(BQ) Part 1 book Essential statistics - Exploring the world through data has contents: Introduction to data, picturing variation with graphs, numerical summaries of center and variation, regression analysis - Exploring associations between variables, modeling variation with probability, modeling random events - The normal and binomial models.
www.downloadslide.com Global edition Essential Statistics Exploring the World through Data For these Global Editions, the editorial team at Pearson has collaborated with educators across the world to address a wide range of subjects and requirements, equipping students with the best possible learning tools This Global Edition preserves the cutting-edge approach and pedagogy of the original, but also features alterations, customization, and adaptation from the North American version Global edition SECoND edition Gould • Ryan • Wong ↜Exploring the World through Data ↜SECoND edition Gould • Ryan • Wong GLOBal edition This is a special edition of an established title widely used by colleges and universities throughout the world Pearson published this exclusive edition for the benefit of students outside the United States and Canada If you purchased this book within the United States or Canada, you should be aware that it has been imported without the approval of the Publisher or Author Essential Statistics Pearson Global Edition Gould_02_1292161221_Final.indd 17/09/16 9:42 AM www.downloadslide.com Available in MyStatLab™ for your Introductory Statistics Courses MyStatLab is the market-leading online resource for learning and teaching statistics Leverage the Power of StatCrunch MyStatLab leverages the power of StatCrunch—powerful, web-based statistics software Integrated into MyStatLab, students can easily analyze data from their exercises and etext In addition, access to the full online community allows users to take advantage of a wide variety of resources and applications at www.statcrunch.com Bring Statistics to Life Virtually flip coins, roll dice, draw cards, and interact with animations on your mobile device with the extensive menu of experiments and applets in StatCrunch Offering a number of ways to practice resampling procedures, such as permutation tests and bootstrap confidence intervals, StatCrunch is a complete and modern solution Real-World Statistics MyStatLab video resources help foster conceptual understanding StatTalk Videos, hosted by fun-loving statistician Andrew Vickers, demonstrate important statistical concepts through interesting stories and real-life events This series of 24 videos includes assignable questions built in MyStatLab and an instructor’s guide A01_GOUL1228_02_GE_FM.indd www.mystatlab.com 07/09/16 2:48 pm www.downloadslide.com This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 24/11/14 5:26 PM www.downloadslide.com CONTENTS Essential Statistics: Exploring the World through Data Second Edition Global Edition Robert Gould University of California, Los Angeles Colleen Ryan California Lutheran University Rebecca Wong West Valley College Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo A01_GOUL1228_02_GE_FM.indd 07/09/16 2:48 pm 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USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF INFORMATION AVAILABLE FROM THE SERVICES THE DOCUMENTS AND RELATED GRAPHICS CONTAINED HEREIN COULD INCLUDE TECHNICAL INACCURACIES OR TYPOGRAPHICAL ERRORS CHANGES ARE PERIODICALLY ADDED TO THE INFORMATION HEREIN MICROSOFT AND/OR ITS RESPECTIVE SUPPLIERS MAY MAKE IMPROVEMENTS AND/OR CHANGES IN THE PRODUCT(S) AND/OR THE PROGRAM(S) DESCRIBED HEREIN AT ANY TIME PARTIAL SCREEN SHOTS MAY BE VIEWED IN FULL WITHIN THE SOFTWARE VERSION SPECIFIED ISBN 10: 1-292-16122-1 ISBN 13: 978-1-292-16122-8 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 Printed and bound by Vivar in Malaysia A01_GOUL1228_02_GE_FM.indd 07/09/16 2:48 pm www.downloadslide.com Dedication To my parents and family, my friends, and my colleagues who are also friends Without their patience and support, this would not have been possible —Rob To my teachers and students, and to my family who have helped me in many different ways —Colleen To my students, colleagues, family, and friends who have helped me be a better teacher and a better person —Rebecca A01_GOUL1228_02_GE_FM.indd 07/09/16 2:48 pm www.downloadslide.com About the Authors Robert Gould Robert L Gould (Ph.D., University of California, San Diego) is a leader in the statistics education community He has served as chair of the American Statistical Association’s Committee on Teacher Enhancement, has served as chair of the ASA’s Statistics Education Section, and served on a panel of co-authors for the Guidelines for Assessment in Instruction on Statistics Education (GAISE) College Report While serving as the associate director of professional development for CAUSE (Consortium for the Advancement of Undergraduate Statistics Education), Rob worked closely with the American Mathematical Association of Two-Year Colleges (AMATYC) to provide traveling workshops and summer institutes in statistics For over ten years, he has served as Vice-Chair of Undergraduate Studies at the UCLA Department of Statistics, and he is director of the UCLA Center for the Teaching of Statistics In 2012, Rob was elected Fellow of the American Statistical Association In his free time, Rob plays the cello and is an ardent reader of fiction Colleen Ryan Colleen N Ryan has taught statistics, chemistry, and physics to diverse community college students for decades She taught at Oxnard College from 1975 to 2006, where she earned the Teacher of the Year Award Colleen currently teaches statistics part-time at California Lutheran University She often designs her own lab activities Her passion is to discover new ways to make statistical theory practical, easy to understand, and sometimes even fun Colleen earned a B.A in physics from Wellesley College, an M.A.T in physics from Harvard University, and an M.A in chemistry from Wellesley College Her first exposure to statistics was with Frederick Mosteller at Harvard In her spare time, Colleen sings, has been an avid skier, and enjoys time with her family Rebecca K Wong Rebecca K Wong has taught mathematics and statistics at West Valley College for more than twenty years She enjoys designing activities to help students actively explore statistical concepts and encouraging students to apply those concepts to areas of personal interest Rebecca earned a B.A in mathematics and psychology from the University of California, Santa Barbara, an M.S.T in mathematics from Santa Clara University, and an Ed.D in Educational Leadership from San Francisco State University She has been recognized for outstanding teaching by the National Institute of Staff and Organizational Development and the California Mathematics Council of Community Colleges When not teaching, Rebecca is an avid reader and enjoys hiking trails with friends A01_GOUL1228_02_GE_FM.indd 07/09/16 2:48 pm www.downloadslide.com Contents Preface 11 Index of Applications 21 CHAPTER Introduction to Data 26 CASE STUDY Deadly Cell Phones? 27 1.1 1.2 1.3 1.4 What Are Data? 28 Classifying and Storing Data 30 Organizing Categorical Data 34 Collecting Data to Understand Causality 39 EXPLORING STATISTICS Collecting a Table of Different Kinds of Data 49 CHAPTER Picturing Variation with Graphs 60 CASE STUDY Student-to-Teacher Ratio at Colleges 61 2.1 2.2 2.3 2.4 2.5 Visualizing Variation in Numerical Data 62 Summarizing Important Features of a Numerical Distribution 67 Visualizing Variation in Categorical Variables 75 Summarizing Categorical Distributions 78 Interpreting Graphs 81 EXPLORING STATISTICS Personal Distance 85 CHAPTER Numerical Summaries of Center and Variation 106 CASE STUDY Living in a Risky World 107 3.1 3.2 3.3 3.4 3.5 Summaries for Symmetric Distributions 108 What’s Unusual? The Empirical Rule and z-Scores 118 Summaries for Skewed Distributions 123 Comparing Measures of Center 130 Using Boxplots for Displaying Summaries 135 EXPLORING STATISTICS Does Reaction Distance Depend on Gender? 142 CHAPTER Regression Analysis: Exploring Associations between Variables 166 CASE STUDY Catching Meter Thieves 167 4.1 4.2 4.3 4.4 Visualizing Variability with a Scatterplot 168 Measuring Strength of Association with Correlation 172 Modeling Linear Trends 180 Evaluating the Linear Model 193 EXPLORING STATISTICS Guessing the Age of Famous People 201 A01_GOUL1228_02_GE_FM.indd 07/09/16 2:48 pm www.downloadslide.com CONTENTS CHAPTER Modeling Variation with Probability 228 CASE STUDY SIDS or Murder? 229 5.1 5.2 5.3 5.4 What Is Randomness? 230 Finding Theoretical Probabilities 233 Associations in Categorical Variables 242 Finding Empirical Probabilities 252 EXPLORING STATISTICS Let’s Make a Deal: Stay or Switch? 257 CHAPTER Modeling Random Events: The Normal and Binomial Models 272 CASE STUDY You Sometimes Get More Than You Pay For 273 6.1 Probability Distributions Are Models of Random Experiments 274 6.2 The Normal Model 279 6.3 The Binomial Model (optional) 292 EXPLORING STATISTICS ESP with Coin Flipping 307 CHAPTER Survey Sampling and Inference 324 CASE STUDY Spring Break Fever: Just What the Doctors Ordered? 325 7.1 7.2 7.3 7.4 7.5 Learning about the World through Surveys 326 Measuring the Quality of a Survey 332 The Central Limit Theorem for Sample Proportions 340 Estimating the Population Proportion with Confidence Intervals 347 Comparing Two Population Proportions with Confidence 354 EXPLORING STATISTICS Simple Random Sampling Prevents Bias 361 CHAPTER Hypothesis Testing for Population Proportions 378 CASE STUDY Dodging the Question 379 8.1 8.2 8.3 8.4 The Essential Ingredients of Hypothesis Testing 380 Hypothesis Testing in Four Steps 387 Hypothesis Tests in Detail 396 Comparing Proportions from Two Populations 403 EXPLORING STATISTICS Identifying Flavors of Gum through Smell 411 CHAPTER Inferring Population Means 428 CASE STUDY Epilepsy Drugs and Children 429 9.1 9.2 9.3 9.4 9.5 9.6 Sample Means of Random Samples 430 The Central Limit Theorem for Sample Means 434 Answering Questions about the Mean of a Population 441 Hypothesis Testing for Means 451 Comparing Two Population Means 457 Overview of Analyzing Means 472 EXPLORING STATISTICS Pulse Rates 476 A01_GOUL1228_02_GE_FM.indd 07/09/16 2:48 pm www.downloadslide.com CONTENTS CHAPTER 10 Analyzing Categorical Variables and Interpreting Researchâ•…500 CASE STUDY Popping Better Popcornâ•… 501 10.1 The Basic Ingredients for Testing with Categorical Variablesâ•… 502 10.2 Chi-Square Tests for Associations between Categorical Variablesâ•… 509 10.3 Reading Research Papersâ•… 518 EXPLORING STATISTICS Skittlesâ•…527 Appendix A Tablesâ•… 543 Appendix B Check Your Tech Answersâ•… 551 Appendix C Answers to Odd-Numbered Exercises â•… 553 Appendix D Creditsâ•… 575 Indexâ•…577 A01_GOUL1228_02_GE_FM.indd 09/09/16 4:16 pm www.downloadslide.com SECTION EXERCISES CHAPTER 309 SECTION EXERCISES SECTION 6.1 TRY 6.9╇ Two Thumbtacks╇ 6.1–6.4╇Directions╇ Determine whether each of the following Â�variables would best be modeled as continuous or discrete a From your answers in Exercise 6.7, find the probability of getting ups, up, or ups when flipping two thumbtacks, and report the distribution in a table 6.1╇ (Example 1) b Make a probability distribution graph of this a Length and breadth of a classroom 6.10╇ Two Dices╇ b Number of students present in a class b.╇ Height of a mountain in the Himalayas (in meters) a From your answers in Exercise 6.8, find the probability of getting no multiples, multiple, or multiples of 3, and report the distribution in a table 6.3╇ a.╇ Weight of a person (kilograms) b Make a probability distribution graph of this 6.2╇ a.╇Number of expeditions to Mount Everest b.╇ Weight of a person (pounds) TRY 6.4╇ a.╇ Number of months in a calendar year b.╇The time taken by earth to complete one revolution around the sun TRY 6.5╇ Loaded Die (Example 2)╇ A magician has shaved an edge off one side of a six-sided die, and as a result, the die is no longer “fair.” The figure shows a graph of the probability density function (pdf) Show the pdf in table format by listing all six possible outcomes and their probabilities 6.11╇ Snow Depth (Example 3)╇ Eric wants to go skiing tomorrow, but only if there are inches or more of new snow According to the weather report, any amount of new snow between inch and inches is equally likely The probability density curve for tomorrow’s new snow depth is shown Find the probability that the new snow depth will be inches or more tomorrow Copy the graph, shade the appropriate area, and calculate its numerical value to find the probability The total area is 0.3 0.25 Density Probability 0.20 0.15 0.2 0.1 0.10 0.05 0.0 0.00 Outcome on Die 6.6╇ Fair Diꕇ Toss a fair six-sided die The probability density function (pdf) in table form is given Make a graph of the pdf for the die Number of Spots Probability 1/6 1/6 1/6 1/6 1/6 1/6 * 6.7╇ Distribution of Two Thumbtacks╇ When a certain type of thumbtack is flipped, the probability of its landing tip up (U) is 0.60 and the probability of its landing tip down (D) is 0.40 Now suppose we flip two such thumbtacks: one red, one blue Make a list of all the possible arrangements using U for up and D for down, listing the red one first; include both UD and DU Find the probabilities of each possible outcome, and record the result in table form Be sure the total of all the probabilities is 6.8╇ Distribution of Two Dices╇ When two dices are thrown, the probability of getting a multiple of (M) is 0.33 and the probability of not getting a multiple of (N) is 0.67 Make a list of all possible arrangements for getting a multiple of 3, using M for multiples and N for numbers that are not Find the probabilities of each arrangement, and record the results in table form Be sure the total of all the probabilities is M06_GOUL1228_02_GE_C06.indd 309 6.12╇ Snow Depth╇ Refer to Exercise 6.11 What is the probability that the amount of new snow will be between and inches? Copy the graph from Exercise 6.11, shade the appropriate area, and report the numerical value of the probability SECTION 6.2 6.13╇ Applying the Empirical Rule with z-Scores╇The Empirical Rule applies rough approximations to probabilities for any unimodal, symmetric distribution But for the Normal distribution we can be more precise Use the figure and the fact that the Normal curve is symmetric to answer the questions Do not use a Normal table or technology 68% Density New Snow Depth (inches) 95% Almost All z –3 –2 –1 05/09/16 4:12 pm www.downloadslide.com 310 CHAPTER MODELING RANDOM EVENTS: THE NORMAL AND BINOMIAL MODELS According to the Empirical Rule: a Roughly what percentage of z-scores are between -2 and 2? a Roughly what percentage of students earn quantitative SAT scores greater than 500? i. almost all i. almost all ii. 95% iii. 68% iv. 50% i. almost all ii. 95% iii. 68% ii. 75% iii. 50% iv. 25% v. about 0% b Roughly what percentage of students earn quantitative SAT scores between 400 and 600? b Roughly what percentage of z-scores are between -3 and 3? iv. 50% c Roughly what percentage of z-scores are between -1 and i. almost all i. almost all c Roughly what percentage of students earn quantitative SAT scores greater than 800? ii. 95% iii. 68% iv. 50% d Roughly what percentage of z-scores are greater than 0? i. almost all ii. 95% iii. 68% i. almost all iv. 50% ii. 13.5% iii. 50% iii. 68% ii. 95% iv. 34% iii. 68% iv. 34% v. about 0% v. about 0% d Roughly what percentage of students earn quantitative SAT scores less than 200? e Roughly what percentage of z-scores are between and 2? i. almost all ii. 95% iv. 2% i. almost all 6.14 IQs Wechsler IQs are approximately Normally distributed with a mean of 100 and a standard deviation of 15 Use the probabilities shown in the figure in Exercise 6.13 to answer the following questions Do not use the Normal table or technology You may want to label the figure with Empirical Rule probabilities to help you think about this question ii. 95% iii. 68% iv. 34% v. about 0% e Roughly what percentage of students earn quantitative SAT scores between 300 and 700? i. almost all ii. 95% iii. 68% iv. 34% v. 2.5% f Roughly what percentage of students earn quantitative SAT scores between 700 and 800? i. almost all ii. 95% iii. 68% iv. 34% v. 2.5% Density 6.16 Women’s Heights Assume that college women’s heights are approximately Normally distributed with a mean of 65 inches and a standard deviation of 2.5 inches On the horizontal axis of the graph, indicate the heights that correspond to the z-scores provided (See the labeling in Exercise 6.14.) Use only the Empirical Rule to choose your answers Sixty inches is feet, and 72 inches is feet –2 70 –1 85 100 115 130 145 Density z –3 IQ 55 a Roughly what percentage of people have IQs more than 100? i. almost all ii. 95% iii. 68% iv. 50% b Roughly what percentage of people have IQs between 100 and 115? i. 34% ii. 17% iii. 2.5% iv. 50% c Roughly what percentage of people have IQs below 55? i. almost all ii. 50% iii. 34% z-Score –3 Height iv. about 0% d Roughly what percentage of people have IQs between 70 and 130? i. almost all ii. 95% iii. 68% ii. 17% iii. 2.5% –1 a Roughly what percentage of women’s heights are greater than 72.5 inches? iv. 50% e Roughly what percentage of people have IQs above 130? i. 34% –2 i. almost all iv. 50% ii. 75% iii. 50% iv. 25% v. about 0% f Roughly what percentage people have IQs above 145? b Roughly what percentage of women’s heights are between 60 and 70 inches? i. almost all i. almost all ii. 50% iii. 34% iv. about 0% 6.15 SAT Scores Quantitative SAT scores are approximately Normally distributed with a mean of 500 and a standard deviation of 100 On the horizontal axis of the graph, indicate the SAT scores that correspond with the provided z-scores (See the labeling in Exercise 6.14.) Answer the questions using only your knowledge of the Empirical Rule and symmetry ii. 95% iii. 68% iv. 34% v. about 0% c Roughly what percentage of women’s heights are between 65 and 67.5 inches? i. almost all ii. 95% iii. 68% iv. 34% v. about 0% d Roughly what percentage of women’s heights are between 62.5 and 67.5 inches? i. almost all ii. 95% iii. 68% iv. 34% v. about 0% e Roughly what percentage of women’s heights are less than 57.5 inches? Density i. almost all i. almost all TRY z-Score –3 SAT Score ii. 95% iii. 68% iv. 34% v. about 0% f Roughly what percentage of women’s heights are between 65 and 70 inches? –2 M06_GOUL1228_02_GE_C06.indd 310 –1 ii. 95% iii. 47.5% iv. 34% v. 2.5% 6.17 Women’s Heights (Example 4) College women have a mean height of 65 inches and a standard deviation of 2.5 inches The distribution of heights for this group is Normal Choose the 03/09/16 3:06 pm www.downloadslide.com SECTION EXERCISES CHAPTER 311 correct StatCrunch output for finding the percentage of college women with heights of less than 63 inches, and report the correct percentage 6.19 Standard Normal Use the table or technology to answer each question Include an appropriately labeled sketch of the Normal curve for each part Shade the appropriate region (A) a Find the area in a standard Normal curve to the left of 1.02 by using the Normal table (See the excerpt provided.) Note the shaded curve b Find the area in a standard Normal curve to the right of 1.02 Remember that the total area under the curve is Table Entry z Format of the Normal Table: The area given is the area to the left of (less than) the given z-score (B) z 00 01 02 03 04 0.9 8159 8186 8212 8238 8264 1.0 8413 8438 8461 8485 8508 1.1 8643 8665 8686 8708 8729 6.20 Standard Normal Use a table or technology to answer each question Include an appropriately labeled sketch of the Normal curve for each part Shade the appropriate region a Find the area to the left of a z-score of -0.50 b Find the area to the right of a z-score of - 0.50 6.21 Standard Normal Use a table or technology to answer each question Include an appropriately labeled sketch of the Normal curve for each part Shade the appropriate region 6.18 ACT Scores ACT scores are approximately Normally distributed with a mean of 21 and a standard deviation of 5, as shown in the figure (ACT scores are test scores that some colleges use for determining admission.) What is the probability that a randomly selected person scores 24 or more? a Find the probability that a z-score will be 1.76 or less b Find the probability that a z-score will be 1.76 or more c Find the probability that a z-score will be between - 1.3 and -1.03 6.22 Standard Normal Use a table or technology to answer each question Include an appropriately labeled sketch of the Normal curve for each part Shade the appropriate region a Find the probability that a z-score will be -1.00 or less b Find the probability that a z-score will be more than -1.00 c Find the probability that a z-score will be between 0.90 and 1.80 6.23 Extreme Positive z-Scores For each question, find the area to the right of the given z-score in a standard Normal distribution In this question, round your answers to the nearest 0.000 Include an appropriately labeled sketch of the N(0, 1) curve a z = 4.00 b z = 10.00 (Hint: Should this tail proportion be larger or smaller than the answer to part a? Draw a picture and think about it.) c z = 50.00 d If you had the exact probability for these tail proportions, which would be the largest and which would be the smallest? e Which is equal to the area in part b: the area below (to the left of) z = - 10.00 or the area above (to the right of) z = -10.00? M06_GOUL1228_02_GE_C06.indd 311 03/09/16 3:06 pm www.downloadslide.com 312 CHAPTER MODELING RANDOM EVENTS: THE NORMAL AND BINOMIAL MODELS 6.24╇ Extreme Negative z-Scores╇ For each question, find the area to the right of the given z-score in a standard Normal distribution In this question, round your answers to the nearest 0.000 Include an appropriately labeled sketch of the N(0, 1) curve a z = - 4.00 6.31╇ Protein Intake: Men╇ The Dietary Reference Intake of proteins is different for men and women For both, the distribution is approximately Normal For men, the middle 95% range from 54 to 58 grams per day, and for women, the middle 95% have protein intake recommendation between 44.8 and 47.4 grams per day b z = - 8.00 a What is the mean for the men? Explain your reasoning c z = - 30.00 b * Find the standard deviation for the men Explain your reasoning d If you had the exact probability for these right proportions, which would be the largest and which would be the smallest? e Which is equal to the area in part b: the area below (to the left of) z = 8.00 or the area above (to the right of) z = 8.00? TRY 6.25╇ Females’ SAT Scores (Example 5)╇ According to data from the College Board, the mean quantitative SAT score for female college-bound high school seniors is 500 Assume that SAT scores are approximately Normally distributed with a population standard deviation of 100 What percentage of female college-bound students had scores above 675? Please include a well-labeled Normal curve as part of your answer See page 317 for guidance 6.26╇ Males’ SAT Scores╇ According to data from the College Board, the mean quantitative SAT score for male collegebound high school seniors is 530 Assume that SAT scores are Â�approximately Normally distributed with a population standard deviation of 100 If a male college-bound high school senior is selected at random, what is the probability that he will score higher than 675? 6.27╇ Stanford–Binet IQs╇ Stanford–Binet IQ scores for children are approximately Normally distributed and have m = 100 and s = 15 What is the probability that a randomly selected child will have an IQ below 115? 6.28╇ Stanford–Binet IQs╇ Stanford–Binet IQs for children are approximately Normally distributed and have m = 100 and s = 15 What is the probability that a randomly selected child will have an IQ of 115 or above? TRY 6.29╇ Birth Length (Example 6)╇ According to National Vital Statistics, the average length of a newborn baby is 19.5 inches with a standard deviation of 0.9 inch The distribution of lengths is approximately Normal Use a table or technology for each question Include an appropriately labeled and shaded Normal curve for each part There should be three separate curves a What is the probability that a baby will have a length of 20.4 inches or more? b What is the probability that a baby will have a length of 21.4 inches or more? c What is the probability that a baby will be between 18 and 21 inches in length? 6.30╇ White Blood Cells╇ The distribution of white blood cell count per cubic millimeter of whole blood is approximately Normal with mean 7500 and standard deviation 1750 for healthy patients Include an appropriately labeled and shaded Normal curve for each part There should be three separate curves * 6.32╇ Protein Intake: Women╇ Answer the previous question for the women 6.33╇ Bilingual Minors╇ A survey shows that in one year, the average number of bilingual minors in a school was 384 Assume that the standard deviation is 98 and the number of bilingual minors is Normally distributed Include an appropriately labeled and shaded Normal curve for each part a What percentage of schools have more than 384 bilingual minors? b What percentage of schools have 350 or less bilingual minors? 6.34╇ Bilingual Employees╇ A survey shows that in one year, the average number of bilingual employees in an office was 12 Assume that the standard deviation is and the number of bilingual employees is Normally distributed Include an appropriately labeled and shaded Normal curve for each part a What percentage of offices have more than 12 bilingual employees? b What percentage of offices have 10 or more bilingual employees? 6.35╇ Bilingual Army Officers╇ A survey shows that in one year, the average number of bilingual officers in an army battalion was 196 Assume that the standard deviation is 22 and the number of bilingual officers in an army battalion is Normally distributed Include an appropriately labeled and shaded Normal curve for each part a What percentage of battalions have between 150 and 200 bilingual officers? b What percentage of battalions have between 200 and 250 bilingual officers? 6.36╇ Bilingual Doctors╇ A survey shows that in one year, the average number of bilingual doctors in a hospital was 42 Assume that the standard deviation is 12 and the number of bilingual Â�doctors in a hospital is normally distributed Include an appropriately labeled and shaded Normal curve for each part a What percentage of hospitals have between 40 and 45 bilingual doctors? b What percentage of hospitals have between 45 and 50 bilingual doctors? 6.37╇ New York City Weather╇ New York City’s mean minimum daily temperature in February is 27°F (http://www.ny.com) Suppose the standard deviation of the minimum temperature is 6°F and the distribution of minimum temperatures in February is approximately Normal What percentage of days in February has minimum temperatures below freezing (32°F)? * 6.38╇ Women’s Heights╇ Assume for this question that college women’s heights are approximately Normally distributed with a mean of 64.6 inches and a standard deviation of 2.6 inches Draw a well-labeled Normal curve for each part a Find the percentage of women who should have heights of 63.5 inches or less a What is the probability that a randomly selected person will have a white blood cell count of between 7000 and 10,000? b In a sample of 123 women, according to the probability obtained in part a, how many should have heights of 63.5 inches or less? b What is the probability that a randomly selected person will have a white blood cell count of between 5000 and 12,000? c The table shows the frequencies of heights for a sample of women, collected by statistician Brian Joiner in his statistics class Count the women who appear to have heights of 63 inches or less by looking at the table They are in the oval c What is the probability that a randomly selected person will have a white blood cell count of more than 10,000? M06_GOUL1228_02_GE_C06.indd 312 07/09/16 2:55 pm www.downloadslide.com SECTION EXERCISES 313 6.43╇ Inverse Normal, Standard╇ Assume a standard Normal distribution Draw a separate, well-labeled Normal curve for each part d Are the answers to parts b and c the same or different? Explain TRY CHAPTER Height Inches Frequency a Find the z-score that gives a left area of 0.8577 59 ╇2 b Find the z-score that gives a left area of 0.0146 5' 60 ╇5 61 ╇7 6.44╇ Inverse Normal, Standard╇ Assume a standard Normal distribution Draw a separate, well-labeled Normal curve for each part 5'2" 62 10 63 16 5'4" 64 23 65 19 5'6" 66 15 67 ╇9 5'8" 68 ╇6 69 ╇6 5'10" 70 ╇3 71 ╇1 6' 72 ╇1 a Find the z-score that gives a left area of 0.9774 b Find the z-score that gives a left area of 0.8225 TRY 6.45╇ Females’ SAT Scores (Example 8)╇ According to the College Board, the mean quantitative SAT score for female collegebound high school seniors in one year was 500 SAT scores are approximately Normally distributed with a population standard deviation of 100 A scholarship committee wants to give awards to college-bound women who score at the 96th percentile or above on the SAT What score does an applicant need? Include a well-labeled Normal curve as part of your answer See page 317 for guidance 6.46╇ Males’ SAT Scores╇ According to the College Board, the mean quantitative SAT score for male college-bound high school seniors in one year was 530 SAT scores are approximately Normally distributed with a population standard deviation of 100 What is the SAT score at the 96th percentile for male college-bound seniors? 6.39╇ Probability or Measurement (Inverse)? (Example 7)╇ The Normal model N(500,100) describes the distribution of critical reading SAT scores in the United States Which of the following questions asks for a probability and which asks for a measurement (and is thus an inverse Normal question)? 6.47╇ Tall Club, Women╇ Suppose there is a club for tall people that requires that women be at or above the 98th percentile in height Assume that women’s heights are distributed as N (64, 2.5) Find what women’s height is the minimum required for joining the club, rounding to the nearest inch Draw a well-labeled sketch to support your answer b What is the probability that a randomly selected person will score 550 or more? 6.48╇ Tall Club, Men╇ Suppose there is a club for tall people that requires that men be at or above the 98th percentile in height Assume that men’s heights are distributed as N (69, 3) Find what men’s height is the minimum required for joining the club, rounding to the nearest inch Draw a well-labeled sketch to support your answer 6.40╇ Probability or Measurement (Inverse)?╇ The Normal model N(65, 2.5) describes the distribution of heights of college women (inches) Which of the following questions asks for a probability and which asks for a measurement (and is thus an inverse Normal question)? 6.49╇ Women’s Heights╇ Suppose college women’s heights are approximately Normally distributed with a mean of 65 inches and a population standard deviation of 2.5 inches What height is at the 20th percentile? Include an appropriately labeled sketch of the Normal curve to support your answer a What reading SAT score is at the 65th percentile? a What is the probability that a random college woman has a height of 68 inches or more? b To be in the Tall Club, a woman must have a height such that only 2% of women are taller What is this height? 6.41╇ Inverse Normal, Standard╇ In a standard Normal distribution, if the area to the left of a z-score is about 0.6986, what is the approximate z-score? First locate, inside the table, the number closest to 0.6986 Then find the z-score by adding 0.5 and 0.02; refer to the table Draw a sketch of the Normal curve, showing the area and z-score 6.50╇ Men’s Heights╇ Suppose college men’s heights are approximately Normally distributed with a mean of 70.0 inches and a population standard deviation of inches What height is at the 20th percentile? Include an appropriately labeled Normal curve to support your answer 6.51╇ Inverse SATs╇ Critical reading SAT scores are distributed as N(500, 100) a Find the SAT score at the 75th percentile b Find the SAT score at the 25th percentile c Find the interquartile range for SAT scores z 00 01 02 03 04 05 0.4 6554 6591 6628 6664 6700 6736 0.5 6915 6950 6985 7019 7054 7088 6.52╇ Inverse Women’s Heights╇ College women have heights with the following distribution (inches): N (65, 2.5) 0.6 7257 7291 7324 7357 7389 7422 a Find the height at the 75th percentile 6.42╇ Inverse Normal, Standard╇ In a standard Normal distribution, if the area to the left of a z-score is about 0.2000, what is the approximate z-score? M06_GOUL1228_02_GE_C06.indd 313 d Is the interquartile range larger or smaller than the standard deviation? Explain b Find the height at the 25th percentile c Find the interquartile range for heights d Is the interquartile range larger or smaller than the standard deviation? Explain 07/09/16 2:55 pm www.downloadslide.com 314 CHAPTER MODELING RANDOM EVENTS: THE NORMAL AND BINOMIAL MODELS 6.53 Girls’ and Women’s Heights According to the National Health Center, the heights of 6-year-old girls are Normally distributed with a mean of 45 inches and standard deviation of inches 0.4 Take a random sample of 15 people opening their accounts within year and 15 people opening their accounts after year The sample chosen is such that either the husband or the wife is included in it Why is the binomial model inappropriate for finding the probability that exactly out of the 30 people in the sample will open joint bank accounts within year? List all of the binomial conditions that are not met a In which percentile is a 6-year-old girl who is 46.5 inches tall? b If a 6-year-old girl who is 46.5 inches tall grows up to be a woman at the same percentile of height, what height will she be? Assume women are distributed as N (64, 2.5) 6.54 Boys’ and Men’s Heights According to the National Health Center, the heights of 5-year-old boys are Normally distributed with a mean of 43 inches and standard deviation of 1.5 inches TRY a In which percentile is a 5-year-old boy who is 46.5 inches tall? b If a 5-year-old boy who is 46.5 inches tall grows up to be a man at the same percentile of height, what height will he be? Assume adult men’s heights (inches) are distributed as N (69, 3) 6.55 Cats’ Birth Weights The average birth weight of domestic cats is about ounces Assume that the distribution of birth weights is Normal with a standard deviation of 0.4 ounce a If 25 males in the age group of 40–49 are chosen at random, what is the probability that 14 of them can read English? b If 25 males in the age group of 40–49 are chosen at random, what is the probability that 17 of them cannot read English? a Find the birth weight of cats at the 90th percentile b Find the birth weight of cats at the 10th percentile 6.64 Identifying n, p, and x For each situation, identify the sample size n, the probability of success p, and the number of successes x When asked for the probability, state the answer in the form b(n, p, x) There is no need to give the numerical value of the probability Assume the conditions for a binomial experiment are satisfied 6.56 Elephants’ Birth Weights The average birth weight of elephants is 230 pounds Assume that the distribution of birth weights is Normal with a standard deviation of 50 pounds Find the birth weight of elephants at the 95th percentile SECTION 6.3 TRY 6.57 Gender of Children (Example 9) A married couple plans to have four children, and they are wondering how many boys they should expect to have Assume none of the children will be twins or other multiple births Also assume the probability that a child will be a boy is 0.50 Explain why this is a binomial experiment Check all four required conditions 6.58 Dice Roll A dice will be rolled four times, and the multiples of that appear will be recorded Explain why this is a binomial experiment Check all four required conditions TRY 6.59 Dice Rolls (Example 10) A group wants to find out whether dice rolls have a 1/3 chance of coming up with multiples of The leader of the group asks all members to roll dices for 10 minutes and then report their results to him Which condition or conditions for use of the binomial model is or are not met? 6.60 Twins In Exercise 6.57 you are told to assume that none of the children will be twins or other multiple births Why? Which of the conditions required for a binomial experiment would be violated if there were twins? 6.61 Joint Bank Account Suppose the probability that a randomly selected person who has a joint bank account with a spouse will close it within 10 years is 0.1 Suppose we follow 20 such persons (40 account holders) for 10 years and record the number of people closing their accounts Why is the binomial model inappropriate for finding the probability that at least 19 of these 40 account holders will close their accounts within 10 years? List all binomial conditions that are not met 6.62 Joint Bank Account Suppose the probability that a randomly selected couple opens a joint bank account within year of marriage is 0.3, and the probability that a randomly selected couple opens a joint bank account after a year of marriage is M06_GOUL1228_02_GE_C06.indd 314 6.63 Identifying n, p, and x (Example 11) For each situation, identify the sample size n, the probability of success p, and the number of successes x When asked for the probability, state the answer in the form b(n, p, x) There is no need to give the numerical value of the probability Assume the conditions for a binomial experiment are satisfied National Bureau of Statistics, Republic of Maldives, reported that 60.3% of males in the age group of 45–49 can read English a According to the Federal Highway Research Institute in Germany, out of persons in an accident get killed In a random sample of 27 persons meeting with an accident, what is the probability that exactly 12 persons would have died? b Twenty-five percent of the persons killed in accidents are pedestrians If we randomly select 27 persons who have died in an accident, what is the probability that 12 persons are pedestrians? TRY 6.65 Stolen Bicycles (Example 12) According to the Sydney Morning Herald, 40% of bicycles stolen in Holland are recovered (In contrast, only 2% of bikes stolen in New York City are recovered.) Find the probability that, in a sample of randomly selected cases of bicycles stolen in Holland, exactly out of bikes are recovered 6.66 Florida Recidivism Rate The three-year recidivism rate of parolees in Florida is about 30%; that is, 30% of parolees end up back in prison within three years (http://www.floridaperforms.com) Assume that whether one parolee returns to prison is independent of whether any of the others returns a Find the probability that exactly out of 20 parolees will end up back in prison within three years, b Find the probability that or fewer out of 20 parolees will end up back in prison within three years 6.67 Harvard Admission The undergraduate admission rate at Harvard University is about 6% a Assuming the admission rate is still 6%, in a sample of 100 applicants to Harvard, what is the probability that exactly will be admitted? Assume that decisions to admit are independent b What is the probability that exactly 95 out of 100 applicants will be rejected? 03/09/16 3:06 pm www.downloadslide.com SECTION EXERCISES 315 6.68╇ Cornell Admission╇ The undergraduate admission rate at Cornell University is about 16% c If 10 households are selected randomly, what is the probability that or fewer have high-speed access? a Assuming the admission rate is still 16%, in a sample of 100 applicants to Cornell, what is the probability that exactly 15 will be admitted? 6.75╇ Drunk Walking╇ You may have heard that drunk driving is dangerous, but what about drunk walking? According to federal information (reported in the Ventura County Star on August 6, 2013), 50% of the pedestrians killed in the United States had a blood-alcohol level of 0.08% or higher Assume that two randomly selected pedestrians who were killed are studied b What is the probability that exactly 85 out of 100 independent Â�applicants will be rejected? 6.69╇ Wisconsin Graduation╇ Wisconsin has the highest high school graduation rate of all states at 90% a In a random sample of 10 Wisconsin high school students, what is the probability that will graduate? a If the pedestrian was drunk (had a blood-alcohol level of 0.08% or higher), we will record a D, and if the pedestrian was not drunk, we will record an N List all possible sequences of D and N b In a random sample of 10 Wisconsin high school students, what is the probability than or fewer will graduate? b For each sequence, find the probability that it will occur, by assuming independence c What is the probability that at least high school students in our sample of 10 will graduate? c What is the probability that neither of the two pedestrians was drunk? d What is the probability that exactly one out of two independent pedestrians was drunk? 6.70╇ Colorado Graduation╇ Colorado has a high school Â�graduation rate of 75% e What is the probability that both were drunk? b In a random sample of 15 Colorado high school students, what is the probability that or fewer will graduate? 6.76╇ Fish Caught in Spain╇ According to the Eurostat Statistics Database, Spain accounts for 19% of the total fishes caught in the European Union (EU) Assume that we randomly sample two fishes caught in the EU c What is the probability that at least high school students in our sample of 15 will graduate? a If a fish is caught in Spain, record Y; if not, record N List all possible sequences of Y and N 6.71╇ Florida Homicide Clearancꕇ The homicide clearance rate in Florida is 60% A crime is cleared when an arrest is made, a crime is charged, and the case is referred to a court b For each sequence, find by hand the probability that it will occur, assuming each outcome is independent a In a random sample of 15 Colorado high school students, what is the probability that exactly will graduate? c What is the probability that neither of the two randomly selected fishes have been caught in Spain? a What is the probability that exactly out of 10 independent homicides are cleared? b Without doing a calculation, state whether the probability that or more out of 10 homicides are cleared will be larger or smaller than the answer to part a? Why? d What is the probability that exactly one out of the two fishes has been caught in Spain? e What is the probability that both have been caught in Spain? TRY 6.77╇ Die Roll (Example 14)╇ A fair die is rolled 60 times c What is the probability that or more out of 10 independent homicides are cleared? a What is the expected number of times that an odd number will turn up? 6.72╇ Virginia Homicide Clearancꕇ The homicide clearance rate in Virginia is 74% A crime is cleared when an arrest is made, a crime is charged, and the case is referred to a court c How many times should you expect odd numbers to turn up, give or take how many times? Based on these numbers, give the range of the number of times odd numbers can turn up b Find the standard deviation for the outcome to be an odd number 6.78╇ Drivers Aged 60–65╇ According to GMAC Insurance, 20% of drivers aged 60–65 fail the written drivers’ test This is the Â�lowest failure rate of any age group (Source: http://www.gmacinsurance.com/SafeDriving/PressRelease.asp) If 200 people aged 60–65 independently take the exam, how many would you expect to pass? Give or take how many? a What is the probability that or fewer out of 10 independent homicides are cleared? b What is the probability that or more out of 10 independent homicides are cleared? c Are the answers to parts a and part b complementary? Why or why not? TRY CHAPTER 6.73╇ DWI Convictions (Example 13)╇ In New Mexico, about 70% of drivers who are arrested for driving while intoxicated (DWI) are convicted (http://www.drunkdrivingduilawblog.com) a If 15 independently selected drivers were arrested for DWI, how many of them would you expect to be convicted? b What is the probability that exactly 11 out of 15 independent selected drivers are convicted? c What is the probability that 11 or fewer are convicted? 6.74╇ Internet Access╇ A 2013 Gallup poll indicated that about 80% of U.S households had access to a high-speed Internet Â�connection Assume this rate has not changed a Suppose 100 households were randomly selected from the United States How many of the households would you expect to have access to a high-speed Internet connection? b If 10 households are selected randomly, what is the probability that exactly have high-speed access? M06_GOUL1228_02_GE_C06.indd 315 TRY 6.79╇ Road Accidents╇ According to the Ministry of Transport in the Republic of South Africa, 80% of road accidents in Johannesburg involve males in the age group of 19–34 Suppose that there are 200 road accidents in a day a What is the number of accidents involving males between the ages of 19 and 34? b What is the standard deviation for the number of accidents involving males between the ages of 19 and 34? c After a great many days, according to the Empirical Rule, on about 95% of these days, the number of accidents involving males between the ages of 19 and 34 will be as low as _ and as high as _ (Hint: Find two standard deviations below and two standard deviations above the mean.) d If you found that on one day, 158 out of 200 accidents involved males between the ages of 19 and 34, would you consider this to be a very high number? 03/09/16 3:06 pm www.downloadslide.com 316 CHAPTER MODELING RANDOM EVENTS: THE NORMAL AND BINOMIAL MODELS 6.80 Road Accidents in Small Towns In smaller cities like Port Elizabeth in South Africa, the rate of road accidents caused by males between the ages of 19 and 34 is 64%, which is much lower than the rate of accidents caused by males between the ages of 19 and 34 in Johannesburg Suppose that there are 200 road accidents in a day a What is the number of accidents caused by males between the ages of 19 and 34? c After a great many days, according to the Empirical Rule, on about 95% of these days, the number of accidents caused by males between the ages of 19 and 34 will be as low as _ and as high as _ d If you found that on one day, 158 out of 200 accidents were caused by males between the ages of 19 and 34, would you consider this to be a very high number? b What is the standard deviation for the number of accidents caused by males between the ages of 19 and 34? CHAPTER REVIEW EXERCISES 6.81 Birth Length A study of U.S births published on the website Medscape from WebMD reported that the average birth length of babies was 20.5 inches and the standard deviation was about 0.90 inch Assume the distribution is approximately Normal Find the percentage of babies with birth lengths of 22 inches or less 6.82 Birth Length A study of U.S births published on the website Medscape from WebMD reported that the average birth length of babies was 20.5 inches and the standard deviation was about 0.90 inch Assume the distribution is approximately Normal Find the percentage of babies who have lengths of 19 inches or less at birth 6.83 Males’ Body Temperatures A study of human body temperatures using healthy men showed a mean of 98.1°F and a standard deviation of 0.70°F Assume the temperatures are approximately Normally distributed a How many of the 800 graduates would you expect to be selfemployed? b What is the standard deviation for the number of self-employed? c According to the Empirical Rule, in 95% of all samples of 800 persons, the number of self-employed would vary between what two values? d In a random sample of 800, how many would you expect not to choose self-employment? e If 500 were self-employed and 300 were not self-employed, would 500 be a surprisingly large number of self-employed? * 6.87 Low Birth Weights, Normal and Binomial Babies a Find the percentage of healthy men with temperatures below 98.6°F (that temperature was considered typical for many decades) weighing 5.5 pounds or less at birth are said to have low birth weights, which can be dangerous Full-term birth weights for single babies (not twins or triplets or other multiple births) are Normally distributed with a mean of 7.5 pounds and a standard deviation of 1.1 pounds b What temperature does a healthy man have if his temperature is at the 76th percentile? a For one randomly selected full-term single-birth baby, what is the probability that the birth weight is 5.5 pounds or less? 6.84 Females’ Body Temperatures A study of human body temperatures using healthy women showed a mean of 98.4°F and a standard deviation of about 0.70°F Assume the temperatures are approximately Normally distributed b For two randomly selected full-term, single-birth babies, what is the probability that both have birth weights of 5.5 pounds or less? a Find the percentage of healthy women with temperatures below 98.6°F (this temperature was considered typical for many decades) d If 200 independent full-term single-birth babies are born at a hospital, how many would you expect to have birth weights of 5.5 pounds or less? Round to the nearest whole number b What temperature does a healthy woman have if her temperature is at the 76th percentile? 6.85 Cremation Rates in Nevada Cremation rates have been increasing In Nevada the cremation rate is 70% Suppose that we take a random sample of 400 deaths in Nevada a How many of these decedents would you expect to be cremated? b What is the standard deviation for the number to be cremated? c How many would you expect not to be cremated? d What is the standard deviation for the number not to be cremated? e * Explain the relationship between the answers to parts b and d * 6.86 Self-employment Rates in India Self-employment rates have been increasing in the rural areas of India In urban areas, this rate is as low as 41.1% Suppose that we take a random sample of 800 fresh graduates in urban areas and they independently make the decision of being self-employed c For 200 random full-term single births, what is the approximate probability that or fewer have low birth weights? e What is the standard deviation for the number of babies out of 200 who weigh 5.5 pounds or less? Retain two decimal digits for use in part f f Report the birth weight for full-term single babies (with 200 births) for two standard deviations below the mean and for two standard deviations above the mean Round both numbers to the nearest whole number g If there were 45 low-birth-weight full-term babies out of 200, would you be surprised? * 6.88 Quantitative SAT Scores, Normal and Binomial The distribution of the math portion of SAT scores has a mean of 500 and a standard deviation of 100, and the scores are approximately Normally distributed a What is the probability that one randomly selected person will have an SAT score of 550 or more? b What is the probability that four randomly selected people will all have SAT scores of 550 or more? c For 800 randomly selected people, what is the probability that 250 or more will have scores of 550 or more? M06_GOUL1228_02_GE_C06.indd 316 03/09/16 3:06 pm www.downloadslide.com GUIDED EXERCISES CHAPTER 317 d For 800 randomly selected people, on average how many should have scores of 550 or more? Round to the nearest whole number c How does your answer to part b compare to the mean birth length? Why should you have expected this? e Find the standard deviation for part d Round to the nearest whole number 6.90 Birth Length and z-Scores, Inverse Babies in the United States have a mean birth length of 20.5 inches with a standard deviation of 0.90 inch The shape of the distribution of birth lengths is approximately Normal f Report the range of people out of 800 who should have scores of 550 or more from two standard deviations below the mean to two standard deviations above the mean Use your rounded answers to part d and e a Find the birth length at the 2.5th percentile g If 400 out of 800 randomly selected people had scores of 550 or more, would you be surprised? Explain b Find the birth length at the 97.5th percentile 6.89 Babies’ Birth Length, Inverse Babies in the United States have a mean birth length of 20.5 inches with a standard deviation of 0.90 inch The shape of the distribution of birth lengths is approximately Normal d Find the z-score for the length at the 97.5th percentile c Find the z-score for the length at the 2.5th percentile a How long is a baby born at the 20th percentile? b How long is a baby born at the 50th percentile? UIDED EXERCISES Step 4 c Add the line, z-score, and shading Draw a vertical line through the curve at the location of 675 Just above the 675 (indicated on the graph with “???”) put in the corresponding z-score We want to find what percentage of students had scores above 675 Therefore, shade the area to the right of this boundary, because numbers to the right are larger 6.25 Females’ SAT Scores According to data from the College Board, the mean quantitative SAT score for female college-bound high school seniors is 500 SAT scores are approximately Normally distributed with a population standard deviation of 100 QUESTION What percentage of the female college-bound high school seniors had scores above 675? Answer this question by following the numbered steps Step 5 c Use the table for the left area Use the following excerpt from the Normal table to find and report the area to the left of the z-score that was obtained from an SAT score of 675 This is the area of the unshaded region Step 1 c Find the z-score To find the z-score for 675, subtract the mean and divide by the standard deviation Report the z-score Step 6 c Answer Because you want the area to the right of the z-score, you will have to subtract the area you obtained in step from This is the area of the shaded region Put it where the box labeled “Answer” is Check to see that the number makes sense For example, if the shading is less than half the area, the answer should not be more than 0.5000 Step 2 c Explain the location of 500 Refer to the Normal curve Explain why the SAT score of 500 is right below the z-score of The tick marks on the axis mark the location of z-scores that are integers from -3 to Density Step 3 c Label with SAT scores Carefully sketch a copy of the curve Pencil in the SAT scores of 200, 300, 400, 600, and 700 in the correct places Answer 500 M06_GOUL1228_02_GE_C06.indd 317 ??? 675 800 Step 7 c Sentence Finally, write a sentence stating what you found z 00 01 02 03 04 05 06 07 08 09 1.6 9452 9463 9474 9484 9495 9505 9515 9525 9535 9545 1.7 9554 9564 9573 9582 9591 9599 9608 9616 9625 9633 1.8 9641 9649 9656 9664 9671 9678 9686 9693 9699 9706 6.45 Females’ SAT scores According to the College Board, the mean quantitative SAT score for female college-bound high school seniors in one year was 500 SAT scores are approximately Normally distributed with a population standard deviation of 100 A scholarship committee wants to give awards to college-bound 03/09/16 3:06 pm www.downloadslide.com 318 CHAPTER MODELING RANDOM EVENTS: THE NORMAL AND BINOMIAL MODELS Density women who score at the 96th percentile or above on the SAT What score does an applicant need to receive an award? Include a welllabeled Normal curve as part of your answer 500 Answer 800 QUESTION What is the SAT score at the 96th percentile? Answer this question by following the numbered steps Step 1 c Think about it Will the SAT test score be above the mean or below it? Explain Step 2 c Label z-scores Label the curve with integer z-scores The tick marks represent the position of integer z-scores from -3 to M06_GOUL1228_02_GE_C06.indd 318 Step 3 c Use the table The 96th percentile has 96% of the area to the left because it is higher than 96% of the scores The table above gives the areas to the left of z-scores Therefore, we look for 0.9600 in the interior part of the table Use the excerpt of the Normal table given on page 317 for Exercise 6.25 to locate the area closest to 0.9600 Then report the z-score for that area Step 4 c Add the z-score, line, and shading to the sketch Add that z-score to the sketch, and draw a vertical line above it through the curve Shade the left side because the area to the left is what is given Step 5 c Find the SAT score Find the SAT score that corresponds to the z-score The score should be z standard deviations above the mean, so x = m + zs Step 6 c Add the SAT score to the sketch Add the SAT score on the sketch where it says “Answer.” Step 7 c Write a sentence Finally, write a sentence stating what you found 03/09/16 3:06 pm www.downloadslide.com TechTips For All Technology All technologies will use the two examples that follow EXAMPLE A: NORMAL c Wechsler IQs have a mean of 100 and standard deviation of 15 and are Normally distributed a. Find the probability that a randomly chosen person will have an IQ between 85 and 115 b. Find the probability that a randomly chosen person will have an IQ that is 115 or less c. Find the Wechsler IQ at the 75th percentile Note: If you want to use technology to find areas from standard units (z-scores), use a mean of and a standard deviation of EXAMPLE B: BINOMIAL c Imagine that you are flipping a fair coin (one that comes up heads 50% of the time in the long run) a. Find the probability of getting 28 or fewer heads in 50 flips of a fair coin b. Find the probability of getting exactly 28 heads in 50 flips of a fair coin TI-84 NORMAL a. Between Two Values Press 2ND DISTR (located below the four arrows on the keypad) Select 2:normalcdf and press ENTER Enter lower: 85, upper: 115, M: 100, S: 15 For Paste, press ENTER Then press ENTER again Your screen should look like Figure 6A, which shows that the probability that a randomly selected person will have a Wechsler IQ between 85 and 115 is equal to 0.6827 m Figure 6B TI-84 normalcdf with indeterminate left boundary (If you have an indeterminate upper, or right boundary, then to find the probability that the person’s IQ is 85 or more, for example, use a upper, or right boundary (such as 1000000) that is clearly above all the data.) m FIGURE 6A TI-84 normalcdf (c stands for “cumulative”) b. Some Value or Less Press 2ND DISTR Select 2:normalcdf and press ENTER Enter: − 1000000, 115, 100, 15, press ENTER and press ENTER again Caution: The negative number button (−) is to the left of the ENTER button and is not the same as the minus button that is above the plus button The probability that a person’s IQ is 115 or less has an indeterminate lower (left) boundary, for which you may use negative 1000000 or any extreme value that is clearly out of the range of data Figure 6B shows the probability that a randomly selected person will have an IQ of 115 or less c Inverse Normal If you want a measurement (such as an IQ) from a proportion or percentile: Press 2ND DISTR Select 3:invNorm and press ENTER Enter (left) area: 75, M: 100, S: 15 For Paste, press ENTER Then ENTER again Figure 6C shows the Wechsler IQ at the 75th percentile, which is 110 Note that the 75th percentile is entered as 75 m Figure 6C TI-84 Inverse Normal 319 M06_GOUL1228_02_GE_C06.indd 319 03/09/16 3:06 pm www.downloadslide.com BINOMIAL a Cumulative (or Fewer) Press 2ND DISTR Select B:binomcdf (you will have to scroll down to see it) and press ENTER (On a TI-83, it is A:binomcdf.) Enter trials: 50, p: 5, x value: 28 For Paste, press ENTER Then press ENTER again The answer will be the probability for x or fewer Figure 6D shows the probability of 28 or fewer heads out of 50 flips of a fair coin (You could find the probability of 29 or more heads by subtracting your answer from 1.) b Individual (Exact) Press 2ND DISTR Select A:binompdf and press ENTER (On a TI-83, it is 0:binompdf.) Enter trials: 50, p: 5, x value: 28 For Paste, press ENTER Then press ENTER again Figure 6E shows the probability of exactly 28 heads out of 50 flips of a fair coin m Figure 6E TI-84 binompdf (individual) m Figure 6D TI-84 binomcdf (cumulative) MINITAB Normal a Between Two Values Enter the upper boundary, 115, in the top cell of an empty column; here we use column C1, row Enter the lower boundary, 85, in the cell below; here column C1, row 2 Calc > Probability Distributions > Normal See Figure 6F: Choose Cumulative probability Enter: Mean, 100; Standard deviation, 15; Input column, C1; Optional storage, C2 Click OK Subtract the lower probability from the larger shown in column C2 0.8413 - 0.1587 = 0.6836 is the probability that a Wechsler IQ is between 85 and 115 m Figure 6F Minitab Normal b Some Value or Less The probability of an IQ of 115 or less, 0.8413, is shown in column C2, row (In other words, as in part a above, except not enter the lower boundary, 85.) c Inverse Normal If you want a measurement (such as an IQ or height) from a proportion or percentile: Enter the decimal form of the left proportion (.75 for the 75th percentile) into a cell in an empty column in the spreadsheet; here we used column C1, row Calc > Probability Distributions > Normal See Figure 6G: Choose Inverse cumulative probability Enter: Mean, 100; Standard deviation, 15; Input column, c1; and Optional storage, c2 (or an empty column) m Figure 6G Minitab Inverse Normal 320 M06_GOUL1228_02_GE_C06.indd 320 03/09/16 3:06 pm www.downloadslide.com Click OK You will get 110, which is the Wechsler IQ at the 75th percentile Binomial a Cumulative (or Fewer) Enter the upper bound for the number of successes in an empty column; here we used column C1, row Enter 28 to get the probability of 28 or fewer heads Calc > Probability Distributions > Binomial See Figure 6H Choose Cumulative probability Enter: Number of trials, 50; Event probability, 5; Input column, c1; Optional storage, c2 (or an empty column) Click OK Your answer will be 0.8389 for the probability of 28 or fewer heads b Individual (Exact) Enter the number of successes at the top of column 1, 28 for 28 heads Calc > Probability Distributions > Binomial Choose Probability (at the top of Figure 6H) instead of Cumulative Probability and enter: Number of trials, 50; Event probability, 5; Input column, c1; Optional storage, c2 (or an empty column) Click OK Your answer will be 0.0788 for the probability of exactly 28 heads m Figure 6H Minitab Binomial EXCEL Normal Unlike the TI-84, Excel makes it easier to find the probability that a random person has an IQ of 115 or less than to find the probability that a random person has an IQ between 85 and 115 This is why, for Excel, part b appears before part a b Some Value or Less Click fx (and select a category All) Choose NORM.DIST See Figure 6I Enter: X, 115; Mean, 100; Standard_dev, 15; Cumulative, true (for 115 or less) The answer is shown as 0.8413 Click OK to make it show up in the active cell on the spreadsheet a Between Two Values If you want the probability of an IQ between 85 and 115: First, follow the instructions given for part b Do not change the active cell in the spreadsheet You will see =NORMDIST(115,100,15,TRUE) in the fx box Click in this box, to the right of TRUE) and put in a minus sign Now repeat the steps for part b, starting by clicking fx, except enter 85 instead of 115 for X The answer, 0.682689, will be shown in the active cell (Alternatively, just repeat steps 123 for part b, using 85 instead of 115 Subtract the smaller probability value from the larger (0.8413 - 0.1587 = 0.6826).) c Inverse Normal If you want a measurement (such as an IQ or height) from a proportion or percentile: Click fx Choose NORM.INV and click OK See Figure 6J Enter: Probability, 75 (for the 75th percentile); Mean, 100: Standard_dev, 15 You may read the answer off the screen or click OK to see it in the active cell in the spreadsheet The IQ at the 75th percentile is 110 m Figure 6I Excel Normal 321 M06_GOUL1228_02_GE_C06.indd 321 03/09/16 3:06 pm www.downloadslide.com m Figure 6J Excel Inverse Normal Binomial a Cumulative (or Fewer) Click fx Choose BINOM.DIST and click OK See Figure 6K Enter: Number_s, 28; Trials, 50; Probability_s, 5; and Cumulative, true (for the probability of 28 or fewer) The answer (0.8389) shows up in the dialogue box and in the active cell when you click OK m Figure 6K Excel Binomial b Individual (Exact) Click fx Choose BINOM.DIST and click OK Use the numbers in Figure 6K, but enter False in the Cumulative box This will give you the probability of getting exactly 28 heads in 50 tosses of a fair coin The answer (0.0788) shows up in the dialogue box and in the active cell when you click OK STATCRUNCH Normal Unlike the TI-84, StatCrunch makes it easier to find the probability for 115 or less than to find the probability between 85 and 115 This is why part b is done before part a b. Some Value or Less Stat > Calculators > Normal See Figure 6L To find the probability of having a Wechsler IQ of 115 or less, Enter: Mean, 100; Std Dev, 15 Make sure that the arrow to the right of P(X points left (for less than) Enter the 115 in the box above Compute Click Compute to see the answer, 0.8413 a Between Two Values To find the probability of having a Wechsler IQ between 85 and 115, use steps 1, 2, and again, but use 85 instead of 115 in the box above Snapshot When you find that probability, subtract it from the probability found in Figure 6L 0.8413 - 0.1587 = 0.6826 c Inverse Normal If you want a measurement (such as an IQ or height) from a proportion or percentile: Stat > Calculators > Normal See Figure 6M To find the Wechsler IQ at the 75th percentile, enter: Mean, 100; Std Dev., 15 Make sure that the arrow to the right of P(X points to the left, and enter 0.75 in the box to the right of the = sign m Figure 6L StatCrunch Normal m Figure 6M StatCrunch Inverse Normal 322 M06_GOUL1228_02_GE_C06.indd 322 03/09/16 3:06 pm www.downloadslide.com Click Compute and the answer (110) is shown above Compute Binomial a Cumulative (or Fewer) Stat > Calculators > Binomial See Figure 6N To find the probability of 28 or fewer heads in 50 tosses of a fair coin, enter: n, 50; and p, 0.5 The arrow after P(X should point left (for less than) Enter 28 in the box above Compute Click Compute to see the answer (0.8389) b Individual (Exact) Stat > Calculators > Binomial To find the probability of exactly 28 heads in 50 tosses of a fair coin, use a screen similar to Figure 6N, but to the right of P(X choose the equals sign You will get 0.0788 m Figure 6N StartCrunch Binomial 323 M06_GOUL1228_02_GE_C06.indd 323 03/09/16 3:06 pm ... IN THE PRODUCT(S) AND/OR THE PROGRAM(S) DESCRIBED HEREIN AT ANY TIME PARTIAL SCREEN SHOTS MAY BE VIEWED IN FULL WITHIN THE SOFTWARE VERSION SPECIFIED ISBN 10 : 1- 2 9 2 -1 612 2 -1 ISBN 13 : 97 8 -1 -2 9 2 -1 612 2-8 ... 31 distance and time, 207–208 driver’s exam, 262, 265, 315 – 316 drivers aged 84–89, 315 driving accidents, 15 6 15 7 DWI convictions, 315 gas mileage of cars, 220 gas prices, 11 0 11 1, 11 6 11 7, 12 5... 17 1 17 2, 19 2 19 3, 480 used car values, 19 2 19 3 waiting for the bus, 278–279 07/09 /16 2:48 pm www.downloadslide.com CONTENTS 25 Essential Statistics: Exploring the World through Data Second Edition