This page has solutions/answers to the QL/QR Problems from the News. In some cases, only an answer or hints are given rather than a complete solution.
1. Some assumptions are needed. It's a fact that about 75% of the population is over 18 years of age, so let's take that figure as the percentage of the population with jobs. Let's also assume that 75% of workers drive to work. Then in 1990, the number of solo drivers was approximately 249 million X 0.75 X 0.75 0.73 = 102 million people. In 2000, the number of solo drivers was approximately 281 million X 0.75 X 0.75 0.76 = 120 million people. There was an increase of 18 million solo drivers, or an increase of 18%.
2. The percent increase in commute time is (25.5 - 22.4)/22.4 = 0.14 or 14%.
3. Assuming 5 working days per week and 48 working weeks per year, one person spends about 5 X 48 X 3.1 = 744 minutes, or 12.4 hours more commuting in 1990 than in 2000.
Poverty: Good News or Bad News?
1. The poverty rate decreased by 12.4 - 13.1 = 0.7 percentage points, or by 5.3%.
2. The total number of people living in poverty in 1990 was 0.131 X 249 million = 32.6 million. The total number of people living in poverty in 2000 was 0.124 X 281 million = 34.8 million. The percentage or relative picture is good news. The absolute picture is bad news.
1. Doing some unit conversions reveals that 80 gallons per hour amounts to 115,000 gallons per day -- close, but not exactly consistent.
2. Doing a few more unit conversions reveals that the swimming pool holds 70,575 cubic feet or 529,313 gallons of water, which is 4.6 times the amount of water lost in a day from the reservoir.
3. If 115,000 is 10% of the total daily outflow, then the total daily outflow is 115,000/0.1 = 1,150,000 gallons.
4. The total daily outflow is (1.15 million)/(9.5 million) = 0.12 or 12% of the capacity of the reservoir. This seems rather high; is it possible that one-eighth of the reservoir is drained every day?
a. The U. S. population as of the 2000 census was 281 million people. Thus, as percentage of the entire population, the 9 million adults who suffer from depression is 9/281 = 3.2% of the population.
b. If 74% of the population is over 18 (adult), then the adult population is 0.74 X 281 million, or 208 million people. Thus, as percentage of the adult population, the 9 million adults who suffer from depression is 9/208 = 4.3% of the population.
3. Given that 5% of the adult population has taken St. John's Wort and 4.3% of the adult population suffers from depression (in any given year), the numbers of people in the two groups are comparable. However, it does not follow that everyone who has taken St. John's Wort suffered from depression, not that all those suffering from depression took St. John's Wort.
1. Suppose there are 50 men and 50 women in a group. The 10% incidence rate means that 10 of the 100 people could be expected to develop a kidney stone in their lifetime. Because the incidence rate is 4 times higher for men, 8 of the 10 would be expected to be men. So the probability of a man developing a kidney stone in his lifetime is 8/100 = 0.08.
2. Using the same group as in part (1), because the incidence rate is 4 times higher for men, 2 of the 10 would be expected to be women. So the probability of a woman developing a kidney stone in her lifetime is 2/100 = 0.02, four times less that for men.
3. Having developed a kidney stone already, the probability of a recurrence for men and women is 0.5.
Ticket Prices at the Superbowl
To compare prices in two different years, we need a measure of inflation, which is commonly given by the Consumer Price Index (CPI). The CPI for 1967 was about 35 and the CPI for 2002 was about 180. Thus, 1967 prices should be scaled up by a factor of 180/35 = 5.14. Thus, the 1967 ticket price of $12 is about $12 X 5.14 = $62. Clearly, Superbowl ticket prices have increased much faster than the rate of inflation. Even in "real dollars," the price has increased by a factor of about $400/$62 = 6.5.
New Gene Test for Colon Cancer
1. Suppose you take a group of 10,000 people. With a base rate of 5 in 10,000, one would expect 5 people with colon cancer, of which 60%, or 3, would test positive and 40%, or 2, would test negative. Of the 9995 who do not have colon cancer, all would test negative (because the false positive rate is zero). Thus, the probability of having a false report is 2 in 10,000.
2. For the second test, with a base rate of 5 in 10,000, one would expect 5 people with colon cancer, of which 70%, or 3.5 would test positive and 30%, or 1.5, would test negative. Of the 9995 who do not have colon cancer, 95%, or about 9495 would test (true) negative and 5%, or about 500, would test (false) positive. Thus, the probability of having a false report is about 501 in 10,000, or 0.05.
3. Of the false reports, nearly all (500 of 501) are false positives.
4. False positives are expensive, because they call for follow up tests or possibly surgery.
1. With a 25% markup, the retail and wholesale price are related by retail = 1.25 X wholesale. Thus, wholesale = retail/1.25. With a retail price of $55 (in Books in Print), we see that the wholesale price is $55/1.25 = $44.
2. The bookstore is using a 25% margin, rather than a 25% markup. With a 25% margin, the retail price is found by adding 25% of the retail price to the wholesale price. This means that retail = wholesale + 0.25 X retail, or wholesale = 0.75 X retail, or retail = wholesale/0.75. Thus, with a $44 wholesale price, the retail price with a 25% margin is $44/0.75 = $58.66.
1. The unemployment rate is the percentage of people who are unemployed among all those people who are either employed or seeking employment. The tricky point is defining who is unemployed (see Department of Labor web site). For example, stay-at-home moms and dads are not unemployed, and people who are not actively seeking work are not unemployed.
2. The unemployment rate increased by 0.2 percentage points (absolute change); however, the percentage change is (5.8 - 5.6)/5.6 = 0.36 or 3.6 %. Thus the unemployment rate increased by 0.2 percentage points or by 3.6%.
3. Yes, they are consistent: 1.4 million jobs were lost since March in all businesses, while 1.3 million jobs were lost by manufacturing alone since the beginning of 2001.
4. Job losses in October, November, and December were roughly 400,000, 400,000, and 124,000, respectively. Total job losses in March through December were 1.4 million. Therefore, job losses in March through September were 1.4 million - 924,000 = 476,000.
5. The 1.3 million manufacturing jobs lost were 7% of the entire work force. Thus, the entire work force is roughly (1.3 million)/0.07 = 18.6 million people.
6. We are told that 200,000 retail jobs were added between January and July, but by the end of the year, retail jobs were down by a total of 73,000 jobs. Thus, 273,000 jobs must have been lost between July and the end of the year (200,000 - 273,000 = -73,000).
7. We are told that 5.6% of the work force was unemployed in November, 5.8% of the work force was unemployed in December, and 124,000 jobs were lost in December. If we assume that the size of the work force, N, is constant during the two months, then we have 0.058N - 0.056N = 124,000. Solving for N, we have that the size of the work force is 62,000, which is a plausible figure.
1. Because 1 euro = 0.7876 pound, we see that 1 euro is worth less than 1 pound. It takes more than 1 euro to equal 1 pound.
2. Using units to convert drachmas to euros, we have that 1200 drachmas is equal to
3. Converting $1000 directly to euros, we have
4. Converting $1000 to francs, we have
Converting this quantity of francs to euros, we have
Comparing, the results of parts 3 and 4, we see that there is very little difference between the two exchange processes. Currency exchange rates (and the use of buy and sell rates) are usually determined to discourage currency traders from making money.
Consider the weather years over the 14 decades since 1860. The probability that any particular weather year (for example, warmest or coldest) occurred in the 1990s is 1/14.
1. The probability that the ten warmest years all occurred in the 1990s is (1/14)^10 = 0.0000000000035 (extremely unlikely).
2. The probability that at least one of the ten warmest years occurred in the 1990s is 1 - (13/14)^10 = 0.52 (reasonably likely).
3. The probability that nine of the ten warmest years on record occurred in the 1990s (using the binomial distribution) is 10 (1/14)^9(13/14) = 0.00000000045, which is again is quite unlikely. This result suggests that the assumption of independence of weather years may not be warranted; that is, there may be a pattern of warming.
1. Percentages are used to measure the fraction of 8th, 10th, and 12th graders who smoke now and one year ago. Percentages used in this way are often called rates: the smoking rate for 8th graders is currently 12.2%.
2. We can also describe the change in smoking rates using percentages. The percent (or relative) change in the 8th grade smoking rate is (12.2% - 14.6%)/14.6% = -16.4%; that is, the smoking rate for 8th graders decreased by 16.4%. Similarly, the smoking rate for 10th graders and 12th graders decreased by 10.9% and 6.1%, respectively. Because the improvement in smoking rates declined with age, it would appear that smoking habits are more difficult to break as teens get older.
3. Percentages give only one measure of smoking habits. While the percentage of smokers decreased, the number of teens also increased. For example, suppose there were 1 million 8th graders last year (a hypothetical figure), which means there were approximately 14.6% X 1 million = 146,000 8th grade smokers. If the population of 8th graders increased to 1.19 million over the past year, then the current number of 8th grade smokers would be 12.2% X 1.19 million = 146,000 -- the same as the year before. If the population of 8th graders increased to 1.25 million over the past year, then the current number of 8th grade smokers would be 12.2% X 1.25 million = 152,000, which is an increase in the total number of smokers, even though the percentage of smokers decreased.
There are many ways to determine the outcome of an election in which candidates are ranked. The Heisman Trophy is determined by the point system or Borda count method, in which a first-place vote is worth 3 points, a second-place vote is worth 2 points, and a third-place vote is worth 1 point. Adding the points for the five candidates, we see that Eric Crouch won with 770 points over Rex Grossman with 708 points, which is a relatively small margin.
1. Percentages are used for two purposes: to measure the fraction of overweight children among all children and to express the change in the fraction of overweight children. Thus, we could say that the percentage of overweight white children increased by 50%, from 8% to 12%.
2. The statements are not entirely consistent with the given data. The percentage of overweight black children increased from 8% to 22%, which is an increase of (22%-8%)/8% = 175%; this is significantly more than a doubling (which would be a 100% increase). The percentage of overweight Hispanic children increased from 10% to 22%, which is an increase of (22%-10%)/10% = 120%; this increase is certainly "more than a doubling." The percentage of overweight white children increased from 8% to 12%, which is an increase of (12%-8%)/8% = 50%; the statement about white children is accurate.
3. It does not follow that the total numbers of overweight children increased by the same percentages as the fractions of overweight children increased. The total number of children (both overweight and not) increased over the same period. If the total number of white children remained constant between 1986 and 1998, then a 50% increase in the fraction of overweight white children would result in a 50% increase in the total number of overweight white children.
4. Suppose the number black children in 1986 was N. The percent increase in the number of overweight black children between 1986 and 1998 was (0.22 X 1.3 X N - 0.08 X N)/(0.08 X N) = 258%. Similarly, the percent increase in the number of overweight children was 186% for Hispanic children and 95% for white children.
5. Because the percent increases are for fractions of all children, they cannot increase forever. At any rate of increase, eventually all children will be overweight. For example, if the fraction of overweight white children increases by 50% every 12 years starting with 8% in 1986, it will be 12% in 1998 (as the study found), 18% in 2010, and eventually 100% in 2060.
American Education Going Two Directions
There are many factors that contribute to these seemingly contradictory figures. First, the figures overlook students who graduate from high school or receive equivalency degrees after age 18; approximately 1.1 million 18-year olds are still in high school. Looking ahead one age cohort, the graduation rate for 25-29 year olds is 90%, which is more realistic. Some claim that the national graduation rate of 81% is inflated. It counts equivalency degrees and it doesn't account for 2 million prisoners, many of whom are high school dropouts. The national graduation rate will continue to climb as long as people born before 1940 (when the graduation rate was at a low of about 25%) continue to die, while people entering the over-25 age group have higher graduation rates. This trend may continue for two decades and the national rate could reach 90% before declining.
Reading Statistical Tea Leaves
1. The question is somewhat ambiguous. In terms of total numbers (an absolute comparison), there are more white poor children than black poor children (8.9 million vs. 4.0 million). In terms of percentage of their racial population (a relative comparison), there are more black poor children than white poor children (37% vs. 16%).
2. The percentage of black or Hispanic poor children is (4.0 + 3.9)/(8.9 + 4.0 + 3.9) = 47.0%.
3. We are told that 8.9 million poor white children is 16% of all white children. Therefore, the number of white children is (8.9 million)/0.16 = 55.6 million.
1. If a 15% decrease amounts to 1 million crimes, then (1 million crimes)/(number of 1999 crimes) = 0.15, which means that in 1999 there were about 6.7 million crimes.
2. If a 15% decrease reduced the crime rate to 28 crimes per 1000 in 2000, then 0.85 X (1999 crime rate) = 28 crimes per 1000. This means that the 1999 crime rate was about 33 crimes per 1000.
3. The overall crime rate of 28 crimes per 1000 is a weighted average of the men's rate (33) and the women's rate (23). Letting p be the fraction of crimes committed by men, we can write 33p + 23(1 - p) = 28. Solving for p, we see that p = 0.5. According to the figures given in the report, 50% of crimes were committed by men (and women). This conclusion seems suspicious!
4. The FBI crime rates were based on crimes actually reported to police. The BJS figures were obtained from a random survey of 160,000 people over the age of 12. It is difficult to explain how the FBI figures could be overestimates (unless crimes were reported to police departments that did not actually occur). On the other hand, the BJS survey could have underestimated crime rates (through sampling error or respondents' reluctance to admit to being victims).
Low Iron Means Low Math Scores
Although the full AP story was not given, it is a good example of sloppy or inaccurate journalism.
1. The only feasible way to interpret this statement is that 3% of the 5,398 children in the sample were iron deficient. This rate of iron deficiency in the population of all American children implies that there are 1.2 million iron deficient children in the United States.
2. The use of percentages is ambiguous here. We are told that 8.7% of the 12-16 year old girls in the sample are iron deficient. We might conclude that 7% of the 12-16 year old girls in the sample are iron-deficient, but not anemic. Or we might conclude that 7% of the 8.7% of the 12-16 year old girls in the sample are iron-deficient, but not anemic. The first is probably the more likely interpretation because anemia genearlly accounts for a small fraction of iron deficientcy cases.
3. The test scores quoted in the article are rather disorganized. We are told that among all children, the test scores are 93.7 (normal children), 87.4 (iron-deficient without anemia), and 86.4 (iron-deficient with anemia). Then we are told that for all children, scores for iron deficient children (with or without anemia) are about six points lower than for normal children. This statement seems inconsistent with the figures 93.7 (normal children) and 87.4 (iron-deficient without anemia), in which we see roughly a six-point difference. The eight-point difference mentioned for adolescent girls is not related to any of the other test scores cited. A table would have been extremely helpful in presenting test scores for the various groups.
This article illustrate a common mistake (particularly among journalists). The article compares the percentage of middle school students who smoke in Colorado and nationally. The absolute difference between the two percentages is 39.4% - 33.5% = 5.9 percentage points. The percentage difference is 5.9/33.5 = 0.176. Thus, the comparison should read that the Colorado smoking rate is 17.6% higher than the national average.The reporter might also have said that the Colorado smoking rate is 5.9 percentage points higher than the national average. A question (perhaps not an error) is how only 32% of middle school smokers have used cigarettes. If true, this implies that over two-thirds of middle school smokers use chewing tobacco, marijuana, cigars, pipes, ... This figure seems high.
1. The number of combinations of 6 numbers drawn from a barrel of 40 numbers is 3,838,380. So your chances of winning the lottery is 1 in 3,838,380 or about 1 in 4 million.
2. Using an annuity or savings plan formula, you would save $286,255.
Phantom Numbers and the War on Drugs
1. The figure 85 is called the relative risk; it is the ratio of the percentage of marijuana users who later used cocaine to the percentage of non-marijuana smokers who later used cocaine: 17%/0.2% = 85.
2. The percentage of marijuana smokers who did not later use cocaine is 100% - 17% = 83%.
3. The percentage of non-marijuana smokers who did not use cocaine is 100% - 0.2% = 99.8%.
4. It is not possible to determine the percentage of cocaine users who once used marijuana. The figures in the study give the percentage of marijuana and non-marijuana users who later used cocaine, not the percentage of cocaine users who did and did not use marijuana.
1. At the end of the first month, you receive $3 X 5 = $15.
2. At the end of the second month you receive $3 (5 + 25) = $90.
3. At the end of the third month you receive $3 (5 + 25 + 125) = $465.
4. At the end of the fourth month you receive $3 (5 + 25 + 125 + 625) = $2340.
5. At the end of the fifth month you receive $3 (5 + 25 + 125 + 625 + 3125) = $9375.
6. Apart from the illegality of the scheme, it depends entirely on the reliability of everyone else in the system; all participants must submit their $20 each month. Some participants might also notice that the real winners or the organizers, whose identity is kept secret!
1. We are told that there were 4% more new cases in 1997 than in 1996. Therefore,
Dividing by 1.04, we find that the number of new cases in 1996 was 40,300/1.04 = 38,750.
2. Repeating this argument, the number of new cases in 1995 was 40,300/(1.04^2) = 37,260.
3. In general, the number of new cases n years before 1997 was 40,300/(1.04^n). So in 1973, the number fof new cases was 40,300/(1.04^24) = 15,721.
4. Projecting ahead in time, the number of new cases n years after 1997 is 40,300 X 1.04^n. In 2010, the number of new cases will be 40,300 X 1.04^13 = 67,102 Assuming the same 4% growth rate persists.
1. Letting Y be the percentage of votes received by Yeltsin and A be the percentage of votes received by Zyuganov, we have Y = Z + 0.13 and Y = 1.33Z. The solution is Y = 52.4% and Z = 39.4%.
2. The erratum does not fully clarify the situation: if the election is a runoff between only two candidates, it should follow that Y + Z = 100%. Nevertheless, Yeltsin was declared the winner.
Several unit conversions are needed. One can work either in cubic feet or cubic meters. The fraction of the total reservoir volume that was drained in one week was 0.013 (1.3%). This estimate neglects water that flowed into the reservoir during the flood. If the inflow were taken into account, this fraction would be even smaller. It would seem that the flood had a negligible effect.
If we assume annual compounding at a rate r, then we have that $22 million = $5000(1 + r)^50. Using a calculator to test various values of r or solving directly for r, we see that r = 0.18 = 18%. Ms. Scheiber invested her money very wisely!
1. 10.% is the national divorce rate; 24.4% is the Beijing divorce rate.
2. The Beijing divorce rate increased from 12% to 24.4%, which is an increase of slightly more than 100%.
3. We are told that the divorce rate is computed by comparing the number of marriages and divorces in a given year. Without more information, this measure seems unreliable because it compares newly married people in a given year with divorced people, who were married in all previous years.