Data, Statistics, and...

KEYWORD: MT8CA Ch11

526 Chapter 11

Data, Statistics, andProbabilityData, Statistics, andProbability

11A Collecting,Describing, andDisplaying Data

LAB Use a Spreadsheet toMake Graphs

11-1 Line Plots and Stem-and-Leaf Plots

11-2 Mean, Median, Mode,and Range

11-3 Box-and-Whisker Plots

LAB Make a Box-and-WhiskerPlot

11-4 Scatter Plots

LAB Make a Scatter Plot

11B Probability11-5 Probability

11-6 Experimental Probability

11-7 Theoretical Probability

11-8 Independent andDependent Events

Scientists can use data tomake predictions aboutpopulations of animals,such as these elephant

seals which were once hunted nearly to extinction.

Data, Statistics, and Probability 527

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

1. A __?__ is a uniform measure where equal distances aremarked to 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 Bar GraphsUse 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

number line

period

place value

rounding

scale

Activity Enrollment

Softball

2000 2001

Soccer Basketball HockeyNum

ber

of s

tude

nts 70

6050403020100

528 Chapter 11

The information below “unpacks” the standards. The Academic Vocabulary ishighlighted and defined to help you understand the language of the standards.Refer to the lessons listed after each standard for help with the math terms andphrases. The Chapter Concept shows how the standard is applied in this chapter.

represent show or describe

relationships connections

You discover how data setsrelate to one another.

SDAP1.1 Know formsof display for data sets, including astem-and-leaf plot or box-and-whisker plot; use the forms

a single set of data or totwo sets of data.

(Lessons 11-1, 11-3; Lab 11-3)compareto display

various various different from each other

to display to show

compare determine how items are the same ordifferent

You learn different ways todisplay data in graphs.

SDAP1.2 Represent twonumerical variables on a scatterplotand describe how thedata points are distributed and any

relationship that existsbetween the two variables (e.g.,between time spent on homeworkand grade level).(Lesson 11-4; Lab 11-4)

apparent

informally

informal casual

apparent visible; easily seen

You describe how two data setsrelate to each other bygraphing them on a coordinateplane.

SDAP1.3 Understand themeaning of, and be able to

, the minimum, the lowerquartile, the median, the upperquartile, and the maximum of adata set.(Lessons 11-2, 11-3; Lab 11-3)

compute

compute determine by using mathematicaloperations such as addition and subtraction

You learn how to find differentvalues that describe a data set.These values will be used tomake box-and-whisker plots.

California Standard

Chapter Concept

Academic Vocabulary

SDAP1.0 Students collect,organize, and data setsthat have one or more variablesand identify amongvariables within a data set by handand through the use of anelectronic spreadsheet softwareprogram.(Labs 11-1, 11-4)

relationships

represent

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 40: 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 10-5 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

Data, Statistics, and Probability 529

Use a Spreadsheet to Make Graphs

Use with Lesson 11-1

11-1

KEYWORD: MT8CA Lab11

A spreadsheet allows you to model and compare different situations easily.

Suppose a farmer has 22 pigs, 2 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

530 Chapter 11 Data, Statistics, and Probability

CaliforniaStandards

SDAP1.0 Students collect, organize, and represent data setsthat have one or more variables and identify relationships amongvariables within a data set by hand and through the use of anelectronic spreadsheet software program.

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

11-1 Technology Lab 531

11-1

CaliforniaStandards

SDAP1.1 Know various formsof display for data sets, includinga stem-and-leaf plot or box-and-whisker plot; use the forms todisplay a single set of data or tocompare two sets of data.

Why learn this? You can use a line plotto organize and display fitness data.

Vocabularyline plotstem-and-leaf plotback-to-back

stem-and-leaf plot

A seventh-grade class participated in a month-long fitness challenge. Students in the class recorded the number of miles that they ran, walked, or biked during the first week.

Organizing raw data such as these can help you see patterns and trends. One way to organize data is to use a line plot.A uses a number line and X’s toshow how often a value occurs in a data set. A line plot can help you see how the data are distributed.

Organizing Data in Line Plots

Use a line plot to organize the data for the seventh-grade fitnesschallenge.

Step 1: Find the least and greatest values in the data set. Then draw a number line that includes these values.

least value: 0

greatest value: 10

Step 2: Place an X above the number on the number line thatcorresponds to the number of miles each student ran,walked, or biked.

There are 30 numbers in the data set and 30 X’s above the number line.

line plot

Number of Miles Run, Walked, or Biked

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

E X A M P L E 1


0

xxxx

xxxx

xxx

xxx

xxxx

xxxxx xx

xxx

xx

1 2 3 4 5 6 7 8 9 10

Line Plots and Stem-and-Leaf Plots

Miles Run, Walked, or Biked

A uses the digits of each number in a data set to group the individualnumbers, usually in increasing or decreasingorder. Each leaf on the plot represents the right-hand digit in a data value, and each stemrepresents the remaining left-hand digits. The key shows the values of the data on the plot.

Organizing and Interpreting Data in a Stem-and-Leaf Plot

The table shows the number of minutes students spent doing theirSpanish homework. Make a stem-and-leaf plot of the data. Thenfind the number of students who studied longer than 45 minutes.

Step 1: Order the data from least to greatest. Since the data valuesrange from 21 to 64, use tens digits for the stems and onesdigits for the leaves.

Step 2: List the stems from least to greatest on the plot.

Step 3: List the leaves for each stem from least to greatest.

Step 4: Add a key, and title the graph.

One student studied for 47 minutes, 2 students studied for 48 minutes, and 1 student studied for 64 minutes.

A total of 4 students studied longer than 45 minutes.

A can be used to compare two setsof data. The stems are in the center, and the leaves for one set of dataare to the right of the stems. The leaves for the other set of data are tothe left of the stems.

back-to-back stem-and-leaf plot

stem-and-leaf plot

Minutes Spent Doing Homework

38 48 45 32 29 48

32 45 36 22 21 6435 45 47 26 43 29

The stems arethe tens digits.

The stem 5 has noleaves, so thereare no data valuesin the 50’s.

The leaves are the ones digits.

The entries in the second rowrepresent the datavalues 32, 32, 35,36, and 38.

2E X A M P L E

Minutes Spent Doing Homework

Key: 3�2 means 32

Stems Leaves

23456

1 2 6 9 92 2 5 6 83 5 5 5 7 8 8

4

Key: 2�7 means 27

Stems Leaves

23

4 7 90 6

11-1 Line Plots and Stem-and-Leaf Plots 533

To represent 5minutes in the stem-and-leaf plot in Example 2, youwould use 0 as thestem and 5 as the leaf.

Think and Discuss

1. Describe how the numbers 4 and 89 can be shown on a stem-and-leaf plot.

2. Tell which number is always the same as the number of data values in a stem-and-leaf plot: the number of stems or the number of leaves. Explain.

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

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

E X A M P L E 3

11-1 ExercisesExercisesKEYWORD: MT8CA Parent

KEYWORD: MT8CA 11-1

Number of Electoral Votes for Select States (2004)

CA 55 GA 15 IN 11 MI 17 NY 31 PA 21

NJ 15 IL 21 KY 8 NC 15 OH 20 TX 34

GUIDED PRACTICE


1. Make a line plot of the data. For the states shown, what was the mostcommon number of electoral votes in 2004?

2. Make a stem-and-leaf plot of the data. How many of the states had morethan 30 electoral votes in 2004?

See Example 2

See Example 1

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

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

CaliforniaStandards Practice

SDAP1.1

Super Bowl Scores, 1995–2005

Use the table for Exercises 1 and 2.

The table shows the ages of the first 18 U.S. presidents when they took office.Use the table for Exercises 4 and 5.

4. Make a line plot of the data. What was the most common age at which thepresidents took office?

5. Make a stem-and-leaf plot of the data. How many of the presidents were intheir 40’s when they took office?

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

Use the stem-and-leaf plot for Exercises 7–9.

7. What is the least data value? What is the greatest data value?

8. Which data value occurs most often?

9. Reasoning Which of the following is most likely the source of the data in the stem-and-leaf plot?

Shoe sizes of 12 middle school students

Number of hours 12 adults exercised in one month

Number of boxes of cereal per household at one time

Monthly temperatures in degrees Fahrenheit in Chicago, IllinoisD

C

B

A

PRACTICE AND PROBLEM SOLVING

INDEPENDENT PRACTICE

Extra PracticeSee page EP22.

See Example 2

See Example 3

See Example 1

Washington

Adams

Jefferson

Madison

Monroe

Adams

57

61

57

57

58

57

President Age

Jackson

Van Buren

Harrison

Tyler

Polk

Taylor

61

54

68

51

49

64

President Age

Fillmore

Pierce

Buchanan

Lincoln

Johnson

Grant

50

48

65

52

56

46

President Age

Stems Leaves

0 4 6 6 91 2 5 8 8 82 0 33 1

Key: 1⏐2 means 12

11-1 Line Plots and Stem-and-Leaf Plots 535

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

3. Use the data below to make a back-to-back stem-and-leaf plot.See Example 3

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

NS1.5, MG2.1, SDAP1.1

18. Multiple Choice How many stems would astem-and-leaf plot of the data in the table have?

1 3

2 4

19. Extended Response Make a stem-and-leaf plot and a line plot of the datain the table. Which data display best shows the distribution of data? Explain.

Write each decimal as a fraction in simplest form. (Lesson 2-1)

20. 0.32 21. 0.025 22. 0.06 23. 0.9

Find the area of each figure with the given dimensions. (Lesson 9-2)

24. trapezoid: b1 � 3, b2 � 5, h � 8 25. triangle: b � 16, h � 9

DB

CA

20 30 9 25 28

8 11 12 7 18

33 26 10 9 2

The stem-and-leaf plot shows the scores for a recent math test. Use the plot for Exercises 10–13.

10. What was the highest score received?

11. What was the lowest score received?

12. Did more boys or girls take the test?

13. How many students took the test?

14. Marty’s and Bill’s bowling scores are given below.

Marty: 137, 149, 167, 134, 121, 127, 143, 123, 168, 162

Bill: 129, 138, 141, 124, 139, 160, 149, 145, 128, 130

a. Make a back-to-back stem-and-leaf plot to compare the data.

b. Whose scores were higher overall? Explain.

15. What’s the Error? Two students made stem-and-leaf plots for the following data: 530, 545, 550, 555, 570. Which plot is incorrect? Explain.

16. Write About It Explain why it is important for a stem-and-leaf plot tohave a key.

17. Challenge Make a line plot and a stem-and-leaf plot for the followingdata: 9.9, 8.2, 10, 8.5, 9.3, 8, 10, 9.7, 8.4, 8.8, 9.3, 10.

Math Test Scores

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

Key: 5|6| means 65|7|1 means 71


CaliforniaStandards

SDAP1.3 Understand themeaning of, and be able tocompute, the minimum, the lowerquartile, the median, the upperquartile, and the maximum of a data set.

Why learn this? You can use the mean to determine theaverage number of hours that people exercise in one week.(See Example 2.)

The mean, median, mode, and range are commonly used to describea set of numerical data.

• The is the sum of the data values divided by the number ofdata items.

• The is the middle value of an odd number of data itemsarranged in order. For an even number of data items, the median isthe mean of the two middle values.

• The is the value or values that occur most often. When all ofthe data values occur the same number of times, there is no mode.

• The is the difference between the greatest and least values inthe set.

Finding the Mean, Median, Mode, and Range of a Data Set

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

2, 1, 8, 0, 2, 4, 3, 4

mean:

2 � 1 � 8 � 0 � 2 � 4 � 3 � 4 � 24 Add the values.

�284� � 3 Divide the sum by the

The mean is 3. number of items.

median:

0, 1, 2, 2, 3, 4, 4, 8 Arrange the values in order.

�2 �

23

� � 2.5

The median is 2.5.

mode:

0, 1, 2, 2, 3, 4, 4, 8 The values 2 and 4 occur twice.

The modes are 2 and 4.

range:8 � 0 � 8 Subtract the least value from

The range is 8. the greatest value.

range

mode

median

mean

Vocabularymeanmedianmoderangeoutlier

E X A M P L E 1

11-2 Mean, Median, Mode, and Range 537

Mean, Median, Mode,and Range

The mean issometimes called the average.

There are two middle values,so find the mean of thesevalues.

11-2

The mean and median are measures of central tendency used torepresent the “middle” of a data set. To decide which measure is mostappropriate for describing a set of data, think about what eachmeasure tells you about the data. The measure that you choose maydepend on how the information in the data set is being used.

Choosing the Best Measure to Describe a Set of Data

The line plot shows the number of hours 15 people exercised inone week. Find the mean and median of the data. Which measurebest describes the typical amount of time a person exercised?Justify your answer.

mean:

� �61

35� � 4.2

The mean is 4.2.

The mean is greater than most of the data values, so it is not the bestchoice to describe the typical amount of time exercised.

median:

0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 5, 7, 7, 14, 14

The median is 2.

Since the majority of the data is clustered around the data value 2,the median is a more useful description of the typical amount oftime exercised.

0 � 1 � 1 � 1 � 1 � 2 � 2 � 2 � 3 � 3 � 5 � 7 � 7 � 14 � 1415

2E X A M P L E

0

xxx

xxx

xxxx

xxx

xx

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

Number of hours


In the data set in Example 2, the value 14 is much greater than theother values in the set. A value such as this that is very different fromthe other values in the set is called an .

Outliers can greatly affect the mean of a data set. For this reason, themean may not be the best measure to describe a set of data with anoutlier.

outlier

Most of data Mean Outlier

10 200 706050 9080 1004030

One way to helpidentify an outlier isby making a line plot.

Reasoning

Exploring the Effects of Outliers

The table shows the number of artpieces created by students in aglass-blowing workshop. Identifythe outlier in the data set. Thendetermine how the outlier affectsthe mean, median, and mode of the data.

The number of art pieces created byHermann is much greater than theother values in the set. The outlier is 14.

Without the Outlier With the Outlier

mean: mean:

� 3 � 4.8

The mean is 3. The mean is about 4.8.

The outlier increases the mean of the data by about 1.8.

median: median:

1, 2, 3, 4, 5 1, 2, 3, 4, 5, 14

�3 �

24

� � 3.5

The median is 3. The median is 3.5.

The outlier increases the median of the data by 0.5.

mode: mode:

No value occurs most often. No value occurs most often.

There is no mode. There is no mode.

The outlier does not change the mode of the data.

The mean is most affected by the outlier.

5 � 1 � 3 � 4 � 14 � 2��

65 � 1 � 3 � 4 � 2��

5

Think and Discuss

1. Describe a situation in which the mean would best describe adata set.

2. Give an example of a data set in which the mean, median, andmode have the same value.

3. Explain how an outlier affects the mean, median, and mode of adata set.

Name Number of Pieces

Suzanne 5

Glen 1

Charissa 3

Eileen 4

Hermann 14

Tom 2

E X A M P L E 3


Since all the datavalues occur the samenumber of times, theset has no mode.

11-2 ExercisesExercises

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

1. 5, 30, 35, 20, 5, 25, 20 2. 44, 68, 48, 61, 59, 48, 63, 49

3. The line plot shows cookingtemperatures required bydifferent recipes. Find the meanand median of the data. Which measure best describes thetypical cooking temperature? Justify your answer.

4. The table shows the number of glasses of water consumed in one day. Identifythe outlier in the data set. Then determine how the outlier affects the mean,median, and mode of the data.

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

5. 92, 88, 65, 68, 76, 90, 84, 88, 93, 89 6. 23, 43, 5, 3, 4, 14, 24, 15, 15, 13

7. 2.0, 4.4, 6.2, 3.2, 4.4, 6.2, 3.7 8. 13.1, 7.5, 3.9, 4.8, 17.1, 14.6, 8.3, 3.9

9. The line plot shows the number of letters in the spellings of the 12 months. Find the mean andmedian of the data. Whichmeasure best describes the typical number of letters? Justify your answer.

Identify the outlier in each data set. Then determine how the outlier affectsthe mean, median, and mode of the data.

10. 13, 18, 20, 5, 15, 20, 13, 20 11. 45, 48, 63, 85, 151, 47, 88, 44, 68

KEYWORD: MS7 Parent

KEYWORD: MS7 7-2

GUIDED PRACTICE


See Example 2

See Example 3

See Example 1

See Example 2

See Example 3

See Example 1

Water Consumption

Name Randy Lori Anita Jana Sonya Victor Mark Jorge

Glasses 4 12 3 1 4 7 5 4

150°F

x xxx

xxx

xxx

200°F 250°F 300°F 350°F 400°F 450°F

0

x

x x

x x

x

x x

x x x

x

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


12. Health Based on the data from three annual checkups, Jon’s mean height is62 in. At the first two checkups Jon’s height was 58 in. and 61 in. What washis height at the third checkup?

PRACTICE AND PROBLEM SOLVINGExtra Practice

See page EP22.

KEYWORD: MT8CA Parent

KEYWORD: MT8CA 11-2


SDAP1.3

Cooking Temperatures

Number of Letters

NS1.7, SDAP1.1, SDAP1.3

13. Find the mean, median, and mode of the data displayed in the line plot.Then determine how the outlier affects the mean and the median.

14. Reasoning The values in a data set are 95, 93, 91, 95, 100, 99, and 92.What value can be added to the set so that the mean, median, and moderemain the same?

15. Sports The ages of the participants in a mountain bike race are 14, 23, 20,24, 26, 17, 21, 31, 27, 25, 14, and 28. Find the mean and median of the data.Which measure best describes the typical age of the participants? Explain.

16. Estimation The table shows the monthlyrainfall in inches for six months. Estimate themean, median, and range of the data.

17. What’s the Question? The values in a data set are 10, 7, 9, 5, 13, 10, 7, 14, 8, and 11. What question gives the answer 9.5 for the data set?

18. Write About It Which measure of centraltendency is most often affected by including an outlier? Explain.

19. Challenge If 4��x �3y � z�� 8, what is the mean of x, y, and z?

Month Rainfall (in.)

Jan 4.33

Feb 1.62

Mar 2.17

Apr 0.56

May 3.35

Jun 1.14

0

x x xx

x xx

x x xxxx

2 4 6 8 10 12 14 16 18 20 22

Sports

Mountain bikesaccount for more than50% of bicycle salesin the United States.


20. Multiple Choice What is the mean of the winning scores shown in the table?

276 282.1

276.8 285

21. Multiple Choice In which data set are the mean, median, and mode all thesame number?

6, 2, 5, 4, 3, 4, 1 2, 3, 7, 3, 8, 3, 2

4, 2, 2, 1, 3, 2, 3 4, 3, 4, 3, 4, 6, 4

22. Brett deposits $4,000 in an account that earns 4.5% simple interest. Howlong will it be before the total amount is $4,800? (Lesson 6-7)

23. Make a stem-and-leaf plot of the following data: 48, 60, 57, 62, 43, 62, 45, and 51. (Lesson 11-1)

DB

CA

DB

CA

Masters Tournament Winning Scores

Year 2001 2002 2003 2004 2005

Score 272 276 281 279 276

Number of Books Read

Vocabularylower quartileupper quartilebox-and-whisker plotminimummaximum

E X A M P L E 1


11-3

CaliforniaStandards

SDAP1.1 Know various forms ofdisplay for data sets, including astem-and-leaf plot or box-and-whisker plot; use the forms todisplay a single set of data or tocompare two sets of data.

SDAP1.3 Understand themeaning of, and be able tocompute, the minimum, the lowerquartile, the median, the upperquartile, and the maximum of adata set.

As you saw in Lesson 11-2, measures of central tendency describe how data tends toward one value. You may also need to know how data is spread out across several values.

The table below summarizes a cat breeder’s records for kitten littersborn in a given year. You can divide the data into four equal parts using quartiles.

You know that the median of a data set divides the data into a lower halfand an upper half. The median of the lower half is the ,and the median of the upper half is the .

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

Lower quartile: 3 Median: 4 Upper quartile: 5median of lower half median of upper half

Finding Quartiles

Find the lower and upper quartiles for each data set.

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

78 85 88 88 89 90 92 Arrange the values inorder.

lower quartile: 85 upper 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 Arrange the values inorder.

lower quartile: �13 �

213

� � 13 upper quartile: �18 �2

19� � 18.5

upper quartile

lower quartile

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.

Who uses this? Animal breedersinterpret data distributions.

Box-and-Whisker Plots

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 of the data and upper fourth of the data arerepresented by “whiskers” that extend to the (least) and

(greatest) 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 from least to greatest. Then find theminimum, lower quartile, median, upper quartile, andmaximum.

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

minimum: 15

lower quartile: �22 �

222

� � 22

median: 38

upper quartile: �50 �

251

� � 50.5

maximum: 63

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

Step 3: Draw the box and whiskers.

maximumminimum

box-and-whisker plot

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

Lower quartile

Median

Upper quartile

11-3 Box-and-Whisker Plots 543



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: MT8CA 11-3

Comparing Data Sets Using Box-and-Whisker Plots

The number of touchdownpasses that Brett Favre and Dan Marino threw during each of the first 14 years of their careers is shown in thebox-and-whisker plots.

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 least to greatestbefore 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


SDAP1.1, SDAP1.3

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


See page EP22.


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.

11-3 Box-and-Whisker Plots 545

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


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

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

21.

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

a. range: 49 b. range: 15 c. range: 36lower quartile: 11 lower quartile: 23 lower quartile: 18upper quartile: 39.5 upper quartile: 29.5 upper quartile: 35

23. Reasoning 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, and make anew box-and-whisker plot. How did the addition of an outlier affect thebox-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?


10 200 30 5040

Data set C

Data set A

Data set B

NS2.4, SDAP1.1, SDAP1.3

27. Multiple Choice Find the lower 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 lower and upper quartiles. Make a box-and-whiskerplot of the data.

Find the two square roots of each number. (Lesson 4-6)

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

Find the mean, median, and mode of each data set to the nearest tenth. (Lesson 11-2)

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

Make a Box-and-WhiskerPlot

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 (or lower)quartile: Q1, median: MED, third (or upper) quartile: Q3, andmaximum: MaxX). The minimum value in the data set is 50 in., thefirst quartile is 52 in., the median is 56 in., the third quartile is 59 in., and the maximum is 68 in.

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

1. The data below shows the number of hours slept one night for each 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


11-3

KEYWORD: MT8CA Lab11


CaliforniaStandards

SDAP1.1 Know various forms of display for data sets,including a stem-and-leaf plot or box-and-whisker plot;use the forms to display a single set of data or tocompare two sets of data.Also covered: SDAP1.3

11-4

A is a graph with pointsplotted to show a possiblerelationship between 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.

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

Use the table to make orderedpairs for the scatter plot.

The points are (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) indicatesthat the point occurs twice.

Hours

BMI

30

25

20

15

10

5

00 8 10 124 16 20

2

Activity and Body Mass Index


Scatter Plots

Vocabularyscatter plot correlationpositive correlationnegative correlationno correlationline of best fit

CaliforniaStandards

SDAP1.2 Represent twonumerical variables on ascatterplot and informally describehow the data points are distributedand any apparent relationship thatexists between the two variables(e.g., between time spent onhomework and grade level).

Who uses this? Health careprofessionals can use scatter plots tohelp them analyze the benefits ofphysical activity.

11-4 Scatter Plots 549

The values in bothdata sets increase atthe same time.

The values in onedata set increase asthe values in theother set decrease.

There is norelationship betweenthe data sets.

No CorrelationNegative CorrelationPositive Correlation

Width (cm)

Len

gth

(cm

)

0

6

4

2

0 1 2 3 4

Rectangle Dimensions

The graph shows that as widthincreases, length increases.

The graph shows that as anautomobile’s engine size increases, fuel economy decreases.

2E X A M P L E Determining Relationships Between Two Sets of Data

Write positive correlation, negative correlation, or no correlationto describe each relationship. Explain.

The graph shows a positive correlation between the data sets.

The graph shows a negative correlation between the data sets.

A , or trend line, is a straight line that approximates therelationship between the data on a scatter plot. It can help show thecorrelation between the data more clearly. You can also use a line ofbest fit when making predictions based on data.

line of best fit

A describes a relationship between two data sets. Thecorrelation can help you analyze trends and make predictions. Thereare three types of correlations between data.

correlation

Engine size (L)

Fuel

eco

no

my

(mi/

gal

)

0

60

40

20

01 2 3 4

Automobile Fuel Economy


KEYWORD: MT8CA 11-4


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

GUIDED PRACTICESee 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

Think and Discuss

1. Describe the type of correlation you would expect between thenumber of absences in a class and the grades in the class.

2. Give an example of a relationship between two sets of data thatyou would expect to show a negative correlation. Then give anexample that you would expect to show a positive correlation.

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 weekshould earn a score of about 96.

E X A M P L E 3

Hours Studied 5 9 3 12 1

Exam Grade 80 95 75 98 70

Hours studied

Gra

de

65

75

85

95

10 122 4 6 8

Hours Studied and Exam GradeDraw a line that has aboutthe same number of pointsabove and below it. Your linemay or may not go throughdata points.

Find the point on the linewhose x-value is 10. Thecorresponding y-value is 96.


SDAP1.2

Reasoning

11-4 Scatter Plots 551

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

See Example 2

See Example 3

See Example 1

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

Use the given data to make a scatter plot.

5. 6.

Write positive correlation, negative correlation, or no correlation to describeeach relationship. Explain.

7. 8.

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


Car Brand Cost Fuel Economy($1000) (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

See Example 3

Write positive correlation, negative correlation, or no correlation to describeeach relationship. Explain.

2. 3.

See Example 2

Math score

Math Score and Shoe Size

Shoe

siz

eAge

Work Experience

Year

s of

wor

k

Advertising cost

Sales

Wee

kly

sale

s

Mileage (thousands)

Car’s Mileage and Value

Val

ue

($)

Name Length Weight(ft) (tons)

Triceratops 30 6

Tyrannosaurus 39 7

Euhelopus 50 25

Brachiosaurus 82 50

Supersaurus 100 55

Identify the correlation that you would expect to see between each pair ofdata sets. Explain.

10. the number of hours of daylight and the amount of rainfall in a day

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

12. the number of students in a district and the number of buses in the district

13. the number of knots tied in a rope and the length of the rope

14. Life Science The table shows pollen levels measured in grains per cubic meter. Use the data to make a scatter plot. Then describe the relationship between the data sets.

15. Write About It Conduct a survey of your classmates to find their age in months and their shoe size. Predict whether there will be a positive correlation, a negative correlation, or no correlation. Then graph the data in a scatter plot. What is the relationship between the two data sets? Was your prediction correct?

16. Challenge A location’s elevation is negatively correlated to its averagetemperature and positively correlated to the amount of snow the locationreceives. What type of correlation would you expect to see betweentemperature and the amount of snowfall? Explain.


Pollen Levels

Day

1234567

Grass Pollen

1619329312

Weed Pollen

35051493094883065

MG2.1, MG2.3, SDAP1.2

17. Multiple Choice The scatter plot shows the amount of money spent on advertising and the amount earned in sales. Which statement is best supported by the scatter plot?

Weekly sales increase as advertising costs increase.

Weekly sales decrease as advertising costs increase.

Weekly sales remain constant as advertising costs increase.

There is no relationship between the amount spent on advertising and the amount earned in sales.

18. Short Response What type of correlation would you expect to see between aperson’s height and his or her birthday? Explain.

Find the area of each circle to the nearest tenth. Use 3.14 for π. (Lesson 9-4)

19. circle with diameter 10 cm 20. circle with radius 5.2 yd

21. A 9 cm cube is built from 1 cm cubes. Compare the ratio of the length of an edge of the large cube to the length of an edge of a small cube. (Lesson 10-7)

D

C

B

A



Advertising cost(thousands of dollars)

0 20 40 60 80

Wee

kly

sale

s(t

ho

usa

nd

s o

f d

olla

rs)

0

100

200

300

400

500


Make a Scatter Plot


11-4KEYWORD: MT8CA Lab11

You can use a graphing calculator to display relationships between data sets in 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. Describe the relationship between height and weight that is shown in the scatter plot.

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.

TRACEZOOM

2nd

2nd

ENTER

ENTERY=STAT PLOT

2nd

ENTERSTAT

Activity

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

CaliforniaStandards

SDAP1.2 Represent two numerical variables on a scatterplotand informally describe how the data points are distributed andany apparent relationship that exists between the two variables(e.g., between time spent on homework and grade level).Also covered: SDAP1.0

554 Chapter 11 Data, Statistics, and Probability554 Chapter 11 Data, Statistics, and Probability

Quiz for Lessons 11-1 Through 11-4

11-1 Line Plots and Stem-and-Leaf Plots

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

2. The list below shows the top speeds of various land animals. Make astem-and-leaf plot of the data.

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

72 78 76 75 79 70 74 80 72 78

71 69 70 72 68 70 69 75 75 74

42 55 62 48 65 51 47 59 67 61 49 54 55 52 44

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

11-2 Mean, Median, Mode, and Range

The list shows the life spans in years of vampire bats in captivity.

18 22 5 21 19 21 17 3 19 20 29 18 17

4. Find the mean, median, mode, and range of the data set. Round your answers tothe nearest tenth of a year.

5. Which measure, mean or median, best describes the typical life span of a vampirebat in captivity? Justify your answer.

11-3 Box-and-Whisker Plots


6. 43, 36, 25, 22, 34, 40, 18, 32, 43 7. 21, 51, 36, 38, 45, 52, 28, 16, 41

11-4 Scatter Plots

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

Identify the correlation you would expect to see between each pair of data sets.Explain.

9. a person’s date of birth and eye color

10. the number of miles on a used car and the value of the car

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

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?

Mrs. Wong wants to put a fence around hervegetable garden. One side of her gardenmeasures 8 ft. Another side measures 5 ft.What length of fencing does Mrs. Wong needto enclose her garden?

Focus on Problem Solving 555

1 4

5

2

3

Number of Marriages in the United States

Year 1940 1950 1960 19701980 1990

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.

CaliforniaStandards

MR1.1 Analyze problems byidentifying relationships,distinguishing relevant fromirrelevant information, identifyingmissing information, sequencingand prioritizing information, andobserving patterns.Also covered: NS1.2

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.

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

11-5


Probability

Vocabularyexperimenttrialoutcomesample spaceeventprobability

An is an activity in which results are observed. Eachobservation is called a , and each result is called an . The

is the set of all possible outcomes of an experiment.

An is any set of one or more outcomes. The

of an event is anumber from 0 (or 0%) to 1 (or100%) that tells you how likelythe event is to happen. You canwrite probability as a fraction, a decimal, or a percent.

• A probability of 0 means the event is impossible, or can never happen.

• A probability of 1 means the event is certain, or will always happen.

• The probabilities of all the outcomes in the sample space add up to 1.

Finding Probabilities of Outcomes in a Sample Space

Give the probability for each outcome.

The weather forecast shows a 30% chance of snow.P(snow) � 30% � 0.3

P(no snow) � 1 � 0.3 � 0.7, or 70%

probability

event

sample spaceoutcometrial

experiment

0.25

25%

0.5

50%

0.75

75%

1

1

100%

0

0

0%

Neverhappens

Happens abouthalf the time

Alwayshappens

14

12

34

E X A M P L E 1

Outcome Snow No snow

ProbabilityThe probability of anevent can be writtenas P(event).

Sample space Event of rollingan odd number

Outcome ofrolling a 6

1 2 3

4 56

CaliforniaStandards

Review of Grade 6 SDAP3.3 Represent

probabilities as ratios, proportions,decimals between 0 and 1, andpercentages between 0 and 100and verify that the probabilitiescomputed are reasonable; knowthat if P is the probability of anevent, 1-P is the probability of anevent not occurring.

Who uses this? Newscasters use probability whenreporting the weather. (See Example 1.)

Experiment Sample Space

Flipping a coin Heads, tails

Rolling a number cube 1, 2, 3, 4, 5, 6

Score 0 1 2 3 4 5

Probability 0.105 0.316 0.352 0.180 0.043 0.004


P(red) � �12� One-half of the spinner is red,

so �12� is a reasonable estimate.

P(yellow) � �14� One-fourth of the spinner is yellow,


P(blue) � �14� One-fourth of the spinner is blue,


Check The probabilities of all the outcomes must add to 1.

�12� � �

14� � �

14� � 1 ✔

To find the probability of an event, add the probabilities of all theoutcomes included in the event.

Finding Probabilities of Events

A quiz contains 3 multiple-choice questions and 2 true-falsequestions. Suppose you guess randomly on every question. Thetable below gives the probability of each score.

What is the probability of guessing 4 or more correct?

The event “4 or more correct” consists of the outcomes 4 and 5.P(four or more correct) � 0.043 � 0.004

� 0.047, or 4.7%

What is the probability of guessing fewer than 3 correct?

The event “fewer than 3 correct” consists of the outcomes 0, 1,and 2.

P(fewer than 3 correct) � 0.105 � 0.316 � 0.352

� 0.773, or 77.3%

Outcome Red Yellow Blue

Probability

2E X A M P L E

11-5 Probability 557

Think and Discuss

1. Give a probability for each of the following: usually, sometimes,always, never. Compare your values with the rest of your class.

2. Explain the difference between an outcome and an event.



KEYWORD: MT8CA 11-5

1. The weather forecast calls for a 60% chance of rain. Give the probabilityfor each outcome.

A game consists of randomly selecting 4 colored ducks from a pond andcounting the number of green ducks. The table gives the probability of each outcome.

2. What is the probability of selecting at most 1 green duck?

3. What is the probability of selecting more than 1 green duck?

4. Give the probability for each outcome.

Customers at Pizza Palace can order up to 5 toppings on a pizza. The tablegives the probabilities for the number of toppings ordered on a pizza.

5. What is the probability that at least 2 toppings are ordered?

6. What is the probability that fewer than 3 toppings are ordered?

GUIDED PRACTICE


Outcome Rain No rain

Probability

See Example 2

See Example 1

See Example 2

See Example 1

Number of Green Ducks 0 1 2 3 4

Probability 0.043 0.248 0.418 0.248 0.043

Number of Toppings 0 1 2 3 4 5

Probability 0.205 0.305 0.210 0.155 0.123 0.002

Outcome Red Blue Yellow Green

Probability


See page EP23.Use the table to find the probability of each event.

7. A, C, or E occurring 8. B or D occurring

9. A, B, D, or E occurring 10. A not occurring

Outcome A B C D E

Probability 0.306 0 0.216 0.115 0.363


Review of Grade 6 SDAP3.3

6SDAP3.3, 7NS2.1, 7AF3.3

11-5 Probability 559

Identify the sample space and the outcome shown for each experiment.

11. spinning a spinner 12. tossing two coins 13. rolling a number cube

14. Reasoning You are told there are 4 possible events that may occur. EventA has a 25% chance of occurring, event B has a probability of �

15�, and

events C and D have an equal likelihood of occurring. What steps wouldyou take in order to find the probabilities of events C and D?

15. Consumer Math A cereal company puts “prizes” in some of its boxes toattract shoppers. There is a 0.005 probability of getting two tickets to amovie theater, �

18� probability of finding a watch, 12.5% probability of

getting an action figure, and 0.2 probability of getting a sticker. What isthe probability of not getting any prize?

16. Give an example of an event that has 0 probability of occurring.

17. What’s the Error? Two people are playing a game. One of them says,“Either I will win or you will. The sample space contains two outcomes, sowe each have a probability of one-half.” What is the error?

18. Write About It Suppose an event has a probability of p. What can yousay about the value of p? What is the probability that the event will notoccur? Explain.

19. Challenge List all possible events in the sample space with outcomes A,B, and C.

20. Multiple Choice The local weather forecaster said there is a 60% chance of rain tomorrow. What is the probability that it will NOT rain tomorrow?

0.6 0.4 60 40

21. Gridded Response A sports announcer states that a runner has an 84% chance of winning a race. Give the probability, as a fraction in lowest terms, that the runner will NOT win the race.

Simplify the powers of 10. (Lesson 4-2)

22. 10�4 23. 10�1 24. 10�5 25. 10�7

Find the slope of the line through the given points. (Lesson 7-7)

26. R(8, 4), S(10, 1) 27. G(4, �3), H(5, 2) 28. A(�2, 5), B(�2, 4) 29. J(3, 2), K(�1, 2)

DCBA

11-6


In , the likelihood of an event is estimated by repeating anexperiment many times and observing thenumber of times the event happens. That number is divided by the total number of trials. The more times the experiment is repeated, the more accurate the estimate is likely to be.

Estimating the Probability of an Event

After 1000 spins of the spinner, the following information wasrecorded. Estimate the probability of the spinner landing on red.

probability � � �1206070� � 0.267

The probability of landing on red is about 0.267, or 26.7%.

A marble is randomly drawn out of a bag and then replaced. Thetable shows the results after 100 draws. Estimate the probabilityof drawing a yellow marble.

probability � � �12010� � 0.21

The probability of drawing a yellow marble is about 0.21, or 21%.

number of yellow marbles drawn��total number of draws

number of spins that landed on red��total number of spins

experimental probability

Experimental Probability

Vocabularyexperimental

probability

E X A M P L E 1

Outcome Blue Red Yellow

Spins 448 267 285

Outcome Green Red Yellow Blue White

Draws 12 35 21 18 14

CaliforniaStandards

Review of Grade 6SDAP3.2 Use data to estimatethe probability of future events(e.g., batting averages or numberof accidents per mile driven).Also covered: 6SDAP3.3

Who uses this? Insurance companiesestimate the probability of accidents bystudying accident rates for different typesof vehicles. (See Example 2.)

EXPERIMENTAL PROBABILITY

probability � number of times the event occurs��total number of trials

11-6 Experimental Probability 561

A researcher has been observing the types of vehicles passingthrough an intersection. Of the last 50 cars, 29 were sedans,9 were trucks, and 12 were SUVs. Estimate the probability thatthe next vehicle through the intersection will be an SUV.

probability � � �15

20� � 0.24 � 24%

The probability that the next vehicle through the intersection willbe an SUV is about 0.24, or 24%.

Safety Application

Use the table to compare the probability of being involved in afatal traffic crash in an SUV with being in a fatal traffic crash in a mid-size car.

probability �

probability of SUV � �751,75293� � 0.0023

probability of mid-size car � �20406,4433� � 0.0023

In 2004, an accident involving an SUV was just as likely to be fatal asan accident involving a mid-size car.

number of fatal crashes��total number of crashes

number of SUVs��total number of vehicles

Think and Discuss

1. Compare the probability in Example 1A of the spinner landing onred to what you think the probability should be.

2. Give a possible number of marbles of each color in the bag inExample 1B. Explain your reasoning.

Outcome Sedan Truck SUV

Observations 29 9 12

2E X A M P L E

Traffic Crashes in Ohio, 2004

Number of Total Number of Vehicle Class Fatal Crashes Crashes

Sub-compact cars 23 7,962

Compact cars 266 110,598

Mid-size cars 464 200,433

Full-size cars 161 76,570

Minivan 97 45,043

SUVs 172 75,593Source: Ohio Department of Public Safety



KEYWORD: MT8CA 11-6

1. A game spinner was spun 500 times. The spinner landed on A 170 times, B 244times, and C 86 times. Estimate the probability that the spinner will land on A.

2. A coin was randomly drawn from a bag and then replaced. After 300draws, it was found that 45 pennies, 76 nickels, 92 dimes, and 87 quartershad been drawn. Estimate the probability of drawing a quarter.

3. Use the table to compare the probability that astudent walks to school to the probability thata student bikes to school.

4. Use the table to compare the probability that a student takes the bus to school to theprobability that a student rides in a car toschool.

5. A researcher polled 260 students at a university and found that 83 of themowned a laptop computer. Estimate the probability that a randomlyselected college student owns a laptop computer.

6. A soccer player made 12 out of her last 58 shots on goal. Estimate theprobability that she will make her next shot on goal.

7. Stefan polled 113 students about the number ofsiblings they have. Use the table to compare theprobability that a student has one sibling to theprobability that a student has two siblings.

8. Use the table to compare the probability that astudent has no siblings to the probability that astudent has three siblings.

Use the table for Exercises 9–13.Estimate the probability of each event.

9. The batter hits a single.

10. The batter hits a double.

11. The batter hits a triple.

12. The batter hits a home run.

13. The batter makes an out.

GUIDED PRACTICE


See Example 2

See Example 1

See Example 1

See Example 2

Single

Result Number

20

Double 12

Home run 8

Walk 10

Out 28

Total 80

Triple 2


See page EP23.


Review of Grade 6SDAP3.2, SDAP3.3

Mode of Number ofTransportation Students

Bus 265

Car 313

Walk 105

Bike 87

Number of Number ofSiblings Students

0 14

1 45

2 27

3 15

4� 12

11-6 Experimental Probability 563

The strength of an earthquake is measured on the Richter scale. A major earthquake measures between 7 and 7.9 on the Richter scale, and a great earthquake measures 8 or higher. The table shows the number of major and great earthquakes per year during a 20-year period.

14. Estimate the probability that there willbe more than 15 major earthquakes inYear 21.

15. Estimate the probability that there will be fewer than 12 major earthquakes in Year 21.

16. Estimate the probability that there willbe no great earthquakes in Year 21.

17. Challenge Estimate the probability that there will be more than one major earthquake in the next month.

Number of Earthquakes

Year Major Great Year Major Great

1 13 1 11 22 3

2 5 1 12 14 1

3 11 0 13 16 0

4 8 0 14 11 1

5 6 1 15 18 0

6 12 0 16 14 1

7 11 0 17 15 1

8 23 0 18 13 0

9 15 1 19 14 1

10 13 2 20 14 2KEYWORD: MT8CA Quake

Science

6SDAP3.2, 7NS1.2, 7SDAP1.2

18. Multiple Choice A spinner was spun 220 times. The outcome was red 58 times. Estimate the probability of the spinner landing on red.

about 0.126 about 0.225 about 0.264 about 0.32

19. Short Response A researcher observed students buying lunch in a cafeteria. Of the last 50 students, 22 bought an apple, 17 bought a banana, and 11 bought apear. If 150 more students buy lunch, estimate the number of students who willbuy a banana. Explain.

Evaluate each expression for the given value of the variable. (Lesson 2-3)

20. 45.6 � x for x � �11.1 21. 17.9 � b for b � 22.3 22. r � (�4.9) for r � 31.8

23. What type of correlation would you expect to see between the number ofjackets purchased at a department store and the outside temperature?Explain. (Lesson 11-4)

DCBA

11-7

Vocabularytheoretical probabilityequally likelyfairmutually exclusivedisjoint events

THEORETICAL PROBABILITY FOR EQUALLY LIKELY OUTCOMES

Suppose there are n equally likely outcomes in the sample space of anexperiment.

• The probability of each outcome is �n1

�.

• The probability of an event is .number of outcomes in the event��n

E X A M P L E 1

Theoretical Probability


is used to estimate probabilities by making certain assumptions about an experiment. Suppose a sample space has 5 outcomes that are —that is, they all have the same probability x. The probabilities must add to 1.

x � x � x � x � x � 1

5x � 1

x � �15� The probability of any one outcome is �

15�.

A coin, die, or other object is called if all outcomes are equally likely.

Calculating Theoretical Probability

An experiment consists of rolling a fair number cube. Find theprobability of each event.

P(5)

The number cube is fair, so all 6 outcomes in the sample spaceare equally likely: 1, 2, 3, 4, 5, and 6.

P(5) � � �16�

P(even number)

There are 3 possible even numbers: 2, 4, and 6.

P(even number) � � �36� � �

12�

number of possible even numbers��6

number of ways to roll a 5��6

fair

equally likely

Theoretical probability

CaliforniaStandards

Review of Grade 6SDAP3.1 Represent all

possible outcomes for compoundevents in an organized way (e.g.,tables, grids, tree diagrams) andexpress the theoretical probabilityof each outcome.Also covered: 6SDAP3.3,6SDAP3.4

Why learn this? Theoretical probability canbe used to determine the likelihood of specificoutcomes in board games. (See Example 4.)

Suppose you roll two fair number cubes. Are all outcomes equally likely? If you look at the total shown on the number cubes, all outcomes are notequally likely. For example, there is only one way to get a total of 2, 1 � 1,but a total of 5 can be 1 � 4, 2 � 3, 3 � 2, or 4 � 1.

Calculating Probability for Two Fair Number Cubes

An experiment consists of rolling two fair number cubes. Find the probability of each event.

P(total shown � 1)

First find the sample space that has all outcomes equally likely.

There are 36 possible outcomes in the sample space. Then find the number of outcomes in the event “total shown � 1.”There is no way to get a total of 1. P(total shown � 1) � �3

06� � 0

P(at least one 6)

There are 11 outcomes in the event rolling “at least one 6.”

P(at least one 6) � �13

16�

Altering Probability

A bag contains 5 blue chips and 8 green chips. How many yellowchips should be added to the bag so that the probability ofdrawing a green chip is �

25�?

Adding chips will increase the number of possible outcomes.

�138� x� � �

25�

2(13 � x) � 8(5) Find the cross products.

26 � 2x � 40 Multiply.

� 26 � 26 Subtract 26 from both sides.��

�22x� � �

124� Divide both sides by 2.

x � 7

Seven yellow chips should be added to the bag.

2E X A M P L E

3E X A M P L E

11-7 Theoretical Probability 565

Set up a proportion. Let x be the number ofyellow chips.

Two events are , or , if they cannotboth occur in the same trial of an experiment. For example, rolling a 5and an even number on a number cube are mutually exclusive eventsbecause they cannot both happen at the same time.

Finding the Probability of Mutually Exclusive Events

Suppose you are playing a game and have just rolled doubles twotimes in a row. If you roll doubles again, you will lose a turn. Youwill also lose a turn if you roll a total of 3 because you are 3 spacesaway from the “Lose a Turn” square. What is the probability thatyou will lose a turn?

It is impossible to roll doubles and a total of 3 at the same time, so theevents are mutually exclusive. Add the probabilities of the events tofind the probability of losing a turn on the next roll.

The event “doubles” consists of six outcomes—(1, 1), (2, 2), (3, 3), (4, 4),(5, 5), and (6, 6).

P(doubles) � �366�

The event “total shown � 3” consists of two outcomes—(1, 2) and (2, 1).

P(total shown � 3) � �326�

P(losing a turn) � P(doubles) � P(total shown � 3)

� �366� � �3

26�

� �386�

The probability that you will lose a turn is �386� � �

29�, or about 22.2%.

disjoint eventsmutually exclusive

Think and Discuss

1. Describe a sample space for tossing two coins that has alloutcomes equally likely.

2. Give an example of an experiment in which it would not bereasonable to assume that all outcomes are equally likely.

3. Give an example of a fair experiment.

E X A M P L E 4


You can write theoutcome of a 1 onone cube and a 1 ona second cube as theordered pair (1, 1).

PROBABILITY OF MUTUALLY EXCLUSIVE EVENTS

Suppose A and B are two mutually exclusive events.

• P(both A and B will occur) � 0• P(either A or B will occur) � P(A) � P(B)


An experiment consists of rolling a fair number cube. Find the probability of each event.

1. P(odd number) 2. P(2 or 4)

An experiment consists of rolling two fair number cubes. Find theprobability of each event.

3. P(total shown � 10) 4. P(rolling two 2’s)

5. P(rolling two odd numbers) 6. P(total shown � 8)

7. What color should you shade the blank region so that theprobability of the spinner landing on that color is about �

12�?

8. Suppose you are playing a game in which two fair dice arerolled. To make the first move, you need to roll doubles or asum of 3 or 11. What is the probability that you will be able tomake the first move?

An experiment consists of rolling a fair number cube. Find the probability of each event.

9. P(9) 10. P(not 6)

11. P(� 5) 12. P(� 3)


13. P(total shown � 3) 14. P(at least one even number)

15. P(total shown � 0) 16. P(total shown � 9)

17. A bag contains 20 pennies, 25 nickels, and 15 quarters. How many dimesshould be added so that the probability of drawing a quarter is �

16�?

18. Suppose you are playing a game in which two fair dice are rolled. You need 9 to land on the finish by an exact count or 3 to land on a “roll again”space. What is the probability of landing on the finish or rolling again?

Three fair coins are tossed: a penny, a dime, and a quarter. Find the samplespace with all outcomes equally likely. Then find each probability.

19. P(TTH) 20. P(THH) 21. P(dime heads)

22. P(exactly 2 tails) 23. P(0 tails) 24. P(at most 1 tail)


KEYWORD: MT8CA 11-7

GUIDED PRACTICE


See Example 1

See Example 1

See Example 2

See Example 3

See Example 4

See Example 2

See Example 4

See Example 3


See page EP23.

11-7 Theoretical Probability 567


Review of Grade 6 SDAP3.1, SDAP3.3

Life Science

What color are your eyes? Can you roll your tongue? These traits are determined by the genes you inherited from your parents. A Punnett square shows all possible gene combinationsfor two parents whose genes are known.

To make a Punnett square, draw a two-by-two grid. Write the genes of one parent above the top row and the other parent along the side. Then fill in the grid as shown.

25. In the Punnett square above, one parent has the genecombination Bb, which represents one gene for brown eyesand one gene for blue eyes. The other parent has the genecombination bb, which represents two genes for blue eyes. If all outcomes in the Punnett square are equally likely, what is the probability of a child with the gene combination bb?

26. Make a Punnett square for two parents who both have the gene combination Bb.

a. If all outcomes in the Punnett square are equally likely, what is theprobability of a child with the gene combination BB?

b. The gene combinations BB and Bb will result in brown eyes, and thegene combination bb will result in blue eyes. What is the probabilitythat the couple will have a child with brown eyes?

27. Challenge The combinations Tt and TT represent the ability to roll your tongue, while tt means you cannot roll your tongue. Draw aPunnett square that results in a probability of �

12� that the child can roll

his or her tongue. Explain whether the parents can roll their tongues.

B b

b Bb bb

b Bb bb


6SDAP3.3, 7AF1.1, 7SDAP1.3

28. Multiple Choice A bag has 3 red marbles and 6 blue marbles in it. What is the probability of drawing a red marble?

1 �23� �

13� �

12�

29. Gridded Response On a fair number cube, what is the probability, written as a fraction, of rolling a 2 or higher?

30. The temperature in an oven is increasing at a rate of 50°F per minute, sothe temperature in the oven is 50 times the number of minutes. Write alinear function that describes the temperature in the oven over time.(Lesson 7-3)

Find the upper and lower quartiles and the median for each data set. (Lesson 11-3)

31. 42, 65, 45, 20, 60, 46, 56 32. 51, 21, 36, 68, 64, 42, 48, 66, 31

DCBA

A is made up of two or more separate events. Tofind the probability of a compound event, you need to know if theevents are independent or dependent.

Events are if the occurrence of one event doesnot affect the probability of the other. Events are if the occurrence of one does affect the probability of the other.

Classifying Events as Independent or Dependent

Determine if the events are dependent or independent.

a coin landing heads on one toss and tails on another toss

The result of one toss does not affect the result of the other, so the events are independent.

drawing a 6 from a deck of cards without replacing the card andthen drawing a 7

Once one card is drawn, the sample space changes. The events are dependent.

Finding the Probability of Independent Events

The spinner is divided into 5 equal sectors. Anexperiment consists of spinning the spinner 3 times.

What is the probability of spinning a 2 all 3 times?

The result of each spin does not affect the results of the otherspins, so the spin results are independent.

For each spin, P(2) � �15�.

P(2, 2, 2) � �15� � �

15� � �

15� � �1

125� � 0.008 Multiply.

dependent eventsindependent events

compound event

Vocabularycompound eventindependent eventsdependent events

E X A M P L E 1

FINDING THE PROBABILITY OF INDEPENDENT EVENTS

If A and B are independent events, then P(A and B) � P(A) � P(B).

5

4

3

21E X A M P L E 2

11-8 Independent and Dependent Events 569

CaliforniaStandards

Review of Grade 6SDAP3.5 Understand the

difference between independentand dependent events.Also covered: 6SDAP3.3,6SDAP3.4

Why learn this? You can use dependent events tofind the probability of selecting a specific type ofgranola bar from a box. (See Exercises 10 and 11.)

11-8 Independent and Dependent Events

An experiment consists of spinning thespinner 3 times. For each spin, all outcomesare equally likely.

What is the probability of spinning an evennumber all 3 times?

For each spin, P(even) � �25�.

P(even, even, even) � �25� � �

25� � �

25� � �1

825� � 0.064 Multiply.

What is the probability of spinning a 2 at least once?

Think: P(at least one 2) � P(not 2, not 2, not 2) � 1.

For each spin, P(not 2) � �45�.

P(not 2, not 2, not 2) � �45� � �

45� � �

45� � �1

6245� � 0.512 Multiply.

Subtract from 1 to find the probability of spinning at least one 2.

1 � 0.512 � 0.488

To calculate the probability of two dependent events occurring, dothe following:

1. Calculate the probability of the first event.

2. Calculate the probability that the second event would occur if the firstevent had already occurred.

3. Multiply the probabilities.

Suppose you draw 2 marbles without replacement from a bag that contains 3 purple and 3 orange marbles. On the first draw,

P(purple) � �36� � �

12�.

The sample space for the second draw depends on the first draw.

If the first draw was purple, then the probability of the second draw being purple is

P(purple) � �25�.

So the probability of drawing two purple marbles is

P(purple, purple) � �12� � �

25� � �

15�.

5

4

3

21

FINDING THE PROBABILITY OF DEPENDENT EVENTS

If A and B are dependent events, then P(A and B) � P(A) � P(B after A).

Before first draw

After first draw

Outcome of first draw Purple Orange

Sample space for 2 purple 3 purplesecond draw 3 orange 2 orange


Finding the Probability of Dependent Events

A jar contains 16 quarters and 10 nickels.

If 2 coins are chosen at random, what is the probability of getting2 quarters?

Because the first coin is not replaced, the sample space isdifferent for the second coin, so the events are dependent. Findthe probability that the first coin chosen is a quarter.

P(quarter) � �12

66� � �1

83�

If the first coin chosen is a quarter, then there would be 15 quarters and a total of 25 coins left in the jar. Find theprobability that the second coin chosen is a quarter.

P(quarter) � �12

55� � �

35�

�183� � �

35� � �

26

45� Multiply.

The probability of getting two quarters is �26

45�.

If 2 coins are chosen at random, what is the probability of getting2 coins that are the same?

There are two possibilities: 2 quarters or 2 nickels. The probabilityof getting 2 quarters was calculated in Example 3A. Now find theprobability of getting 2 nickels.

P(nickel) � �12

06� � �1

53� Find the probability that the first coin

chosen is a nickel.

If the first coin chosen is a nickel, there are now only 9 nickels and 25 total coins in the jar.

P(nickel) � �295� Find the probability that the second coin

chosen is a nickel.

�153� � �2

95� � �6

95� Multiply.

The events of 2 quarters and 2 nickels are mutually exclusive, soyou can add their probabilities.

�26

45� � �6

95� � �

36

35� P(quarters) � P(nickels)

The probability of getting 2 coins the same is �36

35�.

Think and Discuss

1. Give an example of a pair of independent events and a pair ofdependent events.

2. Tell how you could make the events in Example 1B independentevents.

E X A M P L E 3

Two mutuallyexclusive eventscannot both happenat the same time.




1. drawing a red and a blue marble at the same time from a bag containing 6 red and 4 blue marbles

2. drawing a heart from a deck of cards and a coin landing on tails

Each spinner is divided into 8 equal sectors. Anexperiment consists of spinning each spinner one time.

3. Find the probability that the first spinner landson yellow and the second spinner lands on 8.

A sock drawer contains 10 white socks, 6 black socks, and 8 blue socks.

4. If 2 socks are chosen at random, without replacing the first sock, what isthe probability of getting a pair of white socks?

5. If 3 socks are chosen at random, without replacing the first sock, what isthe probability of getting first a black sock, then a white sock, and then ablue sock?


6. drawing the name Roberto from a hat without replacing the name andthen drawing the name Paulo from the hat

7. rolling 2 fair number cubes and getting both a 1 and a 6

An experiment consists of tossing 2 fair coins, a penny and a nickel.

8. Find the probability of heads on the penny and tails on the nickel.

9. Find the probability that both coins will land the same way.

A box contains 4 berry, 3 cinnamon, 4 apple, and 5 oat granola bars.

10. If Dawn randomly selects 2 bars, without replacing the first bar, what isthe probability that they will both be cinnamon?

11. If two bars are selected randomly, without replacing the first bar, what isthe probability that they will be the same kind?

A box contains 6 red marbles, 4 blue marbles, and 8 yellow marbles.

12. Find P(yellow then red) if a marble is selected and then a second marbleis selected without replacing the first marble.

13. Find P(yellow then red) if a marble is selected and replaced and then asecond marble is selected.


KEYWORD: MT8CA 11-8

GUIDED PRACTICE




See Example 2

See Example 2

See Example 3

See Example 3

See Example 1

See Example 1

12

345

6

78



Review of Grade 6SDAP3.3, SDAP3.4,SDAP3.5

14. You roll a fair number cube twice. What is the probability of rolling two 3’sif the first roll is a 5? Explain.

15. School On a quiz, there are 5 true-false questions. A student guesses on all 5 questions. What is the probability that the student gets all 5 questions right?

16. Games The table shows the number of letter tilesavailable at the start of a word-making game. Thereare 100 tiles: 42 vowels and 58 consonants. To beginplay, each player draws a tile. The player with the tileclosest to the beginning of the alphabet goes first.

a. If you draw first, what is the probability that youwill select an A?

b. If you draw first and do not replace the tile, whatis the probability that you will select an E andyour opponent will select an I?

c. If you draw first and do not replace the tile, whatis the probability that you will select an E andyour opponent will win the first turn?

17. Write a Problem Write a problem about the probability of an event in aboard game, and then solve the problem.

18. Write About It In an experiment, two cards are drawn from a deck.How is the probability different if the first card is replaced before thesecond card is drawn than if the first card is not replaced?

19. Challenge Suppose you deal yourself 7 cards from a standard 52-carddeck. What is the probability that you will deal all red cards?

Letter Distribution

A-9 B-3 C-2

D-4 E-12 F-2

G-2 H-2 I-9

J-1 K-2 L-4

M-2 N-6 O-7

P-2 Q-1 R-6

S-5 T-6 U-5

V-2 W-2 X-1

Y-2 Z-1

SDAP1.3


20. Multiple Choice If A and B are independent events such that P(A) � 0.14and P(B) � 0.28, what is the probability that both A and B will occur?

0.0392 0.0784 0.24 0.42

21. Multiple Choice A number cube is rolled twice. What is the probabilityof getting a 2 on both rolls?

�13� �

14� �

19� �3

16�

22. Gridded Response A bag contains 8 red marbles and 2 blue marbles.What is the probability, written as a fraction, of choosing a red marble and then choosing a blue marble without replacing the first marble?

Find the lower and upper quartiles for each data set. (Lesson 11-3)

23. 19, 24, 13, 18, 21, 8, 11 24. 56, 71, 84, 66, 52, 11, 80

25. Find the volume, in square feet, of a sphere with radius 2 feet. Use 3.14 for π. (Lesson 10-6)

DCBA

DCBA

Quiz for Lessons 11-5 Through 11-8

11-5 Probability

Use the table to find the probability of each event.

1. P(C) 2. P(not B) 3. P(A or D) 4. P(A, B, or C)

11-6 Experimental Probability

A colored chip is randomly drawn from a box and then replaced.The table shows the results after 400 draws.

5. Estimate the probability of drawing a red chip.

6. Estimate the probability of drawing a green chip.

7. Use the table to compare the probability of drawing a blue chip to the probability of drawing a yellow chip.

11-7 Theoretical Probability

An experiment consists of rolling two fair number cubes. Find the probability of each event.

8. P(total shown � 7) 9. P(two 5’s) 10. P(two even numbers)

11-8 Independent and Dependent Events

11. An experiment consists of tossing 2 fair coins, a penny and a nickel. Find the probability of tails on the penny and heads on the nickel.

12. A jar contains 5 red marbles, 2 blue marbles, 4 yellow marbles, and 4 green marbles. If two marbles are chosen at random, what is the probability that they will be the same color?

Outcome Red Green Blue Yellow

Draws 76 172 84 68


Outcome A B C D

Probability 0.3 0.1 0.4 0.2

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 either themean, median, or mode to describe the typicalnumber of years that the club members have beenbowling. Then choose a measure to describe theirtypical score. Justify your choices.

2. Make a line plot of the number of years bowling.

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

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

Concept Connection 575


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: MT8CA Games

Buffon’s NeedleIf you drop a needle of a given length onto a woodenfloor with evenly spaced cracks, what is the probabilitythat it will land across a crack?

Comte de Buffon (1707–1788) posed this geometricprobability problem. To answer his question, Buffondeveloped a formula using � to represent the length of theneedle and d to represent the distance between the cracks.

probability � �π2

d��

To re-create this experiment, you need a paper clip andseveral evenly spaced lines drawn on a piece of paper. Make sure that the distance between the lines is greaterthan the length of the paper clip. Toss the paper clip ontothe piece of paper at least a dozen times. Divide thenumber of times the paper clip lands across a line by the number of times you toss the paper clip. Compare this quotient to the probability given by the formula.

The other interesting result of Buffon’s discovery is that youcan use the probability of the needle toss to estimate pi.

π ��probab

2�ility � d�

Toss the paper clip 20 times to find the experimentalprobability. Use this probability in the formula above, and compare the result to 3.14.

A

B

Probability Post-Up

PROJECT

Materials • 7 large sticky

notes• glue• markers

Fold sticky notes into an accordion booklet. Thenuse the booklet to record notes about probability.

DirectionsMake a chain of seven overlapping sticky notesby placing the sticky portion of one note on thebottom portion of the previous note. Figure A

Glue the notes together to make sure they stayattached.

Fold the notes in such a way that they look likean accordion. The folds should occur at thebottom edge of each note in the chain. Figure B

Write the name and number of the chapter onthe first sticky note.

Taking Note of the MathUse the sticky-note booklet to record keyinformation from the chapter. Be sure to include definitions, examples of probability experiments, and anything else that will help you review the material in the chapter.

4

3

2

1

It’s in the Bag! 577


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. The ___?___ of an event tells you how likely the event is to happen.

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

. . . . .533

. .543

. . . . . .569

. . . . . . . . . . . .549

. . . . .569

. . . . . . . .566

. . . . . . . .564

. . . . . . . . . . . . . . . .556

. . . . . . . . . . .556

. . . . . . . . .560

. . . . . . . . . . . . . . . . . .564

. . . .569

. . . . . . . . .549

. . . . . . . . . . . . . .532

. . . . . . . . .542

. . . . . . . . . . . .543

. . . . . . . . . . . . . . . . .537

. . . . . . . . . . . . . . .537

. . . . . . . . . . . .543

. . . . . . . . . . . . . . . . .537

. . . . .566

. . .549

. . . . . . . . .549

. . . . . . . . . . . . .556

. . . . . . . . . . . . . . .538

. . . .549

. . . . . . . . . . .556

. . . . . . . . . . . . . . . . .537

. . . . . . . . .556

. . . . . . . . . . . .548

. . . . .533

. . . . . . . . .564

. . . . . . . . . . . . . . . . .556

. . . . . . . . .542upper quartile

trial

theoreticalprobability

stem-and-leaf plot

scatter plot

sample space

range

probability

positive correlation

outlier

outcome

no correlation

negative correlation

mutually exclusive

mode

minimum

median

mean

maximum

lower quartile

line plot

line of best fit

independent events

fair

experimentalprobability

experiment

event

equally likely

disjoint events

dependent events

correlation

compound event

box-and-whisker plot

back-to-backstem-and-leaf plot

SDAP1.1Line Plots and Stem-and-Leaf Plots (pp. 532–536)11-1

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.

4. Ages of People at a Skate Park

12 13 13 14 12 11

14 15 13 13 12 13

0

x

xxx

xx

xxxxxxx x

x

1 2 3 4 5 6 7 8 9 10

5 10 15 200

SDAP1.3Mean, Median, Mode, and Range (pp. 537–541)11-2

E X A M P L E EXERCISES

■ Find the mean, median, mode, and rangeof the data set 3, 7, 10, 2, and 3.

Mean: 3 � 7 � 10 � 2 � 3 � 25 �255� � 5

Median: 2, 3, 3, 7, 10

Mode: 3

Range: 10 � 2 � 8

Find the mean, median, mode, and rangeof each data set.

5. 324, 233, 324, 399, 233, 299

6. 48, 39, 27, 52, 45, 47, 49, 37

Box-and-Whisker Plots (pp. 542–546)11-3


■ 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

minimum: 7

lower quartile: �10 �2

14� � 12

median: 17

upper quartile: �18 �2

20� � 19

maximum: 20


7. 56, 56, 56, 59, 63, 68, 68, 73, 73, 73

8. 87, 87, 80, 72, 85, 82, 53, 65, 65

9. 80, 80, 80, 82, 85, 87, 87, 90, 90, 90

SDAP1.2Scatter Plots (pp. 548–552)11-4


■ Identify the correlation you wouldexpect to see between the age of abattery in a flashlight and the intensityof the flashlight beam. Explain.

Negative correlation; as the age of thebattery increases, the intensity of theflashlight beam decreases.

Identify the correlation you would expectto see between each pair of data sets.Explain.

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

11. height from which an object is droppedand time the object takes to hit theground

Study Guide: Review 579

SDAP1.1, SDAP1.3


6SDAP3.3

6SDAP3.2, 6SDAP3.3

Probability (pp. 556–559)11-5


■ Of the garbage collected in a city, it isexpected that about �

15� of the garbage

will be recycled. Give the probability foreach outcome.

P(recycled) � �15� � 0.2 � 20%

P(not recycled) �1 � �15� � �

45� � 0.8 � 80%


12. About 85% of the people attending aband’s CD signing have already heardthe CD.

Outcome Recycled Not Recycled

Probability Outcome Heard Not Heard

Probability

Experimental Probability (pp. 560–563)11-6


■ The table shows the results of spinning aspinner 72 times. Estimate the probabilityof the spinner landing on red.

probability � �27

82� � �1

78� � 0.389 � 38.9%

13. The table shows the result of rolling anumber cube 80 times. Estimate theprobability of rolling a 4.

Theoretical Probability (pp. 564–568)11-7


■ A fair number cube is rolled once. Findthe probability of getting a 4.

P(4) � �16�

14. A marble is drawn at random from abox that contains 8 red, 15 blue, and 7 white marbles. What is the probabilityof getting a red marble?

Independent and Dependent Events (pp. 569–573)11-8


■ Two marbles are drawn from a jarcontaining 5 blue marbles and 4 green.What is P(blue, green) if the first marbleis not replaced?

15. A fair number cube is rolled four times.What is the probability of getting a 6 allfour times?

16. Two cards are drawn at random from adeck that has 26 red and 26 black cards.What is the probability that the firstcard is red and the second card is black?

P(blue) P(green) P(blue, green)

Not replaced �59� �

48� �

2702� � 0.28

6SDAP3.1, 6SDAP3.3, 6SDAP3.4

6SDAP3.3,6SDAP3.4, 6SDAP3.5

Outcome White Red Blue Black

Spins 18 28 12 14

Outcome 1 2 3 4 5 6

Rolls 13 15 10 12 5 25

Chapter 11 Test 581

Use the data set 12, 18, 12, 22, 28, 23, 32, 10, 29, and 36 for problems 1–4.

1. Make a line plot of the data.

2. Make a stem-and-leaf plot of the data.

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

4. How would the outlier 57 affect the mean of the data set?


5. 62, 63, 62, 64, 68, 62, 62 6. 104, 68, 90, 96, 101, 106, 95, 88


7. 62, 60, 77, 66, 92, 87, 62, 60, 64 8. 2.2, 6.8, 6.4, 8, 6.5, 4.2, 6.5, 5, 8

9. Use the given data to make a scatter plot. Write positive correlation,negative correlation, or no correlation to describe the relationship. Explain.

10. Identify the correlation that you would expect to see between the speed of a car and the time required to travel a certain distance. Explain.

Use the table to find the probability of each event.

11. P(D)

12. P(not A)

13. P(B or C)

A coin is randomly drawn from a box and then replaced. The tableshows the results.

14. Estimate the probability of each outcome.

15. Estimate P(penny or nickel).

16. Estimate P(not dime).


17. P(total shown � 3) 18. P(rolling two 6’s) 19. P(total � 2)

20. A jar contains 6 red tiles, 2 blue, 3 yellow, and 5 green. If two tiles are chosen at random, what is the probability that they both will be green?

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

Outcome Penny Nickel Dime Quarter

Probability 26 35 19 20

Outcome A B C D

Probability 0.2 0.2 0.1 0.5


Multiple Choice Which statement is true for the given spinner?

The probability of spinning green is �13�.

The probability of spinning blue is �16�.

The probability of spinning white is the same asthe probability of spinning green.

The probability of spinning green is the same asthe probability of spinning yellow or white.

D

C

B

A

Multiple Choice: Answering Context-Based Test ItemsFor some test items, you cannot answer just by reading the problemstatement. You will need to read each option carefully to determine thecorrect response. Review each option and eliminate those that are false.

Read each option carefully. Eliminate options that are false.

Option A: Find the probability of spinning green.

P(green) � �36�, or �

12�

Option A is false.

Option B: Find the probability of spinning blue.

P(blue) � �06�, or 0

Option B is false.

Option C: Find the probabilities and compare.

P(white) � �26�, or �

13� P(green) � �

36�, or �

12�

�13� � �

12�, so P(white) � P(green)

Option C is false.

Option D: Find the probabilities and compare.

P(green) � �36�, or �

12� P(white or yellow) � �

26� � �

16� � �

36�, or �

12�

�12� � �

12�, P(green) � P(white or yellow)

Option D is true. It is the correct response.

Read each test item and answer thequestions that follow.

1. What property do you have to use tosolve each equation?

2. What two methods could you use todetermine if x � 3 is a solution of oneof the equations?

3. Which is the correct option? Explain.

4. What does multiple mean? What aremultiples of 3? What are multiples of 2?

5. How many numbers are less than 4 onthe number cube? How many numbersare greater than 5?


Item AWhich equation has a solution of x � 3?

� 6 � 3(x � 1)

�12 � �32� (�2x � 2)

2(x � 6) � 3

�2(x � 6) � 3D

C

B

A

Item CWhich inequality has 0 as a part of itssolution set?

�3y � �6 4 � 9y � 13

8a � 3 � 7 ��56t� � 5DB

CA

Item BAn experiment consists of rolling a fairnumber cube labeled 1 to 6. Whichstatement is true?

P(odd) � P(even)

P(multiple of 3) � P(multiple of 2)

P(7) � 1

P(less than 4) � P(greater than 5)D

C

B

A

Item DA poll was taken at Jefferson MiddleSchool. Which statement is true for thegiven data?

The probability that a student atJefferson Middle School does notlike dramas best is �

45�.

The probability that a student likescomedies best is �

12

75�.

Out of a population of 1200students, you can predict that 280students will like science fictionmovies best.

The probability that a student likesaction movies best is �3

81�.

D

C

B

A

9. How can you find the probability of anevent not occurring?

10. How can you use probability to make aprediction?


Favorite Type Number ofof Movie Students

Drama 25

Comedy 40

Science fiction 28

Action 32

Be sure to review all of the answeroptions carefully before you make your choice.

Strategies for Success 583

7. What must you remember to do if youmultiply or divide both sides of aninequality by a negative number?



1. In a box containing marbles, 78 areblue, 24 are orange, and the rest aregreen. If the probability of selecting agreen marble is �

25�, how many green

marbles are in the box?

30 102

68 150

2. 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

3. 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 in. 3 in.

2 in. 4 in.

4. What is the value of (�2 � 4)3 � 30?

�215 217

�8 219

5. About what percent of 75 is 55?

25% 75%

66% 135%

6. Which does NOT describe ��

255�

�?

real integer

rational irrational

7. The figure formed by the vertices (�2, 5), (2, 5), (4, �1), and (0, �1) canbe best described by which type ofquadrilateral?

square parallelogram

rectangle trapezoid

8. How many vertices are in the prismbelow?

7 10

8 12

9. A triangular reflecting pool has an areaof 350 ft2. If the height of the triangleis 25 ft, what is the length of thehypotenuse to the nearest tenth?

28.7 ft 38.9 ft

37.5 ft 42.3 ft

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

x � �15 x � �3

x � �7 x � �1DB

CA

DB

CA

DB

CA

DB

CA

DB

CA

DB

CA

DB

CA

DB

CA

DB

CA

DB

CA

Cumulative Assessment, Chapters 1–11Multiple Choice

25 ft x

KEYWORD: MT8CA Practice

Cumulative Assessment, Chapters 1–11 585

11. If the diameter of the dartboard is 10 in., what is the area of the 50-pointportion, to the nearest tenth of asquare inch?

3.1 in2 9.4 in2

6.3 in2 25.1 in2

Gridded Response

12. Emma buys a refrigerator on sale for$665. This is a 30% discount. What isthe original price, in dollars, of therefrigerator?

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

14. What is the probability of rolling aneven number on a number cube andtossing heads on a coin?

15. The function y � �16t2 � 180 modelsthe distance an object falls when it isdropped from the top of a building 180 ft tall in t seconds. How many feetdoes the stone fall after 2 seconds?

16. What is the value of x for the equation 7 � �

23�x � 3?

Short Response

17. Two numbers have a sum of 58. Twicethe first number is 8 more than thesecond number.a. Write an equation that can be used

to find the two numbers.b. What are the two numbers? Show

your work.

18. A function is graphed below.

a. Write an inequality expressing theinput values and an inequalityexpressing the output values.

b. Determine whether the rate ofchange is constant or variable.

Extended Response

19. 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, and 5 miles.

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

DB

CA

2 in.

25 points 50 points

100 points 2 in.

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

16

15.14.82r � 1

VZ

J

Q

T

Distance (mi)

Spee

d (m

i/h)

87654321

10 2 3 4 5 6

Data, Statistics, and...

Documents

Transcript of Data, Statistics, and...