Computer Solutions Printouts and Instructions for Excel and Minitab Visual Description 2.1 The Histogram 2.2 The Stem-And-Leaf Display* 2.3 The Dotplot 2.4 The Bar Chart 2.5 The Line Chart 2.6 The Pie Chart 2.7 The Scatter Diagram 2.8 The Cross-Tabulation 2.9 Cross-Tabulation with Cell Summary Information Statistical Description 3.1 Descriptive Statistics: Central Tendency 3.2 Descriptive Statistics: Dispersion 3.3 The Box Plot* 3.4 Standardizing the Data 3.5 Coefficient of Correlation Sampling 4.1 Simple Random Sampling Discrete Probability Distributions 6.1 Binomial Probabilities 6.2 Hypergeometric Probabilities 6.3 Poisson Probabilities 6.4 Simulating Observations From a Discrete Probability Distribution Continuous Probability Distributions 7.1 Normal Probabilities 7.2 Inverse Normal Probabilities 7.3 Exponential Probabilities 7.4 Inverse Exponential Probabilities 7.5 Simulating Observations From a Continuous Probability Distribution Sampling Distributions 8.1 Sampling Distributions and Computer Simulation Confidence Intervals 9.1 Confidence Interval For Population Mean, Known* 9.2 Confidence Interval For Population Mean, Unknown* 9.3 Confidence Interval For Population Proportion* 9.4 Sample Size Determination Hypothesis Tests: One Sample 10.1 Hypothesis Test For Population Mean, Known* 10.2 Hypothesis Test For Population Mean, Unknown* Page 21 26 27 29 30 32 40 45 46 65 75 77 81 88 122 180 185 191 195 219 220 230 231 233 259 278 285 289 296 Computer Solutions Printouts and Instructions for Excel and Minitab Page 10.3 Hypothesis Test For Population Proportion* 340 10.4 The Power Curve For A Hypothesis Test 349 Hypothesis Tests: Comparing Two Samples 11.1 Pooled-Variances t-Test for (1 Ϫ 2), Population Variances Unknown but Assumed Equal 11.2 Unequal-Variances t-Test for (1 Ϫ 2), Population Variances Unknown and Not Equal 11.3 The z-Test for (1 Ϫ 2) 11.4 Comparing the Means of Dependent Samples 11.5 The z-Test for Comparing Two Sample Proportions* 11.6 Testing for the Equality of Population Variances Analysis of Variance 12.1 One-Way Analysis of Variance 12.2 Randomized Block Analysis of Variance 12.3 Two-Way Analysis of Variance Chi-Square Applications 13.1 Chi-Square Test for Goodness of Fit 13.2 Chi-Square Goodness-of-Fit Test for Normality* 13.3 Chi-Square Test for Independence of Variables* 13.4 Chi-Square Test Comparing Proportions From Independent Samples* 13.5 Confidence Interval for a Population Variance 13.6 Hypothesis Test for a Population Variance Nonparametric Methods 14.1 Wilcoxon Signed Rank Test for One Sample* 14.2 Wilcoxon Signed Rank Test for Comparing Paired Samples* 14.3 Wilcoxon Rank Sum Test for Two Independent Samples* 14.4 Kruskal-Wallis Test for Comparing More Than Two Independent Samples* 14.5 Friedman Test for the Randomized Block Design* 14.6 Sign Test for Comparing Paired Samples* 14.7 Runs Test for Randomness 14.8 Kolmogorov-Smirnov Test for Normality 14.9 Spearman Coefficient of Rank Correlation* 368 374 380 386 391 397 422 436 451 473 475 481 486 492 493 510 513 518 522 527 532 536 539 541 324 333 Simple Linear Regression 15.1 Simple Linear Regression 556 Computer Solutions Printouts and Instructions for Excel and Minitab 15.2 Interval Estimation in Simple Linear Regression* 15.3 Coefficient of Correlation 15.4 Residual Analysis Multiple Regression 16.1 Multiple Regression 16.2 Interval Estimation in Multiple Regression* 16.3 Residual Analysis in Multiple Regression Model Building 17.1 Fitting a Polynomial Regression Equation, One Predictor Variable 17.2 Fitting a Polynomial Regression Equation, Two Predictor Variables 17.3 Multiple Regression With Qualitative Predictor Variables 17.4 Transformation of the Multiplicative Model Page 563 568 578 605 612 626 648 655 659 663 Computer Solutions Printouts and Instructions for Excel and Minitab 17.5 The Correlation Matrix 17.6 Stepwise Regression* Page 666 669 Models for Time Series and Forecasting 18.1 Fitting a Linear or Quadratic Trend Equation 18.2 Centered Moving Average For Smoothing a Time Series 18.3 Excel Centered Moving Average Based On Even Number of Periods 18.4 Exponentially Smoothing a Time Series 18.5 Determining Seasonal Indexes* 18.6 Forecasting With Exponential Smoothing 18.7 Durbin-Watson Test for Autocorrelation* 18.8 Autoregressive Forecasting 694 697 704 708 718 721 Statistical Process Control 20.1 Mean Chart* 20.2 Range Chart* 20.3 p-Chart* 20.4 c-Chart 776 779 785 788 689 692 * Data Analysis Plus™ 5.0 add-in Seeing Statistics Applets Applet 10 11 12 13 14 15 16 17 18 19 20 21 Key Item Title Influence of a Single Observation on the Median Scatter Diagrams and Correlation Sampling Size and Shape of Normal Distribution Normal Distribution Areas Normal Approximation to Binomial Distribution Distribution of Means—Fair Dice Distribution of Means—Loaded Dice Confidence Interval Size Comparing the Normal and Student t Distributions Student t Distribution Areas z-Interval and Hypothesis Testing Statistical Power of a Test Distribution of Difference Between Sample Means F Distribution Interaction Graph in Two-Way ANOVA Chi-Square Distribution Regression: Point Estimate for y Point-Insertion Scatter Diagram and Correlation Regression Error Components Mean Control Chart Text Section 3.2 3.6 4.6 7.2 7.3 7.4 8.3 8.3 9.4 9.5 9.5 10.4 10.7 11.4 12.3 12.5 13.2 15.2 15.4 15.4 20.7 Applet Page 99 100 132 240 241 242 267 268 307 308 308 359 360 408 462 463 502 596 597 598 797 Location Computer setup and notes Follows preface t-table Precedes rear cover z-table Inside rear cover Other printed tables Appendix A Selected odd answers Appendix B Seeing Statistics applets, Thorndike video units, case and exercise data sets, On CD accompanying text Excel worksheet templates, and Data Analysis PlusTM 5.0 Excel add-in software with accompanying workbooks, including Test Statistics.xls and Estimators.xls Chapter self-tests and additional support http://www.thomsonedu.com/bstatistics/weiers Introduction to Business Statistics Sixth Edition Ronald M Weiers Eberly College of Business and Information Technology Indiana University of Pennsylvania WITH BUSINESS CASES BY J Brian Gray University of Alabama Lawrence H Peters Texas Christian University Australia • Brazil • Canada • Mexico • Singapore • Spain • United Kingdom • United States Introduction to Business Statistics, Sixth Edition Ronald M Weiers VP/Editorial Director: Jack W Calhoun Manager of Editorial Media: John Barans Marketing Coordinator: Courtney Wolstoncroft VP/Editor-in-Chief: Alex von Rosenberg Technology Project Manager: John Rich Art Director: Stacy Jenkins Shirley Sr Acquisitions Editor: Charles McCormick Marketing Communications Manager: Libby Shipp Cover and Internal Designer: Craig Ramsdell, Ramsdell Design Developmental Editor: Michael Guendelsberger Editorial Assistant: Bryn Lathrop Sr Marketing Manager: Larry Qualls Content Project Manager: Tamborah Moore COPYRIGHT © 2008, 2005 Thomson South-Western, a part of The Thomson Corporation Thomson, the Star logo, and South-Western are trademarks used herein under license Printed in the United States of America 10 09 08 07 Student Edition: ISBN 13: 978-0-324-38143-6 ISBN 10: 0-324-38143-3 Instructor’s Edition: ISBN 13: 978-0-324-65057-0 ISBN 10: 0-324-65057-4 Sr Manufacturing Print Buyer: Diane Gibbons Production House: ICC Macmillan Inc Printer: RRD Willard Willard, OH Cover Images: Getty Images/Photodisc Photography Manager: John Hill Photo Researcher: Seidel Associates ALL RIGHTS RESERVED No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means— graphic, electronic, or mechanical, including photocopying, recording, taping, Web distribution or information storage and retrieval systems, or in any other manner—without the written permission of the publisher Library of Congress Control Number: 2006935967 For permission to use material from this text or product, submit a request online at http://www.thomsonrights.com Thomson Higher Education 5191 Natorp Boulevard Mason, OH 45040 USA For more information about our products, contact us at: Thomson Learning Academic Resource Center 1-800-423-0563 To Connor, Madeleine, Hugh, Christina, Aidan, Mitchell, Owen, and Mr Barney Jim This page intentionally left blank Brief Contents Part 1: Business Statistics: Introduction and Background A Preview of Business Statistics Visual Description of Data 15 Statistical Description of Data 57 Data Collection and Sampling Methods 101 Part 2: Probability Probability: Review of Basic Concepts 133 Discrete Probability Distributions 167 Continuous Probability Distributions 205 Part 3: Sampling Distributions and Estimation Sampling Distributions 243 Estimation from Sample Data 269 Part 4: Hypothesis Testing 10 11 12 13 14 Hypothesis Tests Involving a Sample Mean or Proportion 309 Hypothesis Tests Involving Two Sample Means or Proportions 361 Analysis of Variance Tests 409 Chi-Square Applications 465 Nonparametric Methods 503 Part 5: Regression, Model Building, and Time Series 15 16 17 18 Simple Linear Regression and Correlation 549 Multiple Regression and Correlation 599 Model Building 643 Models for Time Series and Forecasting 685 Part 6: Special Topics 19 Decision Theory 735 20 Total Quality Management 755 21 Ethics in Statistical Analysis and Reporting (CD chapter) Appendices A Statistical Tables 799 B Selected Answers 835 Index/Glossary 839 v This page intentionally left blank Contents PART 1: BUSINESS STATISTICS: INTRODUCTION AND BACKGROUND Chapter 1: A Preview of Business Statistics 1.1 Introduction 1.2 Statistics: Yesterday and Today 1.3 Descriptive versus Inferential Statistics 1.4 Types of Variables and Scales of Measurement 1.5 Statistics in Business Decisions 11 1.6 Business Statistics: Tools Versus Tricks 11 1.7 Summary 12 Chapter 2: Visual Description of Data 15 2.1 Introduction 16 2.2 The Frequency Distribution and the Histogram 16 2.3 The Stem-and-Leaf Display and the Dotplot 24 2.4 Other Methods for Visual Representation of the Data 28 2.5 The Scatter Diagram 37 2.6 Tabulation, Contingency Tables, and the Excel PivotTable Wizard 43 2.7 Summary 48 Integrated Case: Thorndike Sports Equipment (Meet the Thorndikes: See Video Unit One.) 53 Integrated Case: Springdale Shopping Survey 54 Chapter 3: Statistical Description of Data 57 3.1 Introduction 58 3.2 Statistical Description: Measures of Central Tendency 59 3.3 Statistical Description: Measures of Dispersion 67 3.4 Additional Dispersion Topics 77 3.5 Descriptive Statistics from Grouped Data 83 3.6 Statistical Measures of Association 86 3.7 Summary 90 Integrated Case: Thorndike Sports Equipment 96 Integrated Case: Springdale Shopping Survey 97 Business Case: Baldwin Computer Sales (A) 97 vii statistics statistics in in action action 9.1 9.1 Sampling Error in Survey Research When survey results are published, they are sometimes accompanied by an explanation of how the survey was conducted and how much sampling error could have been present Persons who have had a statistics course need only know the size of the sample and the sample proportion or percentage for a given question For the general public, however, the explanation of survey methods and sampling error needs a bit of rephrasing The following statement accompanied the results of a survey commissioned by the Associated Press this description For 95% confidence, z ϭ 1.96 Since some questions may have a population proportion of 0.5 (the most conservative value to use when determining the required sample size), this is used in the following calculations: nϭ The first sentence in this description briefly describes the size and nature of the sample The second sentence describes the confidence level as 19͞20, or 95%, and the sampling error as plus or minus percentage points Using the techniques of this chapter, we can verify the calculations in where p ϭ estimated population proportion and eϭ How Poll Was Conducted The Associated Press poll on taxes was taken Feb 14–20 using a random sample of 1009 adult Americans No more than one time in 20 should chance variations in the sample cause the results to vary by more than percentage points from the answers that would be obtained if all Americans were polled z 2p(1 Ϫ p) e2 ϭ ͙ ͙ z2p(1 Ϫ p) n (1.96)2(0.5)(1 Ϫ 0.5) ϭ 0.0309 1009 As these calculations indicate, the Associated Press rounded down to the nearest full percentage point (from 3.09 to 3.0) in its published explanation of the sampling error This is not too unusual, since the general public would probably have enough to digest in the description quoted above without having to deal with decimal fractions Source: “How Poll Was Conducted,’’ in Howard Goldberg, “Most Not Ready to Scrap Tax System,” Indiana Gazette, February 26, 1996, p example Sample Size, Estimating a Proportion A tourist agency researcher would like to determine the proportion of U.S adults who have ever vacationed in Mexico and wishes to be 95% confident that the sampling error will be no more than 0.03 (3 percentage points) SOLUTION Assuming the Researcher Has No Idea Regarding the Actual Value of the Population Proportion, What Sample Size Is Necessary to Have 95% Confidence That the Sample Proportion Will Be within 0.03 (3 Percentage Points) of the Actual Population Proportion? For the 95% level of confidence, the z value will be 1.96 The maximum acceptable error is e ϭ 0.03 Not wishing to make an estimate, the researcher will use p ϭ 0.5 in calculating the necessary sample size: nϭ z2p(1 Ϫ p) 1.962(0.5)(1 Ϫ 0.5) ϭ ϭ 1067.1 persons, rounded up to 1068 e2 0.032 Chapter 9: Estimation from Sample Data 295 If the Researcher Believes the Population Proportion Is No More Than 0.3, and Uses p ؍0.3 as the Estimate, What Sample Size Will Be Necessary? Other factors are unchanged, so z remains 1.96 and e is still specified as 0.03 However, the p(1 Ϫ p) term in the numerator will be reduced due to the assumption that the population proportion is no more than 0.3 The required sample size will now be nϭ z2p(1 Ϫ p) 1.962(0.3)(1 Ϫ 0.3) ϭ ϭ 896.4 persons, rounded up to 897 e2 0.032 As in determining the necessary size for estimating a population mean, lower values of e lead to greatly increased sample sizes For example, if the researcher estimated the population proportion as being no more than 0.3, but specified a maximum likely error of 0.01 instead of 0.03, he would have to include nine times as many people in the sample (8068 instead of 897) Computer Solutions 9.4 (page 296) shows how we can use Excel to determine the necessary sample size for estimating a population mean or proportion With these procedures, it is very easy to examine “what-if” scenarios and instantly see how changes in confidence level or specified maximum likely error will affect the required sample size exercises 9.58 “If we want to cut the maximum likely error in half, we’ll have to double the sample size.” Is this statement correct? Why or why not? 9.59 In determining the necessary sample size in making an interval estimate for a population mean, it is necessary to first make an estimate of the population standard deviation On what bases might such an estimate be made? 9.60 From past experience, a package-filling machine has been found to have a process standard deviation of 0.65 ounces of product weight A simple random sample is to be selected from the machine’s output for the purpose of determining the average weight of product being packed by the machine For 95% confidence that the sample mean will not differ from the actual population mean by more than 0.1 ounces, what sample size is required? 9.61 Based on a pilot study, the population standard deviation of scores for U.S high school graduates taking a new version of an aptitude test has been estimated as 3.7 points If a larger study is to be undertaken, how large a simple random sample will be necessary to have 99% confidence that the sample mean will not differ from the actual population mean by more than 1.0 points? 9.62 A consumer agency has retained an independent testing firm to examine a television manufacturer’s claim that its 25-inch console model consumes just 110 watts of electricity Based on a preliminary study, the population standard deviation has been estimated as 11.2 watts for these sets In undertaking a larger study, and using a simple random sample, how many sets must be tested for the firm to be 95% confident that its sample mean does not differ from the actual population mean by more than 3.0 watts? 9.63 A national political candidate has commissioned a study to determine the percentage of registered voters who intend to vote for him in the upcoming election To have 95% confidence that the sample percentage will be within percentage points of the actual population percentage, how large a simple random sample is required? 9.64 Suppose that Nabisco would like to determine, with 95% confidence and a maximum likely error of 0.03, the proportion of first graders in Pennsylvania who had Nabisco’s Spoon-Size Shredded Wheat for breakfast at least once last week In determining the necessary size of a simple random sample for this purpose: a Use 0.5 as your estimate of the population proportion b Do you think the population proportion could really be as high as 0.5? If not, repeat part (a) using an estimated proportion that you think would be more likely to be true What effect does your use of this estimate have on the sample size? 296 Part 3: Sampling Distributions and Estimation computer solutions 9.4 Sample Size Determination These procedures determine the necessary sample size for estimating a population mean or a population proportion EXCEL 10 11 A B C D Sample size required for estimating a population mean: Estimate for sigma: Maximum likely error, e: 400.00 50.00 Confidence level desired: alpha = (1 - conf level desired): The corresponding z value is: 0.95 0.05 1.960 The required sample size is n = 245.9 Sample size for estimating a population mean, using Excel worksheet template TMNFORMU.XLS that accompanies the text To determine the necessary sample size for estimating a population mean within $50 and with 95% confidence, assuming a population standard deviation of $400: Open Excel worksheet TMNFORMU.XLS Enter the estimated sigma (400), the maximum likely error (50), and the specified confidence level as a decimal fraction (0.95) The required sample size (in cell D11) should then be rounded up to the nearest integer (246) This procedure is also described within the worksheet template Caution Do not save any changes when exiting Excel Sample size for estimating a population proportion, using Excel worksheet template TMNFORPI.XLS that accompanies the text To determine the necessary sample size for estimating a population proportion within 0.03 (3 percentage points) and with 95% confidence: Open Excel worksheet TMNFORPI.XLS Enter the estimate for pi (0.50), the maximum likely error (0.03), and the specified confidence level as a decimal fraction (0.95) The required sample size (in cell D11) should then be rounded up to the nearest integer (1068) (Note: If you have knowledge about the population and can estimate pi as either less than or greater than 0.50, use your estimate and the necessary sample size will be smaller Otherwise, be conservative and use 0.50 as your estimate.) This procedure is also described within the worksheet template Caution Do not save any changes when exiting Excel 9.65 The Chevrolet dealers of a large county are conduct- ing a study to determine the proportion of car owners in the county who are considering the purchase of a new car within the next year If the population proportion is believed to be no more than 0.15, how many owners must be included in a simple random sample if the dealers want to be 90% confident that the maximum likely error will be no more than 0.02? 9.66 In Exercise 9.65, suppose that (unknown to the dealers) the actual population proportion is really 0.35 If they use their estimated value ( Յ 0.15) in determining the sample size and then conduct the study, will their maximum likely error be greater than, equal to, or less than 0.02? Why? 9.67 In reporting the results of their survey of a simple random sample of U.S registered voters, pollsters claim 95% confidence that their sampling error is no more than percentage points Given this information only, what sample size was used? Chapter 9: Estimation from Sample Data 297 9.8 WHEN THE POPULATION IS FINITE Whenever sampling is without replacement and from a finite population, it may be necessary to modify slightly the techniques for confidence-interval estimation and sample size determination in the preceding sections As in Chapter 8, the general idea is to reduce the value of the standard error of the estimate for the sampling distribution of the mean or proportion As a rule of thumb, the methods in this section should be applied whenever the sample size (n) is at least 5% as large as the population When n Ͻ 0.05N, there will be very little difference in the results Confidence-Interval Estimation Whether we are dealing with interval estimation for a population mean () or a population proportion (), the confidence intervals will be similar to those in Figure 9.2 The only difference is that the “Ϯ” term will be multiplied by the “finite population correction factor” shown in Table 9.3 As in Chapter 8, this correction depends on the sample size (n) and the population size (N) As an example of how this works, we’ll consider a situation such as that in Section 9.5, where a confidence interval is to be constructed for the population mean, and the sample standard deviation (s) is used as an estimate of the population standard deviation () In this case, however, the sample will be relatively large compared to the size of the population example Interval Estimates, Finite Population According to the Bureau of the Census, the population of Kent County, Texas, is 812 persons.2 For purposes of our example, assume that a researcher has interviewed a simple random sample of 400 persons and found that their average TABLE 9.3 Confidence Interval Estimate for the Population Mean, Known Infinite Population Finite Population xϮz xϮz ͙n ( ͙ ) ͙n NϪn NϪ1 ؒ Confidence Interval Estimate for the Population Mean, Unknown Infinite Population xϮt Finite Population s xϮt ͙n ( ͙ ) s ͙n ؒ NϪn NϪ1 Confidence Interval Estimate for the Population Proportion Infinite Population ͙ pϮz 2Source: p(1 Ϫ p) n The World Almanac and Book of Facts 2003, p 459 Finite Population pϮz (͙ p(1 Ϫ p) ؒ n ͙ NϪn NϪ1 ) Summary of confidence interval formulas when sampling without replacement from a finite population As a rule of thumb, they should be applied whenever the sample is at least 5% as large as the population The formulas and terms are similar to those in Figure 9.2 but include a “finite population correction factor,” the value of which depends on the relative sizes of the sample (n) and population (N) 298 Part 3: Sampling Distributions and Estimation number of years of formal education is x ϭ 11.5 years, with a standard deviation of s ϭ 4.3 years SOLUTION Considering That the Population Is Finite and n Ն 0.05N, What Is the 95% Confidence Interval for the Population Mean? Since the number of degrees of freedom (df ϭ n Ϫ ϭ 400 Ϫ 1, or 399) exceeds the limits of our t distribution table, the t distribution and normal distribution can be considered to be practically identical, and we can use the infinity row of the t table The appropriate column in this table will be 0.025 for a 95% confidence interval, and the entry in the infinity row of this column is a t value of 1.96 Since s is being used to estimate , and the sample is more than 5% as large as the population, we will use the “ unknown” formula of the finite population expressions in Table 9.3 The 95% confidence interval for the population mean can be determined as The finite population correction term: this will be smaller than 1.0 so the standard error will be less than if an infinite population were involved xϮt ( ( ͙ s ؒ ͙n NϪn NϪ1 ͙ ) ) 4.3 812 Ϫ 400 ؒ 812 Ϫ ͙400 ϭ 11.5 Ϯ 1.96(0.215 ؒ 0.713) ϭ 11.5 Ϯ 0.300 or from 11.200 to 11.800 ϭ 11.5 Ϯ 1.96 As expected, the finite correction term (0.713) is less than 1.0 and leads to a 95% confidence interval that is narrower than if an infinite population had been assumed (Note: If the population had been considered infinite, the resulting interval would have been wider, with lower and upper limits of 11.079 and 11.921 years, respectively.) Sample Size Determination As in confidence-interval estimation, the rule of thumb is to change our sample size determination procedure slightly whenever we are sampling without replacement from a finite population and the sample is likely to be at least 5% as large as the population Although different in appearance, the following formulas are applied in the same way that we used their counterparts in Section 9.7 N O T E If you were to substitute an N value of infinity into each of the following equations, you would find that the right-hand term in the denominator of each would be eliminated, and the result would be an expression exactly the same as its counterpart in Section 9.7 Chapter 9: Estimation from Sample Data Required sample size for estimating the mean of a finite population: nϭ 2 e2 2 ϩ z2 N where n ϭ required sample size N ϭ population size z ϭ z value for which Ϯz corresponds to the desired level of confidence ϭ known (or, if necessary, estimated) value of the population standard deviation e ϭ maximum likely error that is acceptable Required sample size, estimating the proportion for a finite population: nϭ p(1 Ϫ p) e2 p(1 Ϫ p) ϩ z2 N where n ϭ required sample size N ϭ population size z ϭ z value for which Ϯz corresponds to the desired level of confidence p ϭ the estimated value of the population proportion (As a conservative strategy, use p ϭ 0.5 if you have no idea as to the actual value of .) e ϭ maximum likely error that is acceptable example Sample Size, Finite Population The Federal Aviation Administration (FAA) lists 8586 pilots holding commercial helicopter certificates.3 Suppose the FAA wishes to question a simple random sample of these individuals to find out what proportion are interested in switching jobs within the next years Assume the FAA wishes to have 95% confidence that the sample proportion is no more than 0.04 (i.e., percentage points) away from the true population proportion SOLUTION Considering That the Population Is Finite, What Sample Size Is Necessary to Have 95% Confidence That the Sample Proportion Will Not Differ from the Population Proportion by More Than 0.04? Since the actual population proportion who are interested in switching jobs has not been estimated, we will be conservative and use p ϭ 0.5 in deciding on the necessary sample size For the 95% confidence level, z will be 1.96 Applying the 3Source: General Aviation Manufacturers Association, General Aviation Statistical Database, 2005 Edition, p 37 299 300 Part 3: Sampling Distributions and Estimation finite population formula, with N ϭ 8586, the number of pilots who should be included in the sample is nϭ p(1 Ϫ p) 0.5(1 Ϫ 0.5) ϭ ϭ 561.0 e2 p(1 Ϫ p) 0.042 0.5(1 Ϫ 0.5) ϩ ϩ z2 N 1.962 8586 Had the population been infinite, the required sample size would have been calculated as in Section 9.7 This would have resulted in n ϭ 600.25, rounded up to 601 By recognizing that the population is finite, we are able to achieve the desired confidence level and maximum error with a sample size that includes only 561 pilots instead of 601 exercises 9.68 As a rule of thumb, under what conditions should the finite population correction be employed in determining confidence intervals and calculating required sample sizes? homes might exceed the Environmental Protection Agency’s recommended limit of 15 parts per billion of lead 9.69 Compared to situations where the population is either ulation of 800 In order to have 95% confidence that the sampling error in estimating is no more than 0.03, what sample size will be necessary? infinite or very large compared to the sample size, what effect will the finite population correction tend to have on a the width of a confidence interval? b the required size of a sample? 9.70 The personnel manager of a firm with 200 employ- ees has selected a simple random sample of 40 employees and examined their health-benefit claims over the past year The average amount claimed during the year was $260, with a standard deviation of $80 Construct and interpret the 95% confidence interval for the population mean Was it necessary to make any assumptions about the shape of the population distribution? Explain 9.71 Of 1200 undergraduates enrolled at a univer- sity, a simple random sample of 600 have been surveyed to measure student support for a $5 activities fee increase to help fund women’s intercollegiate athletics at the NCAA division 1A level Of those who were polled, 55% supported the fee increase Construct and interpret the 95% and 99% confidence intervals for the population proportion Based on your results, comment on the possibility that the fee increase might lose when it is voted on at next week’s university-wide student referendum 9.72 A local environmental agency has selected a simple random sample of 16 homes to be tested for tap-water lead Concentrations of lead were found to have a mean of 12 parts per billion and a standard deviation of parts per billion Considering that the homes were selected from a community in which there are 100 homes, construct and interpret the 95% confidence interval for the population mean Based on your results, comment on the possibility that the average lead concentration in this community’s 9.73 A simple random sample is to be drawn from a pop- 9.74 A simple random sample is to be drawn from a pop- ulation of 2000 The population standard deviation has been estimated as being 40 grams In order to have 99% confidence that the sampling error in estimating is no more than grams, what sample size will be necessary? 9.75 There are 100 members in the United States Senate A political scientist wants to estimate, with 95% confidence and within percentage points, the percentage who own stock in foreign companies How many senators should be interviewed? Explain any assumptions you used in obtaining your recommended sample size 9.76 A transportation company operates 200 trucks and would like to use a hidden speed monitor device to record the maximum speed at which a truck is operated during the period that the device is installed The trucks are driven primarily on interstate highways, and the company wants to estimate the average maximum speed for its fleet with 90% confidence and within miles per hour Using (and explaining) your own estimate for the population standard deviation, determine the number of trucks on which the company should install the hidden speed-recording device 9.77 A research firm supports a consumer panel of 2000 households that keep written diaries of their weekly grocery expenditures The firm would like to estimate, with 95% confidence and within percentage points, the percentage of its panel households who would be interested in providing more extensive information in return for an extra $50 per week remuneration How many of the Chapter 9: Estimation from Sample Data 301 households should be surveyed? Explain any assumptions you used in obtaining your recommended sample size (and explain) your own estimate for the population standard deviation 9.78 A university official wants to estimate, with 99% 9.79 A quality-management supervisor believes that no confidence and within $2, the average amount that members of fraternities and sororities spend at local restaurants during the first week of the semester If the total fraternity/sorority membership is 300 people, how many members should be included in the sample? Use more than 5% of the items in a recent shipment of 2000 are defective If she wishes to determine, within percentage point and with 99% confidence, the percentage of defective items in the shipment, how large a simple random sample would be necessary? SUMMARY • Inferential statistics: point and interval estimates for a population parameter Chapter examined the sampling distribution of a sample mean or a sample proportion from a known population In this chapter, the emphasis has been on the estimation of an unknown population mean () or proportion () on the basis of sample statistics Point estimates involve using the sample mean (x) or proportion (p) as the single best estimate of the value of the population mean or proportion Interval estimates involve a range of values that may contain the actual value of the population parameter When interval estimates are associated with a degree of certainty that they really include the true population parameter, they are referred to as confidence intervals a confidence interval for a population mean or a population • Constructing proportion The procedure appropriate to constructing an interval estimate for the population mean depends largely on whether the population standard deviation is known Figure 9.2 summarizes these procedures and their underlying assumptions Although the t-interval is often associated with interval estimates based on small samples, it is appropriate for larger samples as well Using computer statistical packages, we can easily and routinely apply the t distribution for interval estimates of the mean whenever is unknown, even for very large sample sizes A trade-off exists between the degree of confidence that an interval contains the population parameter and the width of the interval itself The more certain we wish to be that the interval estimate contains the parameter, the wider the interval will have to be • Sample size determination Accuracy, or sampling error, is equal to one-half of the confidence interval width The process of sample size determination anticipates the width of the eventual confidence interval, then determines the required sample size that will limit the maximum likely sampling error to an acceptable amount • When the sample is large compared to the population As in Chapter 8, when sampling is without replacement from a finite population, it is appropriate to use a finite population correction factor whenever the sample is at least 5% of the size of the population Such corrections are presented for both interval estimation and sample size determination techniques within the chapter • Computer-generated confidence intervals Most computer statistical packages are able to construct confidence interval estimates of the types discussed in the chapter Examples of Excel and Minitab outputs are provided for a number of chapter examples in which such confidence intervals were developed 9.9 302 Part 3: Sampling Distributions and Estimation equations Confidence Interval Limits for the Population Mean, Known xϮz ͙n where x ϭ sample mean ϭ population standard deviation n ϭ sample size z ϭ z value for desired confidence level ͙͞n ϭ standard error of the sampling distribution of the mean (Assumes that either (1) the underlying population is normally distributed or (2) the sample size is n Ն 30.) Confidence Interval Limits for the Population Mean, Unknown s where x ϭ sample mean xϮt ͙n s ϭ sample standard deviation n ϭ sample size t ϭ t value corresponding to the level of confidence desired, with df ϭ n Ϫ s͙͞n ϭ estimated standard error of the sampling distribution of the mean (If n Ͻ 30, this requires the assumption that the underlying population is approximately normally distributed.) Confidence Interval Limits for the Population Proportion ͙ pϮz p(1 Ϫ p) n where p ϭ sample proportion ϭ ͙ number of successes number of trials n ϭ sample size z ϭ z value corresponding to desired level of confidence (e.g., z ϭ 1.96 for 95% confidence) p(1 Ϫ p) ϭ estimated standard error of the sampling n distribution of the proportion Required Sample Size for Estimating a Population Mean nϭ z2 ؒ 2 e2 where n ϭ required sample size z ϭ z value for which Ϯz corresponds to the desired level of confidence ϭ known (or, if necessary, estimated) value of the population standard deviation e ϭ maximum likely error that is acceptable Required Sample Size for Estimating a Population Proportion nϭ z2p(1 Ϫ p) e2 where n ϭ required sample size z ϭ z value for desired level of confidence p ϭ estimated value of the population proportion (if not estimated, use p ϭ 0.5) e ϭ maximum likely error that is acceptable Chapter 9: Estimation from Sample Data 303 Confidence Interval Estimates When the Population Is Finite • For the population mean, known: xϮz • ( ͙ ؒ ͙n NϪn NϪ1 where n ϭ sample size N ϭ population size For the population mean, unknown: xϮt • ) ( ͙ s ؒ ͙n NϪn NϪ1 ) For the population proportion: pϮz (͙ p(1 Ϫ p) ؒ n ͙ NϪn NϪ1 ) Required Sample Size for Estimating the Mean of a Finite Population nϭ 2 2 ϩ z N e2 where n ϭ required sample size N ϭ population size z ϭ z value for desired level of confidence ϭ known (or estimated) value of the population standard deviation e ϭ maximum likely error that is acceptable Required Sample Size for Estimating the Proportion for a Finite Population nϭ p(1 Ϫ p) p(1 Ϫ p) ϩ z2 N e2 where n ϭ required sample size N ϭ population size z ϭ z value for desired level of confidence p ϭ the estimated population proportion (if not estimated, use p ϭ 0.5) e ϭ maximum likely error that is acceptable chapter exercises 9.80 In a destructive test of product quality, a briefcase manufacturer places each of a simple random sample of the day’s production in a viselike device and measures how many pounds it takes to crush the case From past experience, the standard deviation has been found to be 21.5 pounds For 35 cases randomly selected from today’s production, the average breaking strength was 341.0 pounds Construct and interpret the 99% confidence interval for the mean breaking strength of the briefcases produced today 9.81 Working independently, each of two researchers has devised a sampling plan to be carried out for the purpose of constructing a 90% confidence interval for the mean of a certain population What is the probability that neither of their confidence intervals will include the population mean? 9.82 The accompanying data represent one-way commuting times (minutes) for a simple random sample of 15 persons who work at a large assembly plant The data are also in file XR09082 Assuming an approximately normal distribution of commuting times for those who work at the plant, construct and interpret the 90% and 95% confidence intervals for the mean 21.7 39.0 30.0 26.8 28.0 33.6 33.1 24.7 33.3 27.9 28.4 34.1 23.5 28.9 35.1 304 Part 3: Sampling Distributions and Estimation 9.83 A torque wrench used in the final assembly of cylinder heads has a process standard deviation of 5.0 lb-ft The engineers have specified that a process average of 135 lb-ft is desirable For a simple random sample of 30 nuts that the machine has recently tightened, the sample mean is 137.0 lb-ft Construct and interpret the 95% confidence interval for the current process mean Discuss the possibility that the machine may be in need of adjustment to correct the process mean 9.84 There are approximately 109 million television households in the United States A ratings service would like to know, within percentage points and with 95% confidence, the percentage of these households who tune in to the first episode of a network miniseries How many television households must be included in the sample? SOURCE: The World Almanac and Book of Facts 2006, p 278 9.85 In Exercise 9.84, a small-scale preliminary survey has indicated that no more than 20% of the television households will tune in to the first episode of the miniseries Given this information, how large must the sample be? 9.86 In a survey of 500 U.S adults, 45% of them said that lounging at the beach was their “dream vacation.” Assuming this to be a simple random sample of U.S adults, construct and interpret the 95% and 99% confidence intervals for the proportion of U.S adults who consider lounging at the beach to be their dream vacation 9.87 For the following simple random sample of household incomes (thousands of dollars) from a large county, construct and interpret the 90% and 95% confidence intervals for the population mean The data are also in file XR09087 58.3 50.0 58.1 33.5 51.1 38.1 42.3 60.4 55.8 46.2 40.4 52.5 51.3 47.5 48.5 59.3 40.9 37.1 39.1 43.6 55.3 42.3 48.2 42.8 61.1 34.7 35.5 52.9 44.7 51.5 9.88 For a new process with which the production personnel have little experience, neither the standard deviation nor the mean of the process is known Twenty different simple random samples, each with n ϭ 50, are to be drawn from the process, and a 90% confidence interval for the mean is to be constructed for each sample What is the probability that at least of the confidence intervals will not contain the population mean? 9.89 There were 904 new Subway Restaurants franchises opened during 2002 Suppose that Subway wished to survey a simple random sample of the new franchisees to find out what percentage of them were totally pleased with their relationship with the company If Subway wanted to have 90% confidence in being within percentage points of the population percentage who are pleased, how many of the new franchisees would have to be included in the sample? SOURCE: Subway.com June 13, 2003 9.90 In Exercise 9.89, suppose Subway has carried out the study, using the sample size determined in that exercise, and 27.5% of the franchisees say they are pleased with their relationship with Subway Construct and interpret the 95% confidence interval for the population percentage 9.91 A research firm wants to be 90% confident that a population percentage has been estimated to within percentage points The research manager calculates the necessary sample size with 0.5 as his estimate of the population proportion A new business school graduate who has just joined the firm questions the research manager further, and they agree that the population proportion is no more than 0.3 If interviews cost $10 each, how much money has the new graduate just saved the company? 9.92 The activities director of a large university has surveyed a simple random sample of 100 students for the purpose of determining approximately how many students to expect at next month’s awards ceremony to be held in the gymnasium Forty of the students said they plan to attend What are the upper and lower 95% confidence limits for the number of the university’s 10,000 students who plan to attend the awards ceremony? 9.93 A research firm has found that 39% of U.S adults in the over-$75,000 income category work at least 51 hours per week Assuming this was a simple random sample of 500 adults in this income group, construct and interpret the 95% and 99% confidence intervals for the proportion who work at least 51 hours per week For each of the confidence intervals, identify and explain the maximum likely error in the study 9.94 For a process having a known standard deviation, a simple random sample of 35 items is selected If the width of the 95% confidence interval is identified as y, express the width of the 99% confidence interval as a multiple of y 9.95 The makers of Count Chocula breakfast cereal would like to determine, within percentage points and with 99% confidence, the percentage of U.S senior citizens who have Count Chocula for breakfast at least once a week What sample size would you recommend? 9.96 In a work-sampling study, an industrial engineer has observed the activities of a clerical worker on 121 randomly selected times during a workweek On 32 of these occasions, the employee was talking on the telephone For an 8-hour day, what are the upper and lower 95% confidence limits for the number of minutes this employee talks on the phone? 9.97 A researcher would like to determine, within percentage points and with 90% confidence, the percentage of Americans who have a certain characteristic If she feels certain that the percentage is somewhere between 20% and 40%, how many persons should be included in the sample? Chapter 9: Estimation from Sample Data 9.98 In a survey of 1320 executives who oversee corporate data systems, 24% said they had experienced losses caused by computer viruses during the past year Assuming the executives were a simple random sample of all such executives, construct and interpret the 90% confidence interval for the population proportion who were monetarily harmed by computer viruses that year 9.99 An airline would like to determine, within percentage points and with 95% confidence, the percentage of next month’s customers who judge the courtesy of its employees as being “very good to excellent.” What sample size would you recommend? 9.100 A consultant conducts a pilot study to estimate a population standard deviation, then determines how large a simple random sample will be necessary to have a given level of confidence that the difference between x and will be within the maximum error specified by her client The necessary sample size has been calculated as n ϭ 100 If the client suddenly decides that the maximum error must be only one-fourth that originally specified, what sample size will now be necessary? 9.101 There are 1733 machinery rebuilding and repairing companies in the United States A tool manufacturer wishes to survey a simple random sample of these firms to find out what proportion of them are interested in a new tool design If the tool manufacturer would like to be 95% confident that the sample proportion is within 0.01 of the actual population proportion, how many machinery rebuilding and repairing companies should be included in the sample? SOURCE: American Business Information, Sales Leads & Mailing Lists, August 1999, p 22 9.102 In Exercise 9.101, suppose the tool manufacturer has carried out the study, using the sample size determined in that exercise, and 39.0% of the machinery rebuilding and repairing companies are interested in the new tool design Construct and interpret the 95% confidence interval for the population percentage 9.103 The Colgate-Palmolive Company has 37,700 employees If the company wishes to estimate, within percentage points and with 99% confidence, the percentage of employees who are interested in participating in a new stock option benefits program, how large a simple random sample will be necessary? SOURCE: Colgate-Palmolive Company, 2002 Annual Report, p 9.104 To gain information about competitors’ products, companies sometimes employ “reverse engineering,” which consists of buying the competitor’s product, then taking it apart and examining the parts in great detail Engaging in this practice, a bicycle manufacturer intends to buy two or more of a leading competitor’s mountain bikes and measure the tensile strength of the crossbar portion of the frame Past experience has shown these strengths to be approximately normally distributed with a standard deviation of 20 pounds per square inch (psi) 305 If the bike purchaser wants to have 90% confidence that the sampling error will be no more than psi, how many of the competitor’s mountain bikes should be purchased for destructive testing? 9.105 A survey of business travelers found that 40% of those surveyed utilize hotel exercise facilities during their stay Under the assumption that a simple random sample of 1000 business travelers were surveyed, construct and interpret the 90% and 95% confidence intervals for the proportion of business travelers who use their hotel’s exercise facilities 9.106 A researcher, believing to be no more than 0.40, calculates the necessary sample size for the confidence level and maximum likely error he has specified Upon completing the study, he finds the sample proportion to be 0.32 Is the maximum likely error greater than, equal to, or less than that originally specified? Explain 9.107 A truck loaded with 8000 electronic circuit boards has just pulled into a firm’s receiving dock The supplier claims that no more than 3% of the boards fall outside the most rigid level of industry performance specifications In a simple random sample of 300 boards from this shipment, 12 fall outside these specifications Construct the 95% confidence interval for the percentage of all boards in this shipment that fall outside the specifications, then comment on whether the supplier’s claim would appear to be correct 9.108 A researcher has estimated that U.S college students spend an average of 17.2 hours per week on the Internet Assuming a simple random sample of 500 college students and a sample standard deviation of 1.4 hours per week, construct and interpret the 99% confidence interval for the population mean / data set / Note: Exercises 9.109–9.111 require a computer and statistical software 9.109 According to the National Restaurant Association, the average check for a sit-down dinner is $25 Such a finding could have been based on data like the 800 sample checks in file XR09109 Using the data in this file, construct and interpret the 95% confidence interval for the population mean SOURCE: “U.S Dining-Out Tab: $1B a Day,” USA Today, May 25, 2000, p 1D 9.110 For taxpayers having an adjusted gross income of $1 million or more, the Internal Revenue Service reports that the average deduction for gifts to charity was $144,700 Curious to see how his state compares, a legislator surveys a simple random sample of 200 taxpayers from his state who are in this gross income category, with the data as shown in file XR09110 Using the data in this file, construct and interpret the 90% confidence interval for the mean charitable-gifts deduction for all of the state’s taxpayers who are in the $1 million or more adjusted gross income category Is $144,700 within the 306 confidence interval? Given the answer to the preceding question, comment on whether the state’s taxpayers who are in this income group might not be typical of those in the nation as a whole in terms of their tax-deductible charitable contributions SOURCE: “Brilliant Deductions, Taxing Questions,” USA Today, March 3, 2000, p 3B 9.111 To avoid losing part of their federal highway fund allocation, state safety administrators must ensure that interstate speed limits are adequately enforced within their state In an upcoming test, federal researchers will be randomly selecting and clocking a very large sample of vehicles on a given section of the state’s portion of an Part 3: Sampling Distributions and Estimation interstate highway that has historically had a relatively high accident rate In anticipation of the upcoming study, state administrators randomly select and clock 100 vehicles along this route, obtaining the speeds shown in data file XR09111 Construct and interpret the 95% confidence interval for the population mean vehicle speed along this stretch of highway Based on this interval, comment on whether the mean speed for the population of vehicles using this part of the highway might be 70 mph, the cutoff above which federal highway funds become endangered integrated cases Thorndike Sports Equipment (Thorndike Video Unit Four) Seeing the fishing pole in his grandfather’s office, Ted Thorndike’s first thought is that old Luke is going to go fishing again and leave him to manage the store He is quite surprised to learn the fishing pole is actually an inspiration for a new series of ads that Luke has in mind The elder Thorndike explains, “Ted, this fishing pole is made of graphite, the same stuff that goes into our GrafPro racquetball racquets It’s so flexible and strong that it can be bent so the two ends actually touch each other They even show this in the ads.” Although Luke realizes that you can’t exactly the same thing with a racquetball racquet, he’d like to put some of his racquets into a horizontal mounting device, then see how much weight they’ll take before they break If the amount of weight is impressive enough, Luke plans to include this kind of test in the television advertisements he’s planning for the firm’s racquetball racquets However, he wants to be careful not to brag about the racquet being able to hold too much weight, since the firm could get into trouble with the government and other truth-in-advertising advocates He asks Ted to set up a test in which racquets are mounted horizontally, then the weight on the end is gradually increased until they break Based on the test results, a weight value would be selected such that the average racquet would almost certainly be able to withstand this amount Although accuracy is important, Ted has been instructed not to break more than 15 or 20 racquets in coming up with an average for all the racquets For 20 racquets subjected to this severe test, the weight (in pounds) at which each one failed was as follows The data are also in file THORN09 221 208 224 217 228 220 222 230 223 217 229 236 218 224 215 222 218 225 221 234 Ted believes it’s reasonable to assume the population of breaking strengths is approximately normally distributed Because of Luke’s concern about being able to support the advertising claim, he wants to be very conservative in estimating the population mean for these breaking strengths Ted needs some help in deciding how conservative he would like to be, and in coming up with a number that can be promoted in the ads Springdale Shopping Survey The case in Chapter listed 30 questions asked of 150 respondents in the community of Springdale The coding key for these responses was also provided in this earlier exercise The data are in file SHOPPING In this exercise, some of the estimation techniques presented in the chapter will be applied to the survey results You may assume that (continued) seeing statistics: applet Chapter 9: Estimation from Sample Data 307 Confidence Interval Size This applet allows us to construct and view z-intervals for the population mean by using the slider to specify the confidence level As in Figure 9.3, the sample mean is 1.400 inches, the sample size is 30, and the population standard deviation is known to be 0.053 inches Note that the confidence interval limits shown in the graph may sometimes differ slightly from those we would calculate using the pocket calculator and our standard normal distribution table This is because the applet is using more exact values for z than we are able to show within printed tables like the one in the text Applet Exercises 9.1 With the slider positioned so as to specify a 95% confidence interval for , what are the upper and lower confidence limits? 9.2 Move the slider so that the confidence interval is now 99% Describe how the increase in the confidence level has changed the width of the confidence interval 9.3 Move the slider so that the confidence interval is now 80% Describe how the decrease in the confidence level has changed the width of the confidence interval 9.4 Position the slider at its extreme left position, then gradually move it to the far right Describe how this movement changes the confidence level and the width of the confidence interval these respondents represent a simple random sample of all potential respondents within the community and that the population is large enough that application of the finite population correction would not make an appreciable difference in the results Managers associated with shopping areas like these find it useful to have point estimates regarding variables describing the characteristics and behaviors of their customers In addition, it is helpful for them to have some idea as to the likely accuracy of these estimates Therein lies the benefit of the techniques presented in this chapter and applied here Item C in the description of the data collection instrument lists variables 7, 8, and 9, which represent the respondent’s general attitude toward each of the three shopping areas Each of these variables has numerically equal distances between the possible responses, and for purposes of analysis they may be considered to be of the interval scale of measurement a Determine the point estimate, then construct the 95% confidence interval for 7 ϭ the average attitude toward Springdale Mall What is the maximum likely error in the point estimate of the population mean? b Repeat part (a) for 8 and 9, the average attitudes toward Downtown and West Mall, respectively Given the breakdown of responses for variable 26 (sex of respondent), determine the point estimate, then construct the 95% confidence interval for 26 ϭ the population proportion of males What is the maximum likely error in the point estimate of the population proportion? Given the breakdown of responses for variable 28 (marital status of respondent), determine the point estimate, then construct the 95% confidence interval for 28 ϭ the population proportion in the “single or other” category What is the maximum likely error in the point estimate of the population proportion? seeing statistics: applet 10 Comparing the Normal and Student t Distributions In this applet, we use a slider to change the number of degrees of freedom and shape for the Student t distribution and then observe how the resulting shape compares to that of the standard normal distribution The standard normal distribution is fixed and shown in red, and the Student t distribution is displayed in blue Applet Exercises 10.1 Move the slider so that df ϭ Describe the shape of the t distribution compared to that of the standard normal distribution 10.2 Move the slider downward so that df ϭ How has this decrease changed the shape of the t distribution? 10.3 Gradually move the slider upward so that df increases from to 10 Describe how the shape of the t distribution changes along the way 10.4 Position the slider so that df ϭ 2, then gradually move it upward until df ϭ 100 Describe how the shape of the t distribution changes along the way seeing statistics: applet 11 Student t Distribution Areas In this applet, we use a slider to change the number of degrees of freedom for the t distribution, and text boxes allow us to change the t value or the two-tail probability for a given df When changing a text-box entry, be sure the cursor is still within the box before pressing the enter or return key Applet Exercises 11.1 With the slider set so that df ϭ and the left text box containing t ϭ 3.25, what is the area beneath the curve between t ϭ Ϫ3.25 and t ϭ ϩ3.25? 11.2 Gradually move the slider upward until df ϭ 89 What effect does this have on the t value shown in the text box? 11.3 Position the slider so that df ϭ 2, then gradually move it upward until df ϭ 100 Describe how the value in the t text box and the shape of the t distribution change along the way 11.4 With the slider set so that df ϭ 9, enter 0.10 into the two-tail probability text box at the right What value of t now appears in the left text box? To what right-tail area does this correspond? Verify the value of t for df ϭ and this right-tail area by using the t table immediately preceding the back cover of the book