Elementary statistics looking at the big picture part 1

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Screenshots shown here are for illustrative purposes only Find out more and see a sample assignment at www.webassign.net/brookscole 04950165427_Pfenning_EWA_IFC.indd 6/12/09 8:44:55 AM Elementary Statistics This page intentionally left blank Elementary Statistics LO O K I N G AT T H E B I G P I C T U R E Nancy pfenning University of Pittsburgh Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Elementary Statistics: Looking at the Big Picture Nancy Pfenning Publisher: Richard Stratton Senior Sponsoring Editor: Molly Taylor Associate Editor: Daniel Seibert Editorial Assistant: Shaylin Walsh Senior Marketing Manager: Greta Kleinert Marketing Coordinator: Erica O’Connell Marketing Communications Manager: Mary Anne Payumo Content Project Manager: Susan Miscio Art Director: Linda Helcher © 2011 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions Further permissions questions can be emailed to permissionrequest@cengage.com Senior Print Buyer: Diane Gibbons Library of Congress Control Number: 2009935400 Senior Rights Acquisition Account Manager, Text: Katie Huha ISBN-13: 978-0-495-01652-6 Production Service: S4Carlisle Publishing Services ISBN-10: 0-495-01652-7 Rights Acquisition Account Manager, Images: Don Schlotman Brooks/Cole 20 Channel Center Street Boston, MA 02210 USA Photo Researcher: Jennifer Lim Interior and Cover Designer: KeDesign Cover Image: © Veer Incorporated Compositor: S4Carlisle Publishing Services Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan Locate your local office at: international.cengage.com/region Cengage Learning products are represented in Canada by Nelson Education, Ltd For your course and learning solutions, visit www.cengage.com Purchase any of our products at your local college store or at our preferred online store www.ichapters.com Printed in the United States of America 12 11 10 09 To Frank, Andreas & Mary, Marina, and Nils This page intentionally left blank Contents Preface xv Introduction: Variables and Processes in Statistics Types of Variables: Categorical or Quantitative Students Talk Stats: Identifying Types of Variables Handling Data for Two Types of Variables Roles of Variables: Explanatory or Response Statistics as a Four-Stage Process Summary 11 / Exercises 11 PA R T I Data Production 16 Sampling: Which Individuals Are Studied 18 Sources of Bias in Sampling: When Selected Individuals Are Not Representative Probability Sampling Plans: Relying on Randomness The Role of Sample Size: Bigger Is Better If the Sample Is Representative From Sample to Population: To What Extent Can We Generalize? Students Talk Stats: Seeking a Representative Sample 18 20 21 22 23 Summary 25 / Exercises 25 Design: How Individuals Are Studied 30 3.1 Various Designs for Studying Variables Identifying Study Design Observational Studies versus Experiments: Who Controls the Variables? Errors in Studies’ Conclusions: The Imperfect Nature of Statistical Studies 30 32 33 35 3.2 Sample Surveys: When Individuals Report Their Own Values Sources of Bias in Sample Surveys 38 38 3.3 Observational Studies: When Nature Takes Its Course Confounding Variables and Causation Paired or Two-Sample Studies Prospective or Retrospective Studies: Forward or Backward in Time 46 46 48 49 3.4 Experiments: When Researchers Take Control Randomized Controlled Experiments Double-Blind Experiments “Blind” Subjects “Blind” Experimenters Pitfalls in Experimentation Modifications to Randomization 51 52 53 53 54 55 57 vii viii Contents Students Talk Stats: Does Watching TV Cause ADHD? Considering Study Design 63 Summary 63 / Exercises 65 PA R T I I Displaying and Summarizing Data 70 Displaying and Summarizing Data for a Single Variable 4.1 72 Single Categorical Variable Summaries and Pie Charts The Role of Sample Size: Why Some Proportions Tell Us More Than Others Do Bar Graphs: Another Way to Visualize Categorical Data Mode and Majority: The Value That Dominates Revisiting Two Types of Bias Students Talk Stats: Biased Sample, Biased Assessment 74 75 77 77 78 4.2 Single Quantitative Variables and the Shape of a Distribution Thinking about Quantitative Data Stemplots: A Detailed Picture of Number Values Histograms: A More General Picture of Number Values 82 83 85 89 4.3 Center and Spread: What’s Typical for Quantitative Values, and How They Vary Five-Number Summary: Landmark Values for Center and Spread Boxplots: Depicting the Key Number Values Mean and Standard Deviation: Center and Spread in a Nutshell 93 93 95 98 4.4 Normal Distributions: The Shape of Things to Come The 68-95-99.7 Rule for Samples: What’s “Normal” for a Data Set From a Histogram to a Smooth Curve Standardizing Values of Normal Variables: Storing Information in the Letter z Students Talk Stats: When the 68-95-99.7 Rule Does Not Apply “Unstandardizing” z-Scores: Back to Original Units The Normal Table: A Precursor to Software 72 72 108 110 113 114 117 118 119 Summary 125 / Exercises 127 Displaying and Summarizing Relationships 133 5.1 Relationships between One Categorical and One Quantitative Variable Different Approaches for Different Study Designs Displays Summaries Notation Data from a Two-Sample Design Data from a Several-Sample Design Data from a Paired Design Students Talk Stats: Displaying and Summarizing Paired Data Generalizing from Samples to Populations: The Role of Spreads The Role of Sample Size: When Differences Have More Impact 133 133 134 134 134 134 137 138 139 141 143 5.2 Relationships between Two Categorical Variables 150 Summaries and Displays: Two-Way Tables, Conditional Percentages, and Bar Graphs 151 The Role of Sample Size: Larger Samples Let Us Rule Out Chance 156 118 Chapter 4: Displaying and Summarizing Data for a Single Variable Students Talk Stats continued Brittany: “Look around How many 1612 -year-olds you see on campus? Do you know anybody in college who’s that young?” Carlos: “Overall the standard deviation might be 2.9 years, but most of the spread is coming from students who are a lot older than the average.” Dominique: “This is a trick question It doesn’t say the distribution is normal, so Adam shouldn’t be talking about the z-score as if it’s from a normal distribution We have to use our common sense and say it is an unusually young age, even though its standardized value isn’t so extreme That’s because some of the students are really old, like Carlos said, so the distribution isn’t normal, it’s skewed right.” This example was designed to remind students of the importance of the requirement that the shape of the distribution must be normal if we want to apply the guidelines of the 68-95-99.7 Rule The shape of ages of college students is quite skewed to the right The rule does not apply because being a certain distance below the mean could be next to impossible, whereas being that same distance above the mean is not so unusual “Unstandardizing” z-Scores: Back to Original Units If we know the mean and standard deviation of a variable’s values, and are told the z-score for a particular value, it is quite easy to “unstandardize,” keeping in mind that the z-score tells us how many standard deviations below or above the mean a value is EXAMPLE 4.32 From z-Score to Original Value Background: An IQ test has a mean of 100 The standard deviation is 15 Question: If a student’s z-score is ϩ1.2, what is the student’s IQ? Response: The z-score tells us that the student scored 1.2 standard deviations above the mean, where a standard deviation is 15 and the mean is 100 We calculate 100 ϩ 1.2(15) ϭ 118 Practice: Try Exercise 4.55(g) on page 121 In general, z = value - mean , so we can express a value in the original standard deviation units by finding value = mean + z(standard deviation) being careful to get the correct sign for z If the mean and standard deviation are given for the population, we can express a value x as x = m + zs Because ␴ is always positive, a positive z results in a value that is greater than the mean, whereas a negative z results in a value that is less than the mean Section 4.4: Normal Distributions: The Shape of Things to Come 119 The Normal Table: A Precursor to Software Before software was widely available, tables were essential for finding proportions or percentages involving normal variables This book does not require the use of tables, but they are included at the end of the book, and a section of “Using the Normal Table” exercises is featured in some of the chapters that involve normal variables Most statistical applications involving normal distributions are carried out with software that automatically provides the needed proportions Still, using normal tables is one way for beginners to become familiar with how these distributions behave EXERCISES FOR S E C T I O N 4.4 Normal Distributions: The Shape of Things to Come Note: Asterisked numbers indicate exercises whose answers are provided in the Solutions to Selected Exercises section, on page 689 *4.46 The histogram in Exercise 4.36 on page 106 shows the distribution of self-rated school ability scores, on a scale of to a What percentage of 12th graders rated themselves less than 4? b What percentage of 12th graders rated themselves or less? c What percentage of 12th graders rated themselves at least 4? d What percentage of 12th graders rated themselves more than 4? 4.47 The histogram in Exercise 4.37 on page 106 shows the distribution of evenings out per week for a large group of students Was the percentage of students who went out fewer than nights per week about 10% to 15%, 15% to 20%, 20% to 25%, or 25% to 30%? *4.48 Data for vertical drop (difference in feet between top and bottom of a continuous slope or trail) of 23 ski slopes in the Middle Atlantic states are summarized below Variable Vertical Variable Vertical N 23 Minimum 180.0 Mean 627.0 Maximum 1500.0 Median 610.0 Q1 360.0 TrMean 606.7 Q3 850.0 StDev 317.4 SE Mean 66.2 a Use the output to tell how many standard deviations above the mean the maximum value is b Did all of the 23 values fall within standard deviations of the mean (between 627 Ϫ 2(317.4) ϭ Ϫ7.8 and 627 ϩ 2(317.4) ϭ 1261.8)? c The minimum drop is 180 feet Would its z-score be positive or negative? d One of the ski slopes has a z-score of ϩ0.86 What is its vertical drop, to the nearest foot? e The mean drop is 627 feet If this is taken as a summary of all the region’s ski slopes, is it a statistic or a parameter? Do we refer to it as x or m? f The mean drop is 627 feet If this is taken as a summary of a sample of some of the nation’s ski slopes, is it a statistic or a parameter? Do we refer to it as x or m? g How would a histogram of the data appear if the standard deviation were only instead of 317.4? 4.49 Here is a stemplot for ages at death (in years) of 20 dinosaur specimens, ranging from years up to 28 years 0 1 2 22 57 0444 5567888 1224 a The mean age at death is 15 years and the standard deviation is years What percentage of the dinosaurs’ ages were within standard deviation of the mean? b What percentage of the dinosaurs’ ages were within standard deviations of the mean? c The units of standard deviation s are years What would be the units of variance s2? 120 Chapter 4: Displaying and Summarizing Data for a Single Variable d The shape of the stemplot is fairly normal, with only a bit of left-skewness Tell how the shape of the distribution for age at death of humans would differ e If paleontologists want to use the mean age, 15 years, to estimate the mean age at death of all dinosaurs, are they mainly concerned with data production, displaying and summarizing data, probability, or statistical inference? f Based on the information from this sample, which of these would be the most plausible value for mean age at death of all dinosaurs: 10 years, 14 years, or 18 years? 4.50 Here are data for per capita beer consumption (in liters per person per year) in 18 countries during 2002.18 Country Beer Consumption Country Beer Consumption Ireland 155 Finland 79 Germany 119 New Zealand 78 Austria 106 Canada 70 Belgium 98 Switzerland 57 Denmark 98 Norway 56 United Kingdom 97 Sweden 56 Australia 89 Japan 55 United States 85 France 41 Netherlands 80 Italy 29 a Display the data with a histogram, using rectangles with intervals of width 20 liters b Explain why the shape of the data suggests that the 68-95-99.7 Rule should work reasonably well c The mean per capita consumption is approximately 80, and the standard deviation is approximately 30 What are the endpoints of the interval that extends within standard deviation of the mean? d The 68-95-99.7 Rule estimates about 68% of values should fall in the interval you reported in part (c); what percentage of those 18 values actually fall within the interval? e What are the endpoints of the interval that extends within standard deviations of the mean? f What percentage of those 18 values actually fall within the interval you reported in part (e)? g What are the endpoints of the interval that extends within standard deviations of the mean? h What percentage of those 18 values actually fall within the interval you reported in part (g)? i Find the z-score for Ireland and report how many standard deviations below or above the mean its value is j Which country has a z-score of ϩ0.3? k Countries with consumption values even less than Italy’s were not included Would their z-scores be positive or negative? 4.51 Total viewerships, in millions, for all the presidential debates held every years from 1976 to 2004 are summarized in this output (Note that there were typically two or three debates held in each of those eight election years.) Variable Debate Viewership N 19 Mean 57.66 Median 62.70 TrMean 57.58 StDev 13.58 SE Mean 3.12 Section 4.4: Normal Distributions: The Shape of Things to Come 121 a How many presidential debates were there altogether from 1976 to 2004? b The viewership for the first Bush-Kerry debate in 2004 was 62.5 million What is the z-score for this value? c Based on the z-score, how does the viewership for the Bush-Kerry debate compare to the data set as a whole—pretty low, pretty high, or fairly typical? d There was only one Carter-Reagan debate in 1980, and the viewership was 80.6 million What is the z-score for this value? e Based on the z-score, how does the viewership for the Carter-Reagan debate compare to the data set as a whole—pretty low, pretty high, or fairly typical? f Explain why the z-score for the number of viewers watching the very first televised debate (between Kennedy and Nixon in 1960) might be misleading as a measure of public interest *4.52 Weights of male Gadwall ducks at the “gawky downy” age follow a distribution like the one shown What are the mean and standard deviation? Weight of Male Gawky-Downy Gadwall Ducks 44 92 140 188 236 284 Weight (grams) 332 4.53 Weights of 1-year-old boys (in kilograms) follow a distribution like the one shown What are the mean and standard deviation? 18 Weight of 1-year-old boys (kilograms) *4.54 Waist circumference of women in their twenties follows a somewhat normal distribution with mean 32 inches, standard deviation inches a Use the 68-95-99.7 Rule to sketch a histogram for a large data set of women’s waist circumferences, marking off the circumferences that are within 1, 2, and standard deviations of the mean b Sketch a normal curve for the waist circumferences of all women c Approximately what is the percentage of waists that are larger than 42 inches? d The smallest 16% of waists are less than how many inches? e More precisely, the mean for women in their twenties is 32.2, and their median is 30.6 The mean for women in their eighties is 36.7, and their median is 36.4 Which age group has more skewness in the distribution of waist circumferences? f Is the skewness observed in part (e) left or right? *4.55 Waist circumference of women in their twenties follows a somewhat normal distribution with mean 32 inches, standard deviation inches a How we denote the number 32? b How would we denote the mean waist circumference for a sample of women in their twenties? c The mean, 32 inches, is actually just an estimate made by researchers Is it closer to the truth if it was based on information from a well-designed study of 1,000 women, or of 10,000 women, or doesn’t it matter? d How many standard deviations away from the mean is a waist circumference of 32? e If the z-score for a woman’s waist circumference is ϩ2.4, should we report it to be somewhere in the top 50%, the top 16%, the top 2.5%, or the top 0.15%? f If the z-score for a woman’s waist circumference is ϩ2.4, would we consider it to be fairly common, unusually small, or unusually large? g Find the waist circumference for a woman whose z-score is ϩ2.4 h Find the waist circumference for a woman whose z-score is Ϫ1.2 122 Chapter 4: Displaying and Summarizing Data for a Single Variable 4.56 Weights of 1-year-old male children are fairly normally distributed with mean 26 pounds, standard deviation pounds a Use the 68-95-99.7 Rule to sketch a curve for this distribution, marking off the weights that are within 1, 2, and standard deviations of the mean b Find the z-score for a male 1-year-old who weighs 20 pounds c Find the weight (to the nearest pound) of a 1-year-old male whose z-score is Ϫ1.67 d What percentage of male 1-year-olds weigh less than 20 pounds? e Would a 19-pound, 1-year-old male be considered pretty light, pretty heavy, or fairly typical? f Would a 28-pound, 1-year-old male be considered pretty light, pretty heavy, or fairly typical? 4.57 Suppose you are told the mean and standard deviation for amounts that a sample of students have in their bank accounts Why is this not enough to sketch the distribution of amounts? 4.58 Students’ scores on a final exam had a mean of 79% If a student had a z-score of Ϫ0.3, which one of these could be her exam score: 75%, 85%, or 95%? *4.59 On an exam with scores that followed a normal distribution, a student’s z-score was ϩ1 What percentage of students scored better than he did? 4.60 On an exam with scores that followed a normal distribution, a student’s z-score was ϩ2 What percentage of students scored worse than he did? 4.61 In a certain population of smokers, the number of cigarettes smoked per day followed a fairly normal curve with a mean of 15 and a standard deviation of a Use the 68-95-99.7 Rule to sketch a curve for this distribution, marking off the numbers of cigarettes smoked daily that are within 1, 2, and standard deviations of the mean b Explain why the lower end of the curve is not perfectly consistent with reality c Find the z-score for a person who smokes one pack (20 cigarettes) a day d Characterize one pack a day as being unusually low, unusually high, or not unusual e Find the z-score for a person who smokes two packs a day f Characterize two packs a day as being unusually low, unusually high, or not unusual *4.62 Keep in mind that certain distributions, such as exam scores or some physical characteristics for specific age and gender groups, naturally follow a normal shape Other distributions are naturally skewed left or have low outliers if there tend to be relatively few values that are unusually low compared to the rest Yet other distributions are naturally skewed right or have high outliers if there tend to be relatively few values that are unusually high compared to the rest Decide if each of the following variables is most likely to have a shape that is approximately normal, or skewed left/low outliers, or skewed right/high outliers: a Math SAT scores b Number of credits taken by a large group of students, including part-time and full-time c Number of minutes that students spend exercising on a given day d Number of minutes that students spend watching TV on a given day e Ages of students’ fathers f Heights of students’ mothers g Number of siblings that students have h How much money students earned the previous year 4.63 Keep in mind that certain distributions, such as exam scores or some physical characteristics for specific age and gender groups, naturally follow a normal shape Other distributions are naturally skewed left or have low outliers if there tend to be relatively few values that are unusually low compared to the rest Yet other distributions are naturally skewed right or have high outliers if there tend to be relatively few values that are unusually high compared to the rest Decide if each of the following variables is most likely to have a shape that is Section 4.4: Normal Distributions: The Shape of Things to Come approximately normal, or skewed left/low outliers, or skewed right/high outliers: a Number of minutes that students spend on the computer on a given day b Number of minutes that students spend on the phone on a given day c Verbal SAT scores d Number of cigarettes smoked in a day by a large group of students who smoke e Amount of cash carried by students f Heights of students’ fathers g Ages of students’ mothers h Number of hours that students slept on a given night in a large group of students that includes a few who got little or no sleep 4.64 “Remains Found of Downsized Human Species” reports that “once upon a time, on a tropical island midway between Asia and Australia, there lived a race of little people, whose adults stood just 3.5 feet high.”19 The 123 article goes on to tell about the excavation in 2003 of an adult female skeleton, the first of several whose size was small enough to warrant identification as a new human species, called Homo floresiensis after the island of Flores where they were discovered a If heights of adult females today have mean 64.5 inches and standard deviation 2.5 inches, what would be the z-score of a 3.5-foot-tall woman? b The article explains that “because the downsizing is so extreme, smaller than that in modern human pygmies, they [scientists] assigned it to a new species.” The smallest race of pygmies is said to have an average height of 54 inches If their standard deviation is about inches, find how many standard deviations below average the Floresian would be compared to pygmies c Do we refer to the standard deviation of all pygmy heights as x, m, s, or s? Using Software [see Technology Guide] *4.65 Here are estimated weights (in thousands of kilograms) of 20 dinosaur specimens: 5,654 5,040 3,230 2,984 1,807 1,761 30 1,105 747 607 229 127 1,142 1,282 1,013 762 50 1,791 1,518 496 a Use software to calculate the mean and standard deviation b If a distribution is normal, about 16% of values are less than the value that is one standard deviation below the mean What percentage of these 20 weights are less than the value that is one standard deviation below the mean? c If a distribution is normal, only about 2.5% of values are more than two standard deviations above the mean What percentage of these 20 weights are more than two standard deviations above the mean? d Comment on how well or how badly this data set conforms to the 68-95-99.7 Rule 4.66 This data set reports the number of McDonald’s restaurants in various European countries as of 2004: 1115 1091 857 290 276 227 205 181 148 133 119 99 93 91 76 64 62 60 55 48 10 a Use software to find the mean and standard deviation for number of restaurants in those countries b For what percentage of countries is the number of restaurants at or below the mean? c If a distribution is normal, about 16% of values are less than the value that is standard deviation below the mean What percentage of these 23 values are less than the value that is standard deviation below the mean? d If a distribution is normal, only about 2.5% of values are more than standard deviations above the mean What percentage of these 23 values are more than standard deviations above the mean? e Comment on how well or how badly this data set conforms to the 68-95-99.7 Rule 124 Chapter 4: Displaying and Summarizing Data for a Single Variable 4.67 Running times (in minutes) of 29 movies currently showing in early December 2004 are listed here, from a 73-minute European film called “The Big Animal”, about a couple that unwittingly adopts a camel into their home when the circus leaves town, to the 173-minute Oliver Stone film about Alexander the Great 73 90 90 93 94 97 98 99 100 100 102 105 105 106 106 107 108 113 115 115 115 120 123 130 130 140 150 152 173 a Use software to find the mean and standard deviation b Report the z-score for the shortest film, about the camel c Report the z-score for the longest film, about Alexander the Great Using the Normal Table [see end of book] or Software 4.68 Part (j) of Exercise 4.50 on page 120 refers to the country whose z-score for per capita beer consumption is ϩ0.3 Find the probability of z being greater than ϩ0.3, if the distribution were perfectly normal *4.69 According to Exercise 4.52 on page 121, weights of male Gadwall ducks at the “gawky downy” age follow a normal distribution with mean 188 grams, standard deviation 48 grams a The lightest 1% weigh less than how many grams? b The lightest 10% weigh less than how many grams? c The heaviest 20% weigh more than how many grams? d The heaviest 2% weigh more than how many grams? 4.70 According to Exercise 4.53 on page 121, weights of 1-year-old boys follow a normal distribution with mean 12 kilograms, standard deviation kilograms a The lightest 5% weigh less than how many kilograms? b The lightest 25% weigh less than how many kilograms? c The heaviest 50% weigh more than how many kilograms? d The heaviest 3% weigh more than how many kilograms? *4.71 Part (h) of Exercise 4.55 on page 121 refers to the waist circumference of a woman whose z-score is Ϫ1.2, where the mean is 32 inches and standard deviation is inches a Find the proportion of z values less than Ϫ1.2 b What proportion of z values are greater than Ϫ1.2? 4.72 According to Exercise 4.56 on page 122, weights of 1-year-old boys are normal with mean 26 pounds, standard deviation pounds a Part (c) refers to the weight of a 1-yearold boy whose z-score is Ϫ1.67 Find the proportion of 1-year-old boys who weigh less than this amount b Part (e) refers to a 1-year-old boy whose weight is 19 pounds Find the proportion of 1-year-old boys who weigh less than this c What proportion of 1-year-old boys weigh more than 19 pounds? d Part (f) refers to a 1-year-old boy whose weight is 28 pounds Find the proportion of 1-year-old boys who weigh less than this 4.73 Exercise 4.58 on page 122 refers to a student whose exam score had z ϭ Ϫ0.3 If the distribution of scores is normal, what proportion of z-scores are less than Ϫ0.3? *4.74 Exercise 4.59 on page 122 refers to a student whose exam score had z ϭ ϩ1 According to the normal table or a computer, exactly what proportion of z-scores are greater than ϩ1? 4.75 Exercise 4.60 on page 122 refers to a student whose exam score had z ϭ ϩ2 According to the normal table or a computer, exactly what proportion of z-scores are greater than ϩ2? 4.76 Exercise 4.61 on page 122 states that in a certain population of smokers, daily number of cigarettes smoked is approximately normal with mean 15, standard deviation a Find the proportion smoking at least one pack a day (20 cigarettes) b Find the proportion smoking at least two packs a day (40 cigarettes) Chapter 4: Summary Chapter 125 Summ ary Often we are interested in sampled values of a single variable The variable may be categorical (such as whether or not a student eats breakfast), or quantitative (such as how much money the student earned) Which displays and summaries are appropriate depends on whether the variable is categorical or quantitative In this chapter, we concentrated on displaying and summarizing sampled values of a variable We did, however, make an exception in the case of a normal curve, which shows a bell-shaped distribution of number values for an entire population Single Categorical Variable ࡯ Summarize results for all individuals in a categorical data set with counts, percents, or proportions in the various categories ࡯ Display categorical values with a pie chart or bar graph ࡯ Categorical variables may have any number of possible values, but much of the theory developed later in this book will be restricted to situations where just two values are possible ࡯ A proportion that summarizes categorical values for a sample is a statistic, written pN If it summarizes values for a population, it is a parameter, written p ࡯ The mode is the most common category value If there are only two possibilities, then the mode is the same as the value taken by the majority ࡯ The percentage that summarizes a categorical variable may be biased because the sample was not representative, or because of flaws in the design for assessing the variable’s values, or both ࡯ If a distribution has one major peak it is called unimodal ࡯ A particular shape that frequently occurs is called normal It is symmetric and unimodal, often called bell-shaped because of the way it bulges at the center and tapers at the ends ࡯ The shape is called skewed if it is lopsided It is skewed left if there is a longer left tail, indicating there are a few values that are relatively low compared to the rest of the values It is skewed right if there is a longer right tail, indicating there are a few values that are relatively high compared to the rest of the values Outliers are values that are far from the bulk of displayed values Single Quantitative Variable When we are interested in values of a single quantitative variable, we can summarize the general pattern of those values by reporting shape, center, and spread Shape ࡯ Shape is best assessed by producing a display of the data A stemplot is one display for a quantitative data set Each stem in a vertical list is followed by a horizontal list of one-digit leaves, representing all observations with that stem ࡯ Stems may be split if there are too few stems for the number of leaves ࡯ Leaves may be truncated or rounded if there are too many stems for the number of leaves ࡯ The list of stems must form a consistent numerical scale, and stems that happen not to have any leaves must still be included Center and Spread ࡯ ࡯ A histogram is a display consisting of rectangles with heights that correspond to the count or percentage or proportion of values occurring in the given horizontal interval ࡯ The shape of a distribution is called symmetric if it is fairly balanced ࡯ The median of a quantitative data set is the single middle value for an odd sample size It is the average of the two middle values for an even sample size ࡯ The first quartile (Q1) has one-fourth of the data values at or below it It is the middle of the values below the median 126 Chapter 4: Displaying and Summarizing Data for a Single Variable ࡯ The third quartile (Q3) has three-fourths of the data values at or below it It is the middle of the values above the median ࡯ The interquartile range is IQR ϭ Q3 Ϫ Q1 We designate as outliers any values below Q1 Ϫ 1.5(IQR) or above Q3 ϩ 1.5(IQR) ࡯ The five-number summary lists the minimum, first quartile, median, third quartile, and maximum ࡯ A boxplot displays the five-number summary, with special treatment for outliers The box spans the quartiles, with a line through the box at the median The whiskers extend to the minimum and maximum non-outliers Outliers are marked “*” Boxplots are skewed left if “bottom heavy,” skewed right if “top heavy.” ࡯ ࡯ The distribution of values of a normal variable may be represented with a smooth bell-shaped curve instead of a histogram The normal curve is generally meant to display values for an entire population, modeled as having an infinite number of individuals, where the variable’s values constitute a continuum of possibilities ࡯ The standardized value of any data value x, or z-score, tells how many standard deviations below or above the mean x is: z = x -s x ࡯ The mean of a population is denoted m and the standard deviation is denoted s A standardized value is z = x -s m ࡯ A z-score can be “unstandardized” to the actual value by finding value ϭ mean ϩ z(standard deviation), or x ϭ m ϩ z(␴) ࡯ For a normal data set, the 68-95-99.7 Rule applied to standardized scores tells us that 68% of z-scores are between Ϫ1 and ϩ1; 95% are between Ϫ2 and ϩ2; 99.7% are between Ϫ3 and ϩ3 ࡯ If a sample data set or population is normal, then the z-score gives us a good idea of how usual or unusual a value is The cut-off for what would be considered unusual is around z ϭ Ϫ2 or z ϭ ϩ2 ࡯ Besides estimating normal percentages with the 68-95-99.7 Rule or finding them with software, a normal table (provided at the end of the book) can be used The sample mean is the arithmetic average x = ࡯ distribution, beyond a certain distance from the mean x1 + Á + xn n The sample standard deviation is the square root of the “average” squared deviation from the mean s = (x1 - x)2 + Á + (xn - x)2 n - D It tells the typical distance of values from their mean Normal Distributions ࡯ If a data set has a normal shape, the 68-95-99.7 Rule tells us that approximately 68% of the values fall within standard deviation of the mean, 95% within standard deviations of the mean, and 99.7% within standard deviations of the mean ࡯ Since 100% of a variable’s values must be accounted for, the 68-95-99.7 Rule can be used to tell percentages on the tails of a normal Chapter 4: Exercises Chapter 127 Exercises Note: Asterisked numbers indicate exercises whose answers are provided in the Solutions to Selected Exercises section, on page 689 Additional exercises appeared after each section: categorical variables (Section 4.1) page 78, quantitative variables (shape, Section 4.2) page 92, quantitative variables (center and spread, Section 4.3) page 103, and quantitative variables (normal, Section 4.4) page 119 Warming Up: Single Categorical Variable 4.77 Use the definition of experiment to explain why we would not expect an experiment to involve just a single categorical variable between males and females, but there are more females in college and they tend to be the ones with pierced ears Carlos: Maybe it’s because the college students feel less inhibited, so they’re the ones who are willing to admit that they have pierced ears Dominique: Inhibitions have nothing to with it unless you’re talking about something more personal like a hidden tattoo I think the difference in proportions with ears pierced just came about by chance Anything’s possible with random samples 4.78 Explain why a random sample of college students has a higher percentage of females, whereas a random sample of high school students is almost evenly divided between males and females 4.79 Four statistics students are answering a homework problem that asks them to explain why a very large sample of college students shows a significantly higher percentage of students with pierced ears than does a sample of high school students Which two answers are best? Adam: Probably because of people like me—I didn’t get my ear pierced until I started college Once you have them pierced, you don’t get them unpierced, so the number with pierced ears accumulates as students get older Brittany: I think it’s because high school students are pretty much evenly divided 4.80 For which one of the following variables would you expect high school and college students may have very different percentages in the various categories: whether they ate breakfast or not; whether they are righthanded, left-handed, or ambidextrous; what number they picked between and 20 when asked to pick a number at random? Exploring the Big Picture: Single Categorical Variable 4.81 In “Compliance with the Item Limit of the Food Supermarket Express Checkout Lane: An Informal Look,” researcher J Trinkaus reported, “75 15-min observations of customers’ behavior at a food supermarket showed that only about 15% of shoppers observed the item limit of the express lane.”20 a Is 15% a statistic or a parameter? Should it be denoted p or pN ? b Can we tell the approximate count of shoppers who observed the item limit? If so, what is it? If not, explain why not c Which type of sample is suggested by the word informal: volunteer, random, convenience, or stratified? d If these results are used to estimate that about 15% of all shoppers observe the item limit of the express lane, are we performing data production, displaying and summarizing data, probability, or inference? 4.82 In November 2004, a newspaper article announced, “Caesarean Deliveries Hit U.S Record” and stated that “roughly 1.13 million, or 27.6% [of births], were caesarean deliveries” in the United States in 2003.21 a Approximately how many births were there altogether in the United States that year? 128 Chapter 4: Displaying and Summarizing Data for a Single Variable b If the proportion 0.276 refers to the entire proportion of births, should it be written as p or pN ? 4.83 An Internet survey first asked a sample of 2,065 men and women how many sexual partners they’d had altogether When subsequently asked about their truthfulness in responding to the question, 5% said they had lied a If the original survey had been used to estimate the mean number of sexual partners for men and women in general, would it result in bias due to a nonrepresentative sample or bias due to inaccurately assessing a variable’s values? b What notation we use for the number 0.05? c How we denote the unknown proportion of all people who would lie when answering such a question? 4.84 “Antarctic Birds Use Scent to Find Their Mates” reports that “Antarctic birds returning to a nesting colony after feeding at sea sniff out their mates, literally Scientists studying the birds [Antarctic prions] found that they have a noticeable odor, which remained behind on the bags used to hold them So the researchers set up a Y-shaped maze On the end of each arm, they placed a bag that had previously held a bird In 17 out of 20 cases, the returning bird selected the bag that smelled like its mate ”22 a Which statistical process concerned the researchers when they decided on a Y-shaped maze to test their theory—data production, displaying and summarizing, probability, or statistical inference? b If the birds could not identify the smell of their mates, approximately what percentage of all returning birds would select the correct bag? c Is your answer to part (b) a statistic or a parameter? d Use percentages to explain why the results seem convincing e What is the largest population for which the researchers may be justified in generalizing their results? Using Software [see Technology Guide]: Single Categorical Variable 4.85 Access the student survey data completed by 446 students in introductory statistics courses at a large university in the fall of 2003 a Use software to report the percentages of males and females b Keeping in mind that introductory statistics is required for most majors, the data suggest that a majority of all students at the university are female? 4.86 Access the student survey data completed by 446 students in introductory statistics courses at a large university in the fall of 2003 a b c d What was the students’ favorite color? What percent preferred the favorite color? What was the students’ least favorite color? What percent preferred the least favorite color? e During that same school year, 267 high school students taking a college statistics course for credit were surveyed Their favorite color was blue, with a percentage of 42%, and their least favorite was yellow, with a percentage of 3% Is taste in colors apparently similar for high school and college students? Discovering Research: Single Categorical Variable 4.87 Find (and hand in) a newspaper article or Internet report about a study that involves just a single categorical variable Besides providing a copy of the article or report, tell how the data were produced (experiment, observational study, or sample survey) Tell if the variable is summarized with counts, percents, or both Tell if results are being reported for a sample or for an entire population, and report the number of individuals studied, if known Tell if there is reason to suspect bias that arises if the sample is not representative, or bias in the design for assessing the variable’s values Chapter 4: Exercises 129 Reporting on Research: Single Categorical Variable 4.88 Use the results of Exercise 4.3 on page 78 and relevant findings from the Internet to make a report on racial profiling that relies on statistical information Warming Up: Single Quantitative Variable 4.89 The United Nations and the World Health Organization expressed alarm regarding the extent to which AIDS has reduced life expectancy in Southern Africa In Swaziland, children born between 2000 and 2005 could expect to live an average of only 34.4 years, compared to a worldwide life expectancy that is about twice that long.23 a Consider age at death for people in a developed country like the United States Would the distribution’s shape be leftskewed, symmetric, or right-skewed? b Consider age at death for people in Swaziland Would the distribution’s shape be more symmetric or less symmetric than that for the United States? 4.90 In an e-mail interview in December 2004, a New York Times reporter asked the renowned physicist Stephen Hawking, “What is your IQ?” to which he replied, “I have no idea People who boast about their IQ are losers,” and went on to say that “there really is a continuous range of abilities with no clear dividing line.”24 Given this comment, would Hawking seem to advocate treating intelligence as a categorical variable, classifying an individual as “moron” for an IQ under 70, “genius” for an IQ over 140, and so on? Exploring the Big Picture: Single Quantitative Variable 4.91 A newspaper article on salaries of college presidents reported compensations for a sample of 11 college presidents in area schools during the year 2002–2003, as well as compensations for the 10 highest-paid leaders of public universities in the entire United States (Leaders of private universities, not included in the latter data set, have been known to receive more than $1 million in yearly pay.) The data, recorded in thousands of dollars, have been summarized with software Variable AreaCollegePres TopUSCollegePres N 11 10 Mean 288.4 643.8 Median 270.0 644.7 TrMean 277.5 641.4 StDev 129.4 82.9 SE Mean 39.0 26.2 a Find the salary of a college president whose z-score for that area was approximately ϩ2 b The same college president was also included in the list of the top 10 salaries for the entire United States; find his z-score in reference to this group c Discuss the drawbacks in using the mean for college presidents in a particular area as an estimate for the mean salary of all college presidents d Explain why the mean salary of top college presidents is useless as an estimate for any larger group 4.92 A New York Times review of the book The State Boys Rebellion by Michael D’Antonio cites this chilling quote: “to make sure every last moron was captured, many states, including Massachusetts, would establish traveling ‘clinics’ to administer IQ tests at public schools.” The review mentions that by 1949, approximately 150,000 Americans were institutionalized.25 a The U.S population in 1949 was about 150,000,000 What proportion of all Americans were institutionalized at that time? b Scores for the Wechsler IQ test are normal with mean 100 and standard deviation 15 Use the 68-95-99.7 Rule to sketch a curve for this distribution, marking off the scores that are within 1, 2, and standard deviations of the mean 130 Chapter 4: Displaying and Summarizing Data for a Single Variable c Apparently, people were being institutionalized in the 1940s if their IQ was approximately how many standard deviations below the mean? d The reviewer cites that of the 150,000 institutionalized Americans at that time, an estimated 12,000 were in fact “of relatively normal intelligence.” Discuss the harmful consequences of using the results of an IQ test to institutionalize people who, in fact, are capable of functioning normally in society e Discuss the harmful consequences of failing to institutionalize people who are unable to function in society 4.93 Typical age, in days, at which various species of ducks begin to fly range from 39.5 (for blue-winged teals) to 62.5 (for canvasbacks) 39.5 47.0 47.0 50.0 50.5 51.0 51.5 56.0 60.5 61.5 62.5 a Display the data with a stemplot b Report the “typical” flying age of the various species by finding the median age and converting to weeks *4.94 In 1954, researchers Gollop and Marshall defined age classes of ducks based on their plumage, from the earliest stages “bright ball of fluff,” and “fading ball of fluff,” to the next-to-last stage “feathered-flightless,” and, finally, “flying.”26 a Suppose female mallards at the “bright ball of fluff” stage have weights (in grams) with mean 32.4 and standard deviation 2.4 Find the z-score of a female mallard at this stage that weighs 40 grams b Suppose female mallards at the “fading ball of fluff” stage have weights (in grams) with mean 115.3 and standard deviation 37.6 Find the z-score of a female mallard at this stage that weighs 40 grams c If a female mallard weighs 40 grams, is she more likely to be at the “bright ball of fluff” stage or the “fading ball of fluff” stage? (Base your answer on the z-scores.) 4.95 In 1954, researchers Gollop and Marshall defined age classes of ducks based on their plumage, from the earliest stages “bright ball of fluff,” and “fading ball of fluff,” to the next-to-last stage “feathered-flightless,” and, finally, “flying.” a Suppose male mallards at the “featheredflightless” stage have weights (in grams) with mean 864 and standard deviation 100 Find the z-score of a male mallard at this stage that weighs 840 grams b Suppose male mallards at the “flying” stage have weights (in grams) with mean 818 and standard deviation 91 Find the z-score of a male mallard at this stage that weighs 840 grams c If a male mallard weighs 840 grams, is he more likely to be at the “featheredflightless” stage or the “flying” stage? Explain 4.96 In 2004, a 59-year-old woman, about to give birth in December, set the record for age when pregnant a If age of all mothers at the time of delivery has a mean of 27 and a standard deviation of 6, tell how many standard deviations above average this woman’s age is b Explain why the distribution of ages at time of delivery cannot be normal c If we incorrectly applied the 68-95-99.7 Rule, it would tell us that half of 100% minus 99.7%, or 0.15%, of mothers are less than years old at the time of delivery Is the actual percentage higher or lower than this? 4.97 Four statistics students discuss the question, “Which of these variables could take a value that is so low as to be surprising: age of a mother at the time of delivery, or age of a mouse at time of death, or both, or neither?” Whose answer is best? Adam: Any data set can have low outliers, so I’d say both Brittany: You shouldn’t be surprised by low outliers, they happen all the time So I’d say neither Carlos: We saw an example of a mouse that died at an unusually old age There must also be mice that die at an unusually young age, so I’d say age of a mouse at time of death could have a surprisingly low value Dominique: Mice die at birth all the time, so the lowest possible value is zero but it wouldn’t surprise us at all On the other hand, mothers are usually at least in their teens when they deliver, but I read that there are a few girls who develop early and give birth when they’re still children themselves Chapter 4: Exercises 131 Using Software [see Technology Guide]: Single Quantitative Variable 4.98 This stemplot shows the number of medals won at the 2008 Olympics by 87 countries, including the 110 medals won by the United States and 100 won by China 1111111111111111111222222222222333333444444455555566666666777889 00013456889 4578 0167 10 11 a Use software to find the mean and standard deviation for number of medals won by the various countries b Find the mean and standard deviation, if the two highest outliers—for China and the United States—are omitted c Which is apparently more affected by outliers: the mean or the standard deviation? 4.99 Rents of one-bedroom apartments near a university were recorded in 2005 475 550 450 350 485 430 650 350 495 425 475 350 Use software to fill in the blanks, to the nearest dollar: the rents average to $ , and they tend to differ from this average by about $ Using the Normal Table [see end of book] or Software: Single Quantitative Variable *4.100 Exercise 4.94 discusses weights of female mallards at various stages of development a The “bright ball of fluff” stage has mean weight 32.4 grams and standard deviation 2.4 grams Find the proportion of female mallards at this stage that weigh more than 40 grams b The “fading ball of fluff” stage has mean weight 115.3 grams and standard deviation 37.6 grams Find the proportion of female mallards at this stage that weigh less than 40 grams 4.101 Exercise 4.95 discusses weights of male mallards at various stages of development a The “feathered-flightless” stage has mean weight 864 grams and standard deviation 100 grams Find the proportion of male mallards at this stage that weigh less than 840 grams b The “flying” stage has mean weight 818 grams and standard deviation 91 grams Find the proportion of male mallards at this stage that weigh more than 840 grams Discovering Research: Single Quantitative Variable 4.102 Find (and hand in) a newspaper article or Internet report about a study that involves just a single quantitative variable Besides providing a copy of the article or report, tell how the data were produced—was it an experiment or an observational study? Tell if the variable is summarized with mean, median, or neither Tell if results are being 132 Chapter 4: Displaying and Summarizing Data for a Single Variable reported for a sample or for an entire population, and the number of individuals studied, if known Tell if there is reason to suspect bias that arises if the sample is not representative, or bias in the design for assessing the variable’s values Reporting on Research: Single Quantitative Variable 4.103 Use results of Exercises 4.90 and 4.92 (page 129) and relevant findings from the Internet to make a report on IQ that relies on statistical information [...]... Samples Get Us Closer to the Truth Time Series: When Time Explains a Response Additional Variables: Confounding Variables, Multiple Regression Students Talk Stats: Confounding in a Relationship between Two Quantitative Variables 16 5 16 6 17 0 17 1 17 3 17 4 17 5 17 6 17 7 18 2 18 2 18 3 18 4 18 5 18 6 18 7 18 8 18 9 19 1 19 1 Summary 204 / Exercises... 313 – 315 curved relationship, 16 6, 17 7, 628 D data, 5, 11 production, 9 10 , 16 –65, 14 3, 15 8, 224, 350, 414 , 416 , 677–682 stacked, 543–544 unstacked, 544 degrees of freedom, 4 81 chi-square, 602–604, 613 – 614 F, 5 51, 568, 569 denominator, 545, 546, 548, 5 51 552, 555 numerator, 545, 546, 548, 5 51 552, 555 paired t, 523 regression, 637–638 t, 4 81 two-sample t, 529 density curve, 313 – 315 , 336 area, 313 – 315 ... sample, 18 , 25, 64 Hawthorne effect, 54–55 histogram, 76, 85, 89– 91, 12 5, 13 4, 13 9 14 0 area, 10 9 11 0, 313 – 315 probability, 270–285, 2 91 307, 311 –330, 335 versus bar graph, 89 hypothesis alternative, 416 –429 See also alternative hypothesis null, 416 –429 See also null hypothesis hypothesis test, 326, 3 51, 363, 387, 389, 3 91 about population mean, 504 with t, 486–489 with z, 474–476 about population proportion,... 233 events, 233–236, 247–2 51 variables, 250–2 51 Independent “And” (multiplication) Rule, 233–236, 238–239, 256, 269 inference, 9 10 , 11 , 73, 83, 15 5, 15 6, 224, 311 , 386–663 influential observation, 18 5 18 7, 206 intercept (of regression line), 18 0 18 1, 205 for population, 18 7 18 8, 205, 6 31 634 for sample, 18 0, 18 7 18 8 IQR (interquartile range), 94–95, 12 6 1. 5 ϫ IQR Rule, 96–97, 12 6 L lack of realism, 55–56,... 6 51 Summary 662 / Exercises 664 14 How Statistics Problems Fit into the Big Picture 677 14 .1 The Big Picture in Problem Solving 677 Students Talk Stats: Choosing the Appropriate Statistical Tools: Question 1 678 Students Talk Stats: Choosing the Appropriate Statistical Tools: Question 2 679 Students Talk Stats: Choosing the Appropriate Statistical Tools:... 2 41 246, 247–250, 293–294, 298–299 variables, 250 design, 16 17 , 30–65 biased, 16 17 paired, 48–49, 58, 13 4, 13 8 14 0, 204, 522–525 several-sample, 13 4, 13 7 13 8, 522, 543–559 two-sample, 48–49, 13 4 13 6, 522, 528–538 unbiased, 16 17 direction (of relationship), 16 6 16 7, 205 negative, 16 6 positive, 16 6 discrete random variable, 268–285, 312 – 315 , 335 disjoint events, 232 distribution, 82 chi-square, 311 ,... population), 18 7 18 8, 206, 632–645, 662 balance point (mean), 2 71 272 bar graph, 72, 75–77, 12 5, 15 4, 204 versus histogram, 89 Bayes Theorem, 246 before-and-after study, 58 bell-shaped distribution, 85, 10 9, 12 5 bias, 16 17 , 19 , 25, 38–43, 77–78, 12 5 nonresponse, 25 selection, 19 biased sample, 16 17 , 83, 350, 352, 363, 422, 554–555 study design, 16 17 summary, 83–84 big picture, 9 10 , 11 , 17 , 71, 224,... variable, 2 91 307, 336, 348, 4 51 blind experimenter, 53, 54–55, 65 subject, 53–54, 65 blocking (in experiment), 57–58, 65 boxplot, 85, 95–98, 12 5 side-by-side, 13 4, 13 6 13 8, 14 2, 204, 5 31 538 C c (number of columns), 600–602, 613 categorical variable, 2–5, 11 , 38, 70, 72–78, 12 5, 15 0 16 0, 312 , 344–345, 388–405, 413 –453 summarizing, 5–6 causation, 7, 33, 46–48, 59, 64, 14 3, 15 0, 15 7 16 0 census, 6, 11 , 346,... 11 , 13 5, 15 1, 16 8, 204–205 extrapolation, 18 0 18 1, 205–206 F F distribution, 311 , 522, 543–559, 568 statistic, 545–559, 568, 569 test, 545 factor, 52 five-number summary, 93–95, 12 6, 204 Index form (of relationship), 16 6 16 7, 205 format (of data), 13 4 13 5 four-stage process, 9 10 frame (sampling), 19 G General “And” (multiplication) Rule, 2 41 245, 256 General “Or” (addition) Rule, 239–2 41, 256 group... Montenegro, “Lifestyles, Dating and Romance: A Study of Midlife Singles,” AARP Knowledge Management, September 2003 14 Personal MD, “Piano Lessons Boost Math Scores.” March 18 , 19 99 15 Nicholas Wade [for New York Times], “Remains Found of Downsized Human Species,” Pittsburgh Post-Gazette, October 28, 2004 Index 1. 5 ϫ IQR Rule, 96–97, 12 6 68–95–99.7 Rule, 11 0 11 3, 11 7, 12 6, 311 , 316 – 319 , 336–337, 345, 350,