The use of probability models and statistical methods for analyzing data has become common practice in virtually all scientific disciplines. This book attempts to provide a comprehensive introduction to those models and methods most likely to be encountered and used by students in their careers in engineering and the natural sciences. Although the examples and exercises have been designed with scientists and engineers in mind, most of the methods covered are basic to statistical analyses in many other disciplines, so that students of business and the social sciences will also profit from reading the book.
SEVENTH EDITION Probability and Statistics for Engineering and the Sciences This page intentionally left blank SEVENTH EDITION Probability and Statistics for Engineering and the Sciences JAY L DEVORE California Polytechnic State University, San Luis Obispo Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Probability and Statistics for Engineering and the Sciences, Seventh Edition, Enhanced Edition Jay L Devore Acquisitions Editor: Carolyn Crockett Assistant Editor: Beth Gershman Editorial Assistant: Ashley Summers Technology Project Manager: Colin Blake © 2009, 2008, 2004 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 Marketing Manager: Joe Rogove Marketing Assistant: Jennifer Liang For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 Marketing Communications Manager: Jessica Perry For permission to use material from this text or product, submit all requests online at cengage.com/permissions Project Manager, Editorial Production: Jennifer Risden Further permissions questions can be e-mailed to permissionrequest@cengage.com Creative Director: Rob Hugel Art Director: Vernon Boes Library of Congress Control Number: 2006932557 Print Buyer: Becky Cross Student Edition: ISBN-13: 978-0-495-55744-9 ISBN-10: 0-495-55744-7 Permissions Editor: Roberta Broyer Production Service: Matrix Productions Text Designer: Diane Beasley Copy Editor: Chuck Cox Illustrator: Lori Heckelman/Graphic World; International Typesetting and Composition Cover Designer: Gopa & Ted2, Inc Cover Image: © Creatas/SuperStock Compositor: International Typesetting and Composition Brooks/Cole 10 Davis Drive Belmont, CA 94002-3098 USA 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 academic.cengage.com Purchase any of our products at your local college store or at our preferred online store www.ichapters.com Printed in Canada 12 11 10 09 08 To my wife, Carol: Your dedication to teaching is a continuing inspiration to me To my daughters, Allison and Teresa: The great pride I take in your accomplishments knows no bounds v This page intentionally left blank Contents Overview and Descriptive Statistics Introduction 1.1 1.2 1.3 1.4 Populations, Samples, and Processes Pictorial and Tabular Methods in Descriptive Statistics 10 Measures of Location 24 Measures of Variability 31 Supplementary Exercises 42 Bibliography 45 Probability Introduction 46 2.1 Sample Spaces and Events 47 2.2 Axioms, Interpretations, and Properties of Probability 51 2.3 Counting Techniques 59 2.4 Conditional Probability 67 2.5 Independence 76 Supplementary Exercises 82 Bibliography 85 Discrete Random Variables and Probability Distributions 3.1 3.2 3.3 3.4 3.5 3.6 Introduction 86 Random Variables 87 Probability Distributions for Discrete Random Variables 90 Expected Values 100 The Binomial Probability Distribution 108 Hypergeometric and Negative Binomial Distributions 116 The Poisson Probability Distribution 121 Supplementary Exercises 126 Bibliography 129 vii viii Contents Continuous Random Variables and Probability Distributions Introduction 130 4.1 Probability Density Functions 131 4.2 Cumulative Distribution Functions and Expected Values 136 4.3 The Normal Distribution 144 4.4 The Exponential and Gamma Distributions 157 4.5 Other Continuous Distributions 163 4.6 Probability Plots 170 Supplementary Exercises 179 Bibliography 183 Joint Probability Distributions and Random Samples Introduction 184 5.1 Jointly Distributed Random Variables 185 5.2 Expected Values, Covariance, and Correlation 196 5.3 Statistics and Their Distributions 202 5.4 The Distribution of the Sample Mean 213 5.5 The Distribution of a Linear Combination 219 Supplementary Exercises 224 Bibliography 226 Point Estimation Introduction 227 6.1 Some General Concepts of Point Estimation 228 6.2 Methods of Point Estimation 243 Supplementary Exercises 252 Bibliography 253 Statistical Intervals Based on a Single Sample Introduction 254 7.1 Basic Properties of Confidence Intervals 255 7.2 Large-Sample Confidence Intervals for a Population Mean and Proportion 263 Contents ix 7.3 Intervals Based on a Normal Population Distribution 270 7.4 Confidence Intervals for the Variance and Standard Deviation of a Normal Population 278 Supplementary Exercises 281 Bibliography 283 Tests of Hypotheses Based on a Single Sample Introduction 284 8.1 8.2 8.3 8.4 Hypotheses and Test Procedures 285 Tests About a Population Mean 294 Tests Concerning a Population Proportion 306 P-Values 311 8.5 Some Comments on Selecting a Test 318 Supplementary Exercises 321 Bibliography 324 Inferences Based on Two Samples 9.1 9.2 9.3 9.4 9.5 Introduction 325 z Tests and Confidence Intervals for a Difference Between Two Population Means 326 The Two-Sample t Test and Confidence Interval 336 Analysis of Paired Data 344 Inferences Concerning a Difference Between Population Proportions 353 Inferences Concerning Two Population Variances 360 Supplementary Exercises 364 Bibliography 368 10 The Analysis of Variance Introduction 369 10.1 Single-Factor ANOVA 370 10.2 Multiple Comparisons in ANOVA 379 10.3 More on Single-Factor ANOVA 385 Supplementary Exercises 395 Bibliography 396 Sample Exams * Point values for each problem are provided in brackets Exam 1-1 INSTRUCTIONS: Show all the work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [10؉10]* Six students took the same test and got 65, 91, 84, 79, 58, and 82 points, respectively a Compute the sample range, sample mean, and sample median of these data b Compute the sample variance and sample standard deviation [15؉5] Consider the following record of the daily temperatures (°F) in Seattle: 65 68 73 66 62 63 61 68 67 70 69 68 68 65 63 61 66 69 68 63 59 64 66 68 69 61 60 62 61 59 57 58 53 55 a Construct a stem-and-leaf display of the data, using the tenth digit as the stem value and the ones digit as the leaf value b What proportion of these days have a temperature in the 50s, 60s, and 70s, respectively? [20] Construct a histogram for the following data using four classes of equal width: 78.8 36.1 78.7 10.4 24.0 74.1 32.8 89.8 32.0 30.7 61.3 82.9 17.0 43.8 51.5 97.7 22.4 16.5 77.6 14.6 Mark the upper limits and lower limits of the class intervals on the horizontal axis of the histogram [5؉10] Jim surveyed his classmates’ reading preferences Eight students responded with the following choices: fiction science science fiction science fiction science science a What is the sample proportion of people who prefer reading fiction? b Suppose seven more questionnaires will be collected How many of these seven people must be in favor of fiction to give a 40% sample proportion of fiction readers for the entire sample? [15؉10] The printing server keeps track of the amount of printing requests submitted by each user A report for last month shows the number of pages printed for the users: 49 59 71 16 20 17 46 44 40 38 57 21 10 38 59 42 49 40 17 37 60 54 16 a Compute the following features of the data: (i) sample median, (ii) upper fourth, (iii) lower fourth, (iv) fourth spread fx b Construct a boxplot using the results of (a) Exam 1-2 INSTRUCTIONS: Show all the work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [10؉10] A set of one- or two-digit numbers are listed in the following stem-and-leaf display, where the stem values are the tenth digits and the leaf values are the ones digits: 111355 1223666689 56 11244778 a Identify the numbers in this stem-and-leaf display, and list them in ascending order What is the sample size of this data set? b Compute the sample range, sample mean, and sample median of these data [10؉10] A laboratory performed twenty tests on the flying distances of footballs filled with helium The footballs are launched by a machine and their flying distances (in yards) are recorded as follows: 25 16 25 14 23 29 25 26 22 26 12 28 28 31 22 29 23 26 35 23 a Using class boundaries 10, 17, 24, 32, and 38, compute the frequencies and relative frequencies of observing flying distances in these classes b Construct a relative frequency histogram using the results of part (a) [20] A biologist measures the body weights (in kg) of different animals and has the following records in his computer: ArcticFox 3.385, OwlMonkey 0.480, MountainBeaver 1.350, Cow 465.0, GreyWolf 36.33, Goat 27.66, RoeDeer 14.83, GuineaPig 1.04, Sheep 55, Chinchilla 0.425, Squirrel 0.101, Donkey 187.1, StarNoseMole 0.06, TreeHyrax 2.0, AsianElephant 2547.0, Horse 521.0, Cat 3.3, Galago 0.2, Genet 1.41 What proportion of the animals in this record have their Sample exams provided by Abram Kagan and Tinghui Yu of University of Maryland 725 726 Sample Exams body weights (a) less than 100 kg? (b) greater than 200 kg? (c) between 10 kg and 500 kg? [10؉10] The math department provides a free walk-in tutoring service to the students The dean needs to determine if they should hire more tutors this semester to match the increasing number of students Last Friday, the tutors marked down the number of students in the tutoring room every hour from 10:00 A.M to 4:00 P.M Here is a copy of the table they submitted to the dean: 10:00 13 11:00 16 12:00 23 13:00 20 14:00 11 15:00 22 16:00 19 a Construct a digidot plot for the preceding data, with the horizontal axis standing for time and the vertical axis listing the number of students in the tutoring room b Compute the sample mean and variance for the number of students in the tutoring room [15؉5] The size of twelve cathedrals in different cities are measured and their lengths (in feet) are 502 522 425 344 407 451 551 530 547 519 225 300 a Compute the following features of the data: (i) sample median, (ii) upper fourth, (iii) lower fourth, (iv) fourth spread fx b Construct a boxplot using the results of (a) Exam 1-3 INSTRUCTIONS: Show all the work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [15؉10] A laboratory performed fifteen tests comparing the flying distances of footballs filled with helium and those filled with air They launched the footballs using a common machine and logged the flying distances (in yards) as follows: Helium: 26 30 29 29 30 29 30 32 26 11 33 27 30 28 14 Air: 28 28 31 25 28 32 28 26 27 20 25 31 22 29 27 a Compute the sample means, variances, and standard deviations for these two groups of observations, respectively b Construct a comparative stem-and-leaf display [20] A phone service center keeps track of the number of incoming calls every day Here is a record from last month: 11 13 12 19 25 13 15 17 18 20 32 11 16 15 18 16 17 23 22 15 10 11 26 14 14 10 11 15 17 18 Construct a dotplot of the data [15؉10] Largemouth bass were studied in 53 different Florida lakes to examine the factors that influence the level of mercury contamination The average mercury concentration (parts per million) in the muscle tissue of the fish sampled from the lakes are as follows: 1.23 0.44 12 12 12 10 0.98 12 0.56 0.5 0.34 10 0.19 12 10 12 0.65 0.27 10 12 14 5.8 11 12 0.73 10 0.87 14 0.10 0.40 11 10 0.19 0.83 10 40 0.34 0.59 0.34 0.56 0.17 13 10 12 12 12 12 a Construct a frequency distribution and histogram of the data using class boundaries 0, 4, 8, and 12 Lump all the observations above 12 into a single category b The cumulative frequency for a particular class interval is the sum of frequencies for that interval and all intervals lying below it Compute the cumulative frequencies for the same data with the same class definitions [15؉15] The sizes of fifteen cathedrals in different cities are measured and their heights (in feet) are: 75 80 68 64 83 80 70 76 74 100 75 52 62 68 86 a Compute the following features of the data: (i) sample median, (ii) upper fourth, (iii) lower fourth, (iv) fourth spread fx b Construct a boxplot using the results of (a) Exam 2-1 INSTRUCTIONS: Show all work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [15؉5؉5] In a community, 60% of the families have at least child, 48% of the families have at least children, and 10% have or more children Let Ai ϭ {a randomly chosen family has at least i children}, i ϭ 1, 2, a Using operations on events, write in terms of A1, A2, and A3 the following events: B ϭ {a family has exactly child}, C ϭ {a family has exactly children}, D ϭ {a family has less than children} b Calculate P(B), P(C ), P(D), and P(A1 | D) c Draw the Venn diagram for the events [10؉5؉10] In a state, 12% of the electorate voted for A1 for the governor and for B1 for the attorney general, 18% voted for A2 for the governor and for B1 for the attorney general, 30% voted for A1 for the governor and for B2 for the attorney general, and 40% voted for A2 for the governor and for B2 for the attorney general a Find the percentage of voters who voted for A1 for the governor b Find the percentage of voters who voted for B2 for the attorney general c Did the voters’ choice of the attorney general depend on their choice of the governor? Sample exams provided by Abram Kagan and Tinghui Yu of University of Maryland Sample Exams [10؉5] Suppose that 6.5% of men and 2.1% of women are color-blind Assume that 51.4% of a population are women and 48.6% are men a Find the probability that a randomly selected person is color-blind b Given that a randomly selected person is not color-blind, what is the (conditional) probability that the selected person was a man? [10؉10] A system consisting of four components, C1 through C4, operates as long as at least one of the following holds true: • Both C1 and C2 operate • Both C1 and C3 operate • C4 operates 727 a Draw a design of such a system b Assuming that the components operate independently and the probability of a failure for each component during a onemonth period is 0.15, find the probability that the system will not fail during the same period [10؉5] An instructor gave her students 12 problems, telling them that of the problems will be on a quiz and that passing the quiz requires solving all of the problems a Given that the instructor chooses the problems at random, what is the probability for a student who knows only 10 problems to pass? b What are the chances to fail for a student who knows only problems? Exam 2-2 INSTRUCTIONS: Show all work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [15؉10] In a community, 50% of the families have at least child, 35% of the families have at least children, and 8% have or more children Let Ai ϭ {a randomly chosen family has at least i children}, i ϭ 1, 2, a Using operations on events, write in terms of A1, A2, and A3 the following events: B ϭ {a family has exactly child}, C ϭ {a family has exactly children}, D ϭ {a family has less than children}, E ϭ {a family has no children} b Calculate P(B), P(C ), P(D), P(E ), and P(A1 | D) [15؉10] Let A, B, and C be three events with P(A) ϭ 0.55, P(B) ϭ 0.49, P(C ) ϭ 0.45, P(A ʝ B) ϭ 0.20, P(A ʝ C) ϭ 0.18, P(B ʝ C ) ϭ 0.19, and P(A ʝ B ʝ C ) ϭ 0.08 a Draw a Venn diagram and calculate P(A ʝ BЈ) b Prove that at least one of the events A, B, or C occurs with probability one a If they take the seats at random, what is the probability that the husbands are in seats and 9? b What is the probability that the husbands are sitting next to their wives? [15؉5] An instructor gave her students 14 problems, telling them that of the problems will be on a quiz and that passing the quiz requires solving all of the problems a Given that the instructor chooses the problems at random, what is the probability that a student who knows only problems will pass? b If passing the quiz requires solving at least problems, what is the probability that a student who knows only 10 problems will pass? [10؉5] Suppose that 40% of cars have anti-theft devices Within a 1-year period, 1.0% of cars with anti-theft devices were stolen, while among cars without anti-theft devices, 2.5% were stolen within the same period a Among the stolen cars, what is the percentage of cars with anti-theft devices? b Among the cars that were not stolen, what is the percentage of cars with anti-theft devices? [10؉5] Jim and Paula and another couple, John and Ann, purchased tickets for seats 7, 8, 9, and 10 in the same row Exam 2-3 INSTRUCTIONS: Show all the work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [15؉5] A student is working on three projects Let Ai, where i ϭ 1, 2, 3, denote the event that the ith project will be completed by the due date a Using the operations of union, intersection, and complementation, describe the following events in terms of A1, A2, and A3: (i) exactly one project will be completed by the due date; (ii) at least one project will be completed by the due date b If P(A1) ϭ 0.52, P(A2) ϭ 0.48, P(A3) ϭ 0.34, P(A1 ʝ A2) ϭ 0.16, P(A1 ʝ A3) ϭ 0.12, P(A2 ʝ A3) ϭ 0.10, and P(A1 ʝ A2 ʝ A3 ) ϭ 0.04, find the probability that none of the projects will be completed by the due date [15؉5] a How many different 7-letter words can be formed using each of the letters a, a, a, b, b, b, and c only once? b How many different 7-letter words can be formed using each of the preceding letters only once, provided that the words not start with an a and not end with a b? Sample exams provided by Abram Kagan and Tinghui Yu of University of Maryland 728 Sample Exams [15؉5] A box contains 13 spare parts, good and defective a If five parts are selected at random, find the probability that at least four selected parts are good b If five parts are selected at random, what is the probability that at least one of them is defective? [10؉10] Suppose that in a certain community 15% of men and 8% of women are color-blind Assume that 65% of the residents are women and 35% are men a Using the Law of Total Probability, find the probability that a randomly selected person is color-blind b Using Bayes’ theorem, find the (conditional) probability that, given that a randomly selected person is color-blind, the person is a woman? [15؉5] Let A1, A2, and A3 be three independent events with P(A1) ϭ 0.70, P(A2) ϭ 0.40, P(A3) ϭ 0.35 a Find the probability of the event B ϭ {none of A1, A2, or A3 occurs} b Given that B did not occur, what is the probability that A1 occurred? Exam 3-1 INSTRUCTIONS: Show all work related to your solution Credit will be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [10؉10] In an exam, each of questions has multiple-choice answers, only one correct A student wanted to try his good luck by randomly choosing answers for each question a What is the distribution of X, the number of questions the student answered correctly? b If passing requires correct answers to at least questions, what is the probability that the student passed the exam? [15؉5] A trial consists of tossing, simultaneously, a fair coin and a fair die so that the outcomes are pairs (H, 4), (T, 3), etc Independent trials are performed until, for the first time, an outcome is (H, 6) Let X be the total number of trials (including the last one) a Find the pmf of X b Calculate E(X ) [15؉10] The number of requests for assistance received by a towing service is a Poisson process with rate ␣ ϭ 1.5 per hour a Compute the probability that at least one request is received during the period 1–3 P.M What is the expected number of requests during this period? b If the operators of the towing service take a lunch break from 11:45 A.M.–12:30 P.M., what is the probability that they won’t miss any calls for assistance? [10؉5] In a class of 18 students, are business majors, are computer science majors, and are math majors The instructor selected students at random for an interview a What is the probability that at least one is a computer science major? b What is the probability that there are exactly students from each major in the group of 6? [10؉10] Five problems are given in a quiz The probability that a student solves an individual problem is 0.7 Assume that solving different problems are independent events Let X be the number of problems the student solved a What is the distribution of X? b What is the probability that the student passes the quiz if passing requires solving (i) problems, (ii) problems? Exam 3-2 INSTRUCTIONS: Show all work related to your solution Credit will be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [10؉5] From past experience, the owner of a small drugstore knows that the (random) demand X for a weekly news magazine is given by the table: x p(x) 10 15 15 20 15 10 10 05 The store owner pays $1.00 for each copy of the magazine, while customers pay $2.50 for a copy The copies left at the end of the week have no salvage value a Compute the expected profit if the owner orders copies of the magazine b Find the probability that the profit exceeds $5.00 [15؉10] A salesman gets a $100 commission for every vacuum cleaner (VC) he sells He can visit four potential customers a day, spending 1.5 hours with each, in which case each customer (acting independently of the others) will buy a VC with a probability 0.2 Or he can visit six potential customers spending only hour with each, in which case each customer will buy a VC with a probability 0.1 Let a random variable X denote the amount of money the salesman makes in a day a Find the pmf of X under each strategy and calculate E(X ) b Calculate P(X Ն 100) under each strategy [10؉10] In an exam, each of questions has multiple-choice answers, with only one of them correct A student wanted to try his good luck by randomly choosing answers for each question a What is the distribution of X, the number of questions the student answered correctly? Sample exams provided by Abram Kagan and Tinghui Yu of University of Maryland Sample Exams b If getting an “A” requires correct answers to all five questions, getting a “B” requires four correct answers, and getting a “C” requires three correct answers, what is the probability that the student gets (i) an “A,” (ii) a “B,” and (iii) a “C”? [15؉5] A trial consists of simultaneously tossing two coins, one fair and the other biased, with P(Head) ϭ 0.6 and P(Tail) ϭ 0.4 Independent trials are performed until, for the first time, both coins land heads Let X be the total number of trials (including the last one) 729 a Find the pmf of X b Calculate E(X ) [15؉5] The number of requests for assistance received by a towing service is a Poisson process with rate ␣ ϭ per hour a Compute the probability that at least two requests are received during the period 2–4 P.M Find the expected number of requests during this period b If an operator of the towing service takes a 45-minute break, what is the probability that he will not miss any calls for assistance? Exam 3-3 INSTRUCTIONS: Show all the work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [15؉10] The owner of a bakery knows that the daily demand X (in dozens) for bagels is given by the table: x p(x) 25 30 30 15 The owner makes $3 on each dozen bagels sold and loses $1.5 on each dozen left unsold a How many dozens of bagels should the owner order to maximize the expected profit? b Calculate the probability that the profit exceeds $15 [10؉10] Let X be the number of successes in a series of independent Bernoulli trials with a probability of an individual success 0.7 and let Y be the number of successes in another series of independent Bernoulli trials with a probability of individual success 0.6 a If your goal is to maximize the expected number of successes, which of the two series would you choose? b If your goal is to maximize the probability of getting at least one success, which of the two series would you choose? [10؉5] A trial consists of tossing simultaneously (biased) coins with P(Head) ϭ 0.6 for one coin and P(Head) ϭ 0.7 for the other Independent trials are performed until two “doubles” (i.e., (Head, Head) or (Tail, Tail)) occur Let X be the total number of trials a Find the pmf of X b Calculate E(X ) [10؉10] Assume that the number of requests for assistance received over the phone by a local AAA office is a Poisson process with rate ␣ ϭ per hour a Compute the probability that at least one request is received during the period 12:30–2:15 P.M What is the expected number of requests during this period? b If the telephone operator in the office takes a 40-minute break, what is the probability that no request for assistance is missed? [15؉5] Of 20 problems on a list given by an instructor to his students, are easy, are medium, and are difficult The instructor chooses problems for an exam at random Let X be the number of easy problems, and Y the number of difficult problems among the chosen a Find P(X ϭ 2, Y ϭ 3) and P(X ϭ 2) b Find P(X ϭ Y ) Exam 4-1 INSTRUCTIONS: Show all work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed 0.04(x Ϫ 10) if 10 Յ x Յ 15 f (x) ϭ 0.04(20 Ϫ x) if 15 Յ x Յ 20 if x Ͻ 10 or x Ͼ 20 [10؉10؉5] Assume the commuting time X (in minutes) of a student has a uniform distribution on the interval (25, 50) a Compute P(X Ͼ 30), P(X Ͻ 35), and P(30 Յ X Յ 40) b If the student leaves home at 8:50 A.M and a class begins at 9:30 A.M., find the probability that the student won’t be late by more than minutes c Find the cdf of X and draw its graph a Find the cdf of X and draw the graphs of the pdf and cdf b Find the probability that the daily demand will exceed 12,000 gallons c Find the median of X and the 75th percentile of X (Note: Geometric arguments suffice; no need to integrate.) [10؉10؉10] The daily demand (in thousands of gallons) X for gas at a gas station can be considered a random variable with pdf: { [20؉10] Assume that the mileage (in miles per gallon) of a certain brand of cars has a normal distribution with mean 28 and standard deviation 1.4 a Find the probability that a randomly selected car will (i) get more than 30 miles per gallon; (ii) get less than Sample exams provided by Abram Kagan and Tinghui Yu of University of Maryland 730 Sample Exams 25 miles per gallon; (iii) get between 25 and 30 miles per gallon b If the tank contains only one gallon of gas and the driver has to drive 31.5 miles, should the driver first stop at the gas station? Justify your answer by calculations distribution, calculate the probability that in a randomly selected group of 60 drivers: a Not more than 30 are a good risk b At least 25 are a good risk c Between 20 and 28 (inclusive) are a good risk [5؉5؉5] Statistics show that 40% of drivers in a certain group are a good risk Using the normal approximation to the binomial Exam 4-2 INSTRUCTIONS: Show all work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [10؉10؉5] Assume that the time X (in milliseconds) it takes a read/write head to locate a desired record on a computer disk memory device is uniformly distributed on the interval (0, 35) a Compute P(X Ն 15), P(X Ͻ 25), and P(10 Յ X Յ 30) b Find the cdf of X and draw its graph c Compute the median and 80th percentile of X [15؉5] The breakdown voltage of a randomly chosen diode of a certain type has a normal distribution with mean 38V and standard deviation 1.5V a Find the probability that the voltage of a single diode is between 35V and 40V b What value is such that only 10% of all diodes have voltages exceeding this value? [5؉5؉5] Suppose that 10% of all steel shafts produced by a certain process are nonconforming Using the normal approximation to the binomial distribution, calculate the probability that in a randomly selected group of 200 shafts: a Not more than 30 are nonconforming b At least 28 are nonconforming c Between 16 and 24 (inclusive) are nonconforming [15؉5] A system consists of two identical components operating independently The lifetime of each component has an exponential distribution with mean days a Find the median and the 80th percentile of the lifetime of each component b Assuming that the system fails when both components fail, calculate the probability that the system will not fail during the first days [10؉10] The daily demand for a certain product is a random variable uniformly distributed on (2.5, 5.0) a Calculate the 75th percentile of the demand What is the meaning of this characteristic? b Assuming that the initial stock of the product was 4.0, calculate the expected value of the leftover of the product at the end of the day Exam 4-3 INSTRUCTIONS: Show all work related to your solution Credit will be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [15؉10] The reaction time X (in seconds) to a certain stimulus is a continuous random variable with pdf 2– (x Ϫ 1) if Յ x Յ f (x) ϭ 1– (4 Ϫ x) if Յ x Յ if x Ͻ or x Ͼ { a Find the cdf F(x) and draw the graphs of f (x) and F(x) Calculate the probability that the reaction time is between 1.5 and 2.5 b Find the median, 25th, and 75th percentile of X (Note: Geometric arguments suffice; no need to integrate.) [10؉10] Assume that the time X (in milliseconds) it takes a read/ write head to locate a desired record on a computer disk memory device is uniformly distributed on the interval (10, 25) a Find the cumulative distribution function (cdf ) of X, draw the graphs of the probability density function (pdf ) and the cdf of X, and compute P(X Ͼ 14), P(X Յ 20), and P (12 Յ X Յ 22) b Find the mean, variance, and 35th and 80th percentiles of X [10؉10] The breakdown voltage of a randomly chosen diode of a certain type has a normal distribution with mean 25V and standard deviation 0.75V a Compute the probability that the voltage of a diode is between 23.5V and 27V b Find the value c such that exactly 7.5% of all diodes have voltages exceeding c [10؉5] Assume that the commuting time (in minutes) of a student is a random variable having normal distribution with mean µ ϭ 30 and standard deviation ϭ 2.5 a If the student leaves her home at 7:40 A.M., find the probability that she won’t be late for a class that starts at 8:00 A.M Sample exams provided by Abram Kagan and Tinghui Yu of University of Maryland Sample Exams b What’s the latest she can leave her home and yet guarantee a probability of 0.90 that she won’t be late for the class? [10؉10] The time X (in hours) it takes auto mechanic A to complete a car inspection may be considered a random variable uniformly distributed on (2, 3.5) The time Y it takes auto mechanic B to the same job is a random variable 731 uniformly distributed on (1.5, 3.0) Assume that A and B started working on different cars at the same time and that X and Y are independent a Find the joint pdf of (X, Y ) and calculate the probability that both A and B finish their jobs in less than 2.5 hours b Find the probability that B finishes the job first Exam 4-4 INSTRUCTIONS: Show all work related to your solution Credit will be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [15؉5] The daily demand for gasoline (in thousands of gallons) at a gas station is a random variable X with probability density function (pdf) x Ϫ if Յ x Յ f (x) ϭ Ϫ x if Յ x Յ otherwise { a Draw the graph of f (x) and calculate the 80th percentile and P(1.4 Յ X Յ 2.7) b Find c such that P(X Ͻ c) ϭ 0.9 [10؉10] The weight distribution (in lb) of parcels sent in a certain manner is normal with mean value µ ϭ 15 and standard deviation ϭ 3.8 A surcharge is applied to parcels weighing more than 21 lb a Find the percentage of parcels under the surcharge weight b Find the probability that among three randomly chosen parcels at least one is above the surcharge weight [15؉5] Peter and Ann agreed to meet for lunch Assume that the arrival time X of Peter is a random variable uniformly distributed on (11:30 A.M., 12:30 P.M.) and the arrival time Y of Ann is a random variable uniformly distributed on (12 P.M., P.M.) and that X and Y are independent a Find the joint pdf of (X, Y) and calculate the probability that both Peter and Ann arrive between 12:15 P.M and P.M b Find the probability that (i) Peter arrives first, (ii) Ann arrives first, (iii) Peter and Ann arrive at the same time [15؉5] The system consisting of two components works as long as both components work Assume that the lifetimes X1, X2 of the components are independent random variables, X1 having an exponential distribution with mean 100 hours, and X2 having an exponential distribution with mean 150 hours a Draw a design of such a system and find the probability density function of X, the lifetime of the system b Calculate E(X ) and P(60 Ͻ X Ͻ 80) [15؉5] The time it takes an instructor to grade an exam paper is a random variable with expected value 7.5 minutes and standard deviation 1.5 minutes Assume that the grading times for different papers are independent a Using the Central Limit Theorem, find the probability that the instructor will grade 70 papers in less than hours b Suppose the instructor grades 40 papers on one day and the remaining 30 papers on another What is the probability that the mean times of grading a paper on both days not exceed 7.8 minutes? Exam 4-5 INSTRUCTIONS: Show all work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [15؉10] Let (X, Y ) be a pair of random variables with P(X ϭ Ϫ2, Y ϭ 0) ϭ P(X ϭ Ϫ2, Y ϭ 1) ϭ 0.1 P(X ϭ Ϫ1, Y ϭ 2) ϭ 0.2 P(X ϭ Ϫ1, Y ϭ 0) ϭ P(X ϭ Ϫ1, Y ϭ 1) ϭ 0.3 P(X ϭ 1, Y ϭ 2) ϭ 0.1 P(X ϭ 2, Y ϭ 0) ϭ P(X ϭ 2, Y ϭ 2) ϭ 0.1 P(X ϭ 2, Y ϭ 1) ϭ a Write the pmf of (X, Y) in the form of a table Are X and Y independent? Justify the answer using the definition of independence Find P(X Y ) b For the random variable Z ϭ |X ϩ Y|, calculate E (Z) [15؉10] Jim and Paula agreed to meet for lunch Assume that Jim’s arrival time X is a random variable uniformly distributed on (11:45, 12:45) and Paula’s arrival time Y is a random variable uniformly distributed on (12:00, 1:00) and that X and Y are independent a Find the joint pdf of (X, Y) and calculate the probability that both Jim and Paula arrive before 12:30 b Find the probability that (i) Paula comes first, (ii) Jim comes first (Hint: Choosing the origin at x ϭ 11:45, y ϭ 11:45 makes calculations easier.) [15؉10] The time it takes an instructor to grade an exam paper is a random variable with expected value minutes and standard deviation 1.2 minutes Assume that the grading times for different papers are independent Sample exams provided by Abram Kagan and Tinghui Yu of University of Maryland 732 Sample Exams a Using the Central Limit Theorem, calculate the probability that the instructor will need more than 8.5 hours to grade 70 papers b Suppose the instructor grades 50 papers on one day and the remaining 20 papers on another What is the probability that the mean times of grading a paper on both days not exceed 7.4 minutes? [15؉10] Two problems are given in a quiz The time it takes a randomly chosen student to solve the first problem is a random variable X1 having normal distribution with mean 14 minutes and standard deviation minutes For the second problem, it is a random variable X2 independent of X1, having normal distribution with mean 12 minutes and standard deviation 1.5 minutes a Find the probability that a student will solve both problems in less than 30 minutes b What is the percentage of students who solve the second problem in less time than the first? Exam 5-1 INSTRUCTIONS: Show all work related to your solution Credit will be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [15؉10] To compare the unknown proportions p1 and p2 of steel shafts produced by two manufacturers, M1 and M2, a sample of size n1 of shafts manufactured by M1 and an independent sample of size n2 of shafts manufactured by M2 were drawn The first sample contained X1 conforming shafts, and the second contained X2 conforming shafts a Show that X1 /n1 Ϫ X2 /n2 is an unbiased estimator of p1 Ϫ p2 and find its standard deviation (Hint: Use the formula for the variance of X Ϫ Y where X, Y are independent random variables.) b For n1 ϭ 100, X1 ϭ 84, n2 ϭ 80, X2 ϭ 65, estimate the standard error of X1 /n1 Ϫ X2 /n2 [15؉10] The following sample of size n ϭ was drawn from a population with mean µ, median µ, ˜ and standard deviation : x1 ϭ 8.9, x2 ϭ 7.1, x3 ϭ 6.6, x4 ϭ 7.8, x5 ϭ 5.8, x6 ϭ 11.2, x7 ϭ 8.1, x8 ϭ 12.6, x9 ϭ 9.2 b What is the estimated standard error of the estimator you used in estimating µ? [10؉15] Let (x1, , xn ) be a sample from a population with pdf f (x; ) ϭ { 3x2 3(1 Ϫ )x2 if Ϫ1 Յ x Յ if Յ x Յ with , Ͻ Ͻ as a parameter a Check that f (x; ) is a pdf and draw its graph for ϭ 0.25 and ϭ 0.5 b Find the estimator of by the method of moments and show that it is unbiased [10؉15] Let x1, , xn be a sample from a population with pdf f (x; ) ϭ 2 xe Ϫx f (x; ) ϭ if x Ն if x Ͻ with Ͼ as a parameter a Check that f (x; ) is a pdf and draw its graph for ϭ (Note: ͐0∞ xeϪx dx ϭ 1.) b Find the maximum likelihood estimator of a Calculate point estimators of the population mean, median, and standard deviation Exam 5-2 INSTRUCTIONS: Show all work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [15؉10] Of n1 randomly selected male smokers, X1 smoked filter cigarettes, whereas of n2 randomly selected female smokers, X2 smoked filter cigarettes Denote by p1 and p2 the probabilities that a randomly selected male and female, respectively, smoke filter cigarettes a Show that X1 /n1 Ϫ X2 /n2 is an unbiased estimator of p1 Ϫ p2 and find its standard error b For n1 ϭ 250, n2 ϭ 150, x1 ϭ 175, x2 ϭ 120, and the estimator from (a), calculate the estimate of p1 Ϫ p2 and estimate the standard error of the estimator [10؉10؉5] Let (x1, , xn ) be a sample from a population with pdf f (x; ) ϭ Ϫ2x if Ϫ1 Յ x Յ 2(1 Ϫ )x if Յ x Յ { with , Ͻ Ͻ as a parameter a Check that f (x; ) is a pdf b Find the estimator ˜ of by the method of moments c Calculate E(˜ ) and show that ˜ is an unbiased estimator of [15؉10] Let (x1, , xn ) be a sample from a population with pdf f (x; ) ϭ {1/ if Յ x Յ otherwise with , Ͼ as a parameter Sample exams provided by Abram Kagan and Tinghui Yu of University of Maryland Sample Exams a Find the estimator of by the method of moments and show that it is unbiased b Find the standard error and the estimated standard error of the method of moments estimator 733 a Check that f (x; ) is a pdf and draw its graph for ϭ b Find the estimator of by the method of maximum likelihood [10؉15] Let (x1, , xn ) be a sample from a population with pdf {(0 Ϫ 1)/x if x Ն otherwise with Ͼ1 as a parameter f (x; ) ϭ Exam 5-3 INSTRUCTIONS: Show all work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [15؉10] In a group of n1 randomly selected patients treated for a common cold by medicine A, X1 were cured within a specified time period, whereas in another group of n2 randomly selected patients treated using medicine B, X2 were cured within the same period Denote by p1 the efficiency of medicine A defined as the probability that a randomly selected patient using A recovers within the specified period, and by p2 the same characteristic for medicine B a Show that X1 /n1 Ϫ X2 /n2 is an unbiased estimator of p1 Ϫ p2 and find its standard error and estimated standard error b For n1 ϭ 85, n2 ϭ 90, x1 ϭ 68, x2 ϭ 77, and the estimator from (a), calculate the estimate of p1 Ϫ p2 and estimate the standard error of the estimator [10؉10؉5] Let (x1, , xn ) be a sample from a population with pdf Ϫ(1/2)x if Ϫ2 Յ x Յ f (x; ) ϭ (1/2)(1 Ϫ )x if Յ x Յ ϩ2 otherwise { with , Ͻ Ͻ as a parameter a Calculate the population mean as a function of the parameter b Find the estimator of by the method of moments and show that it is unbiased c For n ϭ and x1 ϭ Ϫ1.7, x2 ϭ Ϫ1.3, x3 ϭ 1.2, and x4 ϭ 1.6, calculate the method of moments estimate of [15؉10] The following sample of size n ϭ was drawn from a population with mean µ, median µ, ˜ and standard deviation : x1 ϭ 5.1, x2 ϭ 6.4, x3 ϭ 3.5, x4 ϭ 8.0, x5 ϭ 6.1, x6 ϭ 7.6, x7 ϭ 9.2 a Calculate point estimators of the population mean, median, and standard deviation b Calculate the estimated standard error of the estimator used in estimating µ [10؉15] Let (x1, , xn ) be a sample from a population with pdf f (x; ) ϭ { 42 xe Ϫ2x if x Ն 0 otherwise with Ͼ as a parameter a Check that f (x; ) is a pdf and draw its graph for ϭ b Find the estimator of by the method of maximum likelihood Exam 6-1 INSTRUCTIONS: Show all work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [10؉10؉5] A sample of size n is drawn from a normal population with unknown mean µ and known sd ϭ 2.5 a Compute a 95% confidence interval (CI) for µ when n ϭ 25 and the sample mean xෆ ϭ 62.4 b Compute a 95% CI for µ when n ϭ 100 and has the same sample mean as in (a) Explain the difference between the CIs in (a) and (b) c How large must n be if the width of the 95% CI is to be 1.8? [10؉10] A sample of size n is drawn from a normal population with unknown mean µ and unknown standard deviation (sd) a Construct a 95% CI for µ when n ϭ 25, the sample mean xෆ ϭ 62.4 and the sample sd s ϭ 2.5 b Compare this CI with the CI in part (a) of Problem and explain the difference [5؉10؉10] A (large) random sample of 350 spare parts contains 30 defective parts a Estimate the true proportion p of good parts in the population b Construct a 90% confidence interval for p c How large should the sample size be to ensure that the length of the 90% confidence interval for p is less than 0.04? [10؉20] Let X ෆ1, X ෆ2 be the sample means of two independent samples of sizes n1, n2 drawn from two normal populations, one with unknown mean µ1 and known variance 12, the other with unknown mean µ2 and known variance 22 a Show that X ෆ1 Ϫ X ෆ2 is an unbiased estimator of µ1 Ϫ µ2 and find its standard deviation b For n1 ϭ 10, n2 ϭ 8, 12 ϭ 1.2, 22 ϭ 0.94, xෆ1 ϭ 9.0, xෆ2 ϭ 7.6 construct a 95% confidence interval for µ1 Ϫ µ2 Sample exams provided by Abram Kagan and Tinghui Yu of University of Maryland 734 Sample Exams Exam 6-2 INSTRUCTIONS: Show all work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [5؉5؉10] A sample of size n is drawn from a normal population with unknown mean and known variance 2 ϭ 6.1 a Compute a 95% confidence interval (CI) for when n ϭ 15 and the sample mean ෆx ϭ 56.8 b Compute a 95% CI for when n ϭ 30 and ෆx is the same as in (a) Explain the difference between the CIs in (a) and (b) c How large must n be if the length of the 95% CI is to be 0.75? [15؉15؉5] A random sample of size n ϭ 10 from a normal population with an unknown mean and an unknown variance 2 yielded a sample mean of 19.6 and a sample variance of 9.2 a Compute a 95% confidence interval for b Compute a 95% confidence interval for 2 c Compute a 95% confidence interval for a Estimate the true proportion p of customers who incurred the interest charge in year 2008, along with the standard error of the estimator used b Construct a 95% confidence interval for p c How many customers must be selected to ensure that the width of the 95% confidence interval will be 0.06? [10؉5] Two independent samples of sizes n1 and n2 are drawn from normal populations with an unknown mean and known standard deviations 1 and 2, respectively a If 1 ϭ 1.22, is it enough to have n1 Ͼ 1.2n2 to guarantee that the standard 95% confidence interval (CI) for constructed from the first be shorter than the 95% CI constructed from the second sample? b What is the relation between 1, 2, n1, n2 that would guarantee the smaller length of the 95% CI is constructed from the first sample? [5؉10؉15] Among 210 randomly selected credit card customers, 142 incurred an interest charge in year 2008 because of an unpaid balance Exam 7-1 INSTRUCTIONS: Show all work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [15؉15] The sample mean of a sample of size 20 from a normal population with an unknown mean and known standard deviation ϭ 0.7 is ෆx ϭ11.2 a Test H0 : ϭ11.0 versus Ha: 11.0 using a two-tailed level 0.10 test b Compute the type II error probability (Ј) of the test for the alternative Јϭ11.3 [15؉15؉10] The average lifetime (in thousands of miles) of a tire of a certain brand is 60 An inventor claims that tires manufactured on his new technology have a longer average lifetime To support the claim, he brings eight tires manufactured on the new technology for testing The actual lifetimes of the eight tires are a State the null hypothesis and the alternative to test the inventor’s claim b Assuming the lifetimes of the tires to be independent random variables having a normal distribution, use the t test at level 0.10 to test H0 c Determine whether the p-value of the data exceeds 0.10 [10؉10؉10] A (large) sample of 140 students taking STAT 300 reveals that 78 passed the class with a “B” or an “A.” Does this suggest that the actual percentage of students in STAT 300 who got a “B” or an “A” for the class is at least 0.55? a State the appropriate null hypothesis and the alternative to answer the question b Carry out a test using a significance level of ␣ ϭ 0.10 c Compute the p-value of the data x1 ϭ 62, x2 ϭ 64.2, x3 ϭ 58.8, x4 ϭ 60.4, x5 ϭ 61.4, x6 ϭ 59.0, x7 ϭ 63.0, x8 ϭ 62.6 Exam 7-2 INSTRUCTIONS: Show all work related to your solution Credit may be deducted for numerical answers unsupported by valid reasoning or calculations You may use calculators as needed [10؉10؉10] Assume that the mileage (in miles per gallon) of a certain brand of cars has a normal distribution with mean and standard deviation ϭ 1.4 If the actual mileages for n ϭ cars are x1 ϭ 27.8, x2 ϭ 30.0, x3 ϭ 31.8, x4 ϭ 32.6, x5 ϭ 28.4, x6 ϭ 34.0, does this suggest that the mean is at least 30? Sample exams provided by Abram Kagan and Tinghui Yu of University of Maryland Sample Exams a State the appropriate null hypothesis and the alternative to answer the question b Carry out a test using a significance level of ␣ ϭ 0.10 c Compute the type II error probability (Ј) of the test for the alternative Ј ϭ 32 [15؉15؉10] The response time (in seconds) to a certain signal is a random variable having a normal distribution with an unknown mean and unknown standard deviation Seven independent measurements of actual response times are as follows: 735 [10؉10؉10] A (large) sample of 180 drivers insured by a certain company reveals that 105 of them had at least one moving violation in 2008 The company classifies a driver as a good risk if he/she had no moving violations in a previous year Does the preceding data suggest that at least 40% of the drivers insured by the company are a good risk? a State the appropriate null hypothesis and the alternative to answer the question b Carry out a test using a significance level of ␣ ϭ 0.05 c Compute the p-value of the data x1 ϭ 1.80, x2 ϭ 1.62, x3 ϭ 2.00, x4 ϭ 2.02, x5 ϭ 1.78, x6 ϭ 1.82, x7 ϭ 1.94 Some indirect arguments suggest that ϭ 1.90 a State the null hypothesis and the alternative to test the claim b Test the null hypothesis using a two-tailed level 0.10 test c Determine whether the p-value of the data exceeds 0.10 Sample exams provided by Abram Kagan and Tinghui Yu of University of Maryland This page intentionally left blank Φ(z) ϭ P(Z Յ z) Table A.3 Standard Normal Curve Areas Standard normal density function Shaded area = Φ(z) z z 00 01 02 03 04 05 06 07 08 09 Ϫ3.4 Ϫ3.3 Ϫ3.2 Ϫ3.1 Ϫ3.0 0003 0005 0007 0010 0013 0003 0005 0007 0009 0013 0003 0005 0006 0009 0013 0003 0004 0006 0009 0012 0003 0004 0006 0008 0012 0003 0004 0006 0008 0011 0003 0004 0006 0008 0011 0003 0004 0005 0008 0011 0003 0004 0005 0007 0010 0002 0003 0005 0007 0010 Ϫ2.9 Ϫ2.8 Ϫ2.7 Ϫ2.6 Ϫ2.5 0019 0026 0035 0047 0062 0018 0025 0034 0045 0060 0017 0024 0033 0044 0059 0017 0023 0032 0043 0057 0016 0023 0031 0041 0055 0016 0022 0030 0040 0054 0015 0021 0029 0039 0052 0015 0021 0028 0038 0051 0014 0020 0027 0037 0049 0014 0019 0026 0036 0038 Ϫ2.4 Ϫ2.3 Ϫ2.2 Ϫ2.1 Ϫ2.0 0082 0107 0139 0179 0228 0080 0104 0136 0174 0222 0078 0102 0132 0170 0217 0075 0099 0129 0166 0212 0073 0096 0125 0162 0207 0071 0094 0122 0158 0202 0069 0091 0119 0154 0197 0068 0089 0116 0150 0192 0066 0087 0113 0146 0188 0064 0084 0110 0143 0183 Ϫ1.9 Ϫ1.8 Ϫ1.7 Ϫ1.6 Ϫ1.5 0287 0359 0446 0548 0668 0281 0352 0436 0537 0655 0274 0344 0427 0526 0643 0268 0336 0418 0516 0630 0262 0329 0409 0505 0618 0256 0322 0401 0495 0606 0250 0314 0392 0485 0594 0244 0307 0384 0475 0582 0239 0301 0375 0465 0571 0233 0294 0367 0455 0559 Ϫ1.4 Ϫ1.3 Ϫ1.2 Ϫ1.1 Ϫ1.0 0808 0968 1151 1357 1587 0793 0951 1131 1335 1562 0778 0934 1112 1314 1539 0764 0918 1093 1292 1515 0749 0901 1075 1271 1492 0735 0885 1056 1251 1469 0722 0869 1038 1230 1446 0708 0853 1020 1210 1423 0694 0838 1003 1190 1401 0681 0823 0985 1170 1379 Ϫ0.9 Ϫ0.8 Ϫ0.7 Ϫ0.6 Ϫ0.5 1841 2119 2420 2743 3085 1814 2090 2389 2709 3050 1788 2061 2358 2676 3015 1762 2033 2327 2643 2981 1736 2005 2296 2611 2946 1711 1977 2266 2578 2912 1685 1949 2236 2546 2877 1660 1922 2206 2514 2843 1635 1894 2177 2483 2810 1611 1867 2148 2451 2776 Ϫ0.4 Ϫ0.3 Ϫ0.2 Ϫ0.1 Ϫ0.0 3446 3821 4207 4602 5000 3409 3783 4168 4562 4960 3372 3745 4129 4522 4920 3336 3707 4090 4483 4880 3300 3669 4052 4443 4840 3264 3632 4013 4404 4801 3228 3594 3974 4364 4761 3192 3557 3936 4325 4721 3156 3520 3897 4286 4681 3121 3482 3859 4247 4641 (continued) Table A.3 ⌽(z) ϭ P(Z Յ z) Standard Normal Curve Areas (cont.) z 00 01 02 03 04 05 06 07 08 09 0.0 0.1 0.2 0.3 0.4 5000 5398 5793 6179 6554 5040 5438 5832 6217 6591 5080 5478 5871 6255 6628 5120 5517 5910 6293 6664 5160 5557 5948 6331 6700 5199 5596 5987 6368 6736 5239 5636 6026 6406 6772 5279 5675 6064 6443 6808 5319 5714 6103 6480 6844 5359 5753 6141 6517 6879 0.5 0.6 0.7 0.8 0.9 6915 7257 7580 7881 8159 6950 7291 7611 7910 8186 6985 7324 7642 7939 8212 7019 7357 7673 7967 8238 7054 7389 7704 7995 8264 7088 7422 7734 8023 8289 7123 7454 7764 8051 8315 7157 7486 7794 8078 8340 7190 7517 7823 8106 8365 7224 7549 7852 8133 8389 1.0 1.1 1.2 1.3 1.4 8413 8643 8849 9032 9192 8438 8665 8869 9049 9207 8461 8686 8888 9066 9222 8485 8708 8907 9082 9236 8508 8729 8925 9099 9251 8531 8749 8944 9115 9265 8554 8770 8962 9131 9278 8577 8790 8980 9147 9292 8599 8810 8997 9162 9306 8621 8830 9015 9177 9319 1.5 1.6 1.7 1.8 1.9 9332 9452 9554 9641 9713 9345 9463 9564 9649 9719 9357 9474 9573 9656 9726 9370 9484 9582 9664 9732 9382 9495 9591 9671 9738 9394 9505 9599 9678 9744 9406 9515 9608 9686 9750 9418 9525 9616 9693 9756 9429 9535 9625 9699 9761 9441 9545 9633 9706 9767 2.0 2.1 2.2 2.3 2.4 9772 9821 9861 9893 9918 9778 9826 9864 9896 9920 9783 9830 9868 9898 9922 9788 9834 9871 9901 9925 9793 9838 9875 9904 9927 9798 9842 9878 9906 9929 9803 9846 9881 9909 9931 9808 9850 9884 9911 9932 9812 9854 9887 9913 9934 9817 9857 9890 9916 9936 2.5 2.6 2.7 2.8 2.9 9938 9953 9965 9974 9981 9940 9955 9966 9975 9982 9941 9956 9967 9976 9982 9943 9957 9968 9977 9983 9945 9959 9969 9977 9984 9946 9960 9970 9978 9984 9948 9961 9971 9979 9985 9949 9962 9972 9979 9985 9951 9963 9973 9980 9986 9952 9964 9974 9981 9986 3.0 3.1 3.2 3.3 3.4 9987 9990 9993 9995 9997 9987 9991 9993 9995 9997 9987 9991 9994 9995 9997 9988 9991 9994 9996 9997 9988 9992 9994 9996 9997 9989 9992 9994 9996 9997 9989 9992 9994 9996 9997 9989 9992 9995 9996 9997 9990 9993 9995 9996 9997 9990 9993 9995 9997 9998 Table A.5 Critical Values for t Distributions t density curve Shaded area = ␣ t␣, ␣ v 10 05 025 01 005 001 0005 3.078 1.886 1.638 1.533 6.314 2.920 2.353 2.132 12.706 4.303 3.182 2.776 31.821 6.965 4.541 3.747 63.657 9.925 5.841 4.604 318.31 22.326 10.213 7.173 636.62 31.598 12.924 8.610 1.476 1.440 1.415 1.397 1.383 2.015 1.943 1.895 1.860 1.833 2.571 2.447 2.365 2.306 2.262 3.365 3.143 2.998 2.896 2.821 4.032 3.707 3.499 3.355 3.250 5.893 5.208 4.785 4.501 4.297 6.869 5.959 5.408 5.041 4.781 10 11 12 13 14 1.372 1.363 1.356 1.350 1.345 1.812 1.796 1.782 1.771 1.761 2.228 2.201 2.179 2.160 2.145 2.764 2.718 2.681 2.650 2.624 3.169 3.106 3.055 3.012 2.977 4.144 4.025 3.930 3.852 3.787 4.587 4.437 4.318 4.221 4.140 15 16 17 18 19 1.341 1.337 1.333 1.330 1.328 1.753 1.746 1.740 1.734 1.729 2.131 2.120 2.110 2.101 2.093 2.602 2.583 2.567 2.552 2.539 2.947 2.921 2.898 2.878 2.861 3.733 3.686 3.646 3.610 3.579 4.073 4.015 3.965 3.922 3.883 20 21 22 23 24 1.325 1.323 1.321 1.319 1.318 1.725 1.721 1.717 1.714 1.711 2.086 2.080 2.074 2.069 2.064 2.528 2.518 2.508 2.500 2.492 2.845 2.831 2.819 2.807 2.797 3.552 3.527 3.505 3.485 3.467 3.850 3.819 3.792 3.767 3.745 25 26 27 28 29 1.316 1.315 1.314 1.313 1.311 1.708 1.706 1.703 1.701 1.699 2.060 2.056 2.052 2.048 2.045 2.485 2.479 2.473 2.467 2.462 2.787 2.779 2.771 2.763 2.756 3.450 3.435 3.421 3.408 3.396 3.725 3.707 3.690 3.674 3.659 30 32 34 36 38 1.310 1.309 1.307 1.306 1.304 1.697 1.694 1.691 1.688 1.686 2.042 2.037 2.032 2.028 2.024 2.457 2.449 2.441 2.434 2.429 2.750 2.738 2.728 2.719 2.712 3.385 3.365 3.348 3.333 3.319 3.646 3.622 3.601 3.582 3.566 40 50 60 120 ∞ 1.303 1.299 1.296 1.289 1.282 1.684 1.676 1.671 1.658 1.645 2.021 2.009 2.000 1.980 1.960 2.423 2.403 2.390 2.358 2.326 2.704 2.678 2.660 2.617 2.576 3.307 3.262 3.232 3.160 3.090 3.551 3.496 3.460 3.373 3.291 ...SEVENTH EDITION Probability and Statistics for Engineering and the Sciences This page intentionally left blank SEVENTH EDITION Probability and Statistics for Engineering and the Sciences JAY L DEVORE. .. calculate the height of each rectangle using the formula relative frequency of the class rectangle height ϭ ᎏᎏᎏᎏ class width The resulting rectangle heights are usually called densities, and the. .. on the measurement axis) and negative if the observation is smaller than the mean If all the deviations are small in magnitude, then all xi s are close to the mean and there is little variability