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Chapter 12 Resource Masters

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Copyright by The McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such materials be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with the Glencoe Algebra 2 program. Any other reproduction, for sale or other use, is expressly prohibited.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240

ISBN: 978-0-07-890537-7MHID: 0-07-890537-0

Printed in the United States of America.

2 3 4 5 6 7 8 9 10 045 14 13 12 11 10 09

CONSUMABLE WORKBOOKS Many of the worksheets contained in the Chapter Resource Masters are available as consumable workbooks in both English and Spanish.

ISBN10 ISBN13Study Guide and Intervention Workbook 0-07-890861-2 978-0-07-890861-3Homework Practice Workbook 0-07-890862-0 978-0-07-890862-0

Spanish VersionHomework Practice Workbook 0-07-890866-3 978-0-07-890866-8

Answers For Workbooks The answers for Chapter 12 of these workbooks can be found in the back of this Chapter Resource Masters booklet.

StudentWorks PlusTM This CD-ROM includes the entire Student Edition text along with the English workbooks listed above.

TeacherWorks PlusTM All of the materials found in this booklet are included for viewing, printing, and editing in this CD-ROM.

Spanish Assessment Masters (ISBN10: 0-07-890869-8, ISBN13: 978-0-07-890869-9) These masters contain a Spanish version of Chapter 12 Test Form 2A and Form 2C.

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Chapter 12 iii Glencoe Algebra 2

Teachers Guide to Using the Chapter 12 Resource Masters ......................................... iv

Chapter ResourcesStudent-Built Glossary ....................................... 1Anticipation Guide (English) .............................. 3Anticipation Guide (Spanish) ............................. 4

Lesson 12-1Experiments, Surveys, and Observational StudiesStudy Guide and Intervention ............................ 5Skills Practice ................................................... 7Practice ............................................................ 8Word Problem Practice ..................................... 9Enrichment ...................................................... 10

Lesson 12-2Statistical AnalysisStudy Guide and Intervention .......................... 11Skills Practice .................................................. 13Practice ............................................................ 14Word Problem Practice ................................... 15Enrichment ...................................................... 16Graphing Calculator Activity ............................ 17

Lesson 12-3Conditional ProbabilityStudy Guide and Intervention .......................... 18Skills Practice .................................................. 20Practice .......................................................... 21Word Problem Practice ................................... 22Enrichment ...................................................... 23

Lesson 12-4Probability and Probability DistributionsStudy Guide and Intervention .......................... 24Skills Practice .................................................. 26Practice .......................................................... 27Word Problem Practice ................................... 28Enrichment ...................................................... 29

Lesson 12-5The Normal DistributionStudy Guide and Intervention .......................... 30Skills Practice .................................................. 32Practice .......................................................... 33Word Problem Practice ................................... 34Enrichment ...................................................... 35

Lesson 12-6Hypothesis TestingStudy Guide and Intervention .......................... 36Skills Practice .................................................. 38Practice .......................................................... 39Word Problem Practice ................................... 40Enrichment ...................................................... 41TINSpire Calculator Activity .......................... 42

Lesson 12-7Binomial DistributionsStudy Guide and Intervention .......................... 43Skills Practice .................................................. 45Practice .......................................................... 46Word Problem Practice ................................... 47Enrichment ...................................................... 48

AssessmentStudent Recording Sheet ................................ 49Rubric for Scoring Extended Response .......... 50Chapter 12 Quizzes 1 and 2 ........................... 51Chapter 12 Quizzes 3 and 4 ........................... 52Chapter 12 Mid-Chapter Test .......................... 53Chapter 12 Vocabulary Test ........................... 54Chapter 12 Test, Form 1 ................................. 55Chapter 12 Test, Form 2A ............................... 57Chapter 12 Test, Form 2B ............................... 59Chapter 12 Test, Form 2C .............................. 61Chapter 12 Test, Form 2D .............................. 63Chapter 12 Test, Form 3 ................................. 65Chapter 12 Extended-Response Test ............. 67Standardized Test Practice ............................. 68Unit 4 Test ....................................................... 71

Answers ........................................... A1A34

Contents

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Chapter 12 iv Glencoe Algebra 2

Teachers Guide to Using the Chapter 12 Resource Masters

The Chapter 12 Resource Masters includes the core materials needed for Chapter 12. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing on the TeacherWorks PlusTM CD-ROM.

Chapter ResourcesStudent-Built Glossary (pages 12) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. Give this to students before beginning Lesson 12-1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson.

Anticipation Guide (pages 34) This master, presented in both English and Spanish, is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their perceptions have changed.

Lesson ResourcesStudy Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Check Your Progress exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent.

Skills Practice This master focuses more on the computational nature of the lesson. Use as an additional practice option or as homework for second-day teaching of the lesson.

Practice This master closely follows the types of problems found in the Exercises section of the Student Edition and includes word problems. Use as an additional practice option or as homework for second-day teaching of the lesson.

Word Problem Practice This master includes additional practice in solving word problems that apply the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson.

Enrichment These activities may extend the concepts of the lesson, offer an histori-cal or multicultural look at the concepts, or widen students perspectives on the mathematics they are learning. They are written for use with all levels of students.

Graphing Calculator, TINspire, or Spreadsheet Activities These activities present ways in which technology can be used with the concepts in some lessons of this chapter. Use as an alternative approach to some concepts or as an integral part of your lesson presentation.

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Chapter 12 v Glencoe Algebra 2

Assessment OptionsThe assessment masters in the Chapter 12 Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assessment.

Student Recording Sheet This master corresponds with the standardized test practice at the end of the chapter.

Extended Response Rubric This master provides information for teachers and students on how to assess performance on open-ended questions.

Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter.

Mid-Chapter Test This 1-page test provides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions.

Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and 10 questions to assess students knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests.

Leveled Chapter Tests

Form 1 contains multiple-choice questions and is intended for use with below grade level students.

Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

Form 3 is a free-response test for use with above grade level students.

All of the above mentioned tests include a free-response Bonus question.

Extended-Response Test Performance assessment tasks are suitable for all students. Sample answers and a scoring rubric are included for evaluation.

Standardized Test Practice These three pages are cumulative in nature. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and short-answer free-response questions.

Answers The answers for the Anticipation Guide

and Lesson Resources are provided as reduced pages.

Full-size answer keys are provided for the assessment masters.

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Chapter 12 1 Glencoe Algebra 2

12

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 12. As you study the chapter, complete each terms definition or description. Remember to add the page number where you found the term. Add these pages to your Algebra Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term

Found on Page Definition/Description/Example

biased

conditional probability

confidence interval

control group

expected value

experiment

inferential statistics

normal distribution

null hypothesis

observational study

parameter

(continued on the next page)

Student-Built Glossary

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Student-Built Glossary12

Vocabulary Term Foundon Page Definition/Description/Example

population

probability

random variable

relative frequency

sample

skewed distribution

statistic

survey

treated group

unbiased


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Before you begin Chapter 12

Read each statement.

Decide whether you Agree (A) or Disagree (D) with the statement.

Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).

After you complete Chapter 12

Reread each statement and complete the last column by entering an A or a D.

Did any of your opinions about the statements change from the first column?

For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.

Step 2

Step 1

Anticipation GuideProbability and Statistics

12

STEP 1 A, D, or NS Statement

STEP 2 A or D

1. A sample space is a partial list of possible outcomes of an experiment.

2. Two events are called independent if choosing one does not affect choosing the other.

3. Since order is not important in a combination, an outcome ab is the same as an outcome ba.

4. The odds of an event occurring can be expressed as a ratio of the number of successes to the total number of outcomes.

5. If two events are dependent, then the probability of both events occurring is the product of the probabilities of each event.

6. If a set of data contains outliers, the median would be a good choice to represent the set.

7. Measures of variation are the differences between consecutive values in the set.

8. The curve representing a normal distribution is symmetric. 9. The Binomial Theorem can be used to find probabilities only

when there are two possible outcomes.

10. Asking people in a music store how many hours they spend listening to music to determine how many hours people in the city listen to music is an example of an unbiased survey.


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Captulo 12 4 lgebra 2 de Glencoe

Paso 2

Paso 1 Antes de comenzar el Captulo 12

Lee cada enunciado.

Decide si ests de acuerdo (A) o en desacuerdo (D) con el enunciado.

Escribe A o D en la primera columna O si no ests seguro(a) de la respuesta, escribe NS (No estoy seguro(a).

Despus de completar el Captulo 12

Vuelve a leer cada enunciado y completa la ltima columna con una A o una D.

Cambi cualquiera de tus opiniones sobre los enunciados de la primera columna?

En una hoja de papel aparte, escribe un ejemplo de por qu ests en desacuerdo con los enunciados que marcaste con una D.

Ejercicios preparatoriosProbabilidad y estadstica

12

PASO 1 A, D, o NS Statement

PASO 2 A o D

1. Un espacio muestral es una lista parcial de resultados posibles de un experimento.

2. A dos eventos se les llama independientes si al escoger uno, no afecta escoger el otro.

3. Dado que el orden no es importante en una combinacin, un resultado ab es lo mismo que un resultado ba.

4. Las posibilidades de que ocurra un evento se pueden expresar como la razn del nmero de xitos al nmero total de resultados.

5. Si dos eventos son dependientes, entonces la probabilidad de que ocurran ambos eventos es el producto de las probabili-dades de cada evento.

6. Si un conjunto de datos contiene valores atpicos, la mediana sera una buena opcin para representar el conjunto.

7. Las medidas de variacin son las diferencias entre valores consecutivos en el conjunto.

8. La curva que representa una distribucin normal es simtrica. 9. Se puede usar el teorema del binomio para calcular

probabilidades slo cuando existen dos resultados posibles.

10. Preguntarle a la gente en una disquera cuntas horas escuchan msica, para determinar as el nmero de horas que la gente en la ciudad escucha msica, es un ejemplo de una encuesta insesgada.


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12-1

Surveys, Studies, and ExperimentsTerm Definition Example

Survey a means of obtaining information from a population or a sample of the population

taking a poll to learn who people will vote for in an upcoming election

Observational Study an examination in which individuals are observed and no attempt is made to influence the results

observing a group of 100 people, 50 of whom have been taking a treatment; collecting, analyzing, and interpreting the data

Experiment an operation in which something is intentionally done to people, animals, or objects, and then the response is observed

studying the differences between two groups of people, one of which receives a treatment and the other a placebo

State whether the following situation represents an experiment or an observational study. If it is an experiment, then identify the control group and treatment group. Then determine whether there is bias.

Find twenty males and randomly separate them into two groups. One group will receive a new trial medication and the other receives a placebo.

This is an experiment. The group receiving the medication is the treatment group, while the group receiving the placebo is the control group. This is unblased.

Determine whether the following situation calls for a survey, an observational study, or an experiment. Explain the process.

You want to know how students and parents feel about school uniforms.

This calls for a survey. It is best to ask a random sample of students and a random sample of parents to give their opinions.

ExercisesState whether each situation represents an experiment or an observational study. If it is an experiment, then identify the control group and treatment group. Then determine whether there is bias.

1. Find 300 students and randomly split them into two groups. One group practices basketball three times per week and the other group does not practice basketball at all. After three months, you interview the students to find out how they feel about school.

2. Find 100 students, half of whom participated on the school math team, and compare their grade point average.

Study Guide and Intervention Experiments, Surveys, and Observational Studies

Example 1

Example 2

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12-1

Distinguish Between Correlation and Causation

Term Definition Example

correlation When one event happens, the other is more likely.

When the pond is frozen, it is more likely to snow.

causation One event is the direct cause of another event.

Turning on the light makes a room brighter.

Determine whether the following statement shows correlation or causation. Explain your reasoning.

Children who live in very large houses usually get larger allowances than children who live in small houses.

Correlation; there is no reason to assume that the size of their house causes children to receive more allowance. Children living in both a large house and getting a large allowance could be the result of a third factorthe amount of money the parents have.

Determine whether the following statements show correlation or causation. Explain.

1. If I jog in the rain, I will get sick.

3. If you lose a library book, you will have to pay a fine.

5. If I miss a day of school, I will not earn the perfect attendance award.

2. Studies have shown that eating more fish will improve your math grade.

4. Reading a diet book will make you lose weight.

6. Owning an expensive car will make me earn lots of money.

Study Guide and Intervention (continued)Experiments, Surveys, and Observational Studies

Example

Exercises

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12-1

State whether each situation represents an experiment or an observational study. Identify the control group and the treated group. If it is an experiment, determine whether there is bias.

1. Find 200 people at a mall and randomly split them into two groups. One group tries a new pain reliever medicine, and the other group tries a placebo.

2. Find 100 students, half of whom play sports, and compare their SAT scores.

Determine whether the following situation calls for a survey, an observational study, or an experiment. Explain the process.

3. You want to find opinions on the best computer game to buy.

4. You want to see if students who have a 4.0 grade point average study more than those who do not.


5. When a traffic light is red, a driver brings her car to a stop.

6. Studies have shown that students who are confident before a test raises test scores.

7. If I practice the saxophone every day, I will make the school jazz band.

8. If water is heated to 100 Celsius, it will boil.

Skills PracticeExperiments, Surveys, and Observational Studies

Exercises


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12-1 Practice Experiments, Surveys, and Observational Studies

State whether each situation represents an experiment or an observational study. If it is an experiment, then identify the control group and treatment group. Then determine whether there is bias.

1. Find 300 students, half of whom are on the chess team, and compare their grade point averages.

2. Find 1000 people and randomly split them into two groups. Give a new vitamin to one group and a placebo to the other group.

Determine whether each situation call for a survey, an observational study, or an experiment. Explain the process.

3. You want to compare the health of students who walk to school to the health of students who ride the bus.

4. You want to find out if people who eat a candy bar immediately before a math test get higher scores than people who do not.


5. If I jog every day, I can complete a marathon in three hours.

6. When there are no clouds in the sky, it does not rain.

7. Studies show that taking a multivitamin leads to a longer life.

8. If I study for three hours, I will earn a grade of 100% on my history test.


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12-1

1. SURGERY A new technique for knee surgery includes fitting knees with titanium and plastic caps instead of cutting through the muscles and tendons in the knee. A study observes the recovery time after 100 knee surgeries, half of which use the new knee surgery technique. Which group of surgeries is the control group?

2. SPORTS DRINKS A sports drink company gives out free samples of their new sports drink at the mall. They record the number of teens versus the number of adults that take the sample. Does this situation represent an experiment, an observational study, or a survey?

3. ELECTION At the Democratic National Convention, people are asked if they are going to vote for a Democratic or Republican candidate for president.

Who Do You Support for President?

Democratic Candidate

Republican Candidate

Other Candidate

94% 3% 3%

Is this a biased or unbiased survey?

4. COUNTRY CLUB A research study finds that 75% of the members of an exclusive country club are either doctors or lawyers. Does this demonstrate correlation or causation?

5. VIDEO GAMES A behavioral scientist studies the influence of violent video games on teens.

a. Describe an observational study the scientist can perform to study the influence of violent video games on teens.

b. Describe an experiment the scientist can set up to test the influence of violent video games on teens.

c. How can the scientist show that violence is caused by video games, and is not just a correlation?

Word Problem Practice Experiments, Surveys, and Observational Studies


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12-1 EnrichmentStratified Surveys

In some situations, researchers use stratified surveys instead of random surveys. Stratified surveys can sample a very diverse population more accurately than random surveys.

Before conducting a stratified survey, the researchers divide the population into distinct subpopulations called strata. For each stratum, researchers take a sample survey and use the sample to estimate the results for that stratums overall population.

The 2294 eleventh and twelfth grade students at a high school can be classified into the following four subgroups:

Eleventh-grade males = 576 Eleventh-grade females = 530

Twelfth-grade males = 600 Twelfth-grade females = 588

A sample of 500 students will be surveyed. Determine how many students in each subgroup should be surveyed.

Step 1 Calculate the percentage in each group.Eleventh-grade males = 576 2294 = 25.11%Eleventh-grade females = 530 2294 = 23.10%Twelfth-grade males = 600 2294 = 26.16%Twelfth-grade females = 588 2294 = 25.63%

Step 2 Multiply each percentage by the size of the sample.Eleventh-grade males = (25.11%)(500) = 125.55 = 126 studentsEleventh-grade females = (23.10%)(500) = 115.5 = 116 studentsTwelfth-grade males = (26.16%)(500) = 130.8 = 131 studentsTwelfth-grade females = (25.63%)(500) = 128.15 = 128 students

Exercises

For each of the following exercises, determine how many people in each subgroup should be surveyed.

1. Population of students in the Drama Club:Females: 18, Males: 18Total number of the sample to be surveyed = 20

2. Population of voters in the presidential election:Republican females = 100,000, Republican males = 195,000, Democratic females = 150,000, Democratic males = 125,000, Independent females = 52,000, Independent males = 31,000Total number of the sample to be surveyed = 100

Example


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Study Guide and InterventionStatistical Analysis

Measures of Central Tendency

Term Definition Best Used When

Meansum of the data divided by number of items in the data

the data set has no outliers

Medianthe middle number or mean of two middle numbers of the ordered data

the data set has outliers but no big gaps in the middle of the data

Modethe number or numbers that occur most often

the data has many repeated numbers

Margin of Sampling Error 1

n , for a random sample of

n items

estimating how well a sample represents the whole population

Which measure of central tendency best represents the data and why?

{2.1, 21.5, 22.3, 22.8, 23.1, 159.4}

There are outliers, but no large gaps in the middle, the median best represents this data.

What is the margin of sampling error and the likely interval that contains the percentage of the population?

Of 400 people surveyed in a national poll, 51% say they will vote for candidate Gonzlez.

Since 400 people are surveyed, the margin of sampling error is 1 400

or 5%. The

percentage of people who are likely to vote for candidate Gonzlez is the percentage found in the survey plus or minus 5%, so the likely interval is from 46% to 56%.

Exercises

Which measure of central tendency best represents the data, and why?

1. {45, 16, 30, 45, 29, 45} 2. {100, 92, 105, 496, 77, 121, 65, 99, 72}

3. {2.5, 99.5, 110.5, 76, 88.5, 105, 73, 113, 92, 72.5}

4. {60, 50, 55, 62, 44, 65, 51}

5. BOOKS A survey of 28 random people found that 40% read at least three books each month.What is the margin of sampling error? What is the likely interval that contains the percentage of the population that reads at least three books each month?

Example 1

Example 2

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Study Guide and Intervention (continued)Statistical Analysis

Measures of Variation

Standard Deviation Formulas

Variable Formula

For Samples s

k = 1

n

(xn -

x ) 2

n - 1

For Populations

k = 1

n

(xn - ) 2 n

For the following data, determine whether it is a sample or a population. Then find the standard deviation of the data. Round to the nearest hundredth.

The test scores of twelve students in a college mathematics course are displayed below.

Test Scores of Twelve Students Enrolled in a College Mathematics Course

61 99 75 83 92 69

77 94 73 65 98 91

Because the scores of all 12 students enrolled are given, this is a population. Find the mean.

=

1

12

12 = 977

12 = 81.42

Next, take the sum of the squares of the differences between each score and the mean. [ (61 81.42)2 + (99 81.42)2 + (75 81.42)2 + + (91 81.42)2] = 2,095.55

Putting this into the standard deviation formula,

2095.55 12

13.21.

Exercises

1. Determine whether each is a sample or a population. Then find the standard deviation of the data. Round to the nearest hundredth.

a. The Test Scores of Some of the Females in a College History Course88 91 82 95 76 88

75 94 92 85 82 90

b. The Age of All Students in the Chess Club14 17 15 14 15 16

Example

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12-2


1. {10.2, 11.5, 299.7, 15.5, 20}

3. {200, 250, 225, 25, 268, 250,7}

2. {75, 60, 60, 71, 74.5, 60, 67, 72.5}

4. {410, 405, 397, 450, 376, 422, 401}

Determine whether the following represents a population or a sample.

5. a school lunch survey that asks every fifth student that enters the lunch room

7. a list of the test scores of all the students in a class

6. Tenth graders at a high school are surveyed about school athletics.

8. a list of the scores of 1000 students on an SAT test

9. MOVIES A survey of 728 random people found that 72% prefer comedies over romantic movies. What is the margin of sampling error and the likely interval that contains the percent of the population?

10. SPORTS A survey of 3441 random people in one U.S. state found that 80% watched College football games every weekend in the Fall. What is the margin of sampling error and the likely interval that contains the percent of the population?


a. The Shoe Sizes of 12 Students at a High School4 8 5 6 6 5

9 7 10 7 9 8

b. The Number of Sit Ups Completed by All Students in a Gym Class50 28 41 61 54 28

47 33 45 50 50 61

23 41 31 38 42

Skills PracticeStatistical Analysis


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1. {12.1, 14.9, 6.7, 10, 12.8, 14, 18}

3. {10, 14.7, 14.7, 21, 7.4, 14.7, 8, 14.7}

2. {77.9, 101, 78.9, 105, 4.2, 110, 87.9}

4. {29, 36, 14, 99, 16, 15, 12, 30}

Determine whether the following represents a population or a sample.

5. a list of the times every student in gym class took to run a mile

7. friends compare the batting averages of players who are listed in their collections of baseball cards

6. the test scores of seven students in a chemistry class are compared

8. every student in a high school votes in a class president election

9. CARS A survey of 56 random people in a small town found that 14% drive convertibles year-round. What is the margin of sampling error? What is the likely interval that contains the percentage of the population that drives convertibles year-round?

10. BEACHES A survey of 812 random people in Hawaii found that 57% went to the beach at least four times last July. What is the margin of sampling error? What is the likely interval that contains the percentage of the population that went to the beach at least four times last July?


a. The Number of Wins for Each Player on a Tennis Team Last Season10 2 9 17 4 8

9 9 10 15 19 5

b. The Number of Medals Earned by 18 High School Debaters7 10 4 5 10 9

11 5 6 4 4 3

12 7 8 5 3 9

PracticeStatistical Analysis

12-2


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Word Problem Practice Statistical Analysis

1. BASEBALL The following shows the number of wins for Major League Baseballs American League East teams at the end of the 2007 regular season.

Team Number of Wins

Boston 96

New York 94

Toronto 83

Baltimore 69

Tampa Bay 66

The American League also has Central and Western divisions. If the data is used to study the number of wins for all American League baseball teams, is it a sample or a population?

2. VOTING A poll was taken of registered voters to see if they planned to vote in the next presidential election. What is the margin of sampling error if the likely interval of those who plan on voting is between 46.5% and 49.5%?

3. APPLES The following data set depicts the number of apples in 20 different bushel baskets: {80, 75, 68, 82, 77, 74, 81, 85, 73, 79, 75, 73, 80, 71, 82, 81, 77, 80, 78, 84}. What is the median number of apples in a bushel basket?

4. CARS A police officer clocked the following speed of cars (in miles per hour) on the highway: {65, 61, 72, 54, 78, 61, 74, 75, 61, 55, 64, 66, 70}.What is the mode of the data set?

5. SOLAR SYSTEM For much of the 20thcentury, astronomers considered the solar system to have nine planets. The table below lists the masses of these planets.

Planet Mercury Venus Earth Mars Jupiter

Mass (1021 tons)

0.364 5.37 6.58 0.708 2093

Planet Saturn Uranus Neptune Pluto

Mass (1021 tons)

627 95.7 113 0.0138

a. Which measure of central tendency best represents the data, and why?

b. What is the correct number for the measurement that best represents the data?

c. What is the standard deviation of the data?

d. How many of the planets have masses within the range of the central measure of the data plus or minus one standard deviation?

12-2


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12-2 Enrichment

The Harmonic MeanThe harmonic mean, H, is a useful measure of central tendency in special cases of averaging rates.

Recently, Kendra and Bill took a trip of 370 miles and shared the driving. Kendra drove two hours at a rate of 55 mph and then drove the next two hours at 65 mph. Then Bill drove the next 100 miles at 55 mph and he drove the last 100 miles at 65 mph. What was the average speed of each driver?

Kendra drove the same length of time on both portions of her driving,

so her average speed is the mean of the two rates. Her average speed

was 55 + 65 2 or 60 mph.

On the other hand, Bill drove the same distance on both portions of his driving,

but the two length of time varied. Actually, the time he drove was 100 55

+ 100 65

, or

approximately 3.36 hours. His average speed was 200 3.36

, or about 59.6 mph.

Bills average speed may be found by using the formula for the harmonic mean as follows.

Let n = number of rates xi where 1 i n. H = n

i = 1

n

1 xi

We apply the formula to Bills speeds. H = 2 1

55 + 1

65

59.6 mph

The mean, also called the arithmetic mean, is used when equal times are involved. When equal distances are involved, the harmonic mean is used.

Find the harmonic mean of each set of data. Round each answer to the nearest hundredth.

1. {3, 4, 5, 6} 2. {5, 10, 15, 20, 25}

3. Ben, Emily, and Jayden competed in a 375-mile relay race. Ben drove 40 mph, Emily drove 50 mph, and Jayden drove 60 mph. If each drove 125 miles, find the average driving speed of the contestants.

Example


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12-2 Graphing Calculator Activity

Statistical AnalysisThe TI-84 graphing calculator can be used for statistical analysis.

A police officer checks the speed of five cars on a particular highway. The speeds are listed in this table.

52 57 54 58 79

Perform a statistical analysis on the data. Round the numbers to the nearest hundredth.

First, use the STAT key on the calculator and enter the data.

Keystrokes: STAT 1 52 ENTER 57 ENTER 54 ENTER 58 ENTER 79 ENTER 2nd [QUIT]

Next use the STAT key again to perform the analysis.

Keystrokes: STAT 1 2nd 1 ENTER

x = 60 indicates that the mean of the speeds was 60 miles per hour.

x = 300 indicates that the sum of the speed is 300 miles per hour.

Since the five speeds are a sample of speeds of all the cars on the highway, the standard deviation is shown by Sx = 10.885577053. Rounding the standard deviation to the nearest hundredth, the standard deviation is 10.89 miles per hour.

ExercisesUse a graphing calculator to perform a statistical analysis on each set of data. Round the numbers to the nearest hundredth.

1. The number of miles run during the week was recorded for 5 of the students on the Track team.

10 5.5 7 8 4.5

Example


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Conditional Probability The probability of an event, given that another event has already occurred, is called conditional probability. The conditional probability that event B occurs, given that event A has already occurred, can be represented by P(B| A).

SPINNER Naomi is playing a game with the spinner shown. What is the probability that the spinner lands on 7, given that Naomi has spun a number greater than 5?

There are 8 possible results of spinning the spinner shown.Let event A be that she spun a number greater than 5.Let event B be that she spun a 7.

P(A) = 3 8 Three of the eight outcomes are greater than 5.

P(A and B) = 1 8 One out of eight outcomes is both greater than 5 and equal to 7.

P(B| A) = P ( A and B ) P ( B)

Conditional Probability Formula

P(B| A) = 1 8 3

8 Substitute values for P(A) and P(A and B)

P(B| A) = 1 3

The probability of spinning a 7, given that the spin is greater than 5, is 1 3 .

ExercisesCards are drawn from a standard deck of 52 cards. Find each probability.

1. The card is a heart, given that an ace is drawn.

2. The card is the six of clubs, given that a club is drawn.

3. The card is a spade, given that a black card is drawn. (Hint: The black cards are the suits of clubs and spades.)

A six-sided die is rolled. Find each probability.

4. A 4 is rolled, given that the roll is even.

5. A 2 is rolled, given that the roll is less than 6.

6. An even number is rolled, given that the roll is 3 or 4.

Example1 2

3

456

7

8

12-3 Study Guide and InterventionConditional Probability


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Contingency Tables A contingency table is a table that records data in which different possible situations result in different possible outcomes. These tables can be used to determine conditional probabilities.

LACROSSE Find the probability that a student plays lacrosse, given that the student is a junior.

Class Freshman Sophomore Junior Senior

Plays Lacrosse 17 20 34 17

Does Not Play 586 540 510 459

There are a total of 14 + 586 + 20 + 540 + 34 + 510 + 17 + 459 = 2180 students.

P(L|J) = P( L and J ) P ( J )

Conditional Probability Formula

= 34 2180

544 2180

P (L and J) = 34 2180

and P (J ) = 34 + 510 2180

= 34 544

or 1 6 Simplify.

The probability that a student plays lacrosse, given that the student is a junior, is 1 16

.

Exercises

1. WEDDINGS The table shows attendance at a wedding. Find the probability that a person can attend the wedding, given that the person is in the brides family.

2. BASEBALL The table shows the number of students who play baseball. Find the probability that a student plays baseball, given that the student is a senior.

3. SHOPPING Four businesses in town responded to a questionnaire asking how many people paid for their purchases last month using cash, credit cards, and debit cards. Find each probability.

Class Jacobs Gas Gigantomart T.J.s Pet Town

Cash 304 140 102 49

Credit Card 456 223 63 70

Debit Card 380 166 219 28

a. A shopper uses a credit card, given that the shopper is shopping at Jacobs Gas.

b. A shopper uses a debit card, given that the shopper is shopping at Pet Town.

c. A shopper is shopping at T.J.s, given that the shopper is paying cash.

Example

FamilyBrides Family

Grooms Family

Can Attend 104 112

Cannot Attend 32 14

ClassPlays

BaseballDoes Not

Play Baseball

Junior 22 352

Senior 34 306

Study Guide and Intervention (continued)Conditional Probability

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12-3 Skills PracticeConditional Probability

The spinner is numbered from one to eight. Find each probability.

1. The spinner lands on 3, given that the spinner lands on an odd number.

2. The spinner lands on a number greater than 5, given that the spinner lands on an even number.

3. The spinner lands on a number less than 6, given that the spinner does not land on 1 or 2.

4. The spinner lands on 7, given that the spinner lands on a number greater than 4.

5. The spinner does not land on 3, given that the spinner lands on an odd number.

6. The spinner lands on a number less than 6, given that the spinner lands on an odd number.

7. CONCERTS The Panic Squadron is playing a concert. Jennifer surveyed her classmates to see if they were in the Panic Squadron Fan Club and if they were going to a concert. Find the probability that a person surveyed went to the concert, given that they are a fan club member.

8. SPECIAL ELECTIONS When a U.S. congressman vacates their office in the middle of their two-year term, a special election for the remainder of the term is often held to fill the vacancy. The table shows the number of special elections won by each party between 2004 and 2007. Find each probability.

2004 2005 2006 2007

Republican Wins 0 1 2 3

Democrat Wins 3 1 1 2

Source: Clerk of the House of Representatives

a. A Republican wins, given the special election was held in 2007.

b. The election was held in 2005, given the winner was a Democrat.

c. A Democrat wins, given the election was held in 2006.

Fan Club Member

Not in Fan Club

Going to Concert

12 4

Not Going to Concert

2 18

1 23

456

7

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Four dice are thrown. Find each probability.

1. All of the rolls are 5, given that all of the rolls show the same number.

2. One of the rolls is a 4, given that all of the rolls are greater than 3.

3. None of the rolls are 2, given that all of the rolls are even.

4. One of the rolls is a 6, given that one of the rolls is a 2.

5. CHEMISTRY Cheryl and Jerome are testing the pH of 32 unknown substances as part of science class. Cheryl and Jerome split the work between them as shown in the table. Find each probability. a. The substance is acidic, given that Cheryl is testing it.

b. Jerome is testing the substance, given that it is basic.

6. ELECTIONS Rose Heck is running against Joe Coniglio in a district that includes the towns of Hasbrouck, Clinton, Eastwick, and Abletown. The table shows how many votes each candidate received in each town. Find each probability.

Hasbrouck Clinton Eastwick Abletown

Joe Coniglio 1743 1782 886 7790

Rose Heck 2616 2178 1329 5876

a. A voter cast a ballot for Joe Coniglio, given that the voter cast a ballot in Clinton.

b. A voter cast a ballot for Rose Heck, given that the voter cast a ballot in Eastwick.

c. A voter cast a ballot in Hasbrouck, given that the voter cast a ballot for Joe Coniglio.

7. BASEBALL Derek Jeter, a player for the New York Yankees, had 206 hits in the 2007 Major League Baseball season and has 2356 career hits. The table below shows the number of singles, doubles, triples, and home runs Derek Jeter had in the 2007 season and during his career. Find each probability.

Singles Doubles Triples Home Runs

2007 Season 151 39 4 12

Career 1721 386 54 195

a. A hit was a home run, given that the hit happened in the 2007 season.

b. A hit was a double, given that the hit happened during Jeters career.

ResultsCheryls

TestsJeromes

Tests

Acidic 12 8

Basic 9 3

12-3 PracticeConditional Probability


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1. SPINNERS Majorie is playing a game that uses the spinner shown. What is the probability that Majorie spins the number 6, given that she spins a number greater than 2?

1

6

5 4

3

2

2. CARDS Monique is playing a card game. To win the game, she needs to draw a card that is a club. What is the probability that Monique wins the card game if the card she draws is black?

3. PET GROOMING At the CleanPaws pet grooming salon, owners can bring their cats or dogs in to get bathed only, or bathed and groomed. The table shows how many animals the salon serviced this week.

AnimalBathed

OnlyBathed and Groomed

Dog 48 51

Cat 3 12

Find the probability that an animal was bathed and groomed, given that the animal was a cat.

4. DICE Stan rolls a pair of dice. What is the probability that the total of the two dice is 11, given that one of the dice rolls is a 5?

5. WOMEN IN POLITICS The table shows the gender of United States governors who served in 2007, 1997, and 1987.

YearMale

GovernorsFemale

Governors

2007 42 8

1997 47 3

1987 47 3

Source: National Governors Association

a. Find the probability that a governor is female, given that the governor served in 1987.

b. Find the probability that a governor is female, given that the governor served in 2007.

c. Find the probability that a governor listed in the table served in 1997, given that the governor is male.

12-3 Word Problem PracticeConditional Probability


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12-3 Enrichment

The Monty Hall ProblemA famous probability problem is based on the game show Lets Make a Deal with Monty Hall. In the show, a contestant was allowed to choose one of three doors, knowing that a valuable prize was behind one, and worthless prizes were behind the other two.

Once the contestant made his or her choice, Monty Hall often opened one of the other doors to reveal one of the worthless prizes. Then he would offer to allow the contestant to switch their door for the other unopened door. After the contestant chose, he would open the contestants chosen door to reveal his or her prize. It may seem that each of

the two remaining doors has a probability of 1 2 of being the winner. But it has been

mathematically proven that if the contestant switches doors, he or she has twice the probability of winning as when sticking with the original choice!

How can this be? The best way to look at it is to imagine that when the contestant selects a door, the doors are separated into two setsthe chosen doors and the unchosen doors. At this point, each door has a probability

of 1 3 of being the winning door, but the two

sets have different probabilities. Set 1 has a

probability of 1 3 of containing the valuable prize.

Set 2 has twice as many doors, so it has a

probability of 2 3 of containing the valuable prize.

When Monty opens one of the doors in Set 2 to show it isnt the winner, Set 2 still

has a probability of 2 3 of containing the valuable prize. Because there is only one

door with unknown contents, the 2 3 probability is for the unopened door in Set 2,

while Set 1 still has probability 1 3 .

Exercises

1. Work with a partner to simulate the situation 20 or more times. Do your results verify or contradict the theoretical probability? Explain.

2. The Birthday Problem is another classic probability problem. How many people must be in a randomly chosen group for the probability that two of them share a birthday to be at least 50%? Research to answer the problem and explain its solution.

Set 1Chosen Doors

Set 2Unchosen Doors

13

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Probability In probability, a desired outcome is called a success; any other outcome is called a failure.

When 3 coins are tossed, what is the probability that at least 2 are heads?

You can use a tree diagram to find the sample space.

Of the 8 possible outcomes, 4 have at least 2 heads. So the

probability of tossing at least 2 heads is 4 8 or 1

2 .

What is the probability of picking 4 fiction books and 2 biographies from a best-seller list that consists of 12 fiction books and 6 biographies?The number of successes is 12C4 6C2. The total number of selections, s + f, of 6 books is C(18, 6).

P(4 fiction, 2 biography) = 12C4 6C2 18C6

or about 0.40

The probability of selecting 4 fiction books and 2 biographies is about 40%.

Exercises

1. PET SHOW A family has 3 dogs and 4 cats. Find the probability of each of the following if they select 2 pets at random to bring to a local pet show.

a. P(2 cats) b. P(2 dogs) c. P(1 cat, 1 dog)

2. MUSIC Eduardos MP3 player has 10 blues songs and 5 rock songs (and no other music). Find the probability of each of the following if he plays six songs at random and songs maynot repeat. Round to the nearest tenth of a percent.

a. P(6 blues songs) b. P(4 blues songs, 2 rock songs) c. P(2 blues songs, 4 rock songs)

3. CANDY One bag of candy contains 15 red candies, 10 yellow candies, and 6 green candies. Find the probability of each selection.

a. picking a red candy b. not picking a yellow candy

c. picking a green candy d. not picking a red candy

Probability of Success and Failure

If an event can succeed in s ways and fail in f ways, then the probabilities of success, P(S), and of failure, P(F), are as follows.

P(S) = s s + f

and P (F ) = f

s + f

Example 1

HHHHHTHTHHTTTHHTHTTTHTTT

HTHTHTHT

H

T

H

T

H

T

FirstCoin

SecondCoin

ThirdCoin

PossibleOutcomes

Example 2

12-4 Study Guide and InterventionProbability and Probability Distributions


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Probability Distributions A random variable is a variable whose value is the numerical outcome of a random event. A probability distribution for a particular random variable is a function that maps the sample space to the probabilities of the outcomes in the sample space.

Suppose two dice are rolled. The table and the relative-frequency histogram show the distribution of the absolute value of the difference of the numbers rolled. Use the graph to determine which outcome is the most likely. What is its probability?

Difference 0 1 2 3 4 5

probability 1 6 5

18 2

9 1

6 1

9 1

18

The greatest probability in the graph is 5 18

.

The most likely outcome is a difference of 1 and its

probability is 5 18

.

Exercises

1. PROBABILITY Four coins are tossed.

a. Complete the table below to show the probability distribution of the number of heads.

Number of Heads 0 1 2 3 4

Probability

b. Create a relative-frequency histogram.

10

Heads

Heads in Coin Toss

2 3 4

14

Pro

bab

ility

38

18

116

316

516

c. Find P (four heads).

00

Pro

bab

ility

Difference

Numbers Showing on the Dice

1 2 3 4 5

1619

29

118

518

Example

12-4 Study Guide and Intervention (continued)Probability and Probability Distributions


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12-4

1. PHOTOGRAPHY Ahmed is posting 2 photographs on his website. He has narrowed his choices to 4 landscape photographs and 3 portraits. If he chooses the two photographs at random, find the probability of each selection.

a. P(2 portrait) b. P(2 landscape) c. P(1 of each)

2. VIDEOS The Carubas have a collection of 28 movies, including 12 westerns and 16 science fiction. Elise selects 3 of the movies at random to bring to a sleep-over at her friends house. Find the probability of each selection.

a. P(3 westerns) b. P(3 science fiction)

c. P(1 western and 2 science fiction) d. P(2 westerns and 1 science fiction)

e. P(3 comedy) f. P(2 science fiction and 2 westerns)

3. CLASS The chart to the right shows the classand gender statistics for the students taking an Algebra 1 or Algebra 2 class at La Mesa High School. If a student taking Algebra 1 or Algebra 2 is selected at random, find each probability. Express as decimals rounded to the nearest thousandth.

a. P(sophomore/female)

b. P(junior/male)

c. P(freshman/male)

d. P(freshman/female)

4. FAMILY Lisa has 10 cousins. Four of her cousins are older than her; six are younger. Seven of her cousins are boys, and 3 are girls. Find the probability of each when one cousin is chosen at random.

a. P(girl) b. P(younger)

c. P(boy) d. P(older)

Skills PracticeProbability and Probability Distributions

Class/Gender Number

Freshman/Male 95

Freshman/Female 101

Sophomore/Male 154

Sophomore/Female 145

Junior/Male 100

Junior/Female 102


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1. BALLOONS A bag contains 1 green, 4 red, and 5 yellow balloons. Two balloons are selected at random. Find the probability of each selection.

a. P(2 red) b. P(1 red and 1 yellow) c. P(1 green and 1 yellow)

d. P(2 green) e. P(2 red and 1 yellow) f. P(1 red and 1 green)

2. COINS A bank contains 3 pennies, 8 nickels, 4 dimes, and 10 quarters. Two coins are selected at random. Find the probability of each selection.

a. P(2 pennies) b. P(2 dimes) c P(1 nickel and 1 dime)

d. P(1 quarter and 1 penny) e. P(1 quarter and 1 nickel) f. P(2 dimes and 1 quarter)

3. WALLPAPER Henrico visits a home decorating store to choose wallpapers for his new house. The store has 28 books of wallpaper samples, including 10 books of WallPride samples and 18 books of Deluxe Wall Coverings samples. The store will allow Henrico to bring 4 books home for a few days so he can decide which wallpapers he wants to buy. If Henrico randomly chooses 4 books to bring home, find the probability of each selection.

a. P(4 WallPride) b. P(2 WallPride and 2 Deluxe)

c. P(1 WallPride and 3 Deluxe) d. P(3 WallPride and 1 Deluxe)

4. SAT SCORES The table to the right shows the range of verbal SAT scores for freshmen at a small liberal arts college. If a freshman student is chosen at random, find each probability. Express as decimals rounded to the nearest thousandth.

a. P(400449) b. P(550559) c. P(at least 650)

20. CHECKERS The following table shows the wins and losses of the checkers team. If a game is chosen at random, find each probability.

Arthur Lynn Pedro Mei-Mei

Wins 15 7 12 18

Losses 5 13 3 2

a. a game was lost and Arthur was playing

b. Mei-Mei was playing and the game was won

c. a game was won and Lynn or Arthur was playing

d. Pedro or Mei-Mei was playing and the game was lost

e. any of the players was playing and the game was won

Range 400 449 450 499 500 549 550 559 600 649 650+

Number of Students

129 275 438 602 620 412

12-4 PracticeProbability and Probability Distributions


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1. ART The letters A, R, and T are written on three different pieces of paper. The pieces of paper are then put in a bag and mixed up. Logan picks each letter without looking and places them side by side. What is the probability that the letters spell ART?

2. AGE There are 24 students in Miss Masons third grade class, all born on different days. Eleven students are boys. In the morning, the classroom is empty. One student arrives followed by another. What is the probability that when the first two students arrive, one is a boy and the other a girl?

3. DICE Jamal rolls two six-sided dice, one after the other. What is the probability that the second die shows a number larger than the first die?

4. LANGUAGES Noah cannot decide whether to learn French, German, Italian, Russian, or Chinese. He assigns each language a different number from 0 to 4. He then takes four fair coins and flips them. He decided to take the language corresponding to the number of coins that come up heads. Does Noahs method for choosing a language give each language the same chance of being chosen? Explain.

5. ICE CREAM Researchers made this table that shows the flavors of ice cream sold in the United States.

a. One package of ice cream is selected at random from a warehouse. What is the probability that the package is chocolate?

b. One package of ice cream is selected at random from a warehouse. What is the probability that the package is not vanilla?

c. One package of ice cream is selected at random from a warehouse. What is the probability that the package is either vanilla, chocolate, Neapolitan, butter pecan, or chocolate chip?

Flavor Vanilla Chocolate Neapolitan

Percentage 28% 8% 7%

FlavorButter Pecan

Chocolate Chip

Other

Percentage 4.5% 3.5% 49%

12-4 Word Problem PracticeProbability and Probability Distributions


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Geometric ProbabilityIf a dart, thrown at random, hits the triangular board shown at the right, what is the chance that it will hit the shaded region? This chance, also called a probability, can be determined by comparing the area of the shaded region to the area of the board. This ratio indicates what fraction of the tosses should hit in the shaded region.

area of shaded region

area of triangular board

= 1 2 (4)(6)

1 2 (8)(6)

= 12 24

or 1 2

In general, if S is a subregion of some region R, then the probability, P(S), that a point, chosen at random, belongs to subregion S is given by the following.

P(S) = area of subregion S area of region R

Find the probability that a point, chosen at random, belongs to the shaded subregions of the following regions.

1.

3 3

5

5

2.

46

6

64

4

3.

4 4

4

4

The dart board shown at the right has 5 concentric circles whose centers are also the center of the square board. Each side of the board is 38 cm, and the radii of the circles are 2 cm, 5 cm, 8 cm, 11 cm, and 14 cm. A dart hitting within one of the circular regions scores the number of points indicated on the board, while a hit anywhere else scores 0 points. If a dart, thrown at random, hits the board, find the probability of scoring the indicated number of points.

4. 0 points 5. 1 point 6. 2 points

7. 3 points 8. 4 points 9. 5 points

51

2

34

4 4

6

12-4 Enrichment


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Normal and Skewed Distributions A continuous probability distribution is represented by a curve.

Determine whether the data below appear to be positively skewed, negatively skewed, or normally distributed.{100, 120, 110, 100, 110, 80, 100, 90, 100, 120, 100, 90, 110, 100, 90, 80, 100, 90}Make a frequency table for the data.

Value 80 90 100 110 120

Frequency 2 4 7 3 2

Then use the data to make a graph.Since the graph is roughly symmetric, the data appear to be normally distributed.

Exercises

Determine whether the data appear to be positively skewed, negatively skewed, or normally distributed. Make a graph of the data.

1. {27, 24, 29, 25, 27, 22, 24, 25, 29, 24, 25, 22, 27, 24, 22, 25, 24, 22}

2. Shoe Size 4 5 6 7 8 9 10

No. of Students 1 2 4 8 5 1 2

3. Housing Price No. of Houses Sold

less than $100,000 0

$100,00-$120,000 1

$121,00-$140,000 3

$141,00-$160,000 7

$161,00-$180,000 8

$181,00-$200,000 6

over $200,000 12

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The Empirical Rule

Normal Distribution Normal distributions have these properties.The graph is maximized at the mean.The mean, median, and mode are about equal.About 68% of the values are within one standard deviation of the mean.About 95% of the values are within two standard deviations of the mean.About 99% of the values are within three standard deviations of the mean.

The heights of players in a basketball league are normally distributed with a mean of 6 feet 1 inch and a standard deviation of 2 inches.

a. What is the probability that a player selected at random will be shorter than 5 feet 9 inches?Draw a normal curve. Label the mean and the mean plus or minus multiples of the standard deviation.The value of 5 feet 9 inches is 2 standard deviations below the mean, so approximately 2.5% of the players will be shorter than 5 feet 9 inches.

b. If there are 240 players in the league, about how many players are taller than 6 feet 3 inches?

The value of 6 feet 3 inches is one standard deviation above the mean. Approximately 16% of the players will be taller than this height.240 0.16 38About 38 of the players are taller than 6 feet 3 inches.

Exercises

1. EGG PRODUCTION The number of eggs laid per year by a particular breed of chicken is normally distributed with a mean of 225 and a standard deviation of 10 eggs.

a. About what percent of the chickens will lay between 215 and 235 eggs per year?

b. In a flock of 400 chickens, about how many would you expect to lay more than 245 eggs per year?

2. MANUFACTURING The diameter of bolts produced by a manufacturing plant is normally distributed with a mean of 18 mm and a standard deviation of 0.2 mm.

a. What percent of bolts coming off of the assembly line have a diameter greater than 18.4 mm?

b. What percent have a diameter between 17.8 and 18.2 mm?

5'7" 5'9" 5'11" 6'1" 6'3" 6'5" 6'7"

-3

mean

-2 - + +2 +3

Example

12-5 Study Guide and Intervention (continued)The Normal Distribution


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12-5

Determine whether the data appear to be positively skewed, negatively skewed, or normally distributed.

1. Miles Run Track Team Members04 3

59 4

1014 7

1519 5

2023 2

2. Speeches Given Political Candidates05 1

611 2

1217 3

1823 8

2429 8

3. PATIENTS The frequency table to the right shows the average number of days patients spent on the surgical ward of a hospital last year.

a. What percentage of the patients stayed between 4 and 7 days?

b. Does the data appear to be positively skewed, negatively skewed, or normally distributed? Explain.

4. DELIVERY The time it takes a bicycle courier to deliver a parcel to his farthest customer is normally distributed with a mean of 40 minutes and a standard deviation of 4 minutes.

a. About what percent of the couriers trips to this customer take between 36 and 44 minutes?

b. About what percent of the couriers trips to this customer take between 40 and 48 minutes?

c. About what percent of the couriers trips to this customer take less than 32 minutes?

5. TESTING The average time it takes sophomores to complete a math test is normally distributed with a mean of 63.3 minutes and a standard deviation of 12.3 minutes.

a. About what percent of the sophomores take more than 75.6 minutes to complete the test?

b. About what percent of the sophomores take between 51 and 63.3 minutes?

c. About what percent of the sophomores take less than 63.3 minutes to complete the test?

Skills PracticeThe Normal Distribution

Days Number of Patients

03 5

47 18

811 11

1215 9

16+ 6


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Determine whether the data appear to be positively skewed, negatively skewed, or normally distributed.

1. Time Spent at a Museum ExhibitMinutes Frequency

025 27

2650 46

5175 89

75100 57

100+ 24

2. Average Age of High School PrincipalsAge in Years Number

3135 3

3640 8

4145 15

4650 32

5155 40

5660 38

60+ 4

3. STUDENTS The frequency table to the right shows the number of hours worked per week by 100 high school students.

a. What percentage of the students worked between 9 and 17 days?

b. Graph the data. Do the data appear to be positively skewed, negatively skewed, or normally distributed? Explain.

4. TESTING The scores on a test administered to prospective employees are normally distributed with a mean of 100 and a standard deviation of 15.

a. About what percent of the scores are between 70 and 130?

b. About what percent of the scores are between 85 and 130?

c. About what percent of the scores are over 115?

d. About what percent of the scores are lower than 85 or higher than 115?

e. If 80 people take the test, how many would you expect to score higher than 130?

f. If 75 people take the test, how many would you expect to score lower than 85?

5. TEMPERATURE The daily July surface temperature of a lake at a resort has a mean of 82 and a standard deviation of 4.2. If you prefer to swim when the temperature is at least 77.8, about what percent of the days does the temperature meet your preference?

08 917 1825 26+

605040302010F

req

uen

cy

Hours

Weekly Work Hours

Hours Number of Students

08 30

917 45

1825 20

26+ 5

12-5 PracticeThe Normal Distribution


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1. PARKING Over several years, Bertram conducted a study of how far into parking spaces people tend to park by measuring the distance from the end of a parking space to the front fender of a car parked in the space. He discovered that the distribution of the data closely approximated a normal distribution with mean 8.5 inches. He found that about 5% of cars parked more than 11.5 inches away from the end of the parking space. What percentage of cars would you expect parked less than 5.5 inches away from the end of the parking space?

2. HEIGHT Chandras graph of the number of tenth grade students of different heights is shown below.

Is the data positively skewed, negatively skewed, or normally distributed?

3. OVENS An oven manufacturer tries to make the temperature setting on its ovens as accurate as possible. However, if one measures the actual temperatures in the ovens when the temperature setting is 350F, they will differ slightly from 350F. The set of actual temperatures for all the ovens is normally distributed around 350F with a standard deviation of 0.5F. About what percentage of ovens will be between 350F and 351F when their temperature setting is 350F?

Num

ber

of S

tude

nts 10

5

052 7064

Height (in.)58

4. LIGHT BULBS The time that a certain brand of light bulb will last before burning out is normally distributed. About 2.5% of the bulbs last longer than 6800 hours and about 16% of the bulbs last longer than 6500 hours. How long does the average bulb last?

5. DOGS The weights of adult male greyhound dogs are normally distributed. The mean weight is about 68 pounds and the standard deviation is about 10 pounds.

a. Approximately what percentage of adult male greyhound dogs would you expect weigh between 58 and 78 pounds?

b. Approximately what percentage of adult male greyhound dogs would you expect weigh more than 98 pounds?

c. Approximately what percentage of adult male greyhound dogs would you expect weigh less than 48 pounds?

d. What would you expect an adult male greyhound dog to weigh if it weighed less than 0.5% of an average adult greyhound?

12-5 Word Problem PracticeThe Normal Distribution


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The normal distribution is the most important probability distribution. Many physical measurements have distributions approximately normal. Examples include height, weight, and measures of intelligence. More importantly, even if the individual variables are not normally distributed, sums and averages tend to still be normally distributed. Unfortunately, normal probability distribution functions are difficult to calculate. Fortunately, statisticians have compiled a table for a normal distribution with mean of zero and standard deviation of one. This is called the Standard Normal Distribution and is typically denoted by N(0, 1), where the N indicates a normal distribution which has mean, (mu) = 0, and standard deviation, (sigma) = 1.

Suppose the variable x is normally distributed with mean and standard deviation . In order to calculate probabilities of this normal distribution, we must standardize the variable x by an appropriate transformation. The letter Z denotes the transformed variable and is called the Z-score, which is a measure of relative standing. The following steps are needed to complete the transformation.

If the mean and standard deviation are not given, then calculate the mean and standard deviation of the given (population) data.

Define Z = x -

.

Find the standard normal variable Z given and = 15 and = 3. Apply the transform to the variable X using the definition above, that is: Z = X - 15

3 .

1. Suppose that the time, X, to complete an exam is normally distributed. The time, in minutes, of a class of 12 to complete the exam is given in the table. Transform X to a Z-score.

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

Time 35 42 48 33 32 39 40 52 48 34 36 44

2. Suppose that a random variable X is normally distributed with = 20 and = 5. Convert the following probability statements to the equivalent statements by standardizing X.

P(X < 25) = P ( X - 20

5 < 25) = P(Z < 25)

a. P(X > 18)

b. P(17 < X < 23)

c. P(X < 19)

Example 1

Example 2

12-5 EnrichmentCalculating Z-Scores


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Confidence Interval

Term Definition

Confidence Intervalthe estimated range within which a number will fall with a stated degree of certainty

95% Confidence Interval Formula CI =

x 2 s n

A survey asked 100 random people how many minutes they exercised each day. The mean of their answers was 25.3 minutes with a standard deviation of 9.4 minutes. Determine a 95% confidence interval. Round to the nearest tenth.

CI = x 2 s n

Confidence Interval Formula

= 25.3 2 9.4 100

x = 25.3, s = 9.4, n = 100

= 25.3 1.88The 95% confidence interval to the nearest tenth is 23.4 27.2.

Exercises

Find a 95% confidence interval for each of the following. Round to the nearest tenth if necessary.

1. x = 10, s = 6, and n = 100 2. x =100, s = 5.4, and n = 5

3. x = 90, s = 1.8, and n = 170 4. x = 82, s = 4.5, and n = 8000

5. x = 1088, s = 7.8, and n = 200 6. x = 70, s = 10, and n = 50

7. x = 120, s = 8, and n = 1000 8. x = 147, s = 39, and n = 100

9. x = 70.5, s = 5.5, and n = 150 10. x = 788.2, s = 52, and n = 8

Example

12-6 Study Guide and InterventionHypothesis Testing


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Hypothesis TestingSteps in Testing a Null Hypothesis

Step 1 State the null hypothesis H0 and alternative hypothesis H1.

Step 2 Design the experiment.

Step 3 Conduct the experiment and collect the data.

Step 4 Find the confidence interval.

Step 5 Make the correct statistical inference. Accept the null hypothesis if the population parameter falls within the confidence interval.

A team of students claimed the average student at their school studied at least 27.5 hours per week. They designed an experiment using the 5 steps for testing a null hypothesis.

Step 1 The null hypothesis H0: < 27.5 hours per week. The alternative hypothesis H1: 27.5 hours per week.

Step 2 They decided to survey students and wrote a questionnaire.

Step 3 They surveyed 10 students. They found x = 29.2 and s = 5.3.

Step 4 They calculated the confidence interval. CI = x 2 s

n Confidence Interval Formula

= 29.2 2 5.3 10

x = 25.3, s = 9.4, n = 100

= 29.2 3.35 Use a calculator.

Step 5 The null hypothesis H0 overlaps the confidence interval, so the students should accept the null hypothesis. They have not proven that the average student at their school studies at least 27.5 hours per week.

Exercises

Test each null hypothesis. Write accept or reject.

1. H0 = 20, H1 < 20, n = 50,

x = 12, and = 2

2. H0 = 21, H1 < 21, n = 100,

x = 20, and = 5

3. H0 = 80, H1 > 80, n = 50,

x = 80, and = 21.3

4. H0 = 64.5, H1 > 64.5, n = 150,

x = 68, and = 3.5

5. H0 = 87.6, H1 > 87.6, n = 1200,

x = 88, and = 2

6. H0 = 10.4, H1 < 10.4, n = 200,

x = 10, and = 4

7. H0 = 99.44, H1 > 99.44, n = 10,

x = 100, and = 2

8. H0 = 16.2, H1 < 16.2, n = 150,

x = 15.8, and = 3.5

9. H0 = 49.2, H1 < 49.2, n = 100,

x = 49, and = 1

10. H0 = 298, H1 > 298, n = 225,

x = 300, and = 15

Example

12-6 Study Guide and Intervention (continued)Hypothesis Testing

005_048_A2CRMC12_890537.indd 37005_048_

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