3. Thinking about shape. Would you expect distributions of these variables to be uniform, unimodal, or bimodal? Symmetric or skewed? Explain why. a) The number of speeding tickets each student in the senior class of a college has ever had. b) Players' scores (number of strokes) at the US Open golf tournament in a given year. c) Weights of female babies born in a particular hospital over the course of a year. d) The length of the average hair on the heads of stu- dents in a large class. 4. More shapes. Would you expect distributions of these variables to be uniform, unimodal, or bimodal? Symmet- ric or skewed? Explain why. a) Ages of people at a Little League game. b) Number of siblings of people in your class. c) Pulse rates of college-age males. d) Number of times each face of a die shows in 100tosses. 5. Heart attack stays. The histogram shows the lengths of hospital stays (in days) for all the female patients ad- mitted to hospitals in New York in 1993with a primary diagnosis of acute myocardial infarction (heart attack). Write a few sentences describing this distribution (shape, center, spread, unusual features). 600 .!!l c: '" ~ 400 '" <ti E .'E 200 <5 "" 5 10 15 20 25 30 35 40 Stay (days) o 6. E-mails. A university teacher saved every e-mail re- ceived from students in a large Introductory Statistics class during an entire term. He then counted, for each student who had sent him at least one e-mail, how many e-mails each student had sent. Based on the histogram below, describe the distribution of e-mails. '" 60 c '" -g us 40 <5 "" 80 20 6 11 16 # of E-mails 21 Chapter 4• Displaying Quantitative Data 65 7. Sugar in cereals. The histogram displays the sugar content (as a percent of weight) of 49 brands of breakfast cereals. o 16 24 32 40 48 56 Sugar(%) a) Describe this distribution. b) What do you think might account for this shape? 8. Singers. The display shows the heights of some of the singers in a chorus, collected so that the singers could be positioned on stage with shorter ones in front and taller ones in back. 60 68 Height (in.) 76 a) Describe the distribution. b) Can you account for the features you see here? 9. Vineyards. The histogram shows the sizes (in acres) of 36 vineyards in the Finger Lakes region of New York. 15 f- r-- {l 10 f- ~ '" c; ':> o :ki "" 5- - o 120 Size (acres) 240 a) Approximately what percentage of these vineyards are under 60 acres? b) Write a brief description of this distribution (shape, center, spread, unusual features).
Transcript
3. Thinking about shape. Would you expect distributionsof these variables to be uniform, unimodal, or bimodal?Symmetric or skewed? Explain why.a) The number of speeding tickets each student in the
senior class of a college has ever had.b) Players' scores (number of strokes) at the US Open
golf tournament in a given year.c) Weights of female babies born in a particular hospital
over the course of a year.d) The length of the average hair on the heads of stu-
dents in a large class.
4. More shapes. Would you expect distributions of thesevariables to be uniform, unimodal, or bimodal? Symmet-ric or skewed? Explain why.a) Ages of people at a Little League game.b) Number of siblings of people in your class.c) Pulse rates of college-age males.d) Number of times each faceof a die shows in 100tosses.
5. Heart attack stays. The histogram shows the lengths ofhospital stays (in days) for all the female patients ad-mitted to hospitals in New York in 1993with a primarydiagnosis of acute myocardial infarction (heart attack).Write a few sentences describing this distribution(shape, center, spread, unusual features).
600.!!lc:
'"~ 400'"<tiE.'E 200<5""
5 10 15 20 25 30 35 40Stay (days)
o 6. E-mails. A university teacher saved every e-mail re-ceived from students in a large Introductory Statisticsclass during an entire term. He then counted, for eachstudent who had sent him at least one e-mail, how manye-mails each student had sent. Based on the histogrambelow, describe the distribution of e-mails.
'" 60c'"-gus 40<5""
80
20
6 11 16# of E-mails
21
Chapter 4 • Displaying Quantitative Data 65
7. Sugar in cereals. The histogram displays the sugarcontent (as a percent of weight) of 49 brands of breakfastcereals.
o 16 24 32 40 48 56Sugar(%)
a) Describe this distribution.b) What do you think might account for this shape?
8. Singers. The display shows the heights of some of thesingers in a chorus, collected so that the singers could bepositioned on stage with shorter ones in front and tallerones in back.
60 68Height (in.)
76
a) Describe the distribution.b) Can you account for the features you see here?
9. Vineyards. The histogram shows the sizes (in acres) of36 vineyards in the Finger Lakes region of New York.
15 f- r--
{l 10 f-
~'"c;
':>o :ki"" 5 -
-
o 120Size (acres)
240
a) Approximately what percentage of these vineyardsare under 60 acres?
b) Write a brief description of this distribution (shape,center, spread, unusual features).
66 Part I • Exploring and Understanding Data
10. Run times. One of the authors collected the times (inminutes) it took him to run 4 miles on various coursesduring the period 1986 to 1997.Here is a histogram ofthe times.
50
40
~ 30o"0"" 20
10
28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.04-Mile Time (min)
Describe the distribution and summarize the importantfeatures. What is it about running that might account forthe shape you see?
11. Gasoline. In June 2004, 16 gas stations in Ithaca, NY,posted these prices for a gallon of regular gasoline.
2.0292.0792.0692.169
2.1192.0892.269
2.189
2.2592.0792.0992.039
2.0492.0392.1292.079
a) Make a stem-and-leaf display of these gas prices. Usesplit stems; for example, use two 2.1 stems, one forprices between $2.10 and $2.149,the other for prices$2.15to $2.199.
b) Describe the shape, center, and spread of this distri-bution.
c) What unusual feature do you see?
12. The Great One. During his 20 seasons in the NHL,Wayne Gretzky scored 50% more points than anyonewho ever played professional hockey. He accomplishedthis amazing feat while playing in 280 fewer gamesthan Gordie Howe, the previous record holder. Hereare the number of games Gretzky played during eachseason:
a) Create a stem-and-leaf display for these data usingsplit stems.
b) Describe the shape of the distribution.c) Describe the center and spread of this distribution.d) What unusual feature do you see? What might ex-
plain this?
o 13. Home runs. The stem-and-leaf display shows the num-ber of home runs hit by Mark McGwire during the 1986-2001 seasons. Describe the distribution, mentioning itsshape and any unusual features.
7 0655284293223992 2910399
Home Runs(710 means 70)
o 14. Bird species. The Cornell Lab of Ornithology holds anannual Christmas Bird Count, in which birdwatchers atvarious locations around the country see how many dif-ferent species of birds they can spot. Here are some ofthe counts reported from sites in Texas during the 1999event.
228183160
178181160
186206157
162177156
206175153
163162152
166167153
a) Create a stem-and-leaf display of these data.b) Write a brief description of the distribution. Be sure
to discuss the overall shape as well as any unusualfeatures.
o 15. Home runs, again. Students were asked to make a his-togram of the number ofhome runs hit by Mark McGwirefrom 1986to 2001(see Exercise 13). One student submit-ted the following display:
60 l-
I-
I- I~
~ nn
<n§ 40a:~o:c 20
o1990 1995
Year2000
a) Comment on this graph.b) Create your own histogram of the data.
o 16. Return of the birds. Students were given the assign-ment to make a histogram of the data reported in Exer-cise 14, on bird counts. One student submitted the fol-lowing display:
o 18. Population growth. Here is a "back-to-hack" stem-and-leaf display that shows two data sets at once-one goingto the left, one to the right. It compares the percentchange in population for two regions of the UnitedStates (based on census figures for 1990 and 2000).Thefastest growing states were Nevada at 66% and Arizonaat 40%.Write a few sentences describing the difference ingrowth rates for the two regions of the United States. Toshow the distributions better, this display breaks eachstem into two lines, putting leaves 0-4 on one stem andleaves 5-9 on the other.
Chapter 4 • Displaying Quantitative Data 67
o 19. Hurricanes. The data below give the number of hurri-canes that happened each year from 1944 through 2000as reported by Science magazine.3,2,1,2,4,3,7,2,3,3,2,5,2,2,4,2,2,6,0,2,5,1,3,1,0,3, 2, 1, 0, 1,2, 3, 2, 1,2, 2, 2, 3, 1, 1, 1,3, 0, 1,3, 2, 1,2, 1, 1,0,5,6,1,3,5,3
a) Create a dotplot of these data.b) Describe the distribution.
o 20. Hurricanes, again. A bimodal distribution usually in-dicates that there are actually two different behaviorspresent in the data. Investigating those two behaviorsseparately can produce important insights. Here are thedata again, broken into two groups showing the numberof hurricanes recorded annually before and after 1970.Create an appropriate visual display and write a fewsentences comparing the two distributions.
021. Acid rain. Two researchers measured the pH (a scale onwhich a value of 7 is neutral and values below 7 areacidic) of water collected from rain and snow over a6-month period in Allegheny County, Pennsylvania.Describe their data with a graph and a few sentences.
022. Marijuana. In 1995 the Council of Europe published areport entitled The European School Survey Project on Alco-hol and Other Drugs. Among other issues, the survey in-vestigated the percentages of 9th graders who had usedmarijuana. Here are the results for 20 Western Europeancountries. Create an appropriate graph of these data, anddescribe the distribution.
NE/MW States S/W States Austria 10% Italy 19%6 6 Belgium 19% Luxembourg 6%65 Denmark 17% Netherlands 31%5 England 40% No.Ireland 23%4 Finland 5% Norway 6%4 0 France 12% Portugal 7%3 Germany 21% Scotland 53%3 001 Greece 2% Spain 15%2 6 Iceland 10% Sweden 6%2 001134 Ireland 37% Switzerland 27%1 578
2100 1 001134444 23. Hospital stays. The Ll.S. National Center for Health99998876655 0 6999 Statistics compiles data on the length of stay by patients
4431 0 1 in short-term hospitals and publishes its finding in VitalPopulationGrowthrate and Health Statistics. Data from a sample of 39 male pa-(1616means66/',) tients and 35 female patients on length of stay (in days)
are displayed in the histograms on the next page.
68 Part I • Exploring and Understanding Data
15 15
2lcr 10 10'"~a..0"" 5 5
0.0 12.5 0.0 10.0 20.0Men Women
a) What would you suggest be changed about these his-tograms to make them easier to compare?
b) Describe these distributions by writing a few sen-tences comparing the duration of hospitalization formen and women.
c) Can you suggest a reason for the peak in women'slength of stay?
24. Deaths. A National Vital Statistics Report indicated thatnearly 300,000black Americans died in 1999,comparedwith just over 2 million white Americans. Here are calcu-lator histograms displaying the distributions of theirages at death.
P 1:L 1..IIHUE
",in=&5",·]:«75
",in=65""3:<·::75n=18.7772 n=20.'10B1
Most of the bars in these histograms display ten-year agegroups. For example, the first histogram shows that forwhite Americans about 19%of the deaths were of peoplebetween 65and 74 years old. The leftmost bars representthe percentage of total deaths that were children aged 0through 4 years and the rightmost bars people over 85.Write a brief comparison of the distributions.
25. Final grades. A professor (of something other thanStatistics!) distributed the following histogram to showthe distribution of grades on his 200-point final exam.Comment on the display.
50 100 150Final Grade
026. Cities. Here's a histogram of the cost of living in 25 in-ternational cities. Costs are given in U'S, dollars.
~10(3ts"" 5
15
60 100 140Cost of Living ($)
180
a) Write a few sentences describing this distribution.b) Does the most expensive city included here (Tokyo)
appear to be an outlier? Explain.
27. Final grades revisited. After receiving many com-plaints about his final grade histogram from studentscurrently taking a Statistics course, the professor fromExercise 25 distributed the following revised histogram.
a) Comment on this display.b) Describe the distribution of grades.
o 28. Cities revisited. Here is a picture of the costs of livingin 25 international cities resealed to bars that are $12wide, rather than the $20bins you saw in Exercise26.
8
<n6
~(3 40"*'
2
108 132 156Cost of Living ($)
a) Describe what you see in this histogram.b) Now would you consider Tokyo, the most expensive
city, to be an outlier? Defend your opinion.
84
29. Zip codes. Holes-R-Us, an Internet company that sellspiercing jewelry, keeps transaction records on its sales.At a recent sales meeting, one of the staff presented a his-togram of the zip codes of the last 500 customers so thatthey might understand where sales are coming from.
Comment on the usefulness and appropriateness of thedisplay.
80
li3 60Eogj 40oo"" 20
15000 65000Zip
9000040000
30. CEO data revisited. For each CEO, a code is listed thatcorresponds to the industry of the CEO's company. Hereare a few of the codes and the industries to which theycorrespond.
A recently hired investment analyst has been assigned toexamine the industries and the compensations of theCEOs. To start the analysis, he produces the followinghistogram of industry codes.
200
150(3~ 100o""
50
0.00 3.75 7.50 11.25 15.00 18.75Industry Code
a) What might account for the gaps seen in the histogram?b) Is the histogram unimodal?c) What advice might you give the analyst about the
appropriateness of this display?
31. Productivity study. The National Center for Productiv-ity releases information on the efficiency of workers. In arecent report, they included the following graph show-
Chapter 4 • Displaying Quantitative Data 69
ing a rapid rise in productivity. What questions do youhave about this display?
4~--------------------------,
3.5csot5=>'" 3e(L
2.5
32. Productivity revisited. A second report by the NationalCenter for Productivity analyzed the relationship betweenproductivity and wages. Comment on the graph they used.
~Productivity ••• Wages
o 33. Law enforcement. Some federal employees have the au-thority to carry firearms and make arrests. Obviouslysome danger is associated with these jobs, but how much?The table below summarizes the rates of assault and in-jury (or death) for these employees for 5 years, 1995-1999.
Bureau of Alcohol, 31.1 2.2Tobacco, andFirearms (BATF)
Capitol Police 5.0 3.6Customs Service 9.7 5.1Drug Enforcement 17.9 1.1
Agency (DEA)Federal Bureau of 3.9 1.2
Investigation (FBI)Immigration and Naturalization 14.1 2.5
Services (INS)Internal Revenue Service (IRS) 1.7 0.2U.S. Marshal Service 9.7 3.0National Park Service 38.7 15.0Postal Service 5.7 2.9Secret Service 9.7 3.0
70 Part I • Exploring and Understanding Data
a) Create a visual display of these data.b) Describe these data (shape, center, spread, unusual
features).c) Which agencies are outliers?
034. Cholesterol. A study examining the health risks ofsmoking measured the cholesterol levels of people whohad smoked for at least 25 years and people of similarages who had smoked for no more than 5 years and thenstopped. Create histograms for both groups and write abrief report comparing their cholesterol levels.
a) Create a back-to-back stem-and-leaf display for thesedata.
b) Write a few sentences comparing the distributions.
o 36. Baseball. American League baseball teams play theirgames with the designated hitter rule, meaning that pitch-ersdo not bat. The League believes that replacing thepitcher, typically a weak hitter, with another player in thebatting order produces more runs and generates more in-terest among fans. Below are the average number of runsscored in American League and National League stadi-ums for the first half of the 2001season.
a) Create a back-to-back stern-and-leaf display of thesedata.
b) Write a few sentences comparing the average num-ber of runs scored per game in the two leagues. (Re-member: shape, center, spread, unusual features!)
c) Coors Field, in Denver, stands a mile above sea level,an altitude far greater than that of any other majorleague ball park. Some believe that the thinner airmakes it harder for pitchers to throw curve balls andeasier for batters to hit the ball a long way. Do yousee any evidence that the 14 runs scored per gamethere is unusually high? Explain.
037. Nuclear power. For a while in the 20th century, manynuclear-powered electrical generating plants were built,but then growing environmental concerns and construc-tion costs led to increasing reliance on other forms of en-ergy. The table shows the dates of completion (in monthsafter January 1967)and costs (in thousands of dollars permegawatt) of 12nuclear generators.
o 35. MPG. A consumer organization compared gas mileagefigures for several models of cars made in the UnitedStates with autos manufactured in other countries. Thedata are shown in the table.
a) Create a stem-and-leaf display of the costs.b) Describe the distribution.c) Create a timeplot of the costs.d) What information about the construction of nuclear
plants can you see from the timeplot that is not obvi-ous in the stem-and-leaf display?
G gs. Drunk driving. Accidents involving drunk drivers ac-count for about 40% of all deaths on the nation's high-ways. The table tracks the number of alcohol-related fa-talities for 20 years.
a) Why is "acre-feet" a good way to measure theamount of precipitation produced by cloud seeding?
b) Plot these data, and describe the distribution.c) Create a re-expression of these data that produces a
more advantageous distribution .d) Explain what your re-expressed scale means.
( just checking --------------------~-~--
I Answers(Thoughts will vary.)
1. Roughly symmetric, slightly skewed to the right. Centeraround 3 miles? Few over 10 miles.
2. Bimodal. Center between 1 and 2 hours? Many peoplewatch no football, others watch most of one or moregames. Probably only a few values over 5 hours.
3. Strongly skewed to the right, with almost everyone at $0;a few small prizes, with the winner an outlier.
4. Fairly symmetric, somewhat uniform, perhaps slightlyskewed to the right. Center in the 40s? Few ages below25 or above 70.
5. Uniform, symmetric. Center near 5. Roughly equal countsfor each digit 0-9.
