HW  #20:  Ch  7  #10,  18acd,  24,  25,  28,  30    

 

 

162 Part II • Exploring Relationships Between Variables

a) Make a histogram showing the distribution of the number of broken pieces in the 24 batches of pottery examined.

b) Describe the distribution as shown in the histogram. What feature of the problem is more apparent in the histogram than in the scatterplot?

c) What aspect of the company's problem is more ap-parent in the sca tterplot?

10. Coffee sales. Owners of a new coffee shop tracked sales for the first 20 days, and displayed the data in a scatter-plot (by day) :

4 Day

I 12

I 16

a) Make a histogram of the daily sales since the shop has been in business.

b) State one fact that is obvious from the scatterplot, but not from the histogram.

c) State one fact that is obvious from the histogram, but not from the scatterplot.

11. Matching. Here are several scatterplots. The calcu lated correlations are - 0.923, - 0.487,0.006, and 0.777. Which is which?

• • \ • • • ... • • • • • •

••• • • • •• • • • • • • • • • •

(a) (b)

• • • • • •• • • • • • • • • • • • • • • • • • • (e) (d)

12. Matching. Here are several scatterplots. The calculated correlations are - 0.977, -0.021, 0.736, and 0.951. Which is which?

•• •• • •• '. • ., • ...

• • •• .:-• • •• ..... -

• e. e •• e ••• •

• ••• (a) "--t--+---+------j (b) "---+-+---+--l

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • (e) (d) •

o 13. Roller coasters. Roller coasters get all their speed by dropping down a steep initial incline, so it makes sense that the height of that drop might be related to the speed of the coaster. Here's a scatterplot of top speed and largest drop for 75 roller coasters arOlmd the world. a) Does the scatterplot indicate that it is appropriate to

calculate the correlation? Explain. b) In fact, the correlation of speed and drop is 0.91. De-

scribe the association.

87.5! \ tJ 75.0 • ... II

s -.. tI-: -g 62.5 • _:

g- 50.0 .:.' . .. I 75 150 225

Drop (ft)

..

300

o 14. Antidepressants. A study compared the effectiveness of severa l antidepressants by examining the experiments in which they had passed the FDA requirements. Each of those experiments compared the active drug to a placebo, an illert pilJ given to some of the subjects. In each experi-ment some patients treated with the placebo had im-proved, a phenomenon called the placebo effect. Patients' depression levels were evaluated on the Hamilton Depres-sion Rating Scale, where larger numbers indicate greater improvement. The sea tterplot shown compares mean

164 Part II • Exploring Relationships Between Variab les

0 17. Fuel economy. Here are advertised horsepower ratings 0 19. Burgers. Fast food is often considered unhealthy be-and expected gas mileage for several 2001 vehicles. cause much of it is high in both fat and sodium. But are

the two related? Here are the fa t and sodium contents of Vehicle Horsepower Gas Mileage

AudiA4 170 hp 22mpg Buick LeSabre 205 20 Chevy Blazer 190 15 Chevy Pri zm 125 31 Ford Excursion 310 10 GMCYukon 285 13 Honda Civic 127 29 Hyundai Elantra 140 25 Lexus 300 215 21 Lincoln LS 210 23 MazdaMPV 170 18 OldsAlero 140 23 Toyota Camry 194 21 VWBeetle 115 29

a) Make a scatterplot for these data. b) Describe the direction, form, and strength of the plot. c) Find the correlation between horsepower and miles

per gallon. d) Write a few sentences telling what the plot says about

fuel economy. 0 18. Drug abuse. A survey was conducted in the United

States and 10 countries of Western Europe to determine the percentage of teenagers who had used marijuana and other drugs. The results are summarized in the table.

Percent Who Have Used country Marijuana Other Drugs Czech Rep. 22 4 Denmark 17 3 England 40 21 Finland 5 1 Ireland 37 16 Italy 19 8 No. Ireland 23 14 Norway 6 3 Portugal 7 3 Scotland 53 31 USA 34 24

a) Create a scatterplot. b) What is the correlation between the percent of teens

who have used marijuana and the percent who have used other drugs?

c) Write a brief description of the association. d) Do these results confirm that marijuana is a "gate-

way drug," that is, that marijuana use leads to the use of other drugs? Explain.

severa l brands of burgers. Analyze the association be-tween fat content and sodium.

Fat (g) 1 19 1 31 1 34 1 35 1 39 1 39 1 43 Sodium (mg) 920 1500 1310 860 1180 940 1260

o 20. Burgers. In the previous exercise you analyzed the asso-ciation between the amounts of fat and sodium in fast food hamburgers. What about fat and ca lories? Here are data for the same burgers.

Fat (g) 1 19 Calories 410 1

31 1 34 580 590

35 139 139 570 640 680 1

43 660

o 21. Attendance, American League baseball games are p layed under the designated hitter rul e, meaning that wea k-hitting pitchers d o not com e to bat. Baseball own-ers believe that the design ated hitter rule means more runs scored, w hich in turn means higher attendance. Is there evidence that more fans attend games if the teams score more runs? Data collected mid way through the 2001 season indica te a correlation of 0.74 between runs scored and the number of people at the ga me.

0 22.

37500 • • • • •

•

130000

• • 22500 • • , •

• 360 440

Runs a) Does the scatterp lo t indica te that it's appropriate to

calculate a correlation? Explain. b) Describe the association between attendance and

runs scored. c) Does this prove that the owners are right, that more

fans will come to games if the teams score more runs? Second inning. Perhaps fans are just more interested in teams that win. Are the teams that win necessarily those that score the most runs?

CORRELATTON Wins Runs Attend

Wins 1.000 Runs Attend

0.680 0.557

1.000 0.740 1.000

   

 

   

 

Chapter 7 • Scatterplots, Association, and Correlation 165

• 37500 •• • •

1; • 30000

;g •• « 22500 •• •• •

• 30 40 50 60

Wins

a) Do winning teams generally enjoy greater atten-dance at their home games? Describe the association.

b) Is attendance more strongly associated with winning or scoring runs? Explain.

c) How strongly is scoring more runs associated with winning more games?

23. Politics. A candidate for office claims that "there is a correlation between television watching and crime." Criticize this statement in statistical terms.

24. Association. A researcher investigating the association between two variables collected some data and was sur-prised when he calculated the correlation. He had ex-pected to find a fairly strong association, yet the correla-tion was near O. Discouraged, he didn't bother making a scatterplot. Explain to him how the scatterplot could still reveal the strong association he anticipated.

25. Height and reading. A researcher studies children in elementary school and finds a strong positive linear as-sociation between height and reading scores. a) Does this mean that taller children are generally bet-

ter readers? b) What might explain the strong correlation?

26. Cellular telephones and life expectancy. A survey of the world's na tions in 2004 shows a s trong posi tive cor-relation between percentage of the country using cell phones and life expectancy in years at birth. a) Does this mean that cell phones are good for your

health? b) What might exp lain the strong correlation?

27. Hard water. In a s tudy of streams in the Adirondack Mountains, the following relationship was found be-tween the pH of the water and the water's hardness (measured in grains):

• • 8.0 ••••••••• • • • . . • • . • •••• • • • 7.6 • ... • • ••• • •••••• • '5. • • ·Ii·. • 7.2 . ..... .

6.8 I·' • • •••• 6.4 •

125 250 375 500 Hardness (grains)

Is it appropriate to summarize the strength of associa-tion w ith a correlation? Explain.

28. Traffic headaches. A study of traffic delays in 68 U.s. cities found the following relationship between total de-lays (in total hours lost) and mean highway speed:

29.

600000

450000

" o '" 300000 ;2

150000 .. '.

• • •• •

.. :-1.. •. -,_, 4' ...... I I I I

40 45 50 55 60 Highway Speed (mph)

Is it appropriate to summarize the strength of associa-tion with a correlation? Explain.

Correlation errors. Your Economics instructor assigns your class to investigate factors associated with the gross domestic product (GDP) of nations. Each student exam-ines a different factor (such as life expectancy, literacy rate, etc.) for a few countries and reports to the class. Appar-ently some of your classmates do not understand Statis-tics very well because you know several.of their conclu-sions are incorrect. Explain the mistakes in their s ta temen ts below. a) "My correlation of - 0.772 shows that there is almost

no association between GDP and infant mortality rate."

b) "There was a correlation of 0.44 between GDP and continent. "

Chapter 7 • Scatterplots, Association, and Correlation 165

• 37500 •• • •

1; • 30000

;g •• « 22500 •• •• •

• 30 40 50 60

Wins

a) Do winning teams generally enjoy greater atten-dance at their home games? Describe the association.

b) Is attendance more strongly associated with winning or scoring runs? Explain.

c) How strongly is scoring more runs associated with winning more games?

23. Politics. A candidate for office claims that "there is a correlation between television watching and crime." Criticize this statement in statistical terms.

24. Association. A researcher investigating the association between two variables collected some data and was sur-prised when he calculated the correlation. He had ex-pected to find a fairly strong association, yet the correla-tion was near O. Discouraged, he didn't bother making a scatterplot. Explain to him how the scatterplot could still reveal the strong association he anticipated.

25. Height and reading. A researcher studies children in elementary school and finds a strong positive linear as-sociation between height and reading scores. a) Does this mean that taller children are generally bet-

ter readers? b) What might explain the strong correlation?

26. Cellular telephones and life expectancy. A survey of the world's na tions in 2004 shows a s trong posi tive cor-relation between percentage of the country using cell phones and life expectancy in years at birth. a) Does this mean that cell phones are good for your

health? b) What might exp lain the strong correlation?

27. Hard water. In a s tudy of streams in the Adirondack Mountains, the following relationship was found be-tween the pH of the water and the water's hardness (measured in grains):

• • 8.0 ••••••••• • • • . . • • . • •••• • • • 7.6 • ... • • ••• • •••••• • '5. • • ·Ii·. • 7.2 . ..... .

6.8 I·' • • •••• 6.4 •

125 250 375 500 Hardness (grains)

Is it appropriate to summarize the strength of associa-tion w ith a correlation? Explain.

28. Traffic headaches. A study of traffic delays in 68 U.s. cities found the following relationship between total de-lays (in total hours lost) and mean highway speed:

29.

600000

450000

" o '" 300000 ;2

150000 .. '.

• • •• •

.. :-1.. •. -,_, 4' ...... I I I I

40 45 50 55 60 Highway Speed (mph)

Is it appropriate to summarize the strength of associa-tion with a correlation? Explain.

Correlation errors. Your Economics instructor assigns your class to investigate factors associated with the gross domestic product (GDP) of nations. Each student exam-ines a different factor (such as life expectancy, literacy rate, etc.) for a few countries and reports to the class. Appar-ently some of your classmates do not understand Statis-tics very well because you know several.of their conclu-sions are incorrect. Explain the mistakes in their s ta temen ts below. a) "My correlation of - 0.772 shows that there is almost

no association between GDP and infant mortality rate."

b) "There was a correlation of 0.44 between GDP and continent. "

Chapter 7 • Scatterplots, Association, and Correlation 165

• 37500 •• • •

1; • 30000

;g •• « 22500 •• •• •

• 30 40 50 60

Wins

a) Do winning teams generally enjoy greater atten-dance at their home games? Describe the association.

b) Is attendance more strongly associated with winning or scoring runs? Explain.

c) How strongly is scoring more runs associated with winning more games?

23. Politics. A candidate for office claims that "there is a correlation between television watching and crime." Criticize this statement in statistical terms.

24. Association. A researcher investigating the association between two variables collected some data and was sur-prised when he calculated the correlation. He had ex-pected to find a fairly strong association, yet the correla-tion was near O. Discouraged, he didn't bother making a scatterplot. Explain to him how the scatterplot could still reveal the strong association he anticipated.

25. Height and reading. A researcher studies children in elementary school and finds a strong positive linear as-sociation between height and reading scores. a) Does this mean that taller children are generally bet-

ter readers? b) What might explain the strong correlation?

26. Cellular telephones and life expectancy. A survey of the world's na tions in 2004 shows a s trong posi tive cor-relation between percentage of the country using cell phones and life expectancy in years at birth. a) Does this mean that cell phones are good for your

health? b) What might exp lain the strong correlation?

27. Hard water. In a s tudy of streams in the Adirondack Mountains, the following relationship was found be-tween the pH of the water and the water's hardness (measured in grains):

• • 8.0 ••••••••• • • • . . • • . • •••• • • • 7.6 • ... • • ••• • •••••• • '5. • • ·Ii·. • 7.2 . ..... .

6.8 I·' • • •••• 6.4 •

125 250 375 500 Hardness (grains)

Is it appropriate to summarize the strength of associa-tion w ith a correlation? Explain.

28. Traffic headaches. A study of traffic delays in 68 U.s. cities found the following relationship between total de-lays (in total hours lost) and mean highway speed:

29.

600000

450000

" o '" 300000 ;2

150000 .. '.

• • •• •

.. :-1.. •. -,_, 4' ...... I I I I

40 45 50 55 60 Highway Speed (mph)

Is it appropriate to summarize the strength of associa-tion with a correlation? Explain.

Correlation errors. Your Economics instructor assigns your class to investigate factors associated with the gross domestic product (GDP) of nations. Each student exam-ines a different factor (such as life expectancy, literacy rate, etc.) for a few countries and reports to the class. Appar-ently some of your classmates do not understand Statis-tics very well because you know several.of their conclu-sions are incorrect. Explain the mistakes in their s ta temen ts below. a) "My correlation of - 0.772 shows that there is almost

no association between GDP and infant mortality rate."

b) "There was a correlation of 0.44 between GDP and continent. "

166 Part II • Exploring Relationships Between Variables

30. More correlation errors. Students in the Economics class d iscussed in Exercise 29 also wrote these conclu-sions. Explain the mistakes they made. a) "There was a very strong correlation of 1.22 between

life expecta.ncy and GOP" b) "The correlation between literacy rate and GOP was

0.83. This shows that countries wanting to increase their standard of living should invest heavily in edu-cation."

31. Baldness and heart disease. Medical researchers fol-lowed 1435 middle-aged men for a period of 5 years, measuring the amount of baldness present (none = 1, little = 2, some = 3, much = 4, extreme = 5) and presence of heart disease (No = 0, Yes = 1). They fou nd a correlation of 0.089 between the two variables. Comment on their conclusion that this shows that baldness is not a possible cause of heart disease.

32. Sample survey. A polling organization is checking its database to see if the two data sources they used sam-pled the same zip codes. The variable datasollrce = 1 if the data source is MetroMedia, 2 if the data source is DataQwest, and 3 if it's RollingPol1. The organization finds that the correlation between five digit zip code and datasollrce is - 0.0229. It concludes that the correlation is low enough to state that there is no dependency between zip code and SOllrce of data. Comment.

33. Thrills. People who responded to a July 2004 Discovery Chmmel poll named the 10 best roller coasters in the United States. The table below shows the length of the initial drop (in feet) and the duration of the ride (in sec-onds). What do these data indicate about the height of a roller coaster and the length of the ride you can expect?

Roller Coaster State Drop (ft) Duration (sec) Incredible Hulk FL 105 135 Millennium Force OH 300 105 Goliath CA 255 180 Nitro NJ 215 240 Magnum XL-2000 OH 195 120 The Beast OH 141 65 Son of Beast OH 214 140 Thunderbolt PA 95 90 Ghost Rider CA 108 160 Ra ven IN 86 90

o 34. Oil production. The following table shows the oil pro-duction of the Un ited States from 1949 to 2000 (in thou-sands of barrels per year).

Year 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974

Oil 1,841,940 1,973,574 2,247,711 2,289,836 2,357,082 2,314,988 2,484,428 2,617,283 2,616,901 2,448,987 2,574,590 2,574,933 2,621,758 2,676,189 2,752,723 2,786,822 2,848,514 3,027,763 3,215,742 3,329,042 3,371,751 3,517,450 3,453,914 3,455,368 3,360,903 3,202,585

Year 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000

Oil 3,056,779 2,976,180 3,009,265 3,178,216 3,121,310 3,146,365 3,128,624 3,156,715 3,170,999 3,249,696 3,274,553 3,168,252 3,047,378 2,979,123 2,778,773 2,684,687 2,707,039 2,624,632 2,499,033 2,431,476 2,394,268 2,366,017 2,354,831 2,281,919 2,146,732 2,135,062

a) Find the correlation between year and production. b) A reporter concludes that a low correlation between

year and prodllction shows that oil production has re-mained steady over the 50-year period. Do you agree wi th this in terpreta tion? Explain.

0 35. Planets. Is there any pattern to the locations of the plan-ets in our solar system? The table shows the average dis-tance of each of the nine planets from the sun .

Planet Mercury Venus Earth Mars Jupiter Sa turn Uranus Neptune Pluto

Position Number

1 2 3 4 5 6 7 8 9

Distance from Sun (million miles)

36 67 93

142 484 887

1784 2796 3666