ELEMENTARY STATISTICS MARIO F TRIOLA 11TH EDITION Addison-Wesley Editor in Chief: Deirdre Lynch Acquisitions Editor: Christopher Cummings Project Editor: Elizabeth Bernardi Associate Editor: Christina Lepre Assistant Editor: Dana Jones Senior Managing Editor: Karen Wernholm Senior Production Supervisors: Peggy McMahon and Tracy Patruno Interior Design: Leslie Haimes Cover Art Direction: Beth Paquin Cover Design: Lisa Kuhn, Curio Press, LLC Cover Images: Windmills, Art Life Images; Canada, Nunavut Territory, Arctic, Getty Images; Crash Test Dummy, Pea Plant; and Pencil, Shutterstock Senior Marketing Manager: Alex Gay Marketing Assistant: Kathleen DeChavez Photo Researcher: Beth Anderson Media Producers: Christine Stavrou and Vicki Dreyfus MyStatLab Project Supervisor: Edward Chappell QA Manager, Assessment Content: Marty Wright Senior Author Support> Technology Specialist: Joe Vetere Rights and Permissions Advisors: Shannon Barbe and Michael Joyce Manufacturing Manager: Evelyn Beaton Senior Manufacturing Buyers: Ginny Michaud and Carol Melville Production Coordination, Illustrations, and Composition: Nesbitt Graphics, Inc For permission to use copyrighted material, grateful acknowledgment has been made to the copyright holders listed on pages 843–844, which is hereby made part of this copyright page Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks Where those designations appear in this book, and Pearson Education was aware of a trademark claim, the designations have been printed in initial caps or all caps Library of Congress Cataloging-in-Publication Data Triola, Mario F Elementary statistics technology update / Mario F Triola 11th ed p cm Rev ed of: Elementary statistics 11th ed c2010 Includes bibliographical references and index ISBN 0-321-69450-3 I Triola, Mario F Elementary statistics II Title QA276.12.T76 2012 519.5 dc22 2010003324 Copyright © 2012, 2010, 2007 Pearson Education, Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher Printed in the United States of America For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston St., Suite 900, Boston, MA 02116, fax your request to (617) 671-3447, or e-mail at http://www.pearsoned.com/legal/permissions.htm 10—CRK—14 13 12 11 10 www.pearsonhighered.com ISBN-13: 978-0-321-69450-8 ISBN-10: 0-321-69450-3 ✎ To Ginny Marc, Dushana, and Marisa Scott, Anna, Siena, and Kaia This page intentionally left blank Mario F Triola is a Professor Emeritus of Mathematics at Dutchess Community College, where he has taught statistics for over 30 years About the Author Marty is the author of Essentials of Statistics, 4th edition; Elementary Statistics Using Excel, 4th edition; Elementary Statistics Using the TI-83/84 Plus Calculator, 3rd edition; and he is a coauthor of Biostatistics for the Biological and Health Sciences; Statistical Reasoning for Everyday Life, 3rd edition; Business Statistics; and Introduction to Technical Mathematics, 5th edition Elementary Statistics is currently available as an International Edition, and it has been translated into several foreign languages Marty designed the original STATDISK statistical software, and he has written several manuals and workbooks for technology supporting statistics education He has been a speaker at many conferences and colleges Marty’s consulting work includes the design of casino slot machines and fishing rods, and he has worked with attorneys in determining probabilities in paternity lawsuits, identifying salary inequities based on gender, and analyzing disputed election results He has also used statistical methods in analyzing medical data, medical school surveys, and survey results for New York City Transit Authority Marty has testified as an expert witness in New York State Supreme Court The Text and Academic Authors Association has awarded Marty a “Texty” for Excellence for his work on Elementary Statistics v This page intentionally left blank Brief Contents Introduction to Statistics Summarizing and Graphing Data Statistics for Describing, Exploring, and Comparing Data Probability Discrete Probability Distributions Normal Probability Distributions Estimates and Sample Sizes Hypothesis Testing Inferences from Two Samples 10 44 82 136 202 248 326 390 Correlation and Regression 460 516 11 Goodness-of-Fit and Contingency Tables 12 Analysis of Variance 13 Nonparametric Statistics 14 Statistical Process Control 15 Projects, Procedures, Perspectives 584 626 660 714 742 Appendices 747 Appendix A: Appendix B: Appendix C: Appendix D: Credits Index Tables 748 Data Sets 765 Bibliography of Books and Web Sites 794 Answers to odd-numbered section exercises, plus answers to all end-of-chapter Statistical Literacy and Critical Thinking exercises, chapter Quick Quizzes, Review Exercises, and Cumulative Review Exercises 795 843 845 vii This page intentionally left blank Contents Chapter Chapter Chapter Chapter Chapter Introduction to Statistics 1-1 Review and Preview 1-2 Statistical Thinking 1-3 Types of Data 1-4 Critical Thinking 1-5 Collecting Sample Data 4 11 17 26 Summarizing and Graphing Data 2-1 Review and Preview 2-2 Frequency Distributions 2-3 Histograms 2-4 Statistical Graphics 2-5 Critical Thinking: Bad Graphs 44 46 46 55 59 70 Statistics for Describing, Exploring, and Comparing Data 82 3-1 Review and Preview 84 3-2 Measures of Center 3-3 Measures of Variation 3-4 Measures of Relative Standing and Boxplots 84 99 114 Probability 4-1 136 Review and Preview 4-2 Basic Concepts of Probability 4-3 Addition Rule 4-4 Multiplication Rule: Basics 4-5 Multiplication Rule: Complements and Conditional Probability 171 4-6 Probabilities Through Simulations 4-7 Counting 4-8 Bayes’ Theorem (on CD-ROM) 138 138 152 159 178 184 193 Discrete Probability Distributions 202 5-1 Review and Preview 204 5-2 Random Variables 5-3 Binomial Probability Distributions 5-4 Mean, Variance, and Standard Deviation for the Binomial Distribution 229 5-5 The Poisson Distribution 205 218 234 ix 6-7 Assessing Normality Here are some important comments about procedures for determining whether data are from a normally distributed population: • If the requirement of a normal distribution is not too strict, examination of a histogram and consideration of outliers may be all that you need to assess normality • Normal quantile plots can be difficult to construct on your own, but they can be generated with a TI-83>84 Plus calculator or suitable computer software, such as STATDISK, SPSS, SAS, Minitab, and Excel • In addition to the procedures discussed in this section, there are other more advanced procedures for assessing normality, such as the chi-square goodness-of-fit test, the Kolmogorov-Smirnov test, the Lilliefors test, the Anderson-Darling test, and the Ryan-Joiner test (discussed briefly in Part 2) Part 2: Beyond the Basics of Assessing Normality The following is a relatively simple procedure for manually constructing a normal quantile plot, and it is the same procedure used by STATDISK and the TI-83>84 Plus calculator Some statistical packages use various other approaches, but the interpretation of the graph is basically the same Manual Construction of a Normal Quantile Plot Step First sort the data by arranging the values in order from lowest to highest Step With a sample of size n, each value represents a proportion of 1>n of the sample Using the known sample size n, identify the areas of 1>2n, 3>2n, and so on These are the cumulative areas to the left of the corresponding sample values 311 Small Sample The Children’s Defense Fund was organized to promote the welfare of children The group published Children Out of School in America, which reported that in one area, 37.5% of the 16- and 17-year-old children were out of school This statistic received much press coverage, but it was based on a sample of only 16 children Another statistic was based on a sample size of only students (See “Firsthand Report: How Flawed Statistics Can Make an Ugly Picture Look Even Worse,” American School Board Journal, Vol 162.) Step Use the standard normal distribution (Table A-2 or software or a calcula- tor) to find the z scores corresponding to the cumulative left areas found in Step (These are the z scores that are expected from a normally distributed sample.) Step Match the original sorted data values with their corresponding z scores found in Step 3, then plot the points (x, y), where each x is an original sample value and y is the corresponding z score Step Examine the normal quantile plot and determine whether or not the distri- bution is normal Movie Lengths Data Set in Appendix B includes lengths (in minutes) of randomly selected movies Let’s consider only the first movie lengths: 110, 96, 170, 125, 119 With only values, a histogram will not be very helpful in revealing the distribution of the data Instead, construct a normal quantile plot for these values and determine whether they appear to come from a population that is normally distributed The following steps correspond to those listed in the above procedure for constructing a normal quantile plot Step First, sort the data by arranging them in order We get 96, 110, 119, 125, 170 continued 312 Chapter Normal Probability Distributions Step With a sample of size n = 5, each value represents a proportion of 1>5 of the sample, so we proceed to identify the cumulative areas to the left of the corresponding sample values The cumulative left areas, which are expressed in general as 1>2n, 3>2n, 5>2n, 7>2n, and so on, become these specific areas for this example with n = 5: 1>10, 3>10, 5>10, 7>10, and 9>10 The cumulative left areas expressed in decimal form are 0.1, 0.3, 0.5, 0.7, and 0.9 Step We now search in the body of Table A-2 for the cumulative left areas of 0.1000, 0.3000, 0.5000, 0.7000, and 0.9000 to find these corresponding z scores: -1.28, -0.52, 0, 0.52, and 1.28 Step We now pair the original sorted movie lengths with their corresponding z scores We get these (x, y) coordinates which are plotted in the accompanying STATDISK display: (96, -1.28), (110, -0.52), (119, 0), (125, 0.52), and (170, 1.28) STATDISK We examine the normal quantile plot in the STATDISK display Because the points appear to lie reasonably close to a straight line and there does not appear to be a systematic pattern that is not a straight-line pattern, we conclude that the sample of five movie lengths appears to come from a normally distributed population In the next example, we address the issue of an outlier in a data set Movie Lengths Let’s repeat Example after changing one of the values so that it becomes an outlier Change the highest value of 170 in Example to a length of 1700 (The actual longest movie is Cure for Insomnia with a length of 5220 min, or 87 hr.) The accompanying STATDISK display shows the normal quantile plot of these movie lengths: 110, 96, 1700, 125, 119 Note how that one outlier affects the graph This normal quantile plot does not result in points with an approximately straight-line pattern This STATDISK display suggests that the values of 110, 96, 1700, 125, 119 are from a population with a distribution that is not a normal distribution 6-7 Assessing Normality Is Parachuting Safe? STATDISK The Ryan-Joiner test is one of several formal tests of normality, each having their own advantages and disadvantages STATDISK has a feature of Normality Assessment that displays a histogram, normal quantile plot, the number of potential outliers, and results from the Ryan-Joiner test Information about the Ryan-Joiner test is readily available on the Internet Ryan-Joiner Test Tobacco in Children’s Movies Data Set in Appendix B includes the times (in seconds) that the use of tobacco was shown in 50 different animated children’s movies Shown below is the STATDISK display summarizing results from the feature of Normality Assessment All of these results suggest that the sample is not from a normally distributed population: (1) The histogram is far from being bell-shaped; (2) the points in the normal quantile plot are far from a straight-line pattern; (3) there appears to be one or more outliers; (4) results from the Ryan-Joiner test indicate that normality should be rejected The evidence against a normal distribution is strong and consistent STATDISK 313 About 30 people die each year as more than 100,000 people make about 2.25 million parachute jumps In comparison, a typical year includes about 200 scuba diving fatalities, 7000 drownings, 900 bicycle deaths, 800 lightning deaths, and 1150 deaths from bee stings Of course, these figures don’t necessarily mean that parachuting is safer than bike riding or swimming A fair comparison should involve fatality rates, not just the total number of deaths The author, with much trepidation, made two parachute jumps but quit after missing the spacious drop zone both times He has also flown in a hang glider, hot air balloon, ultralight, sailplane, and Goodyear blimp 314 Chapter Normal Probability Distributions Many data sets have a distribution that is not normal, but we can transform the data so that the modified values have a normal distribution One common transformation is to replace each value of x with log(x + 1) If the distribution of the log(x + 1) values is a normal distribution, the distribution of the x values is referred to as a lognormal distribution (See Exercise 22.) In addition to replacing each x value with log(x + 1), there are other transformations, such as replacing each x value with 1x, or 1>x, or x In addition to getting a required normal distribution when the original data values are not normally distributed, such transformations can be used to correct other deficiencies, such as a requirement (found in later chapters) that different data sets have the same variance U S I N G T E C H N O LO GY Data Transformations STATDISK can be used to generate a normal S TAT D I S K quantile plot, and the result is consistent with the procedure described in this section Enter the data in a column of the Sample Editor window Next, select Data from the main menu bar at the top Select Normal Quantile Plot to generate the graph Better yet, select Normality Assessment to obtain the normal quantile plot included in the same display with other results helpful in assessing normality Proceed to enter the column number for the data, then click Evaluate Minitab can generate a graph similar to the norM I N I TA B mal quantile plot described in this section Minitab’s procedure is somewhat different, but the graph can be interpreted by using the same criteria given in this section That is, normally distributed data should lie reasonably close to a straight line, and points should not reveal a pattern that is not a straight-line pattern First enter the values in column C1, then select Stat, Basic Statistics, and Normality Test Enter C1 for the variable, then click on OK Minitab can also generate a graph that includes boundaries If the points all lie within the boundaries, conclude that the values are normally distributed If the points lie beyond the boundaries, conclude that the values are not normally distributed To generate the graph that includes the boundaries, first enter the values in column C1, select the main menu item of Graph, select Probability Plot, then select the option of Simple Proceed to enter C1 for the variable, then click on OK The accompanying Minitab display is based on Example 2, and it includes the boundaries MINITAB First enter the data in column A If using Excel E XC E L 2010 or Excel 2007, click on Add-Ins, then click on DDXL; if using Excel 2003, click on DDXL Select Charts and Plots, then select the function type of Normal Probability Plot Click on the pencil icon for “Quantitative Variable,” then enter the range of values, such as A1:A36 Click OK The TI-83>84 Plus calculator can be TI-83/84 PLUS used to generate a normal quantile plot, and the result is consistent with the procedure described in this section First enter the sample data in list L1 Press F E (for STAT PLOT), then press [ Select ON, select the “type” item, which is the last item in the second row of options, and enter L1 for the data list The screen should appear as shown here After making all selections, press B, then 9, and the points in the normal quantile plot will be displayed TI-83/84 PLUS 6-7 Assessing Normality 6-7 Basic Skills and Concepts Statistical Literacy and Critical Thinking Normal Quantile Plot What is the purpose of constructing a normal quantile plot? Rejecting Normality Identify two different characteristics of a normal quantile plot, where each characteristic would lead to the conclusion that the data are not from a normally distributed population Normal Quantile Plot If you select a simple random sample of M&M plain candies and construct a normal quantile plot of their weights, what pattern would you expect in the graph? Criteria for Normality Assume that you have a data set consisting of the ages of all New York City police officers Examination of a histogram and normal quantile plot are two different ways to assess the normality of that data set Identify a third way Interpreting Normal Quantile Plots In Exercises 5–8, examine the normal quantile plot and determine whether it depicts sample data from a population with a normal distribution Old Faithful The normal quantile plot represents duration times (in seconds) of Old Faithful eruptions from Data Set 15 in Appendix B STATDISK Heights of Women The normal quantile plot represents heights of women from Data Set in Appendix B STATDISK 315 316 Chapter Normal Probability Distributions Weights of Diet Coke The normal quantile plot represents weights (in pounds) of diet Coke from Data Set 17 in Appendix B STATDISK Telephone Digits The normal quantile plot represents the last two digits of telephone numbers of survey subjects STATDISK Determining Normality In Exercises 9–12, refer to the indicated data set and determine whether the data have a normal distribution Assume that this requirement is loose in the sense that the population distribution need not be exactly normal, but it must be a distribution that is roughly bell-shaped Space Shuttle Flights The lengths (in hours) of flights of NASA’s Space Transport Sys- tem (Shuttle) as listed in Data Set 10 in Appendix B 10 Astronaut Flights The numbers of flights by NASA astronauts, as listed in Data Set 10 in Appendix B 11 Heating Degree Days The values of heating degree days, as listed in Data Set 12 in Appendix B 12 Generator Voltage The measured voltage levels from a generator, as listed in Data Set 13 in Appendix B Using Technology to Generate Normal Quantile Plots In Exercises 13–16, use the data from the indicated exercise in this section Use a TI-83>84 Plus calculator or computer software (such as STATDISK, Minitab, or Excel) to generate Review a normal quantile plot Then determine whether the data come from a normally distributed population 13 Exercise 14 Exercise 10 15 Exercise 11 16 Exercise 12 17 Comparing Data Sets Using the heights of women and the cholesterol levels of women, as listed in Data Set in Appendix B, analyze each of the two data sets and determine whether each appears to come from a normally distributed population Compare the results and give a possible explanation for any notable differences between the two distributions 18 Comparing Data Sets Using the systolic blood pressure levels and the elbow breadths of women, as listed in Data Set in Appendix B, analyze each of the two data sets and determine whether each appears to come from a normally distributed population Compare the results and give a possible explanation for any notable differences between the two distributions Constructing Normal Quantile Plots In Exercises 19 and 20, use the given data values to identify the corresponding z scores that are used for a normal quantile plot Then construct the normal quantile plot and determine whether the data appear to be from a population with a normal distribution 19 Braking Distances A sample of braking distances (in feet) measured under standard conditions for an Acura RL, Acura TSX, Audi A6, BMW 525i, and Buick LaCrosse: 131, 136, 129, 127, 146 20 Satellites A sample of the numbers of satellites in orbit: 158 (United States); 17 (China); 18 (Russia); 15 ( Japan); (France); (Germany) 6-7 Beyond the Basics 21 Transformations The heights (in inches) of men listed in Data Set in Appendix B have a distribution that is approximately normal, so it appears that those heights are from a normally distributed population a If inches is added to each height, are the new heights also normally distributed? b If each height is converted from inches to centimeters, are the heights in centimeters also normally distributed? c Are the logarithms of normally distributed heights also normally distributed? 22 Lognormal Distribution The following values are the times (in days) it took for prototype integrated circuits to fail Test these values for normality, then replace each x value with log (x + 1) and test the transformed values for normality What can you conclude? 103 1396 547 307 106 362 662 1091 329 102 510 3822 1169 547 267 725 1894 4337 1065 339 Review In this chapter we introduced the normal probability distribution—the most important distribution in the study of statistics Section 6-2 In Section 6-2 we worked with the standard normal distribution, which is a normal distribution having a mean of and a standard deviation of The total area under the density curve of a normal distribution is 1, so there is a convenient correspondence between areas and probabilities We presented methods for finding areas (or probabilities) that correspond to standard z scores, and we presented important methods for finding standard z scores that correspond to known areas (or probabilities) Values of areas and z scores can be found using Table A-2 or a TI-83>84 Plus calculator or computer software 317 318 Chapter Normal Probability Distributions Section 6-3 In Section 6-3 we extended the methods from Section 6-2 so that we could work with any normal distribution, not just the standard normal distribution We presented the standard score z = (x - m)>s for solving problems such as these: • Given that IQ scores are normally distributed with m = 100 and s = 15, find the probability of randomly selecting someone with an IQ above 90 • Given that IQ scores are normally distributed with m = 100 and s = 15, find the IQ score separating the bottom 85% from the top 15% Section 6-4 In Section 6-4 we introduced the concept of a sampling distribution of a statistic The sampling distribution of the mean is the probability distribution of sample means, with all samples having the same sample size n The sampling distribution of the proportion is the probability distribution of sample proportions, with all samples having the same sample size n In general, the sampling distribution of any statistic is the probability distribution of that statistic Section 6-5 In Section 6-5 we presented the following conclusions associated with the central limit theorem: The distribution of sample means x will, as the sample size n increases, approach a normal distribution The mean of the sample means is the population mean m The standard deviation of the sample means is s> 1n Section 6-6 In Section 6-6 we noted that a normal distribution can sometimes approximate a binomial probability distribution If both np Ú and nq Ú 5, the binomial random variable x is approximately normally distributed with the mean and standard deviation given as m = np and s = 1npq Because the binomial probability distribution deals with discrete data and the normal distribution deals with continuous data, we apply the continuity correction, which should be used in normal approximations to binomial distributions Section 6-7 In Section 6-7 we presented procedures for determining whether sample data appear to come from a population that has a normal distribution Some of the statistical methods covered later in this book have a loose requirement of a normally distributed population In such cases, examination of a histogram and outliers might be all that is needed In other cases, normal quantile plots might be necessary because of factors such as a small sample or a very strict requirement that the population must have a normal distribution Statistical Literacy and Critical Thinking Normal Distribution What is a normal distribution? What is a standard normal distribution? Normal Distribution In a study of incomes of individual adults in the United States, it is observed that many people have no income or very small incomes, while there are very few people with extremely large incomes, so a graph of the incomes is skewed instead of being symmetric A researcher states that because incomes are a normal occurrence, the distribution of incomes is a normal distribution Is that statement correct? Why or why not? Distribution of Sample Means In each of the past 50 years, a simple random sample of 36 new movies is selected, and the mean of the 36 movie lengths (in minutes) is calculated What is the approximate distribution of those sample means? Large Sample On one cruise of the ship Queen Elizabeth II, 17% of the passengers became ill from Norovirus America Online conducted a survey about that incident and received 34,358 responses Given that the sample is so large, can we conclude that this sample is representative of the population? Review Exercises Chapter Quick Quiz Find the value of z 0.03 A process consists of rolling a single die 100 times and finding the mean of the 100 outcomes If that process is repeated many times, what is the approximate distribution of the resulting means? (uniform, normal, Poisson, binomial) What are the values of m and s in the standard normal distribution? For the standard normal distribution, find the area to the right of z = 1.00 For the standard normal distribution, find the area between the z scores of -1.50 and 2.50 In Exercises 6–10, assume that IQ scores are normally distributed with a mean of 100 and a standard deviation of 15 Find the probability that a randomly selected person has an IQ score less than 115 Find the probability that a randomly selected person has an IQ score greater than 118 Find the probability that a randomly selected person has an IQ score between 88 and 112 If 25 people are randomly selected, find the probability that their mean IQ score is less than 103 10 If 100 people are randomly selected, find the probability that their mean IQ score is greater than 103 Review Exercises Heights In Exercises 1–4, assume that heights of men are normally distributed with a mean of 69.0 in and a standard deviation of 2.8 in Also assume that heights of women are normally distributed with a mean of 63.6 in and a standard deviation of 2.5 in (based on data from the National Health Survey) Bed Length A day bed is 75 in long a Find the percentage of men with heights that exceed the length of a day bed b Find the percentage of women with heights that exceed the length of a day bed c Based on the preceding results, comment on the length of a day bed Bed Length In designing a new bed, you want the length of the bed to equal or exceed the height of at least 95% of all men What is the minimum length of this bed? Designing Caskets The standard casket has an inside length of 78 in a What percentage of men are too tall to fit in a standard casket, and what percentage of women are too tall to fit in a standard casket? Based on those results, does it appear that the standard casket size is adequate? b A manufacturer of caskets wants to reduce production costs by making smaller caskets What inside length would fit all men except the tallest 1%? Heights of Rockettes In order to have a precision dance team with a uniform appearance, height restrictions are placed on the famous Rockette dancers at New York’s Radio City Music Hall Because women have grown taller over the years, a more recent change now requires that a Rockette dancer must have a height between 66.5 in and 71.5 in What percentage of women meet this height requirement? Does it appear that Rockettes are taller than typical women? Genetics Experiment In one of Mendel’s experiments with plants, 1064 offspring consisted of 787 plants with long stems According to Mendel’s theory, 3>4 of the offspring plants should have long stems Assuming that Mendel’s proportion of 3>4 is correct, find the probability of getting 787 or fewer plants with long stems among 1064 offspring plants continued 319 320 Chapter Normal Probability Distributions Based on the result, is 787 offspring plants with long stems unusually low? What does the result imply about Mendel’s claimed proportion of 3>4? Sampling Distributions Assume that the following sample statistics were obtained from a simple random sample Which of the following statements are true? a The sample mean x targets the population mean m in the sense that the mean of all sample means is m b The sample proportion pN targets the population proportion p in the sense that the mean of all sample proportions is p c The sample variance s targets the population variance s2 in the sense that the mean of all sample variances is s2 d The sample median targets the population median in the sense that the mean of all sample medians is equal to the population median e The sample range targets the population range in the sense that the mean of all sample ranges is equal to the range of the population High Cholesterol Levels The serum cholesterol levels in men aged 18–24 are normally distributed with a mean of 178.1 and a standard deviation of 40.7 Units are in mg>100 mL, and the data are based on the National Health Survey a If man aged 18–24 is randomly selected, find the probability that his serum cholesterol level is greater than 260, a value considered to be “moderately high.” b If man aged 18–24 is randomly selected, find the probability that his serum cholesterol level is between 170 and 200 c If men aged 18–24 are randomly selected, find the probability that their mean serum cholesterol level is between 170 and 200 d The Providence Health Maintenance Organization wants to establish a criterion for recommending dietary changes if cholesterol levels are in the top 3% What is the cutoff for men aged 18–24? Identifying Gender Discrimination Jennifer Jenson learns that the Newport Temp Agency has hired only 15 women among its last 40 new employees She also learns that the pool of applicants is very large, with an equal number of qualified men and women Find the probability that among 40 such applicants, the number of women is 15 or fewer Based on the result, is there strong evidence to charge that the Newport Temp Agency is discriminating against women? Critical Values a Find the standard z score with a cumulative area to its left of 0.6700 b Find the standard z score with a cumulative area to its right of 0.9960 c Find the value of z 0.025 10 Sampling Distributions A large number of simple random samples of size n = 85 are obtained from a large population of birth weights having a mean of 3420 g and a standard deviation of 495 g The sample mean x is calculated for each sample a What is the approximate shape of the distribution of the sample means? b What is the expected mean of the sample means? c What is the expected standard deviation of the sample means? 11 Aircraft Safety Standards Under older Federal Aviation Administration rules, airlines had to estimate the weight of a passenger as 185 lb (That amount is for an adult traveling in winter, and it includes 20 lb of carry-on baggage.) Current rules require an estimate of 195 lb Men have weights that are normally distributed with a mean of 172 lb and a standard deviation of 29 lb a If adult male is randomly selected and is assumed to have 20 lb of carry-on baggage, find the probability that his total is greater than 195 lb b If a Boeing 767-300 aircraft is full of 213 adult male passengers and each is assumed to have 20 lb of carry-on baggage, find the probability that the mean passenger weight (including carry-on baggage) is greater than 195 lb Based on that probability, does a pilot have to be concerned about exceeding this weight limit? Cumulative Review Exercises 12 Assessing Normality Listed below are the weights (in grams) of a simple random sample of United States one-dollar coins (from Data Set 20 in Appendix B) Do those weights appear to come from a population that has a normal distribution? Why or why not? 8.1008 8.1281 8.1072 8.0271 8.0813 8.0241 8.0510 7.9817 8.0954 8.0658 8.1238 8.0307 8.0719 8.0345 8.0775 8.1384 8.1041 8.0894 8.0538 8.0342 Cumulative Review Exercises Salaries of Coaches Listed below are annual salaries (in thousands of dollars) for a sim- ple random sample of NCAA Division 1-A head football coaches (based on data from the New York Times) 235 159 492 530 138 125 128 900 360 212 a Find the mean x and express the result in dollars instead of thousands of dollars b Find the median and express the result in dollars instead of thousands of dollars c Find the standard deviation s and express the result in dollars instead of thousands of dollars d Find the variance s and express the result in appropriate units e Convert the first salary of $235,000 to a z score f What level of measurement (nominal, ordinal, interval, ratio) describes this data set? g Are the salaries discrete data or continuous data? Sampling a What is a simple random sample? b What is a voluntary response sample, and why is it generally unsuitable for statistical purposes? Clinical Trial of Nasonex In a clinical trial of the allergy drug Nasonex, 2103 adult pa- tients were treated with Nasonex and 14 of them developed viral infections a If two different adults are randomly selected from the treatment group, what is the probability that they both developed viral infections? b Assuming that the same proportion of viral infections applies to all adults who use Nasonex, find the probability that among 5000 randomly selected adults who use Nasonex, at least 40 develop viral infections c Based on the result from part (b), is 40 an unusually high number of viral infections? Why or why not? d Do the given results (14 viral infections among 2103 adult Nasonex users) suggest that viral infections are an adverse reaction to the Nasonex drug? Why or why not? Graph of Car Mileage The accompanying graph depicts the fuel consumption (in miles per gallon) for highway conditions of three cars Does the graph depict the data fairly, or does it somehow distort the data? Explain 30 25 Ion tu rn 5i Sa 52 W BM ur aR L 20 Ac Highway Miles Per Gallon 35 321 322 Chapter Normal Probability Distributions Left-Handedness According to data from the American Medical Association, 10% of us are left-handed a If three people are randomly selected, find the probability that they are all left-handed b If three people are randomly selected, find the probability that at least one of them is left- handed c Why can’t we solve the problem in part (b) by using the normal approximation to the binomial distribution? d If groups of 50 people are randomly selected, what is the mean number of left-handed people in such groups? e If groups of 50 people are randomly selected, what is the standard deviation for the numbers of left-handed people in such groups? f Would it be unusual to get left-handed people in a randomly selected group of 50 people? Why or why not? Technology Project Assessing Normality This project involves using STATDISK for assessing the normality INTERNET PROJECT of data sets If STATDISK has not yet been used, it can be installed from the CD included with this book Click on the Software folder, select STATDISK, and proceed to install STATDISK The data sets in Appendix B are available by clicking on Datasets on the top menu bar, then selecting the textbook you are using STATDISK can be used to assess normality of a sample by clicking on Data and selecting the menu item of Normality Assessment Use this feature to find a sample that is clearly from a normally distributed population Also find a second sample that is clearly not from a normally distributed population Finally, find a third sample that can be considered to be from a normally distributed population if we interpret the requirements loosely, but not too strictly In each case, obtain a printout of the Normality Assessment display and write a brief explanation justifying your choice Exploring the Central Limit Theorem Go to: http://www.aw.com/triola The central limit theorem is one of the most important results in statistics It also may be one of the most surprising Informally, the central limit theorem says that the normal distribution is everywhere No matter what probability distribution underlies an experiment, there is a corresponding distribution of means that will be approximately normal in shape The CD included with this book contains applets designed to help visualize various concepts Open the Applets folder on the CD and double-click on Start Select the menu item of Sampling Distributions Use The best way to both understand and appreciate the central limit theorem is to see it in action The Internet Project for this chapter will allow you to just that You will examine the central limit theorem from both a theoretical and a practical point of view First, simulations found on the Internet will help you understand the theorem itself Second, you will see how the theorem is key to such common activities as conducting polls and predicting election outcomes the applet to compare the sampling distributions of the mean and the median in terms of center and spread for bell-shaped and skewed distributions Write a brief report describing your results F R O M DATA T O D E C I S I O N From Data to Decision 323 Critical Thinking: Designing aircraft seating In this project we consider the issue of determining the “sitting distance” shown in Figure 6-24(a) We define the sitting distance to be the length between the back of the seat cushion and the seat in front To determine the sitting distance, we must take into account human body measurements Specifically, we must consider the “buttock-to-knee length,” as shown in Figure 6-24(b) Determining the sitting distance for an aircraft is extremely important If the sitting distance is unnecessarily large, rows of seats might need to be eliminated It has been estimated that removing a single row of six seats can cost around $8 million over the life of the aircraft If the sitting distance is too small, passengers might be uncomfortable and might prefer to fly other aircraft, or their safety might be compromised because of their limited mobility In determining the sitting distance in our aircraft, we will use previously collected data from measurements of large numbers of people Results from those measurements are summarized in the given table We can use the data in the table to determine the required sitting distance, but we must make some hard choices If we are to accommodate everyone in the population, we will have a sitting distance that is so costly in terms of reduced seating that it might not be economically feasible Some questions we must ask ourselves are: (1) What percentage of the population are we willing to exclude? (2) How much extra room we want to provide for passenger comfort and safety? Use the available information Distance from the seat back cushion to the seat in front • Buttock-to-knee length plus any additional distance to provide comfort to determine the sitting distance Identify the choices and decisions that were made in that determination Buttock-to-Knee Length (inches) Standard Mean Deviation Distribution Males 23.5 in 1.1 in Normal Females 22.7 in 1.0 in Normal • Buttock-toknee length (b) (a) Figure 6-24 Sitting Distance and Buttock-to-Knee Length 324 Chapter Normal Probability Distributions Cooperative Group Activities In-class activity Divide into groups of three or four students and address these issues affecting the design of manhole covers • Which of the following is most relevant for determining whether a manhole cover diameter of 24 in is large enough: weights of men, weights of women, heights of men, heights of women, hip breadths of men, hip breadths of women, shoulder breadths of men, shoulder breadths of women? • Why are manhole covers usually round? (This was once a popular interview question asked of applicants at IBM, and there are at least three good answers One good answer is sufficient here.) Out-of-class activity Divide into groups of three or four students In each group, develop an original procedure to illustrate the central limit theorem The main objective is to show that when you randomly select samples from a population, the means of those samples tend to be normally distributed, regardless of the nature of the population distribution For this illustration, begin with some population of values that does not have a normal distribution In-class activity Divide into groups of three or four students Using a coin to simulate births, each individual group member should simulate 25 births and record the number of simulated girls Combine all results from the group and record n = total number of births and x = number of girls Given batches of n births, compute the mean and standard deviation for the number of girls Is the simulated result usual or unusual? Why? In-class activity Divide into groups of three or four students Select a set of data from Appendix B (excluding Data Sets 1, 8, 9, 11, 12, 14, and 16, which were used in examples or exercises in Section 6-7) Use the methods of Section 6-7 and construct a histogram and normal quantile plot, then determine whether the data set appears to come from a normally distributed population CHAPTER PROJECT Throw Out Your Normal Distribution Table This chapter made extensive use of normal probability distributions Table A-2 in Appendix B lists many probabilities for the standard normal distribution, but it is usually better to use technology for finding those probabilities StatCrunch is better because it can be used with any normal distribution, not just the standard normal distribution in Table A-2 Consider Example in Section 6-3 In that example, we are given a normal probability distribution of men’s heights with a mean of 69.0 in and a standard deviation of 2.8 in., and we want to find the probability of randomly selecting a man and finding that his height is less than or equal to 80 in StatCrunch Procedure for Finding Normal Probabilities or Values Sign into StatCrunch, then click on Open StatCrunch • The box showing = indicates that the default option is to find the cumulative probability for the value of x and lower You could click on the box labeled ▼ and select = if you want the cumulative probability for the value of x and higher (There are no options for selecting or for selecting because the resulting probabilities would be the same.) Click on Compute • The display given here shows that the probability of 80 in or less is 0.99995726, which can be expressed as 99.995726% (Note: The displayed graph of the normal distribution will include the shaded area that corresponds to the probability being found In this example, the probability of 0.99995726 is so close to that the entire region appears to be shaded.) Click on Stat, then select the menu item of Calculators Project In the window that appears, scroll down and click on Normal You will see a Normal Calculator window similar to the one shown below, but it will have entries different from those shown here Given a normal probability distribution of men’s heights with a mean of 69.0 in and a standard deviation of 2.8 in., use StatCrunch to find the probability of the indicated event In the Normal Calculator window, make the following entries Randomly selecting a man and finding that his height is less than 65 in • Enter the mean in the box labeled “Mean.” • Enter the standard deviation in the box labeled “Std Dev.” • To find a probability, enter the value of x in the box in the middle of the second line; or to find the value x corresponding to a known probability, enter the probability in the box at the far right Randomly selecting a man and finding that his height is at least 70 in Randomly selecting 36 men and finding that their mean height is less than 68.0 in Randomly selecting 25 men and finding that their mean height is greater than 67.5 in 325 [...]... 311 Extrasensory Perception (ESP), (M), 17 1; (CGA), 245, 457 Florence Nightingale (M), 62 Gender in a Family (M), 277 Gender of Children (IE), 14 3, 14 8, 17 1, 283–284; (E), 14 8, 15 0; (BB), 287; (CGA), 324 Gender Selection (IE), 8, 17 8, 17 9, 18 7, 393, 394, 397, 399, 4 01, 404, 405, 407, 413 , 666, 668; (E), 16 , 14 9, 15 6, 16 9, 18 3, 19 1, 225, 232, 233, 305, 306, 3 41, 409, 422, 6 71, 672; (BB), 15 9, 411 , 412 ;... 10 -2 Correlation 518 10 -3 Regression 536 10 -5 Multiple Regression 10 -6 Modeling 432 443 9 -1 Review and Preview 355 370 462 516 518 10 -4 Variation and Prediction Intervals x 2 51 287 Estimates and Sample Sizes 10 -1 Chapter 11 248 5 51 560 570 Goodness-of-Fit and Contingency Tables 11 -1 Review and Preview 586 11 -2 Goodness-of-Fit 11 -3 Contingency Tables 11 -4 McNemar’s Test for Matched Pairs 586 598 611 ... (BB), 17 8 Rolling Dice (E), 9, 18 1, 238; (IE), 277, 279, 280 Roulette (E), 14 8, 217 ; (BB), 15 1; (IE), 212 , 14 7 Schemes to Beat the Lottery (M), 3 01 Slot Machine (E), 9, 227, 594 Solitaire (BB), 15 1 Tossing Coins (BB), 17 8, 344; (IE), 18 1; (E), 18 1; (TP), 19 8; (CGA), 387 Winning the Lottery (E), 14 8, 18 9, 19 0, 19 1; (R), 19 7 General Interest Age of Books (CGA), 387, 457 Alarm Clock Redundancy (E), 17 7... Actresses (DD), 80, 13 3, 512 ; (E), 12 6, 12 7, 368, 494, 534, 550, 672, 705 Deal or No Deal Television Game Show (E), 217 ; (R), 242 iPod Random Shuffle (M), 700; (CGA), 712 Movie Budgets and Gross (E), 16 , 97, 11 2, 11 6, 11 7, 354, 486, 534, 550, 569, 640, 685; (IE), 12 0, 12 1 12 2; (BB), 12 9 Movie Data (E), 17 , 37, 318 , 344, 368, 379, 653; (CR), 40; (IE), 311 , 312 Movie Ratings (E), 17 , 424; (R), 38, 39;... 412 ; (CGA), 387, 457; (CP) 3 91 Height (E), 52, 4 51; (M), 11 4, 289 Height Requirement for Soldiers (E), 12 7, 273 Heights of Girls (R), 13 1 Heights of Men and Women (IE), 10 9, 11 4; (E), 11 3, 12 7, 315 , 317 , 3 51, 495, 502, 505, 557, 650; (BB), 275, 317 ; (CGA), 513 ; (R), 577; (CR), 578, 6 21 Heights of Presidents (E), 12 6, 496, 532, 549 Heights of Rockettes (R), 319 Heights of Statistics Students (E), 54... Shutterstock Page 10 0, AP Wideworld Photo Page 10 2, PhotoDisc Page 11 4, Shutterstock Page 11 5, Shutterstock Page 11 9, Shutterstock Chapter 4 Page 13 6, Everett Collection Page 13 9, Shutterstock Page 14 0, (margin), PhotoDisc Page 14 0 (Figure 4.1a), Shutterstock Page 14 0 (Figure 4.1b), Shutterstock Page 14 0 (Figure 4.1c), NASA/John F Kennedy Space Center Page 14 1 and 16 5, PhotoDisc Page 14 2, PhotoDisc Page 14 4, Shutterstock... 518 , 520 categorical (qualitative, attribute), 12 center, 46, 84–93 See also Center characteristics, 46 collection, 26–34 continuous, 13 definition, 4 discrete, 13 distribution, 46 interval, 14 15 missing, 22, 56 multimodal, 87 nominal, 13 14 , 15 , 667–668 ordinal, 14 , 15 outlier, 46, 11 5 11 6, 12 3, 12 4, 542–543 paired, 518 , 520 procedures, 744–745 process, 716 quantitative (numerical), 12 13 range, 10 0... 420, 440, 4 41, 449, 594, 595 Anchoring Numbers (CGA), 13 4, 513 Area Codes (E), 19 2 Areas of States and Countries (E), 16 , 73 Authors Identified (M), 48 Bed Length (R), 319 Birthdays (E), 14 9, 18 1; (BB), 17 1, 17 8, 18 3; (IE), 16 5; (SCP), 2 01 Coincidences (M), 17 2 Combination Lock (E), 18 9 Comparing Ages (E), 507 Comparing Readability (R), 508; (E), 639; (CR), 709– 710 Cost of Laughing Index (M), 11 5 Deaths... 280–284 standard deviation, 10 4 10 6, 373–377 Student t distribution, 355–357, 359–360 variance, 370–377 Estimator, 10 3 10 4, 280–282, 373 Ethics, 6, 680 Event(s) certain, 14 4 complementary, 14 4 14 6, 15 5 15 6 compound, 15 2 definition, 13 9 dependent, 16 2, 16 4 16 6 disjoint (mutually exclusive), 15 4 impossible, 14 4 independent, 16 2 simple, 13 9 unusual, 14 6 Exacta bet problem, 18 6 Excel See Software/calculator... Detectors (M), 398, 4 01; (E), 422, 607 Murders Cleared by Arrest (IE), 15 6 Percentage of Arrests (E), 4 21, 423 Polygraph Test (CP), 13 7; (E), 14 8 14 9, 15 7, 16 8, 17 6; (IE), 15 3, 15 4, 16 1, 17 3 Prisoners (E), 35; (M), 446 Prosecutor’s Fallacy (M), 17 4 Ranking DWI Judges (E), 698 Sentence Independent of Plea (E), 608 Sobriety Checkpoint (E), 35, 15 7 Solved Robberies (E), 17 6 Speed Limits (IE), 11 9 Speeding Tickets