Chapter 09- Data and Statis - Advisory...

KEYWORD: MT7 Ch9

9A Collecting andDescribing Data

9-1 Samples and Surveys

LAB Explore Samples

9-2 Organizing Data

9-3 Measures of CentralTendency

9-4 Variability

LAB Create Box-and-Whisker Plots

9B Displaying DataLAB Make a Circle Graph

9-5 Displaying Data

LAB Create Histograms

9-6 Misleading Graphs andStatistics

9-7 Scatter Plots

LAB Create a Scatter Plot

9-8 Choosing the BestRepresentation of Data

LAB Use a Spreadsheet toCreate Graphs

458 Chapter 9

Quality Assurance Specialist

How do manufacturers know that their products are well made? It is the job of the quality assurance specialist. QA specialists designtests and procedures that allow the companies to determine how good their products are.Because checking every product or procedure may not be possible, QA specialists use sampling to predict the margin of error.

Errors in Samples

Company Type Sample Size Errors

Software25

2

Stoneworks100

7

Tools 50

4

Pizza 75

3

Data and StatisticsData and Statistics

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Data and Statistics 459

VocabularyChoose the best term from the list to complete each sentence.

1. A __?__ is a uniform measure where equal distances are markedto represent equal amounts.

2. __?__ is the process of approximating to a given __?__.

3. Ordered pairs of numbers are graphed on a __?__.

Complete these exercises to review skills you will need for this chapter.

Round DecimalsRound each number to the indicated place value.

4. 34.7826; nearest tenth 5. 137.5842; nearest whole number

6. 287.2872; nearest thousandth 7. 362.6238; nearest hundred

Compare and Order DecimalsOrder each sequence of numbers from greatest to least.

8. 3.005, 3.05, 0.35, 3.5 9. 0.048, 0.408, 0.0408, 0.48

10. 5.01, 5.1, 5.011, 5.11 11. 1.007, 0.017, 1.7, 0.107

Place Value of Whole NumbersWrite each number in standard form.

12. 1.3 million 13. 7.59 million 14. 4.6 billion 15. 2.83 billion

Read a TableUse the table for problems 16–18.

16. Which activity experienced the greatestchange in participation from 2000 to 2001?

17. Which activity experienced the greatestpositive change in participation from 2000 to 2001?

18. Which activity experienced the least changein participation from 2000 to 2001?

coordinate grid

place value

rounding

scale

Activity Enrollment

Softball

2000 2001

Soccer Basketball HockeyNum

ber

of s

tude

nts 70

6050403020100

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Previously, you

• used an appropriaterepresentation for displayingrelationships among collecteddata.

• described a set of data usingmean, median, mode, and range.

• made inferences based onanalysis of data.

You will study

• selecting an appropriaterepresentation for displayingrelationships among collecteddata.

• selecting the appropriatemeasure of central tendency to describe data.

• making predictions andanalyzing trends in scatter plots.

• recognizing misuses ofgraphical information.

You can use the skillslearned in this chapter

• to make predictions based onsurvey results.

• to conduct advanced researchstudies in science and socialstudies courses.

460 Chapter 9

Key Vocabulary/Vocabulariohistogram histograma

line plot diagrama deacumulación

mean media

median mediana

mode moda

population población

sample muestra

scatter plot diagrama dedispersión

Stu

dy

Gu

ide:

Pre

view

Vocabulary ConnectionsTo become familiar with some of thevocabulary terms in the chapter, consider thefollowing. You may refer to the chapter, theglossary, or a dictionary if you like.

1. The population of an area is the totalnumber of people living in that area. Whatmight mean in the process ofgathering data?

2. The word median is derived from the Latinword medius, meaning “middle.” Whatmight the value in a set of data be?

3. When you sample a food, you taste a smallportion. What might a be in datacollection?

sample

median

population

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Read

ing

and

Writin

g M

ath

Data and Statistics 461

Reading Strategy: Interpret GraphicsKnowing how to interpret figures, diagrams, charts, and graphs will helpyou gather the information you need to solve the problem.

Look up each exercise in the text and answer the corresponding questions.

1. Lesson 1-6 Exercise 36: What is the title of the graph? How deep is thedeepest trench?

2. Lesson 2-3 Exercises 38 and 39: What does each number in the graphrepresent? What source provided the most energy?

3. Lesson 8-8 Exercise 2: What is the slant height of the cone? What is theradius of the base of the cone?

✔ Read all labels.

Diameter = 5 cmSurface Area = 120π cm2

The height of the cylinder is unknown.

✘ Do not assume anything

The height of the cylinder appears tobe about the same as the diameter,but you can’t know this withoutcalculating the height.

✔ Read the title.

“Loan Options”

✔ Read each axis label.

horizontal Indicates the amounts ofthe loans measured indollars

Vertical Indicates the loan options

✔ Determine what information isrepresented.

The amounts of principal and interestfor two loans are shown.

Try This

Loan Options

A

B

10,000

Initial loan

20,000 40,00030,0000 50,000

Amount of loan ($)

Loan

$14,643.75

$11,005.00$35,500

$35,500

Interest

5 cm ? S � 120π cm2

m807_c09_461_461 1/6/06 4:30 PM 61

Learn to identifysampling methods and to recognize biasedsamples.

Vocabulary

biased sample

voluntary-responsesample

convenience sample

stratified sample

systematic sample

random sample

sample

population

E X A M P L E 1


462 Chapter 9 Data and Statistics

A fitness magazine printed a readers’ survey. Statements 1, 2, and 3 are interpretations of the results. Which do you think themagazine would use?

1. The average American exercises 3 times a week.

2. The average reader of thismagazine exercises 3 times a week.

3. The average reader who respondedto the survey exercises 3 times a week.

The is the entire groupbeing considered for a survey. The

is the part of the populationbeing surveyed.

To get accurate information, it isimportant to use a good samplingmethod. In a , eachmember of the population has an equalchance of being selected. A random sample is best, but other methods can also be used.

Identifying Sampling Methods

Identify the sampling method used.

An exit poll taken of every tenth voter.

Systematic The rule is to question every tenth voter.

Listeners are invited to call in to a radio show to voice theiropinions.

Voluntary-response Callers choose to participate.

random sample

sample

population

People who read fitness magazines are likely to be interested in exercise. This could make the sample biased in favor of people who exercise more times per week.

Sampling Method How Members are Chosen

Random By chance

Systematic According to a rule or formula

Stratified At random from randomly chosen subgroups

Convenience Easiest to reach

Voluntary-response Members choose to be in the sample

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In a statewide survey, five counties are randomly chosen, and100 people are randomly chosen from each county.

Stratified The five counties are the randomsubgroups. People are chosen randomlyfrom within the counties.

Sometimes, these sampling methods result in biased samples. A does not accurately represent the population. The

data collected from biased samples is not reliable.

Identifying Biased Samples

Identify the population and sample. Give a reason the samplecould be biased.

A radio station manager chooses 1500 names from the localphone book to survey people about their listening habits.

An advice columnist asks her readers to write in with theiropinions about how to hang the toilet paper on the roller.

Surveyors in a mall choose shoppers to ask about productpreferences.

biased sample

2E X A M P L E

Think and Discuss

1. Describe ways to eliminate the possible bias in Example 2C.

2. Decide which sampling method would be best to find the numberof times a week the average student in your school exercises.

9-1 Samples and Surveys 463

Population Sample Possible Bias

People in the Up to 1500 people Not all people are in the local area who take the survey phone book.


Readers of the Readers who write Only readers with strong column in opinions write in.


All shoppers in The people who are Not all of the shoppers the mall polled in the mall will be near

the surveyors.

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9-1 ExercisesExercises


1. People whose house number ends with a 1 are polled.

2. Students sitting at the same lunch table are polled.

Identify the population and sample. Give a reason the sample couldbe biased.

3. A pet store owner surveys 100 customers to find out which brand of dogfood is most frequently purchased.


4. A surveyor flips through the phone book and selects 30 names.

5. A newspaper columnist asks readers to write in with their 10 favoriterestaurants.


6. A deli owner asks Sunday’s customers to choose their favorite mustard.

7. A baseball team asks season ticket holders their preference of concessionstands.


8. Every fifth name is called from a list of voters.

9. The customers in a hair salon are surveyed.

10. Each student writes a question on a slip of paper and puts it in a box. The teacher draws one question to discuss.

11. A Web site asks visitors to fill out a survey.

12. Fifteen classes are randomly chosen. Ten students are randomly chosenfrom each class.


13. A teacher asks students who buy lunch if they like the cafeteria food.

14. An architecture firm asks people attending a city council meeting whichdesign for a new city hall they prefer.

15. A biologist studying trees samples blossoms of trees along the river.

KEYWORD: MT7 Parent

KEYWORD: MT7 9-1

GUIDED PRACTICE

PRACTICE AND PROBLEM SOLVING

INDEPENDENT PRACTICE

Extra PracticeSee page 798.

See Example 2

See Example 2

See Example 1

See Example 1


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16. Recreation Marvin looked through the baseball cards he collected 20 years ago. Most of the baseball players began their careers in the 1980s.

a. What is the population of this survey?

b. Give a reason why the sample could be biased.

17. Business For an advertising campaign, Jared needs to survey people tofind out why they like to visit the Cincinnati Zoo.

a. How can he select an unbiased sample for the survey?

b. How can he make the sampling method systematic?

c. Why would surveying only families with children be biased?

18. What’s the Error? Kyla wanted to use a stratified sample to find out themost ordered food product at restaurants. She surveyed every tenthcustomer from five randomly chosen restaurants. Why is this not astratified sample?

19. Write About It To plan your class picnic, you survey students aboutwhere they want to have the picnic. Choose a sampling method. Explainyour choice.

20. Challenge The diagrams show the locations where soil samples will be taken to test for pollution. Identify the sampling method used for each diagram.

21. Multiple Choice Every fifth person standing in line is asked a question.Identify the sampling method used.

Convenience Random Stratified Systematic

22. Short Response At a baseball game, a market researcher randomly asks100 people to name their favorite sport. Identify the population andsample. Give a reason why the sample could be biased.

Use a number line to find each sum. (Lesson 1-4)

23. �6 � 11 24. �31 � (�31) 25. �8 � 14

Multiply. Write the product as one power. (Lesson 4-3)

26. 36 � 37 27. 72 � 74 28. 124 � 145 29. x 3 � x 5

DCBA

Business

The CincinnatiZoo has morethan 500 animaland 3000 plantspecies on display.

9-1 Samples and Surveys 465

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Explore Samples

Use with Lesson 9-1

9-1

REMEMBER

• Be organized before starting.

• Be sure that your sample reflects your population.

You can predict data about a population by collecting data from arepresentative sample.

Your school district has been discussing the possibility of school uniforms.Each school will get to choose its uniform and colors. Your class has beenchosen to make the selection for your school. To be fair, you want the otherstudents in the school to have some input. You conduct a survey to seewhat the majority of students in your school want.

Model the survey by following the steps below.

a. Choose your population.

• every student in the school • only your class • all 8th grade students• all girls • all boys • teachers

b. Choose two different sampling methods. Discuss the pros and cons ofeach method listed.

• random • systematic • stratified • convenience • voluntary response

c. Decide what colors and what uniform choices to present to your sample.

• pants • sweaters • school colors • shorts • jackets• navy blue • skirts • vests • forest green

1. Explain why choosing the teachers as your population might not be the best choice.

2. How did you decide which colors to present to your sample?

1. Create forms for your survey listing the different options. Then survey your sample. Make a table of your results. Explain what your table tells you about the population.

Activity

KEYWORD: MT7 Lab9

1

Think and Discuss

Try This


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An eighth-grade class participated in a month- long fitness challenge. Below are the numbersof miles each student ran, walked, or bikedduring the first week.

7 5 6 5 5 10 9 9 9 3 3 10 1 0 8 6 8 2 3 1 5 0 4 6 4 8 9 3 4 4

Organizing raw data can help you see patternsand trends. One way to organize data is to use aline plot. A uses a number line to show how often a value occurs in a data set.

Organizing Data in Line Plots

Use a line plot to organize the data for the eighth-grade fitnesschallenge.Find the least value, 0, and the greatest value, 10, in the data set. Then draw a number line from 0 to 10. Place an “x” above eachnumber on the number line for each time it appears in the data set

A is a graph used to organizeand display data to compare frequencies. Eachleaf on the plot represents the right-hand digit ina data value. Each stem represents the remainingleft-hand digits.

Reading Stem-and-Leaf Plots

List the data values in the stem-and-leaf plot.0 2 5

1 3 3 7 8

2 0 2 6

3 1 7 Key: 3 � 1 means 31

The data values are 2, 5, 13, 13, 17, 18, 20, 22, 26, 31, and 37.

stem-and-leaf plot

There are 30 numbers in thedata set and 30 x’s above thenumber line.0

xxxx

xxxx

xxx

xxx

xxxx

xxxxx xx

xxx

xx

1 2 3 4 5 6 7 8 9 10

line plot

Learn to organize data in tables and stem-and-leaf plots.

Vocabulary

Venn diagram

back-to-back stem-and-leaf plot

line plot stem-and-leaf plot

E X A M P L E 1

E X A M P L E 2

9-2 Organizing Data 467

9-2

Stem � first digit(s)

2 � 5 � 25

Leaf � last digit

Organizing Data

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A can be used to compare two sets ofdata. The stems are in the center, and the left leaves are read in reverse.

Organizing Data in Back-to-Back Stem-and-Leaf Plots

Use the given data to make a back-to-back stem-and-leaf plot.

are used to show relationships between sets.

Organizing Data in Venn Diagrams

In a survey, the genders and ages of people who completed the questions is shown at right. Make a Venn diagram to show thenumber of people who are female and over age 30.Draw two circles. Label one circle“Female” and the other circle “Over30.” The regionthat overlapsrepresents thecharacteristicsthat are sharedby both sets of data.

Venn diagrams

back-to-back stem-and-leaf plot

3E X A M P L E

4E X A M P L E

Survey Results

Gender Age Gender Age

M 18 M 16

F 30 M 35

F 25 F 28

M 50 F 45

M 60 F 35

F 17 M 38

F 42 F 29

F 30 F 46

M 27 M 32

F 48 F 25

Key: �3 �1 means 31 points 1�2� means 21 points

Losses Wins

01234

0 3 4 71 2 4 4 5 8 9

79 7 7 6

9 6 4 1 1 1

Female Over 30

75

5

3


Super Bowl Scores, 1995–2005

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Winning 49 27 35 31 34 23 34 20 48 32 24

Losing 26 17 21 24 19 16 7 17 21 29 21

Think and Discuss

1. Explain what the overlapping region represents in a Venn diagram.

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1. Use a line plot to organize the data of ages of children visiting a park.

List the data values in the stem-and-leaf plot.

2. 0 2 3 3 7 3. 6 3 6 81 1 3 7 7 8 7 3 3 5 72 0 0 7 8 0 0 1 13 4 4 5 5 Key: 3 � 5 means 35 9 0 4 5 9 Key: 9 � 9 means 99

4. Use the given data to make a back-to-back stem-and-leaf plot.

5. Make a Venn diagramto show how manyeighth-grade femalestudents respondedto a recent survey.

6. Use a line plot to organize the data of the number of books read by students over the summer.

List the data values in the stem-and-leaf plot.

7. 5 0 1 4 8 8. 0 1 5 76 2 6 7 1 2 4 6 87 1 4 5 6 6 2 0 1 7 98 2 Key: 6 � 2 means 62 3 3 3 4 6 Key: 2 � 1 means 21

9. Use the data given in the map to make a back-to-back stem-and-leaf plot.

KEYWORD: MT7 Parent

KEYWORD: MT7 9-2

GUIDED PRACTICE


See Example 1

See Example 1

See Example 2

See Example 3

See Example 3

See Example 4

See Example 2

Republicans

Democrats

32

68

36

64

43

57

44

54

42

56

37

61

38

61

41

58

53

46

54

46

Congress 89th 90th 91st 92nd 93rd 94th 95th 96th 97th 98th

Political Divisions of the U.S. Senate

Survey Results

Grade 6th 5th 8th 6th 7th 8th 7th 8th 6th 5th

Gender M F M F M F M F F M

3 0 3 6 0 1 2 5 5 5 23 0 0 1 6 0 7 8 6 5 6

5 7 8 2 3 3 4 5 8 29 10 9 10 8 8 7 6 6 5

Northeast32°F, 44°F

Northwest27°F, 43°F

Southwest36°F, 47°F South

46°F, 59°F

Southeast48°F, 62°F

West NorthCentral

17°F, 50°F

Central36°F, 51°F

West10°F, 52°F

East NorthCentral

25°F, 40°F

Extreme November Temperatures


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10. Make a Venn diagram to show how many unemployed college graduatesresponded to a recent survey.


12. The ages of 20 middle school students are shown in the line plot. List theages in order from the most frequent to the least frequent.

PRACTICE AND PROBLEM SOLVINGExtra Practice

See page 798.

See Example 4

Miles per Gallon Ratings of a Car Company’s Models

Model A B C D E F G H I J

City Miles 11 17 28 19 18 15 18 22 14 20

Highway Miles 15 24 36 28 26 20 23 25 17 29

8

xxxxx

xxx

xxxxxx

xxxxx

x

9 10 11 12 13 14 15


Survey Results

College yes no yes yes yes yes no yes no yes yes noGraduate

Employed yes yes no no yes no yes yes yes no yes no

Over 40

African American Unmarried

A BDC

Boys Girls

1 5 6 96 8 8 9 7 1 5 5 5

1 1 1 5 9 8 2 2 81 8 9 9 1 2 2 7 9

Use the Venn diagram to answer questions 13–16.

13. What does the portion of the Venn diagram labeled “A” represent?

14. What does the portion of the Venn diagram labeled “B” represent?

15. What does the portion of the Venn diagram labeled “C” represent?

16. What does the portion of the Venn diagram labeled “D” represent?

The stem-and-leaf plot shows the scores for a recent math test.Use it to answer questions 17–19.

17. How many students took the test?

18. What was the highest score received? How many students received the highest score?

19. What was the lowest score received?

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22. Multiple Choice For which set of data would it NOT be appropriate tomake a stem-and-leaf plot?

Scores of a baseball league’s games Prices of fruit at a local market

Average high temperatures Instruments played in the band

23. Short Response Use the data to make a back-to-back stem-and-leaf plot of the ages of people who visited an art exhibit. Men: 32, 45, 61, 33, 41, 61; women: 31, 44, 55, 32, 55, 64

Solve each proportion. (Lesson 5-4)

24. �190� � �1

x5� 25. �w

2� � �1

82� 26. �

61� � �

d3� 27. �4

r� � �

336�

Identify the sampling method used. (Lesson 9-1)

28. Shoppers at a store grand opening place their names in a box. One shopperis chosen at random.

29. Every fourth customer at the grand opening receives a gift certificate.

DB

CA

Language Arts

An author’s writing style is as unique as a fingerprint. Punctuation, spelling, and word usage can be used to determine authorship.

Don Foster used this fact to analyze the 350-year-oldpoem “A Funeral Elegy.” The analysis confirmed that thepoem of previously unknown authorship was actually written by William Shakespeare.

20. Act 5 of Shakespeare’s A Midsummer Night’s Dream has the following references to numbers: 1 nine times, 2 three times, 3 six times, 10 two times, 12 one time, and 14 one time. Use the data to make a line plot.

21. Challenge Select two paragraphs from a work by your favorite author and a third paragraph by a differentauthor. Compare word choices orpunctuation use in the threeparagraphs. Explain the similaritiesand differences. Use a line plot orback-to-back stem-and-leaf plot tosupport your argument.

Don Foster’s methodshave also been usedto analyze ransomnotes and evidencein court cases.


Verse

1 2 3 4 5 6 7 8 9 10 11 12 13 14

, 4 8 6 8 10 12 15 10 7 3 5 5 5 11

— 1 1 3 0 1 2 2 0 0 0 1 1 2 2

! 0 0 1 0 0 1 3 1 0 0 0 0 0 1

. 1 1 1 1 1 1 2 1 1 2 2 3 2 1

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Learn to findappropriate measures of central tendency.

Vocabulary

outlier

range

mode

median

mean

Measures of central tendency are used to describe the middle of adata set. Mean, median, and mode are measures of central tendency.

An is a value that is either far less than or far greater than therest of the values in the data.

Finding Measures of Central Tendency and Range

Find the mean, median, mode, and range of the data set.

9, 6, 91, 5, 7, 6, 8, 8, 7, 9

mean: 9 � 6 � 91 � 5 � 7 � 6 � 8 � 8 � 7 � 9 � 156

�11506

� � 15.6 Divide by 10.

median: 5 6 6 7 7 8 8 9 9 91 Order the values.

5 values 5 values

�7 �

28

� � 7.5 Average the two middle values.

mode: 6, 7, 8, 9 Four values occur twice each.

range: 91 � 5 � 86

outlier

Measures of Central Tendency and Range

Description

Mean To find the mean (average), add the values in the data set. Then divide by the number of values in the set.

Use when the data does not have any outliers.

Median The middle value, or the mean of the two middle values, in anordered set of data.

Use when the data does have outliers.

Mode The value(s) that occur most frequently. A data set may have nomode, one mode, or several modes.

Use when you want to show which value(s) occur most often.

Range The difference between the least and the greatest values in a dataset.

Use when you want to show the spread of the data.

E X A M P L E 1


9-3 Measures of CentralTendency

� �

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Choosing the Best Measure of Central Tendency

Determine and find the most appropriate measure of centraltendency or range for each situation. Justify your answer.

The students in an 8th grade math class received the followingscores on a test: 98, 79, 75, 90, 85, 90, 79, 88, 99, 100, 90, 72, 83, 90,95, 98, 85, 69, 90, 82, 97, and 90. What score occurred most often?

Find the mode.

69, 72, 75, 79, 79, 82, 83, 85, 85, 88, 90, 90, 90, 90, 90, 90, 95, 97, 98, 98, 99, 100

Ninety appears most frequently. The mode is 90.

For the 2004 NFL season, the top four player salaries were$35,037,700, $19,004,000, $16,536,500, and $16,000,000. Whatnumber best describes these salaries?

$35,037,700 is an outlier because it is much greater than the othersalaries. Find the median.

$16,000,000, $16,536,500, $19,004,000, $35,037,700

� 17,770,250

The median of the top four NFL salaries is $17,770,250.

Sometimes you may want to choose a measure of central tendency orrange in order to give a certain message about a group of data.

Business Application

A store had sales of $1025, $974, $993, $1001, $1027, $1657, and$1471 during one week. Which measure of data would make thestore’s sales for the week look the best?Find each measure of central tendency and the range of the data.

mean: � 1164

median: 974, 996, 1001, 1025, 1027, 1471, 1657; 1025

mode: There is no mode.

range: 1471 � 974 � 497

The mean makes the sales for the week appear the greatest.

1025 � 974 � 993 � 1001 � 1027 � 1657 � 1471��7

16,536,500 � 19,004,000��2

Think and Discuss

1. Explain how the range is affected by outliers.

2. Give a data set with the same mean, median, and mode.

3E X A M P L E

9-3 Measures of Central Tendency 473

E X A M P L E 2

List the scores in order.Underline the scores thatappear more than once.

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Find the mean, median, mode, and range of each data set.

1. 35, 21, 34, 44, 36, 42, 29 2. 2.0, 4.4, 6.2, 3.2, 4.4, 6.2

3. 7, 5, 4, 6, 8, 3, 5, 2, 5 4. 23, 13, 45, 56, 72, 44, 89, 92, 67

Determine and find the most appropriate measure of central tendency orrange for each situation.

5. The ages of the people attending the Harris family reunion are shown inthe table. What was the spread in ages?

6. The scores on a math test were 80, 79, 90, 95, 85, 82, 96, 94, 81, 49, 92, and 87. What number best describes these scores?

7. The weekly salaries of the employees of a hair salon are $1025, $975, $823,$750, $1400, $823, $1000, and $823. Which measure of data would makethe salaries seem the highest?

63 60 38 35 59 57 40 38 9 4 3535 15 12 10 30 59 59 3 12 4 31

Find the mean, median, mode, and range of each data set.

8. 5, 2, 12, 7, 13, 9, 8 9. 92, 88, 84, 86, 88

10. 6, 8, 6, 7, 9, 2, 4, 22 11. 4.3, 1.3, 4.5, 8.6, 9, 3, 2.1, 14

Determine and find the most appropriate measure of centraltendency or range for each situation.

12. The ages of the students in a middle school choir are 10, 11, 12, 11, 12, 11,10, 11, 13, 12, 13, 12, 11, 13, 14, 12, 14, 12, 11, and 12. What age appearsmost often in the list?

13. The number of customers each day at a deli was 62, 50, 63, 58, 61, 122,and 70. What number best describes this data?

14. The students in one middle school homeroom received the followingscores on their achievement test: 32, 87, 89, 96, 96, 85, 79, 96, 90, 85, 78,72, 96, 99, 91, 82, 77, 82, and 81. Which measure of data would make thescores appear the highest?

KEYWORD: MT7 Parent

KEYWORD: MT7 9-3

GUIDED PRACTICE


See Example 3

See Example 3

See Example 1

See Example 2

See Example 1

See Example 2


Which measure of central tendency gives the lowest value for each data set?

15. 20, 17, 42, 26, 27, 12, 31 16. 8, 5, 3, 75, 7, 3, 4, 7, 9, 8, 2, 8, 5, 7, 8

17. 3.3, 4.0, 3.3, 5.6, 4.6, 3.3, 5.6 18. 15, 10, 12, 10, 13, 13, 13, 10, 3, 13


See page 798.

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19. Astronomy The table shows the approximate distance each planet isfrom the Sun.

a. Find the range of the data.

b. Which measure of central tendency makes the planets appear to becloser to the Sun?

20. School Teresa has taken three tests worth 100 points each. Her scores are85, 93, and 88. She has one test left to take. What score must she get onher last test to get an average of 90?

21. Write a Problem Use your test scores from one course to write a problem about central tendency.

22. Write About It If six friends went to dinner and split the check equally,what measure of central tendency would describe the amount each person paid? Explain.

23. Challenge If 4��x �

3y � z�� 8, what is the mean of x, y, and z?

Pluto

3661

Mercury

36

Neptune

2796

Mars

141

Jupiter

484

Saturn

887

Earth

93

Venus

67

Uranus

1784

Distance from the Sun

Planet

Miles (million)

9-3 Measures of Central Tendency 475

24. Multiple Choice Which measure of central tendency has the smallestvalue for the data set: 11, 11, 4, 15, 18, 22, 24, 7?

Mean Median Mode They are all equal.

25. Gridded Response Kelly recorded the number of sit-ups she did each dayin the table below. Find the mean number of sit-ups Kelly did per day.

Find each number to the nearest tenth. (Lesson 6-3)

26. What number is 55% of 240? 27. What number is 66�23�% of 847?

Identify the population and sample. Give a reason why the sample could be biased. (Lesson 9-1)

28. In December, a store owner asks every third shopper whether they arebuying items for themselves or as gifts.

29. A market researcher pays a group of shoppers at a mall to fill out aquestionnaire about products they are shown.

DCBA

Mon Tue Wed Thur Fri

34 45 66 75 82

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Learn to find measuresof variability.

Vocabulary

box-and-whisker plot

quartile

variability

E X A M P L E 1


9-4 Variability

The table below summarizes a veterinarian’s records for kitten litters born in a given year.

While central tendency describes the middle of a data set, describes how spread out the data is.

divide a data set into four equal parts. The third quartile minus the first quartile is therange for the middle half of the data.

Kitten DataLower half Upper half

2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6

First quartile: 3 Median: 4 Third quartile: 5median of lower half (second median of upper half

quartile)

Finding Measures of Variability

Find the first and third quartiles for each data set.

85, 92, 78, 88, 90, 88, 89

78 85 88 88 89 90 92 Order the values.

first quartile: 85

third quartile: 90

13, 14, 16, 18, 18, 21, 12, 21, 11, 19, 15, 13

11, 12, 13, 13, 14, 15, 16, 18, 18, 19, 21, 21 Order the values.

first quartile: �13 �

213

� � 13

third quartile: �18 �2

19� � 18.5

Quartiles

variability

Litter Size 2 3 4 5 6

Number of Litters 1 6 8 11 1

The term box-and-whisker plot mayremind you of a box of kittens. But itis a way to display data.

m807_c09_476_480 1/6/06 4:31 PM Page 476

A shows the distribution of data. The middlehalf of the data is represented by a “box” with a vertical line at themedian. The lower fourth and upper fourth are represented by“whiskers” that extend to the smallest and largest values.

Making a Box-and-Whisker Plot

Use the given data to make a box-and-whisker plot.

23 16 51 23 56 22 63 51 22 15 19 42 44 50 38 31 47

Step 1: Order the data and find the smallest value, first quartile,median, third quartile, and largest value.

15, 16, 19, 22, 22, 23, 23, 31, 38, 42, 44, 47, 50, 51, 51, 56, 63

smallest value: 15

first quartile: �22 �

222

� � 22

median: 38

third quartile: �50 �

251

� � 50.5

largest value: 63

Step 2: Draw a number line and plot a point above each value fromStep 1.

Step 3: Draw the box and whiskers.


2E X A M P L E

20 30 40 50 55 6010 25 35 4515

20 30 40 50 55 6010 25 35 4515

2 3 4 5 71 6

of the data isin each “whisker.”

14 of the data is

in each “whisker.”

14

of the data is in the “box,”on each side of the line.

12

14

First quartile

Median

Third quartile

9-4 Variability 477

m807_c09_476_480 1/6/06 4:31 PM Page 477



1. 52, 75, 55, 30, 70, 56, 66 2. 4, 1, 3, 0, 6, 3, 5, 4, 3, 2, 6, 2


3. 32, 47, 42, 33, 23, 59, 29, 19, 34 4. 41, 11, 26, 58, 54, 32, 38, 56, 21

Use the box-and-whisker plots tocompare the data sets.

5. Compare the medians and ranges.

6. Compare the ranges of the middlehalf of the data for each set.

GUIDED PRACTICE

See Example 2

See Example 1

KEYWORD: MT7 Parent

KEYWORD: MT7 9-4

Comparing Data Sets Using Box-and-Whisker Plots

The number of touchdownpasses that Brett Favre and Dan Marino threw during thefirst 14 years of their careers isshown in the box-and-whiskerplots at right.

Compare the medians and ranges.

Brett Favre’s median is greater than Dan Marino’s.

Dan Marino’s range is greater than Brett Favre’s.

Compare the ranges of the middle half of the data for each.

The range of the middle of each data set is the length of the “box”.So, the range of the middle half of the data is greater for Brett Favre.

E X A M P L E 3

Think and Discuss

1. Explain why the data must first be ordered from smallest tolargest before making a box-and-whisker plot.

2. Compare the number of data values in the box with the numberof data values in the whiskers.

20 30 40 50 70100 60

Data set A

Data set B

See Example 3


Brett Favre

Dan Marino

10 15 20 25 3550 30 40 45 50

Dan Marino

m807_c09_476_480 1/6/06 4:31 PM Page 478

11. Compare the medians and ranges.

12. Compare the ranges of the middle half of the data for each set.


See Example 2

See Example 3

See Example 1

20 30 40100

Data set X

Data set Y


13. 88, 78, 85, 74, 66, 82, 68 14. 9, 2, 8, 6, 1, 7, 3, 11

15. 46, 53, 67, 29, 35, 54, 49, 61, 35 16. 3.5, 3.4, 3.7, 3.5, 3.4, 3.3, 3.4, 3.4


17. 87, 79, 95, 99, 67, 71, 83, 91 18. 16, 3, 9.3, 11.3, 14, 7, 7, 4.2, 4.5

19. 0, 2, 5, 2, 1, 3, 5, 2, 4, 3, 5, 4 20. 6.4, 8.0, 6.5, 3.0, 5.4, 2.2, 5.3

21. Earth Science Hurricanes and tropical storms form in all seven oceanbasins. Use a box-and-whisker plot to compare the number of tropicalstorms in every ocean basin per year with the number of hurricanes inevery ocean basin per year.


See page 798.


7. 48, 72, 43, 42, 69, 50, 56, 48, 52 8. 18, 17, 13, 7, 6, 25, 55, 3, 6


9. 50, 68, 85, 54, 80, 75, 68 10. 7, 4, 5.7, 1.4, 6.8, 6.3, 11, 3.2

Use the box-and-whisker plots to compare the data sets.

9-4 Variability 479

Number of Storms Per Year

NW Pacific

Ocean Basin Tropical Storms

26

NE Pacific 17

Atlantic 10

N Indian 5

SW Indian 10

SE Indian 7

Hurricanes

16

9

5

3

4

3

SW Pacific 9 4

m807_c09_476_480 1/6/06 4:31 PM Page 479

22. Match each set of data with a box-and-whisker plot.

a. range: 49 b. range: 15 c. range: 36first quartile: 7 first quartile: 18 first quartile: 12third quartile: 35 third quartile: 25 third quartile: 30

23. Critical Thinking Make a box-and-whisker plot of the following data:18, 16, 21, 10, 15, 25, 13, 22, 25, 13, 15, 10. Add 50 to the list of data andmake a new box-and-whisker plot. How did the addition of an outlieraffect the box-and-whisker plot?

24. What’s the Error? A student wrote that the data set 33, 28, 29, 56, 27,43, 33, 25, 40, 65 has a range of 32. What’s the error?

25. Write About It What do box-and-whisker plots tell you about data thatmeasures of central tendency do not?

26. Challenge What would an exceptionally short box with extremely longwhiskers tell you about a data set?

27. Multiple Choice Find the first quartile for the data set shown in thebox-and-whisker plot.

38 44 46 48

28. Extended Response Ken recorded his golf scores during a three weekperiod. His scores were: 85, 76, 83, 99, 83, 74, 75, 81, and 87. Find therange, median, and first and third quartiles. Make a box-and-whisker plotof the data.

Find the square roots of each number. (Lesson 4-5)

29. 16 30. 81 31. 100 32. 1

Find the mean, median, and mode of each data set to the nearest tenth. (Lesson 9-3)

33. 3, 5, 5, 6, 9, 3, 5, 2, 5 34. 17, 15, 14, 16, 18, 13 35. 100, 75, 48, 75, 48, 63, 45

DCBA

38 40 42 44 46 48 50 52 54


10 200 30 5040

Data set C

Data set A

Data set B

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Create Box-and-WhiskerPlots

The data below are the heights in inches of the 15 girls in Mrs. Lopez’s 8th-grade class.

57, 62, 68, 52, 53, 56, 58, 56, 57, 50, 56, 59, 50, 63, 52

Graph the heights of the 15 girls in Mrs. Lopez’s class on a box-and-whisker plot.

Press Edit to enter the values into List 1 (L1). If necessary,

press the up arrow and then to clear old data. Enter

the data from the class into L1. Press after each value.

Use the STAT PLOT editor to obtain the plot setup menu.

Press . Use the arrow keys and to select On and then the fifth type. Xlist should be L1 and Freq should be 1, as shown. Press 9:ZoomStat.

Use the key and the and keys to see all five summary statistical values (minimum: MinX, first quartile: Q1,median: MED, third quartile: Q3, and maximum: MaxX). Theminimum value in the data set is 50 in., the first quartile is 52 in., the median is 56 in., the third quartile is 59 in., and themaximum is 68 in.

1. Explain how the box-and-whisker plot gives information that ishard to see by just looking at the numbers.

1. The data below shows the number of hours slept one night for ach of the 11 boys from Mrs. Lopez’s 8th-grade class.

7.5, 6.5, 5, 6, 8, 7.25, 6.5, 7, 7, 8, 6.75

Make a box-and-whisker plot of this data. What are the minimum, first quartile, median, third quartile, and maximum values of the data set?

TRACE

ZOOM

ENTERENTERY=STAT PLOT

2nd

ENTER

ENTERCLEAR

STAT

Activity

Think and Discuss

Try This

Use with Lesson 9-4

9-4

KEYWORD: MT7 Lab9

9-4 Technology Lab 481

m807_c09_481_481 1/6/06 4:31 PM Page 481

Quiz for Lessons 9-1 Through 9-4



1. Every third student is called from a class roster.

2. Fifty customers are chosen by chance from five random grocery stores.

Identify the population and sample. Give a reason the sample could be biased.

3. A restaurant owner asks Friday’s customers to choose a favorite salsa.

4. A DVD rental manager asks people who rent dramas what their favorite movie is.

9-2 Organizing Data

5. Use a line plot to organize the data of the ages of people playing bridge.


7. Make a Venn diagram to show how many 2-year-old, male cats were adopted.

Rea

dy

to G

o O

n?


72 78 76 75 79 70 74 80 72 78

71 69 70 72 68 70 69 75 75 74

Greatest Number of Home Runs by a Player, 2000–2004

2000 2001 2002 2003 2004

American League 47 52 57 47 43

National League 50 73 49 47 48

Gender F M M F M M M F F M

Age (y) 1 1 2 3 1 1 2 3 2 2

9-3 Measures of Central Tendency

Determine and find the most appropriate measure of central tendency orrange for each situation.

8. The finishing times in minutes for a 5-kilometer run by a group of friends were 21.1, 20.6, 19.7, 20.3, 17.7, and 22.6. What was the spread of the times?

9. The week’s average high temperatures in degrees Fahrenheit were 70, 72, 72, 74, 76, 75, and 74. What number best describes the temperatures?

9-4 Variability


10. 43, 36, 25, 22, 34, 40, 18, 32, 43 11. 21, 51, 36, 38, 45, 52, 28, 16, 41

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On Monday, 20 students took an exam. Therewere 10 students who scored above 85 and10 students who scored below 85. What wasthe average score?

The average elevation in California is about2900 ft above sea level. The highest point,Mt. Whitney, has an elevation of 14,494 ftabove sea level. The lowest point, DeathValley, has an elevation of 282 ft below sealevel. What is the range of elevations inCalifornia?

Use the table to find the median number ofmarriages per year in the United States forthe years between 1940 and 2000.

Aishya is cross-training for a marathon. Sheran for 50 minutes on Monday, 70 minuteson Wednesday, and 45 minutes on Friday. OnTuesday and Thursday, she lifted weights atthe gym for 45 minutes each day. She swamfor 45 minutes over the weekend. What wasthe average amount of time per day Aishyaspent running last week?

Read the problems below and decide if there is too much or toolittle information in each problem. If there is too much information,tell what information you would use to solve the problem. If there istoo little information, tell what additional information you wouldneed to solve the problem.

Make a Plan• Identify too much/too little information

When you read a problem, you must decide if the problem has toomuch or too little information. If the problem has too muchinformation, you must decide what information to use to solve theproblem. If the problem has too little information, then you shoulddetermine what additional information you need to solve theproblem.

Focus on Problem Solving 483

1 4

2

3

Number of Marriages in the United States

Year 1940 1950 1960 19701980 1990

Source: National Center for Health Statistics

Number of Marriages in the United States

Year 1940 1950 1960 1970 1980 1990 2000

Number (thousands) 1596 1667 1523 2159 2390 2443 2329

m807_c09_483_483 1/6/06 4:31 PM Page 483

Make a Circle Graph

Use with Lesson 9-5

9-5

Activity

KEYWORD: MT7 Lab9WHAT YOU NEED:

• Compass • Protractor• Ruler • Paper

REMEMBER

• A circle measures 360°.

• Percent compares a number to 100.

Illegal54%

Legal(no restrictions)

12%

Legal withpermit24%

Otherconditions

6%

Legal insome areas

4%Think and Discuss

Try This


Skunks are legal pets in some states but not in most. Use the information from the table to make acircle graph showing the percents for each category.

a. Use a compass to draw a large circle. Use a ruler to draw a vertical radius.

b. Extend the table to show the percent of states with each category of legality.

c. Use the percents to determine the angle measure of each sector of the graph.

d. Use a protractor to draw each angle clockwise from the radius.

e. Label the graph and each sector. Color thesectors.

1. How many states would need to legalize skunks for the largest sector to be 180°?

1. Make a circle graph to show only the states where skunks are not illegal.

Skunks as Pets by State

Legality Number of States

Legal (no restrictions) 6

Legal with permit 12

Legal in some areas 2

Illegal 27

Other conditions 3

Number Percent ofLegality of States States Angle of Section

Legal (no restrictions) 6 �560� � 12% �1

1020� � 360 � 43.2°

Legal with permit 12 �1520� � 24% �1

2040� � 360 � 86.4°

Legal in some areas 2 �520� � 4% �1

400� � 360 � 14.4°

Illegal 27 �2570� � 54% �1

5040� � 360 � 194.4°

Other conditions 3 �530� � 6% �1

600� � 360 � 21.6°

m807_c09_484_484 1/6/06 4:32 PM Page 484

In 1990, the United Statesqualified for the soccerWorld Cup for the first time in40 years. Since then, popularityin youth soccer in the United States has grown tremendously.

A is used todisplay and compare two sets of data. You can organize data using a by listing items according to the number of times that the items occur.

Displaying Data in a Double-Bar Graph

Make a double-bar graph.

The following are the ages when a randomly chosen soccer group of20 boys and 20 girls began playing in a local youth soccer league.

The frequencies are the heights of the bars in the bar graph. Use a different color torepresent each gender.

A is a bar graph that shows the frequency of data withinequal intervals.

histogram

frequency table

double-bar graph

Learn to display data in bar graphs, histograms, and line graphs.

Vocabulary

double-line graph

histogram

frequency table

double-bar graph

Age 4 5 6 7 8

Boys 5 10 3 2 0

Girls 2 11 5 1 1

E X A M P L E 1

12

10

8

6

4

2

0

Freq

uenc

y

Age (yr)

Starting Ages for Youth Soccer League

4 5 6 7 8

Boys Girls

9-5 Displaying Data 485

9-5 Displaying Data

m807_c09_485_488 1/6/06 4:32 PM Page 485

Displaying Data in a Histogram

Edwina asked 10 classmates how many minutes of sleep they had the previous night. Use the data to make a histogram.460 400 425 440 490 365 435 500 380 505

Make a frequency table with 30-minute intervals. Then make ahistogram.

A is used to show how two related sets of datachange over time.

Displaying Data in a Line Graph

Make a double-line graphof the given data. Use thegraph to estimate thenumber of measles casesand whooping coughcases in 2002.Plot the data. The graphshows about 200,000whooping cough casesand about 650,000measles cases in 2002.

double-line graph

Think and Discuss

1. Describe the difference between a bar graph and a histogram.

2E X A M P L E

3E X A M P L E

Minutes Frequency

360–389 2

390–419 1

420–449 3

450–479 1

480–509 3

Histograms do nothave spaces betweenthe bars.

4

3

2

1

0

Freq

uenc

y

Time (min)

Amount Slept

360–389 390–419 420–449 450–479 480–509


Number of Worldwide Cases

WhoopingYear Measles Cough

1992 1,481,971 255,4751996 870,989 141,4452000 836,407 186,1982004 504,742 235,740

Measles and Whooping CoughCases Worldwide

1500

1150

800

450

1001992 1996 2000 2004

Num

ber

of c

ases

(tho

usan

ds)

YearMeasles Whooping cough

m807_c09_485_488 1/18/06 3:58 PM Page 486

San Francisco Year Philadelphia

715,674 1970 1,948,609 678,974 1980 1,688,210 723,959 1990 1,585,577 776,733 2000 1,517,550

City Population


1. Make a double-bar graph.Data Set 1: 11 10 13 11 12 13 13 9 10 11 12 10Data Set 2: 13 11 12 12 11 9 10 11 12 10 9 11

2. The Freshman National Merit Scholars and their schools are listed for2004. Use the data to make a histogram with intervals of 50.

Vanderbilt, 144; Princeton, 192; Duke, 90; Stanford, 217; Yale, 224;Northwestern, 152; Rice University, 173; Cal Tech, 51; University ofChicago, 198; M.I.T., 134; University of Texas-Austin, 242; WashingtonUniversity, 197

3. Make a double-line graph of the given data. Use thegraph to estimate the lifeexpectancies of a male and a female born in 1997.

4. Make a double-bar graph.

5. Restaurants sometimes organize their menus by the number of items ineach price range. Use the entrée prices to make a histogram with intervalsof $10.

$9 $11 $22 $22 $30 $24 $13 $16 $17 $21 $18 $25 $17 $25$17 $21 $19 $21 $14 $19 $15 $15 $10 $16 $12 $21 $19 $17

6. Make a double-line graph of the given data. Use the graph to estimate thepopulations of Philadelphia and San Francisco in 1995.

KEYWORD: MT7 Parent

KEYWORD: MT7 9-5

GUIDED PRACTICE


See Example 2

See Example 3

See Example 1

See Example 1

See Example 2

See Example 3

Life Expectancy by Birth Year (U.S.)

Year 1980 1985 1990 1995 2000

Age Male 70.0 71.1 71.8 72.5 74.3

Age Female 77.4 78.2 78.8 78.9 79.7Source: National Center for Health Statistics

9-5 Displaying Data 487

Temperature °F 10 15 20 25 30 35 40

Data Set 1 2 6 9 7 4 2 1

Data Set 2 5 7 8 5 4 2 0

m807_c09_485_488 1/6/06 4:32 PM Page 487

7. Organize the data into a frequency table and make a double-bar graph.Data set 1: 1 6 3 1 4 6 4 5 6 1 2 5 5 4 2 3 1 6 2 2 Data set 2: 3 1 3 4 2 1 5 6 1 2 6 5 1 6 4 3 3 2 1 5

8. Make a histogram of honey yield per colony with intervals of 4.

9. Write a Problem You are given the heights of the players on a soccerteam. Determine the number of four different-size jerseys to order. Writea problem using a histogram that would help you find this information.

10. Write About It Which kind of graph would you use to compare theaverage salaries of professional basketballplayers and professional hockey playersfrom 1995 to 2005?

11. Challenge Determine the size of theinterval used in the histogram using the data below.

Time needed to heat a frozen dinner in the microwave (min)4:30 5:30 7:00 4:45 5:20 8:003:45 2:30 6:40 6:00 4:30 5:25


See page 799.

Yield per Colony (pounds) 76.3

1999

83.9

2000

74.0

2001

67.8

2002

70.0

2003

71.8

2004

Honey-Producing Colonies

Year

Source: USDA


7

6

5

4

3

2

1

0Fr

eque

ncy

Time (min)

Time Needed to Heat Dinner

? ? ? ? ?

12. Multiple Choice Which data display is most appropriate to show thechange in sales over a four-month period?

Bar graph Frequency table Line graph Stem-and-leaf plot

13. Short Response A middle school principal asked 12 students the number offootball games each attended during the school year. The results were 1, 3, 5, 2,4, 4, 4, 3, 5, 2, 4, 2. Organize the data into a frequency table.

Find the sum of the angle measures in each polygon. (Lesson 7-4)

14. 18-gon 15. 24-gon 16. heptagon

Find the range of each set of data. (Lesson 9-4)

17. 16, 32, 1, 54, 30, 28 18. 105, 969, 350, 87, 410 19. 0.2, 0.8, 0.65, 0.7, 1.6, 1.1

DCBA

m807_c09_485_488 1/20/06 4:48 PM Page 488

Create Histograms

You can use a graphing calculator to make a histogram.

The frequency table shows the length of the feet of students inMrs. Alvarez’s math class. Use a graphing calculator to make ahistogram of the data.

To enter the data, press .

Then press to select 1:Edit.

Under L1, enter 1, 2, 3, 4, 5, 6, and7 to represent the seven intervals.Interval 1 corresponds to “Lessthan 5,” while interval 7corresponds to “10 or greater.”

In L2, enter the number ofstudents for each interval.

To see a histogram of the data,

press to select“STAT PLOTS 1:”

Scroll and press to select“On” and the histogram icon.

Then scroll to “Freq:” and press 2 to paste the data from L2.

Press 9 to view the histogram. Press and the arrow keys toread the histogram.

1. How would the histogram change if the first interval is left out? Drawthe histogram.

2. Explain how you can use the histogram to find the total number ofstudents who have feet that are at least 7 inches long.

1. Measure the lengths of the right arms of everyone in your classroom.Divide the data into 5 equal intervals. Use a graphing calculator tomake a histogram of the data.

TRACEZOOM

2nd

ENTER

ENTERY=2nd

ENTER

STAT

Activity

Think and Discuss

Try This

Use with Lesson 9-5

9-5

KEYWORD: MT7 Lab9

Number ofFoot Length (in.) Students

Less than 5 0

5 to less than 6 1

6 to less than 7 4

7 to less than 8 11

8 to less than 9 7

9 to less than 10 4

10 or greater 1


m807_c09_489_489 1/6/06 4:32 PM Page 489

Learn to recognizemisleading graphs and statistics.

Graphs and statistics are often usedto make advertisements visuallyappealing. Some advertisements,however, use art to misleadconsumers.

Identifying MisleadingGraphs

Explain why each graph ismisleading.

The heights of the balls are used to represent the ticket prices.However, the areas of the circles and volumes of the balls distortthe comparison. The basketball prices are only about 2�

12� times

greater than the baseball prices, but they look like much more.

Since the horizontal scale does not start at 0, the bar for Brand 1appears to be four times as long as the bar for Brand 3. In fact, the capacity of Brand 1 is only 20% more than Brand 3.

E X A M P L E 1

Ticket Prices

$60

$50

$40

$30

$20

$10

$01991 20011991 2001

Baseball game Basketball game

Average Life Capacity of AA Batteries

Brand 3

Brand 2

Brand 1

1500 1600 1700 18001400 1900

Milliampere-hours


9-6 Misleading Graphs and Statistics

m807_c09_490_493 1/6/06 4:41 PM Page 490

Explain why the graph is misleading.

Different-sized icons represent the same number of vehicles. Thenumber of light trucks looks like it is close to the number of cars,but it is really less than half. The number of heavy trucks is lessthan 5% of the total, but it appears much greater.

Identifying Misleading Statistics

Explain why each statistic is misleading.

A housing development features 5 home models with thestarting prices of $475,000, $500,000, $225,000, $480,000, and$510,000. The developer places an ad that reads:

“New homes—average price $438,000”

Although $438,000 is the average price, only one model sellsbelow that price. It is likely that a new home owner will pay morethan the advertised price of $438,000.

A movie previews for 12 selected viewers. Eight viewers rate themovie highly. The producer tells the production studio:

“The movie will be a hit because test audiences rate the moviefavorably at a rate of 2 to 1.”

The sample size is too small. Twice as many people liked the movie,but the difference between 4 and 8 people is not meaningful.

The revenue for Ski Resort A for November and December was$6,600,000. The revenue for Ski Resort B for January andFebruary was $8,300,000.

The revenues are measured at different times of the year. Weatherconditions can change dramatically from month to month,affecting revenue.

Think and Discuss

1. Give an example of a graph that starts at zero but is stillmisleading.

2. Explain how a statistic can be accurate but still misleading.

Registered Vehicles

= 9 million cars

= 9 million light trucks

= 9 million heavy trucks

2E X A M P L E

9-6 Misleading Graphs and Statistics 491

m807_c09_490_493 1/6/06 4:41 PM Page 491


Explain why each graph is misleading.

1. 2.


3. A stalk of broccoli has 477 mg of potassium. A large carrot has 230 mg ofpotassium. A small head of cauliflower has 803 mg of potassium.

4. The total number of life jackets sold by Water Sports World from April 1 toSeptember 1 was 619. The total number of life jackets sold by Boats andMore from July 1 to September 1 was 153.


5. 6.


7. A survey of 1000 college students found that 110 majored in engineeringand 112 majored in the social sciences. A magazine article reports thatstudents prefer the social sciences over engineering.

8. A reporter asked 90 students if they participate in organized athletics. Ofthe 50 who responded “yes,” 26 played on school teams, 14 played incommunity leagues, and 10 competed in individual competitions. Thereporter said, “Half of all students play on school teams.”

KEYWORD: MT7 Parent

KEYWORD: MT7 9-6

GUIDED PRACTICE


See Example 2

See Example 2

See Example 1

See Example 1

Heights of Students

20

16

12

8

4

Num

ber

of s

tude

nts

5'0" 5'3" 5'8" 5'10" 6'1"

Height

Food Donated

= 100 cans = 50 boxes = 20 cases

Num

ber

of h

ouse

s so

ld

Comparison of Houses Sold

Agent A Agent B Agent C Agent D

Average Female Executive Salaries14013012011010090807060

2001 2002 20032004 2005

Sala

ries

($10

00)

Year


m807_c09_490_493 1/6/06 4:41 PM Page 492

13. Multiple Choice Six dentists are surveyed regarding toothpaste. Fourdentists recommended Brand X. An ad for Brand X states: “Recommendedby 2 out of 3 dentists.” Explain why the statement is misleading.

The sample was too large.

The sample was too small.

The sample should have included construction workers.

The statement should say “Recommended by 1 out of 2 dentists.”

14. Short Response A salesman earns the following commissions: December $965;January $125; February $170; March $100; April $110; May $120. He tells his friends thathe averages $265 per month in commission. Explain why the statistic is misleading.

Find the area of each figure with the given dimensions. (Lesson 8-2)

15. trapezoid: b1 � 3, b2 � 5, h � 8 16. triangle: b � 16, h � 9

17. People responding to a survey had the following ages: 30, 21, 20, 26, 23, 30, 23, 23, 21,20, 27, 20, 24, 23, and 30. Use the data to make a line plot. (Lesson 9-2)

D

C

B

A


9. 10.

11. Write About It When might you want to use a scale on a graph thatdoes not start at 0?

12. Challenge The two graphs show the recent performance of twocompanies’ stocks, A and B. Which graph should be shown to thestockholders of company A?


See page 799.Sailboat Sales

25

20

15

10

5N

umbe

r so

ldYawls Sloops Catboats

BrandC

150

0

50

100

Calories per Serving

BrandB

BrandA

9-6 Misleading Graphs and Statistics 493

Graph 160

40

20

0

Pric

e ($

)

TimeStock A Stock B

Graph 240

36

32

28Pr

ice

($)

TimeStock A Stock B

m807_c09_490_493 1/6/06 4:41 PM Page 493

Learn to create andinterpret scatter plots.

Vocabulary

line of best fit

correlation

scatter plot

Many health care professionals are concerned about the increase in the number of overweight children. Children are strongly encouraged to be more active.

A is a graph with pointsplotted to show a relationshipbetween two sets of data.

Making a Scatter Plot of a Data Set

A teacher surveyed her students about the amount of physicalactivity they get each week. She then had their body mass index(BMI) measured. Use her data to make a scatter plot.

describes the relationship between two data sets. A is a straight line that comes closest to the points on ascatter plot. One way to estimate a line of best fit is to lay a ruler’s edgeover the graph and adjust it until it looks closest to all the points.

line of best fitCorrelation

scatter plot

E X A M P L E 1

Active ActiveHours per Hours per

Student Week BMI Student Week BMI

A 10 16 F 8 18

B 3 25 G 7 21

C 6 24 H 2 28

D 8 20 I 19 9

E 10 16 J 14 12

The points on the scatter plotare (10, 16), (3, 25), (6, 24), (8, 20), (10, 16), (8, 18), (7, 21),(2, 28), (19, 9), and (14, 12). The 2 at (10, 16) indicates thatthe point occurs twice.

Hours

BMI

30

25

20

15

10

5

00 8 10 124 16 20

2


9-7 Scatter Plots

m807_c09_494_497 1/6/06 4:32 PM Page 494

Identifying the Correlation of Data

Do the data sets have a positive, a negative, or no correlation?

The number of hours a plane is in flight and the number of miles flown

Positive correlation: The longer a plane is in flight, the more miles it flies.

The number of hours in flight and the number of passengers

No correlation: The number of hours in flight does not affect thenumber of passengers on the plane.

The number of hours in flight and the gallons of fuel remaining

Negative correlation: The longer a plane is in flight, the less fuel it has.

Using a Scatter Plot to Make Predictions

Use the data to predict the exam grade for a student who studies 10 hours per week.

According to the graph, a student who studies 10 hours per week shouldearn a score of about 95.

Think and Discuss

1. Compare a scatter plot to a line graph.

2. Give an example of each type of correlation.

strong weak14444444244444443

Positive correlation: both data sets increase together.

weak strong144444442444444443

Negative correlation: as one data set increases, the other decreases.

No correlation: changes in one data set do not

affect the other data set.

2E X A M P L E

A strong correlationdoes not mean thereis a cause-and-effectrelationship. Forexample, your ageand the price of aregular movie ticketare both increasing,so they are positivelycorrelated.

E X A M P L E 3

Hours Studied 5 9 3 12 1

Exam Grade 80 95 75 98 70

Hours

Gra

de

50

70

90

10 122 4 6 8

9-7 Scatter Plots 495

m807_c09_494_497 1/6/06 4:32 PM Page 495

9-7 ExercisesExercisesKEYWORD: MT7 Parent

KEYWORD: MT7 9-7


1. Use the given data to make a scatter plot.


2. The square footage of a house in a given neighborhood and its price

3. The age of a house and the number of people living in the house

4. Use the data to predict the wind chill at 35 mi/h.

GUIDED PRACTICE

See Example 2

See Example 2

See Example 3

See Example 3

See Example 1

See Example 1

Honduras

Nicaragua

Panama

Guatemala

El Salvador

Costa Rica

Country Area (mi2) Population

Guatemala 42,467 12,335,580

Honduras 43,715 5,997,327

El Salvador 8,206 5,839,079

Nicaragua 50,503 4,717,132

Costa Rica 19,929 3,674,490

Panama 30,498 2,778,526

Apparent Temperature Due to Wind at 15°F

Wind speed (mi/h) 10 20 30 40 50 60

Wind Chill (°F) 2.7 �2.3 �5.5 �7.9 �9.8 �11.4



6. The number of weeks a CD has been out and weekly sales

7. The number of weeks a CD has been out and total sales

8. Use the data to predict the apparent temperature at 70% humidity.


Car Brand Cost ($1000) Fuel Economy (mi/gal)

A 25 19

B 19 31

C 34 15

D 28 23

E 22 33

Temperature Due to Humidity at a Room Temperature of 72°F

Humidity (%) 0 20 40 60 80 100

Apparent Temperature (°F) 64 67 70 72 74 76

m807_c09_494_497 1/6/06 4:32 PM Page 496

14. Multiple Choice Does the size of a box of cereal and the price of the cerealhave a positive, negative, or no correlation?

Positive Negative Scatterplot No

15. Short Response What type of correlation exists between a person’s birthdayand his or her height? Explain.

Determine the number of lines of symmetry for each polygon. (Lesson 7-8)

16. square 17. equilateral triangle 18. regular pentagon

19. A bookstore sells 2 copies of Sail Away and 4 copies of Race Car Mania. Thebookstore owner concludes that his customers are twice as likely to buy racingbooks than sailing books. Identify why this statistic is misleading. (Lesson 9-6)

DCBA

About 50 million Americans suffer from allergies. Airborne pollen generated by trees, grasses, plants, and weeds is a major cause of illness and disability. Because pollen grains are small and light, they can travel through the air for hundreds of miles. Pollen levels are measured in grains per cubic meter.

9. Use the given data to make a scatterplot. Describe the correlation.

10. Explain how the pollens are compared in the chart at right.

Use the chart at right to determine ifthe pollens have a positive, a negative,or no correlation.

11. mountain cedar, grass 12. fall elm, ragweed

13. Challenge Use the allergy chart to explain the difference between correlation and a cause-and-effect relationship.

Life Science

Pollen Levels

Day

1234567

Grass Pollen

1619329312

Weed Pollen

35051493094883065

Some common substances that cause allergiesinclude pollens, dust mites, and mold spores.

9-7 Scatter Plots 497

Source: Central Texas Allergy and Asthma Center

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Grasses

Mold spores

Fall elmOak

Pecan

Sycamore

Mulberry

Ragweed

Mountaincedar

Mountaincedar

Annual Allergy Chart

KEYWORD: MT7 Pollen

m807_c09_494_497 1/6/06 4:32 PM Page 497

Create a Scatter Plot

Use with Lesson 9-7

9-7

KEYWORD: MT7 Lab9You can use a graphing calculator to make a scatter plot.

The table shows heights and weights of students in Mr. Devany’s class. Use a graphing calculator to create a scatter plot of the data.

To enter the data, press to select “1:Edit”

In L1, enter the heights. In L2, enter the weights.

To see a scatter plot of the data, press to select “STAT PLOTS 1:”

Scroll and press to select “On” and the scatter plot icon. Scroll to “Xlist=” and press 1.

Scroll to “Ylist=” and press 2. Finally, scroll to “Mark:” and choose the box.

To view the scatter plot, press 9. Press and the arrow keys to read the histogram.

1. Explain what happens when you change the window to [0, 100] by [0, 500].

2. Suppose you added a third category: boy or girl. How could the height, weight, and gender data be displayed?

Use a graphing calculator to create a scatter plot of the data.

1.

TRACE

ZOOM

2nd

2nd

ENTER

ENTERY=2nd

ENTERSTAT

Activity 1

Think and Discuss

Try This

x 41 43 46 50 51 55 60 62 66 69 70

y 92 111 105 120 110 107 125 142 152 175 210

Height Weight (in.) (lb)

41 92

43 111

46 105

50 120

51 110

55 107

60 125

62 125

62 125

66 152

69 175

70 210


m807_c09_498_499 1/20/06 4:49 PM Page 498

You can use a graphing calculator to find a line of best fit on a scatter plot.

Create a scatter plot of the data shown. Use a line of best fit to predictthe value of y when x � 11.

Follow the steps in Activity 1 to make a scatter plot of the data.

To find a line of best fit, press 4 to choose “LinReg(ax+b).”

Press 1 2 .

Your calculator will display the y-intercept and slope of the line of best fit.

Press 9 to see the scatter plot and the line of best fit.

You can press to see an equation of the line.

To find y when x � 11, press 11 . The screen

shows that y � 9.68 when x � 11.

1. What uses might a line of best fit have in the real world?

2. What type of correlation does the data have in Activity 2? How doyou know?

Collect at least 6 pieces of string with different lengths. Measure the length of each piece ofstring and record the values in L1. Form a square with each piece of string. Measure thelength of one side of each square and record the values in L2. Make a scatter plot of the dataand find the line of best fit.

1. What should the slope be? (Hint: What is the relationship between the perimeter of asquare and the length of one side?)

2. What is the slope of the line of best fit you found on your calculator?

3. Explain why the slope of your line might not match the slope you predicted.

ENTERENTERTRACE2nd

Y=

ZOOM

ENTERENTERENTERVARS,2nd,2nd

STAT

Think and Discuss

Try This

Activity 2 x y

2 26.1

4 21.5

6 17.4

8 13.2

10 11.7

12 8.5

14 4.2

16 1.9


m807_c09_498_499 1/20/06 4:50 PM Page 499

Learn to select the bestrepresentation for a setof data.

Selecting a Data Display

Which graph is a better display of the data on studentsvolunteering?

In a survey, students were asked, “About how many hours a year do you volunteer?” The responses are shown in the table.

Data can be represented in several different ways, depending both onthe type of data and the message to be conveyed.

Hours Spent Volunteering

Fewer than 20 15%

20–39 35%

40–59 13%

60–79 7%

80 or more 30%

E X A M P L E 1


9-8 Choosing the BestRepresentation of Data

Type of Graph Common Use

Line graph Shows change in data over time.

Bar graph Shows relationships or comparisons between groups.

Circle graph Compares parts to a whole.

Histogram Shows the frequency of data divided into equal groups.

Box-and-whisker plot Shows the distribution and spread of data.

Line plot Shows the distribution of data.

Scatter plot Shows the relationship of two data sets.

Yearly Student Volunteer Work

Less than20 hours,

15%

More than80 hours,

30%

60–80 hours, 7%

40–59 hours,13%

20–39 hours, 35%

40

30

20

10

0

Perc

ent

of s

tude

nts

Hours

Yearly Student Volunteer Work

Less

than

2020

–39

40–5

960

–80

More th

an

80

The data shows how groups of people who responded to the surveycompare to the whole. The circle graph is the better representation.

m807_c09_500_503 1/6/06 4:33 PM Page 500

Which graph shows the distribution of test scores better?

Since the question asks aboutthe distribution of the data,the box-and-whisker plot isthe better representation.

PROBLEM SOLVING APPLICATION

José spent a week camping and hiking. The data of each hike isrecorded in the table. Choose an appropriate data display. Draw thegraph. About how long would it have taken José to hike 12 km?

Understand the Problem

You are looking for the best data display and the estimated time for a 12 km hike.

Make a Plan

You need to find the relationship between time and distance. Sincethe data can be written as ordered pairs, plot them in a scatter plot.

SolvePlot the data points on the scatter plot. Toestimate the time needed for a 12 km hike,draw the line of best fit. Then find t whend � 12. The line of best fit indicates that a12 km hike would take about 5 hours.

Look Back

Look at the table. An 11.2 km hike took5.5 h and a 13.7 km hike took 6 h, so 5 hfor a 12 km hike is reasonable.

2E X A M P L E

Time (h) 1 1.5 2 3 4.5 6 7

Distance (km) 3.2 4.8 8 10.5 11.2 13.7 15.6

1

2

3

4

Test Scores100

90

80

70

60

50

02 4 6 8 10 12 14 16 18 20

Scor

e

Student number

50 60 70 80

Test Scores

90 100

9-8 Choosing the Best Representation of Data 501

16141210

86420

0 1284 1062

Dis

tanc

e (k

m)

Time (h)

Think and Discuss

1. Describe the kind of data that is best represented by a bar graph.

2. Give a situation in which you would use a line graph to display data.

m807_c09_500_503 1/6/06 4:33 PM Page 501

1. Which graph is a better display of the numbers of students participatingin high school sports?

2. The highest elevations for several states arelisted in the table. Choose an appropriatedata display and draw the graph. Which ofthe states shown in the graph has the thirdhighest elevation?

3. Which graph is a better display of the percent of times a coin comes up heads and tails in 80 tosses?

4. Ann spent the day shopping and runningerrands. The table shows her distance fromhome at various times during the day.Choose an appropriate data display anddraw the graph. Approximately how far was Ann from home at 11:00 AM?

KEYWORD: MT7 Parent

KEYWORD: MT7 9-8

GUIDED PRACTICE

PRACTICE AND PROBLEM SOLVING


Extra PracticeSee page 799.

See Example 2

See Example 2

See Example 1

See Example 1

State Highest Elevation

Alaska 6194 m

California 4421 m

Colorado 4399 m

Washington 4392 m

Time Distance (mi)

8:00 AM 0

10:00 AM 5

12:00 noon 12

2:00 PM 8

4:00 PM 3

TailsHeads

50403020100

Heads Tails

40

30

20

10

0

Stud

ents

Sport

Sports Participation

Baseba

ll

Basketb

all

Footb

all

Socce

r

Softb

allTra

ck


40

30

20

10

0

Stud

ents

Sport

Sports Participation

Baseba

ll

Basketb

all

Footb

all

Socce

r

Softb

allTra

ck


Choose the best data display for each situation. Explain.

5. height of a child over time 6. class sizes at a middle school

7. amount of time spent on different 8. comparison of people’s shoe sizestasks during a day to their ages

m807_c09_500_503 1/6/06 4:33 PM Page 502

9. Fitness A survey of exercise habits was conducted.The ages of respondents and the number of minutesthey reported exercising weekly are shown. Chooseand construct a better display for the data.

10. Sports What kind of graph would best show the increase in the number of participants in the Special Olympics World Summer Games since it was founded in 1968?

11. Write a Problem Write a survey question for which a circle graph wouldbest represent the data. Then collect the data and make the circle graph.

12. Write About It Explain how you would decide if a line graph or a scatterplot were a better representation of data.

13. Challenge An appliance store sells four brands oftelevisions. The table shows how many of each brandwere sold last month. Which two kinds of graphs could be used to display this data? What message would eachkind of graph give about the data?

14. Multiple Choice What type of display would you least likely constructfrom data of test scores for a class?

circle graph line graph histogram bar graph

15. Short Answer Find the mean, median, mode, and range of the data in the stem-and-leaf plot. If any of the measures cannot be found, give the reason.

Find the area of each circle. Round to the nearest tenth, if necessary.Use 3.14 for π. (Lesson 8-3)

16. circle with diameter 10 cm 17. circle with radius 5.2 yd

18. A 9 cm cube is built from 1 cm cubes. Compare the ratio of the length of anedge of the large cube to the length of an edge of a small cube. (Lesson 8-10)

DCBA

Sports

In 2003, 7000athletes from 150 countriescompeted in theSpecial OlympicsWorld SummerGames.

Age Time (min)

13 120

17 120

18 100

19 90

22 150

28 135

32 100

35 180

40 160

200

150

100

50

013 18 19 22 28 32 35 4017

Tim

e (m

in)

Age (yr)

9-8 Choosing the Best Representation of Data 503

0 1 4 9

1 3 3 4 7

2 1 2 2 2 3 3

NumberBrand Sold

A 120

B 130

C 100

D 95

m807_c09_500_503 1/18/06 3:59 PM Page 503

Use a Spreadsheet to Create Graphs

Use with Lesson 9-8

9-8

KEYWORD: MT7 Lab9You can use a spreadsheet to make circle graphs, line graphs, and bargraphs. A spreadsheet allows you to model different situations easily.

Suppose a farmer has 22 pigs, 2 milk cows, 4 goats, 3 sheep, and 6 chickens. You can use aspreadsheet to make a circle graph of the data.

In row 2, enter the type of animal.

In row 3, enter the number of each type of animal.

Select the data by clicking in cell B2 and dragging over to cell F3.

Click the Chart Wizard iconin the top toolbar.

Click “Pie” under Chart Typein the Chart Wizard window.(Pie chart is another namefor a circle graph.)

Click the top left circle graphunder the Chart Sub-Type.

Click “Next” until the Finishbutton appears. Click“Finish.”

Now change the number ofpigs to 12 and the number ofgoats to 11. Notice how thecircle graph changes toreflect the new data.

Activity

1

Chart Wizard icon


m807_c09_504_505 1/6/06 4:33 PM Page 504

Use the spreadsheet to draw a line graph of the data.

Right click on the graph andselect “Chart Type . . .”

Click “Line” and make sure thatthe top left graph is selected.

Click “OK.”

Now change the number ofanimals. Notice how the linegraph changes to reflect the new data.

Use the spreadsheet to make a bar graph of the data.

Right click on the graph andselect “Chart Type . . .”

Click “Column” and make surethat the top left graph is selected.

Click “OK.”

Now change the number ofanimals. Notice how the linegraph changes to reflect the new data.

1. Compare the three types of graphs. When might you prefer usingone type over the others? Which is the best representation of theanimal data? Explain.

2. Explain the value of spreadsheets for modeling different situations.

3. Describe a situation when you would want to use a spreadsheet tomake a circle graph.

1. Take a walk in your neighborhood and record the color of the first 30 cars you see. Use a spreadsheet to make a circle graph, a linegraph, and a bar graph of your data. Which represents the data best?Explain.

2. Now record the color of the next 30 cars you see. Modify your datafrom Try This 1. How did each graph change?

Think and Discuss

Try This

2

3


m807_c09_504_505 1/6/06 4:33 PM Page 505

Rea

dy

to G

o O

n?


Num

ber

of

stud

ents

Game

Favorite Games of Second-Graders

Hopscotch Tag Jump rope Kick the can

Quiz for Lessons 9-5 Through 9-8

9-5 Displaying Data

1. Organize the data into a frequency table and make a double-bar graph.

Data set 1: 3, 5, 4, 2, 5, 2, 3, 3, 6, 5, 3, 3, 4, 2, 1Data set 2: 2, 5, 4, 3, 2, 5, 4, 6, 3, 4, 3, 2, 2, 4, 5

2. A fitness group calculated the average number of minutes they exercisedeach day. Use the data to make a histogram with intervals of 10.

29 31 42 42 50 44 33 36 37 41 38 45 37 45

37 41 39 41 34 39 35 35 30 36 32 41 39 37

9-6 Misleading Graphs and Statistics


3. 4.

5. A survey found 39% of students like tacos best, 32% like pizza best, and 29%like hamburgers best. The survey concludes that most students at themiddle school like tacos the best. Explain why the statistic is misleading.

9-7 Scatter Plots

6. Use the given data of the estimated U.S. population to make a scatter plot.

Does the data set have a positive, a negative, or no correlation?

7. The number of miles on a used car and the price of the used car

9-8 Choosing the Best Representation of Data

8. The eighth-grade chorus had 10 altos, 16 sopranos, 4 bass vocalists, and 10 tenors. Choose an appropriate data display and draw the graph. What percent of the chorus were the altos and tenors?

Value 1 2 3 4 5 6

Data Set 1 1 3 5 2 3 1

Data Set 2 0 4 3 4 3 1

Favorite Lunch

= 5 students

= 2 students

= 10 students

Year 1998 1999 2000 2001 2002 2003 2004

Population (in millions) 270.2 272.7 282.2 285.1 287.9 290.8 293.7

m807_c09_506_506 1/6/06 4:33 PM Page 506

Mu

lti-Step Test Prep

Bowled Over A group of middle school studentsforms a bowling club. The table shows the number ofyears each student has been bowling and the scoresfrom the group’s first trip to the bowling alley.

1. The club’s president prepares a newsletter describingthe “typical” student in the club. Choose a measure of central tendency to describe the typical number of years that the club members have been bowling.Choose a measure of central tendency to describetheir typical score. Justify your choices.

2. Make a stem-and-leaf plot of the scores. What canyou say about the scores based on the stem-and-leaf plot?

3. Brian is making a graph showing his score andthose of his friends Jessica and Kendall. He wants to make it seem like his score was much greaterthan those of his friends. Show how he can make amisleading graph.

4. Make a scatter plot of the data.

5. A new student joins the club. She has been bowlingfor 5 years. Use your scatter plot to predict her score the next time the group goes bowling.

YearsStudent Bowling Score

Jessica 3 90

Brian 3 100

Chandra 2 81

Roberto 7 128

Lee 1 84

Flora 3 92

Mike 4 102

Hisako 3 90

Isabel 6 135

Warren 1 65

Kendall 2 90

Mei 1 77

Multi-Step Test Prep 507

m807_c09_507_507 1/6/06 4:33 PM 7

Distribution of PrimesRemember that a prime number is only divisible by 1 and itself. There areinfinitely many prime numbers, but there is no algebraic formula to findthem. The largest known prime number, discovered on November 14, 2001, is 213,466,917 – 1. In standard form, this number would have 4,053,946 digits.

Sieve of Eratosthenes

One way to find prime numbers is calledthe sieve of Eratosthenes. Use a list ofwhole numbers in order. Cross off 1. Thenext number, 2, is prime. Circle it. Thencross off all multiples of 2, because theyare not prime. Circle the next number on the list. Cross off all of its multiples.Repeat this step until all of the numbersare circled or crossed off. The circled numbers will all be primes.

Use the sieve of Eratosthenes to find all prime numbers less than 50.Create a scatter plot of the first 15 prime numbers. Use the primenumbers as the x-coordinates and their positions in the sequence asthe y-coordinates; 2 is the 1st prime, 3 is the 2nd prime, and so on.

Estimate the line of best fit and use it to estimate the number of primes under 100. Use the sieve of Eratosthenes to check your estimate.

3

21

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

Prime Number 2 3 5 7

Position in Sequence 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


This game can be played by two or more players. Onyour turn, roll 5 number cubes. The number of spacesyou move is your choice of the mean, rounded to thenearest whole number; the median; or the mode, if itexists. The winner is the first player to land on theFinish square by exact count.

A complete set of rules and a game board are available online.

Math in the MiddleMath in the Middle

KEYWORD: MT7 Games

m807_c09_508_508 1/7/06 12:14 PM Page 508

Here is a way to take notes on collecting, displaying,and analyzing data that is guaranteed to pop out!

Cut one paper plate in half. You will use the twohalves later to make covers for your pop-up book.

Fold each of the remaining paper plates in half.Cut two 1-inch slits in the middle of the foldededge of each plate. The slits should be about 1 inch apart. Figure A

Bend the paper between the slits back and forth,and then push it inward as you unfold the plate.This will create a pop-up tab. Figure B

Fold the paper plates shut. Glue the bottom ofone paper plate to the top of the next paper plateto form a book. Make covers by gluing one of thepaper-plate halves onto the front of the book andthe other onto the back of the book.

Cut out four small rectangles of decorative paper.After taking notes on these rectangles, you willglue them onto the pop-up tabs in your book.Figure C

Taking Note of the MathUse the rectangles of decorative paper totake notes on collecting, displaying, andanalyzing data. Then glue therectangles to the pop-up tabs insidethe book. You can also take notes bywriting directly on the paper plates.

5

4

3

2

1

A

C

B

Data Pop-UpsPROJECT

Materials• 5 paper plates• scissors• glue• decorative paper• markers

509

m807_c09_509_509 1/6/06 4:34 PM Page 509


Complete the sentences below with vocabulary words from the list above.

1. The ___?___ of a data set is the middle value, while the ___?___ is the value that occurs most often.

2. ___?___ describes how spread out a data set is. One measure of ___?___is the ___?___.

3. The ___?___ is the line that comes closest to all the points on a(n) ___?___. ___?___ describes the type of relationship between two data sets.

Vocabulary

Samples and Surveys (pp. 462–465)9-1

E X A M P L E EXERCISES

■ Identify the population and sample. Givea reason the sample could be biased.

In a community of 1250 people, a pollsterasks 250 people living near a railroadtrack if they want the tracks moved.

Population: 1250 people

Sample: 250 people

Possible bias: People living near tracksare annoyed by the noiseand want tracks moved.

Identify the population and sample. Give areason the sample could be biased.

4. Out of the 125 people in line for a StarWars movie, 25 are asked to name theirfavorite type of movie.

5. A pollster surveyed 100 people whoowned cell phones about whether theyfelt it was safe to use cell phones whiledriving.

6. Fifty parents of children attending localpreschools are asked if the communityshould build a new playground.

Stu

dy

Gu

ide:

Rev

iew

. . . . .467

. . . . . . . . .463

. .477

. . .462

. . . . . . . . . . . .494

. . . . . .485

. . . . .486

. . . . . . . .485

. . . . . . . . . . . . .485

. . . . . . . . .494

. . . . . . . . . . . . . . .467

. . . . . . . . . . . . . . . . .472

. . . . . . . . . . . . . . .472

. . . . . . . . . . . . . . . . .472

. . . . . . . . . . . . . . . .472

. . . . . . . . . . . .462

. . . . . . . . . . . . . . .476

. . . . . . .462

. . . . . . . . . . . . . . . . .472

. . . . . . . . . . . . . . . .462

. . . . . . . . . . . .494

. . . . .467

. . . . . . .462

. . . . .462

. . . . . . . . . . . . .476

. . . . . . . . .468

. . . .462voluntary-responsesample

Venn diagram

variability

systematic sample

stratified sample

stem-and-leaf plot

scatter plot

sample

range

random sample

quartile

population

outlier

mode

median

mean

line plot

line of best fit

histogram

frequency table

double-line graph

double-bar graph

correlation

convenience sample


biased sample

back-to-backstem-and-leaf plot

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Organizing Data (pp. 467–471)9-2

E X A M P L E EXERCISE

■ Use a line plot to organize the data.

7 10 6 9 7 4 8 93 8 2 10 5 9 7

Use a line plot to organize the data.

7.

Stud

y Gu

ide: R

eview

Ages of People at a Skate Park

12 13 13 14 12 11

14 15 13 13 12 13

Measures of Central Tendency (pp. 472–475)9-3


■ The numbers of people to swim in apublic pool each day one week were 50,65, 72, 3, 85, 105, and 120. Explain whichmeasure of central tendency bestdescribes the middle of these numbersand find it.

Because there is an outlier, the median isthe best measure of central tendency. 5, 50, 65, 72, 85, 105, 120

Explain which measure of central tendencyis the most appropriate for the situationand find it.

8. The prices of the cars sold in onemonth were $17,500; $15,300; $16,800;$65,900; $12,800; $16,300. What numberbest describes the middle of thesenumbers?

0

x

xxx

xx

xxxxxxx x

x

1 2 3 4 5 6 7 8 9 10

Study Guide: Review 511

Variability (pp. 476–480)9-4


■ Use the given data to make a box-and-whisker plot.

7, 10, 14, 16, 17, 17, 18, 20, 20

7 10 14 16 17 17 18 20 20

smallest value: 7

first quartile: �10 �2

14� � 12

median: 17

third quartile: �18 �2

20� � 19

largest value: 20


9. 56, 56, 56, 59, 63, 68, 68, 73, 73, 73

10. 87, 87, 80, 72, 85, 82, 53, 65, 65

11. 80, 80, 80, 82, 85, 87, 87, 90, 90, 90

5 10 15 200

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Stu

dy

Gu

ide:

Rev

iew

Displaying Data (pp. 485–488)9-5


■ Make a histogram of the data set.

72, 64, 56, 60, 66, 72, 48, 66, 58, 60,60, 50, 68, 72, 68, 62, 72, 58, 60, 68

Make a histogram of each data set.

12.

13. Computer usage (h/week): 8, 3, 5, 10,11, 12, 10, 7, 8, 7, 7, 22, 13, 15, 18, 6, 3

7654321

Freq

uenc

y

45 50 55 65 7060 75

Weight (lb) Frequency

91–100 5

101–110 7

111–120 10

121–130 4

131–140 2

141–150 3


Misleading Graphs and Statistics (pp. 490–493)9-6


■ Explain why the graph is misleading.

The bar for mixed juice is 7 times longerthan the bar for cherry juice, but it is onlypreferred by 2 times as many people.

14. Explain why the graph is misleading.

Juice Preference

Mixed

Orange

Cherry

6 7 8 9 10 110 12

People

Number of Pizzas Ordered

= 5 pizzasLarge

Medium

Small

Scatter Plots (pp. 494–497)9-7


■ Does the age of a battery in a flashlightand the intensity of the flashlight beamhave a positive, a negative, or nocorrelation? Explain.

Negative: The older the battery is, the lessintense the flashlight beam will be.

■ Choose the best display to comparechildren’s shoe sizes to their heights.

A scatter plot would be the best displaybecause you are comparing two sets of data.

Does the data set have a positive, anegative, or no correlation? Explain.

15. the number of miles on a car’sodometer and the size of the gas tank

Choose the best data display for thesituation below. Explain your answer.

16. the amount of money spent in eachcategory of a budget

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Chapter 9 Test 513


1. Twenty U.S. cities are randomly chosen and 100 people are randomlychosen from each city.

2. A telemarketer flips through the phone book and selects 30 names.

3. A chef asks the first five customers who order the new dessert if they like it.Identify the population and the sample. Why might the sample be biased?

4. The scores on a history test were 79, 82, 85, 100, 82, 83, 78, 84, 80, 82, and 77. What number best describes the middle of these scores?


5. 62, 60, 77, 66, 92, 87, 62, 60, 64

6. 2.2, 6.8, 6.4, 8, 6.5, 4.2, 6.5, 5, 8

7. A middle school class calculated the average number of minutes they spenton the phone each day. Use the data to make a histogram with intervals of 10.18 31 32 42 50 34 33 36 27 41 5 35 27 1537 12 9 31 24 29 10 25 20 66 22 31 9 3


8. 9.


11. In a randomly chosen group of 100 people, 38 have type O positive blood,7 have O negative, 34 have A positive, 6 have A negative, 9 have B positive,2 have B negative, 3 have AB positive, and 1 has AB negative. Choose anappropriate data display and draw the graph. About what fraction of thepopulation has type O blood?

Ch

apter Test

Food Pizza Hamburger Taco Hot Dog Caesar Salad Taco Salad

Fat (g) 11 13 14 12 4 21

Calories 374 310 220 270 90 410

Patients’ Blood Pressures

14121086420

120–129 150–159130–149Num

ber

of p

atie

nts

Blood pressure (mmHg)

Quarter Profits9

8

7

6

5

Month

Prof

it

April May June

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Multiple Choice The box-and-whisker plot shows the number of sales for theyear. What is the range?

46 54 68 80

• Look at the box-and-whisker plot. The whiskers extend to the smallest andlargest values. The range is the difference between these values.

94 � 48 � 46 Find the difference.

• The range is 46, so the correct answer is choice A.

• Sometimes you will need to draw a diagram based on the information givenin a test item. Always read the question carefully to make sure that yourdiagram is properly labeled.

DCBA

All Types: Using a GraphicSometimes a graph or a picture is given with a test item. Look carefully atany drawings on a test. Keep in mind that figures are not always drawn toscale and can be misleading.

Short Answer An ice rink has an area of 3750 ft2 and length of 75 ft. What isthe perimeter of the ice rink? Explain your reasoning and show your work.

Draw a diagram to help you visualize the problem.

A � bh3750 � 75h

50 � h

P � 2(b � h)P � 2(75 � 50)

You know the area and base. Youneed to find the height.

Use the formula for perimeter.Substitute the known values.

48 54 60 66 72 78 84 90

75 ft

3750 ft2 h

P � 2(125)P � 250

The perimeter of the ice rinkis 250 ft.

Test

Tac

kler

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Test Tackler 515

Read each test problem and answer thequestions that follow.

1. Draw a diagram to help you visualizethe problem.

2. Use information from your diagram tosolve the problem.

Draw a diagram if one is not providedto help you visualize the problem.

Item AA pizza restaurant sells a 12-inch smallpizza, a 14-inch medium pizza, and a16-inch large pizza. How much morepizza do you get for a large pizza than a small pizza? Explain your reasoningand show your work.

Item CWhen a rectangle is divided into thirds,three squares are formed, each with aperimeter of 9.6 cm. What is theperimeter of the original rectangle?

Item D A 10 cm section of plastic pipe has aninner diameter 12 cm and an outerdiameter 16 cm. What is the volume of the solid plastic pipe to the nearesttenth?

879.2 cm3 2009.6 cm3

1130.4 cm3 3140 cm3DB

CA

Item B A middle school has 1000 students.According to the circle graph, howmany students are in track?

400 100

300 30JG

HF

5. Draw a diagram to visualize theproblem.

6. What information from yourdiagram do you need to solve theproblem?

7. If your answer is a decimal, what doyou need to remember to do on thegrid?

8. Show how you would grid yourresponse below.

0123456789

0123456789

0123456789

0123456789

0123456789

9. Draw a diagram to help youvisualize the problem.

10. Use information from your diagramto solve the problem.

Percent of Students Participating in Extra curricular Activities

Soccer5%

Basketball30%

Track10%Other

15%

None40%

Test Tackler

3. What percent of the students are intrack? How do you know?

4. How do you find the number ofstudents who are in track?

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Stan

dar

diz

ed T

est

Prep

KEYWORD: MT7 TestPrep

1. Which is NOT true for this data set? 10, 10, 10.5, 9, 9.5

mean � mode

median � mean

median � mean

median � mode

2. In order to participate in after-schoolactivities, a student needs to have agrade point average, g, of 2.0 orbetter. Which inequality represents this requirement?

g � 2.0 g � 2.0

g � 2.0 g � 2.0

3. Which ordered pair is a solution to theequation 2x � 4y � �18?

(0, �9) (�11, 1)

(6, 0) (�3, �4)

4. Which expression is NOT equivalent to4 � 4 � 4 � 4 � 4?

�41�5� 42 � 43

20 1024

5. A 6-inch model is made to represent a30-foot plane. What is the scale?

1 in. � 5 ft 6 in. � 5 ft

5 in. � 1 ft 30 in. � 5 ft

6. The stem-and-leaf plot shows testscores for a teacher’s first and secondperiods. What can you conclude?

1st period 2nd period

7 6 5 86 4 2 7 5 6 9

9 8 6 4 2 0 8 1 3 5 7 7 8 89 7 7 2 1 9 0 6 7 8 9

More first period students scored inthe 90’s.

Fewer first period students scored80 or below.

More second period students scoredin the 70’s.

More second period students scoredin the 80’s.

7. A soup company is producing acylindrical can to package its new soup.The radius of the cylinder is 1.5 in. andthe volume of the cylinder has to be 14 in3. What must the height of the canbe, rounded to the nearest whole inch?

1 inch 3 inches

2 inches 4 inches

8. Emma buys a refrigerator on sale for$665. This is 30% off the original price.What is the original price of therefrigerator?

$200 $1995

$950 $2217JG

HF

DB

CA

J

H

G

F

DB

CA

JG

HF

DB

CA

JG

HF

D

C

B

A

Cumulative Assessment, Chapters 1–9Multiple Choice


Key: � 9 � 0 means 90

7 � 6 � means 67

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Cumulative Assessment, Chapters 1–9 517

9. Which is a solution to the equation�10 � 5x � �25?

x � �15 x � �3

x � �7 x � �1

10. If triangle JQZ � triangle VTZ, what isthe value of r?

7.5 9

8.5 33

Gridded Response

11. The function f(t) � �16t2 � 180 modelsthe distance an object falls when it isdropped from the top of a building 180ft tall in t seconds. How many feet doesthe stone fall after 2 seconds?

Use the box-and-whisker plot to answersquestions 12 and 13.

12. What is the range of the data?

13. What is the first quartile of the data?

14. Monica scored 85, 83, 81, 80, and 81 on her last five assignments. Whatwould Monica need to earn on hernext assignment to bring her averageto an 85?

Short Response

15. Name two ordered pairs (x, y) thatsatisfy these conditions: The mean of 0,x, and y is twice the median; 0 � x � y;and y � nx (y is a multiple of x). What isthe value of n? Show your work orexplain in words how you determinedyour answer.

16. Explain why the graph is misleadingand then redraw it so that it betterrepresents the data.

Extended Response

17. Twenty students in a gym class kept arecord of their jogging. The results areshown in the scatter plot.

a. Describe the correlation of the datain the scatter plot.

b. Find the average speeds of joggerswho run 1, 2, 3, 4, 5, and 6 miles.

c. Explain the relationship betweenyour answer from part a and youranswers from part b.

JG

HF

DB

CA

Stand

ardized

Test Prep

Read a graph or diagram as closely asyou do the actual question. Thesevisual aids contain importantinformation.

Fruit Preference

Apple

Banana

Strawberry

60 70 80 90 100 1100 120

Number of People

8 10 12 14 16 18 20 22 24 26

Distance (mi)

Spee

d (m

i/h)

87654321

10 2 3 4 5 6

16

15.14.82r � 1

VZ

J

Q

T

m807_c09_516_517 1/6/06 4:35 PM Page 517

Chapter 09- Data and Statis - Advisory...

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