking on the weekends.” What percent of the week do Friday, Saturday, and Sunday make up? Are you surprised that 42% of fatalities occur on those days?

9.9 Advertising painkillers. An advertisement for the pain reliever Tylenol was headlined “Why Doctors Recommend Tylenol More Than All Leading Aspirin Brands Combined.” The makers of Bayer Aspirin, in a reply headlined “Makers of Tylenol, Shame on You!,” accused Tylenol of misleading by giving the truth but not the whole truth. You be the detective. How is Tylenol’s claim misleading even if true?

9.10 Advertising painkillers. Anacin was long advertised as containing “more of the ingredient doctors recommend most.” Another over-the-counter pain reliever claimed that “doctors specify Bufferin most” over other “leading brands.” Both advertising claims were literally true; the Federal Trade Commission found them both misleading. Explain why. (Hint: What is the active pain reliever in both Anacin and Bufferin?)

9.11 Deer in the suburbs. Westchester County is a suburban area covering 438 square miles immediately north of New York City. A garden magazine claimed that the county is home to 800,000 deer. Do a calculation that shows this claim to be implausible.

9.12 Suicides among Vietnam veterans. Did the horrors of fighting in Vietnam drive many veterans of that war to suicide? A figure of 150,000 suicides among Vietnam veterans in the 20 years following the end of the war has been widely quoted. Explain why this number is not plausible. To help you, here are some facts: about 20,000 to 25,000 American men commit suicide each year; about 3 million men served in Southeast Asia during the Vietnam War; there were roughly 93 million adult men in the United States 20 years after the war.

9.13 Trash at sea? A report on the problem of vacation cruise ships polluting the sea by dumping garbage overboard said:

On a seven-day cruise, a medium-size ship (about 1,000 passengers) might accumulate 222,000 coffee cups, 72,000 soda cans, 40,000 beer cans and bottles, and 11,000 wine bottles.

Are these numbers plausible? Do some arithmetic to back up your conclusion. Suppose, for example, that the crew is as large as the passenger list. How many cups of coffee must each person drink every day?

9.14 Funny numbers. Here’s a quotation from a book review in a scientific journal:

. . . a set of 20 studies with 57 percent reporting significant results, of which 42 percent agree on one conclusion while the remaining 15 percent favor another conclusion, often the opposite one.

Do the numbers given in this quotation make sense? Can you determine how many of the 20 studies agreed on “one conclusion,” how many favored another conclusion, and how many did not report significant results?

9.15 Airport delays. An article in a midwestern newspaper about flight delays at major airports said:

According to a Gannett News Service study of U.S. airlines’ performance during the past five months, Chicago’s O’Hare Field scheduled 114,370 flights. Nearly 10 percent, 1,136, were canceled.

Check the newspaper’s arithmetic. What percent of scheduled flights from O’Hare were actually canceled?

9.16 How many miles do we drive? Here is an excerpt from Robert Sullivan’s “A Slow-Road Movement?” in the Sunday magazine section of the New York Times on June 25, 2006:

In 1956, Americans drove 628 million miles; in 2002, 2.8 billion. . . . In 1997, according to the Department of Transportation, the Interstate System handled more than 1 trillion ton-miles of stuff, a feat executed by 21 million truckers driving approximately 412 billion miles.

9.17 Battered women? A letter to the editor of the New York Times complained about a Times editorial that said “an American woman is beaten by her husband or boyfriend every 15 seconds.” The writer of the letter claimed that “at that rate, 21 million women would be beaten by their husbands or boyfriends every year. That is simply not the case.” He cited the National Crime Victimization Survey, which estimated 56,000 cases of violence against women by their husbands and 198,000 by boyfriends or former boyfriends. The survey showed 2.2 million assaults against women in all, most by strangers or someone the woman knew who was not her past or present husband or boyfriend.

9.18 We can read, but can we count? The U.S. Census Bureau once gave a simple test of literacy in English to a random sample of 3400 people. The New York Times printed some of the questions under the headline “113% of Adults in U.S. Failed This Test.” Why is the percent in the headline clearly wrong?

9.19 Stocks go down. On September 29, 2008, the Dow Jones Industrial Average dropped 778 points from its opening level of 11,143. This was the biggest one-day decline ever. By what percentage did the Dow drop that day? On October 28, 1929, the Dow Jones Industrial Average dropped 38 points from its opening level of 299. By what percentage did the Dow drop that day? This was the second-biggest one-day percentage drop ever.

9.20 Poverty. The number of Americans living below the official poverty line increased from 24,975,000 to 45,318,000 in the 38 years between 1976 and 2013. What percentage increase was this? You should not conclude from these values alone that poverty became more common during this time period. Why not?

9.21 Reducing CO₂ emissions. An online article reported the following:

In order to limit warming to two degrees Celsius and permit development, several developed countries would have to reduce their CO₂ emissions by more than 100 percent.

Explain carefully why it is impossible to reduce anything by more than 100%.

9.22 Are men more promiscuous? Are men more promiscuous by nature than women? Surveys seem to bear this out, with men reporting more sexual partners than women.

On August 12, 2007, the New York Times reported the following.

One survey, recently reported by the U.S. government, concluded that men had a median of seven female sex partners. Women had a median of four male sex partners. Another study, by British researchers, stated that men had 12.7 heterosexual partners in their lifetimes and women had 6.5.

There is one mathematical problem with these results. What is this problem?

9.23 Don’t dare to drive? A university sends a monthly newsletter on health to its employees. One issue included a column called “What Is the Chance?” that said:

Chance that you’ll die in a car accident this year: 1 in 75.

There are about 310 million people in the United States. About 40,000 people die each year from motor vehicle accidents. What is the chance a typical person will die in a motor vehicle accident this year?

9.24 How many miles of highways? Organic Gardening magazine once said that “the U.S. Interstate Highway System spans 3.9 million miles and is wearing out 50% faster than it can be fixed. Continuous road deterioration adds $7 billion yearly in fuel costs to motorists.” The distance from the east coast to the west coast of the United States is about 3000 miles. How many separate highways across the continent would be needed to account for 3.9 million miles of roads? What do you conclude about the number of miles in the interstate system?

9.25 In the garden. Organic Gardening magazine, describing how to improve your garden’s soil, said, “Since a 6-inch layer of soil in a 100-square-foot plot weighs about 45,000 pounds, adding 230 pounds of compost will give you an instant 5% organic matter.”

9.26 No eligible men? A news report quotes a sociologist as saying that for every 233 unmarried women in their 40s in the United States, there are only 100 unmarried men in their 40s. These numbers point to an unpleasant social situation for women of that age. Are the numbers plausible? (Optional: The Statistical Abstract of the United States has a table titled “Marital status of the population by age and sex” that gives the actual counts.)

9.27 Too good to be true? The late English psychologist Cyril Burt was known for his studies of the IQ scores of identical twins who were raised apart. The high correlation between the IQs of separated twins in Burt’s studies pointed to heredity as a major factor in IQ. (“Correlation” measures how closely two variables are connected. We will meet correlation in Chapter 14.) Burt wrote several accounts of his work, adding more pairs of twins over time. Here are his reported correlations as he published them:

What is suspicious here?

9.28 Where you start matters. When comparing numbers over time, you can slant the comparison by choosing your starting point. Say the Chicago Cubs lose five games, then win four, then lose one. You can truthfully say that the Cubs have lost 6 of their last 10 games (sounds bad) or that they have won 4 of their last 5 (sounds good).

The following example can also be used to make numbers sound bad or good. The median income of American households (in dollars of 2013 buying power) was $51,735 in 1990, $56,800 in 2000, and $51,939 in 2013. All three values are in dollars of 2013 buying power, which allows us to compare the numbers directly. By what percentage did household income increase between 1990 and 2013? Between 2000 and 2013? You see that you can make the income trend sound bad or good by choosing your starting point.

9.29 Being on top also matters. The previous exercise noted that median household income decreased slightly between 2000 and 2013. The top 5% of households earned $196,440 or more in 2000 and $196,000 or more in 2013. It is important to note that these amounts are both in dollars of 2013 buying power, which allows us to compare the numbers directly. By what percentage did the income of top earners decrease between 2000 and 2013? How does this compare with the percentage decrease in median household income between 2000 and 2013?

9.30 Boating safety. Data on accidents in recreational boating in the U.S. Coast Guard’s Recreational Boating Statistics Report show that the number of deaths dropped from 676 in 2004 to 610 in 2013. The number of injuries reported also fell from 3363 in 2004 to 2678 in 2013. Why does it make sense that the number of deaths in these data is less than number of injuries? Which count (deaths or injuries) is probably more accurate?

9.31 Obesity and income. An article in the November 3, 2009, issue of the Guardian reported, “A separate opinion poll yesterday suggested that 50% of obese people earn less than the national average income.” Income has a distribution that is such that more than 50% of all workers would earn less than the national average. Is this evidence that obese people tend to earn less than other workers?