Set Student Name: 1. Answer true or false for each part, and if false, explain your answer. a. The point estimate for the population mean, , of an x distribution is x-bar, computed from a random sample of the x distribution. b. Every random sample of the same size from a given population will produce exactly the same confidence interval for . c. For the same random sample, when the confidence level is reduced, the confidence interval for becomes wider. 2. Use tables or software to find the t-value for a 90% confidence interval when the sample size is 20. 3. A random sample of size 64 has sample mean 24 and sample standard deviation 4. d. Is it appropriate to use the t distribution to compute a confidence interval for the population mean? Why or why not? e. Construct a 95% confidence interval for the population mean. f. Explain the meaning of the confidence interval you just constructed. 4. How much do adult male grizzly bears weigh in the wild? Six adult males were captured, tagged and released in California and here are their weights: 480, 580, 470, 510, 390, 550 g. What is the point estimate for the population mean? h. Construct at 90% confidence interval for the population average weight of all adult male grizzly bears in the wild. i. Interpret the confidence interval in the context of this problem. 5. After going to a fast food restaurant, customers are asked to take a survey. Out of a random sample of 340 customers, 290 said their experience was \"satisfactory.\" Let p represent the proportion of all customers who would say their experience was \"satisfactory.\" j. What is the point estimate for p? k. Construct a 99% confidence interval for p. l. Give a brief interpretation of this interval. 6. In your own words, define each of the following terms that are used in hypothesis testing: a. The null hypothesis. Copyright2011 Cengage Learning. All Rights Reserved. 1 Problem Set b. c. d. e. f. The alternative hypothesis. The test statistic The p-value Type I Error Type II Error 7. Suppose the p-value for a right-tailed test is .0245. a. What would be your conclusion at the .05 level of significance? b. What would the p-value have been if it were a two-tailed test? 8. A random sample has 42 values. The sample mean is 9.5 and the sample standard deviation is 1.5. Use a level of significance of 0.02 to conduct a left-tailed test of the claim that the population mean is 10.0. a. b. c. d. Are the requirements met to run a test like this? What are the hypotheses for this test? Compute the test statistic and the p-value for this test. What is your conclusion at the 0.02 level of significance? 9. MTV states that 75% of all college students have seen at least one episode of their TV show \"Jersey Shore\". Last month, a random sample of 120 college students was selected and asked if they had seen at least one episode of the show. Out of the 120, 85 of them said they had seen at least one episode. Is there enough evidence to claim the population proportion of all college student that have watched at least one episode is less than 75% at the 0.05 level of significance? a. b. c. d. Are the requirements met to run a test like this? What are the hypotheses for this test? Compute the test statistic and the p-value for this test. What is your conclusion at the 0.05 level of significance? Copyright2011 Cengage Learning. All Rights Reserved. 2 Instuctor's Annotated Edition TENTH EDITION Understandable Statistics Concepts and Methods Charles Henry Brase Regis University Corrinne Pellillo Brase Arapahoe Community College Australia Brazil Japan Korea Mexico Singapore Spain United Kingdom United States Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. This book is dedicated to the memory of a great teacher, mathematician, and friend Burton W. Jones Professor Emeritus, University of Colorado Understandable Statistics: Concepts and Methods, Tenth Edition Charles Henry Brase, Corrinne Pellillo Brase Editor in Chief: Michelle Julet Publisher: Richard Stratton Senior Sponsoring Editor: Molly Taylor Senior Editorial Assistant: Shaylin Walsh Media Editor: Andrew Coppola Marketing Manager: Ashley Pickering Marketing Communications Manager: Mary Anne Payumo Content Project Manager: Jill Clark 2012, 2009, 2006 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions. Further permissions questions can be emailed to permissionrequest@cengage.com. Art Director: Linda Helcher Senior Manufacturing Buyer: Diane Gibbons Senior Rights Acquisition Specialist, Text: Katie Huha Rights Acquisition Specialist, Images: Mandy Groszko Text Permissions Editor: Sue Howard Production Service: Elm Street Publishing Services Library of Congress Control Number: 2009942998 Student Edition: ISBN-13: 978-0-8400-4838-7 ISBN-10: 0-8400-4838-6 Annotated Instructor's Edition: ISBN-13: 978-0-8400-5456-2 ISBN-10: 0-8400-5456-4 Cover Designer: RHDG Cover Image: Anup Shah Compositor: Integra Software Services, Ltd. Pvt. Brooks/Cole 20 Channel Center Street Boston, MA 02210 USA Cengage Learning is a leading provider of customized learning solutions with ofce locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil and Japan. Locate your local ofce at international.cengage.com/region Cengage Learning products are represented in Canada by Nelson Education, Ltd. For your course and learning solutions, visit www.cengage.com. Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com. Printed in the United States of America 1 2 3 4 5 6 7 14 13 12 11 10 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 7 7.1 Estimating m When s Is Known Rusty Jarrett/Getty Images for NASCAR/ Getty Images 7.2 Estimating m When s Is Unknown 7.3 Estimating p in the Binomial Distribution 7.4 Estimating m1 m2 and p1 p2 Library of Congress We dance round in a ring and suppose, But the Secret sits in the middle and knows. Robert Frost, \"The Secret Sits\"* In Chapter 1, we said that statistics is the study of how to collect, organize, analyze, and interpret numerical data. That part of statistics concerned with analysis, interpretation, and forming conclusions about the source of the data is called statistical inference. Problems of statistical inference require us to draw a random sample of observations from a larger population. A sample usually contains incomplete information, so in a sense we must \"dance round in a ring and suppose,\" to quote the words of the celebrated American poet Robert Lee Frost (1874-1963). Nevertheless, conclusions about the population can be obtained from sample data by the use of statistical estimates. This chapter introduces you to several widely used methods of estimation. *\"The Secret Sits,\" from The Poetry of Robert Frost, edited by Edward Connery Lathem. Copyright 1942 by Robert Frost, 1970 by Lesley Frost Ballantine, 1969 by Henry Holt and Company, Inc. Reprinted by permission of Henry Holt and Company, Inc. For online student resources, visit The Brase/Brase, Understandable Statistics, 10th edition web site at http://www.cengage.com/statistics/brase. 332 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. P R EVI EW QU ESTIONS How do you estimate the expected value of a random variable? What assumptions are needed? How much condence should be placed in such estimates? (SECTION 7.1) Bobby Deal / RealDealPhoto, 2010/Used under license from Shutterstock.Com Estimation At the beginning design stage of a statistical project, how large a sample size should you plan to get? (SECTION 7.1) What famous statistician worked for Guinness brewing company in Ireland? What has this got to do with constructing estimates from sample data? (SECTION 7.2) How do you estimate the proportion p of successes in a binomial experiment? How does the normal approximation t into this process? (SECTION 7.3) Sometimes even small differences can be extremely important. How do you estimate differences? (SECTION 7.4) FOCUS PROBLEM The National Wildlife Federation published an article entitled \"The Trouble with Wood Ducks\" (National Wildlife, Vol. 31, No. 5). In this article, wood ducks are described as beautiful birds living in forested areas such as the Pacic Northwest and southeast United States. Because of overhunting and habitat destruction, these birds were in danger of extinction. A federal ban on hunting wood ducks in 1918 helped save the species from extinction. Wood ducks like to nest in tree cavities. However, many such trees were disappearing due to heavy timber cutting. For a period of time it seemed that nesting boxes were the solution to disappearing trees. At rst, the wood duck population grew, but after a few seasons, the population declined sharply. Good biology research combined with good statistics provided an answer to this disturbing phenomenon. Cornell University professors of ecology Paul Sherman and Brad Semel found that the nesting boxes were placed too close to each other. Female wood ducks prefer a secluded nest that is a considerable distance from the next wood duck nest. In fact, female wood duck behavior changed when the nests were too close to each other. Some females would lay their Photo Researchers The Trouble with Wood Ducks 333 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 334 Chapter 7 ESTIMATION eggs in another female's nest. The result was too many eggs in one nest. The biologists found that if there were too many eggs in a nest, the proportion of eggs that hatched was considerably reduced. In the long run, this meant a decline in the population of wood ducks. In their study, Sherman and Semel used two placements of nesting boxes. Group I boxes were well separated from each other and well hidden by available brush. Group II boxes were highly visible and grouped closely together. In group I boxes, there were a total of 474 eggs, of which a eld count showed that about 270 hatched. In group II boxes, there were a total of 805 eggs, of which a eld count showed that, again, about 270 hatched. The material in Chapter 7 will enable us to answer many questions about the hatch ratios of eggs from nests in the two groups. (a) Find a point estimate p1 for p1, the proportion of eggs that hatch in group I nest box placements. Find a 95% condence interval for p1. (b) Find a point estimate p2 for p2, the proportion of eggs that hatch in group II nest box placements. Find a 95% condence interval for p2. (c) Find a 95% condence interval for p1 p2. Does the interval indicate that the proportion of eggs hatched from group I nest box placements is higher than, lower than, or equal to the proportion of eggs hatched from group II nest boxes? (d) What conclusions about placement of nest boxes can be drawn? In the article, additional concerns are raised about the higher cost of placing and maintaining group I nest boxes. Also at issue is the cost efciency per successful wood duck hatch. Data in the article do not include information that would help us answer questions of cost efciency. However, the data presented do help us answer questions about the proportions of successful hatches in the two nest box congurations. (See Problem 26 of Section 7.4.) S E C T I O N 7. 1 Estimating M When S Is Known FOCUS POINTS Explain the meanings of condence level, error of estimate, and critical value. Find the critical value corresponding to a given condence level. Compute condence intervals for m when s is known. Interpret the results. Compute the sample size to be used for estimating a mean m. Because of time and money constraints, difculty in nding population members, and so forth, we usually do not have access to all measurements of an entire population. Instead we rely on information from a sample. In this section, we develop techniques for estimating the population mean m using sample data. We assume the population standard deviation s is known. Let's begin by listing some basic assumptions used in the development of our formulas for estimating m when s is known. Assumptions about the random variable x 1. We have a simple random sample of size n drawn from a population of x values. 2. The value of s, the population standard deviation of x, is known. 3. If the x distribution is normal, then our methods work for any sample size n. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 7.1 335 Estimating m When s Is Known 4. If x has an unknown distribution, then we require a sample size n 30. However, if the x distribution is distinctly skewed and denitely not mound-shaped, a sample of size 50 or even 100 or higher may be necessary. An estimate of a population parameter given by a single number is called a point estimate for that parameter. It will come as no great surprise that we use x (the sample mean) as the point estimate for m (the population mean). Point estimate A point estimate of a population parameter is an estimate of the parameter using a single number. x is the point estimate for m. Even with a large random sample, the value of x usually is not exactly equal to the population mean m. The margin of error is the magnitude of the difference between the sample point estimate and the true population parameter value. Margin of error When using x as a point estimate for m, the margin of error is the magnitude of x m or 0 x m 0 . Condence level, c FIGURE 7-1 Condence Level c and Corresponding Critical Value zc Shown on the Standard Normal Curve Finding the critical value We cannot say exactly how close x is to m when m is unknown. Therefore, the exact margin of error is unknown when the population parameter is unknown. Of course, m is usually not known, or there would be no need to estimate it. In this section, we will use the language of probability to give us an idea of the size of the margin of error when we use x as a point estimate for m. First, we need to learn about condence levels. The reliability of an estimate will be measured by the condence level. Suppose we want a condence level of c (see Figure 7-1). Theoretically, we can choose c to be any value between 0 and 1, but usually c is equal to a number such as 0.90, 0.95, or 0.99. In each case, the value zc is the number such that the area under the standard normal curve falling between zc and zc is equal to c. The value zc is called the critical value for a condence level of c. For a condence level c, the critical value zc is the number such that the area under the standard normal curve between zc and zc equals c. The area under the normal curve from zc to zc is the probability that the standardized normal variable z lies in that interval. This means that P(zc 6 z 6 zc) c EX AM P LE 1 Find a critical value Let us use Table 5 of Appendix II to nd a number z0.99 such that 99% of the area under the standard normal curve lies between z0.99 and z0.99. That is, we will nd z0.99 such that P(z0.99 6 z 6 z0.99) 0.99 SOLUTION: In Section 6.3, we saw how to nd the z value when we were given an area between z and z. The rst thing we did was to nd the corresponding area Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 336 Chapter 7 ESTIMATION FIGURE 7-2 Area Between z and z Is 0.99 to the left of z. If A is the area between z and z, then (1 A)/2 is the area to the left of z. In our case, the area between z and z is 0.99. The corresponding area in the left tail is (1 0.99)/2 0.005 (see Figure 7-2). Next, we use Table 5 of Appendix II to nd the z value corresponding to a left-tail area of 0.0050. Table 7-1 shows an excerpt from Table 5 of Appendix II. TABLE 7-1 Excerpt from Table 5 of Appendix II z .00 3.4 ... .07 .08 .09 .0003 .0003 .0003 .0002 .0062 .0051 .0049 .0048 o 2.5 0 z c .0050 From Table 7-1, we see that the desired area, 0.0050, is exactly halfway between the areas corresponding to z 2.58 and z 2.57. Because the two area values are so close together, we use the more conservative z value 2.58 rather than interpolate. In fact, z0.99 2.576. However, to two decimal places, we use z0.99 2.58 as the critical value for a condence level of c 0.99. We have P(2.58 6 z 6 2.58) 0.99 The results of Example 1 will be used a great deal in our later work. For convenience, Table 7-2 gives some levels of condence and corresponding critical values zc. The same information is provided in Table 5(b) of Appendix II. An estimate is not very valuable unless we have some kind of measure of how \"good\" it is. The language of probability can give us an idea of the size of the margin of error caused by using the sample mean x as an estimate for the population mean. Remember that x is a random variable. Each time we draw a sample of size n from a population, we can get a different value for x. According to the central limit theorem, if the sample size is large, then x has a distribution that is approximately normal with mean mx m, the population mean we are trying to estimate. The standard deviation is sx s/ 1n. If x has a normal distribution, these results are true for any sample size. (See Theorem 6.1.) This information, together with our work on condence levels, leads us (as shown in the optional derivation that follows) to the probability statement z Pa c s s 6 x m 6 zc bc 1n 1n TABLE 7-2 (1) Some Levels of Condence and Their Corresponding Critical Values Level of Condence c Critical Value zc 0.70, or 70% 1.04 0.75, or 75% 1.15 0.80, or 80% 1.28 0.85, or 85% 1.44 0.90, or 90% 1.645 0.95, or 95% 1.96 0.98, or 98% 2.33 0.99, or 99% 2.58 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 7.1 Estimating m When s Is Known FIGURE 7-3 zc Distribution of Sample Means x 337 n Equation (1) uses the language of probability to give us an idea of the size of the margin of error for the corresponding condence level c. In words, Equation (1) states that the probability is c that our point estimate x is within a distance zc(s/ 1n) of the population mean m. This relationship is shown in Figure 7-3. In the following optional discussion, we derive Equation (1). If you prefer, you may jump ahead to the discussion about the margin of error. Optional derivation of Equation (1) For a c condence level, we know that P(zc 6 z 6 zc) c (2) This statement gives us information about the size of z, but we want information about the size of x m. Is there a relationship between z and x m? The answer is yes since, by the central limit theorem, x has a distribution that is approximately normal, with mean m and standard deviation s/ 1n. We can convert x to a standard z score by using the formula z s/ 1n xm s/ 1n (3) Multiplying all parts of the inequality in (4) by s/ 1n gives us Substituting this expression for z into Equation (2) gives P azc 6 P azc xm 6 zc b c s s 6 x m 6 zc bc 1n 1n (4) (1) Equation (1) is precisely the equation we set out to derive. Maximal margin of error, E The margin of error (or absolute error) using x as a point estimate for m is 0x m 0 . In most practical problems, m is unknown, so the margin of error is also unknown. However, Equation (1) allows us to compute an error tolerance E that serves as a bound on the margin of error. Using a c% level of condence, we can say that the point estimate x differs from the population mean m by a maximal margin of error s 1n Note: Formula (5) for E is based on the fact that the sampling distribution for x is exactly normal, with mean m and standard deviation s/ 1n. This occurs whenever the x distribution is normal with mean m and standard deviation . If the x E zc Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. (5) 338 Chapter 7 ESTIMATION distribution is not normal, then according to the central limit theorem, large samples (n 30) produce an x distribution that is approximately normal, with mean m and standard deviation s/ 1n. Using Equations (1) and (5), we conclude that P(E 6 x m 6 E) c (6) Equation (6) states that the probability is c that the difference between x and m is no more than the maximal error tolerance E. If we use a little algebra on the inequality E 6 x m 6 E (7) for m, we can rewrite it in the following mathematically equivalent way: Condence interval for m with s known xE 6 m 6 xE (8) Since formulas (7) and (8) are mathematically equivalent, their probabilities are the same. Therefore, from (6), (7), and (8), we obtain P(x E 6 m 6 x E) c (9) Equation (9) states that there is a chance c that the interval from x E to x E contains the population mean m. We call this interval a c condence interval for m. A c condence interval for m is an interval computed from sample data in such a way that c is the probability of generating an interval containing the actual value of m. In other words, c is the proportion of condence intervals, based on random samples of size n, that actually contain m. We may get a different condence interval for each different sample that is taken. Some intervals will contain the population mean m and others will not. However, in the long run, the proportion of condence intervals that contain m is c. P ROCEDU R E HOW TO FIND A is known CONFIDENCE INTERVAL FOR m when s Requirements Let x be a random variable appropriate to your application. Obtain a simple random sample (of size n) of x values from which you compute the sample mean x. The value of s is already known (perhaps from a previous study). If you can assume that x has a normal distribution, then any sample size n will work. If you cannot assume this, then use a sample size of n 30. Condence interval for m when s is known where x sample mean of a simple random sample s E zc 1n c confidence level (0 6 c 6 1) xE 6 m 6 xE (10) zc critical value for confidence level c based on the standard normal distribution (see Table 5(b) of Appendix II for frequently used values). Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 7.1 EX AM P LE 2 Estimating m When s Is Known 339 Condence interval for m with s known Julia enjoys jogging. She has been jogging over a period of several years, during which time her physical condition has remained constantly good. Usually, she jogs 2 miles per day. The standard deviation of her times is s 1.80 minutes. During the past year, Julia has recorded her times to run 2 miles. She has a random sample of 90 of these times. For these 90 times, the mean was x 15.60 minutes. Let m be the mean jogging time for the entire distribution of Julia's 2-mile running times (taken over the past year). Find a 0.95 condence interval for m. SOLUTION: Check Requirements We have a simple random sample of running times, and the sample size n 90 is large enough for the x distribution to be approximately normal. We also know s. It is appropriate to use the normal distribution to compute a condence interval for m. To compute E for the 95% condence interval x E to x E, we use the fact that zc 1.96 (see Table 7-2), together with the values n 90 and s 1.80. Therefore, Myrleen Ferguson Cate/PhotoEdit E zc s 1n E 1.96 a 1.80 b 190 E 0.37 Using Equation (10), the given value of x, and our computed value for E, we get the 95% condence interval for m. xE 6 m 6 xE 15.60 0.37 6 m 6 15.60 0.37 15.23 6 m 6 15.97 Interpretation We conclude with 95% confidence that the interval from 15.23 minutes to 15.97 minutes is one that contains the population mean m of jogging times for Julia. CR ITICAL TH I N KI NG Interpreting Condence Intervals A few comments are in order about the general meaning of the term condence interval. Since x is a random variable, the endpoints x E are also random variables. Equation (9) states that we have a chance c of obtaining a sample such that the interval, once it is computed, will contain the parameter m. After the condence interval is numerically xed for a specic sample, it either does or does not contain m. So, the probability is 1 or 0 that the interval, when it is xed, will contain m. A nontrivial probability statement can be made only about variables, not constants. Equation (9), P(x E 6 m 6 x E) c, really states that if we draw many random samples of size n and get lots of condence intervals, then the proportion of all intervals that will turn out to contain the mean m is c. For example, in Figure 7-4, on the next page, the horizontal lines represent 0.90 condence intervals for various samples of the same size from an x distribution. Some of these intervals contain m and others do not. Since the Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 340 Chapter 7 ESTIMATION intervals are 0.90 condence intervals, about 90% of all such intervals should contain m. For each sample, the interval goes from x E to x E. Once we have a specic condence interval for m, such as 3 6 m 6 5, all we can say is that we are c% condent that we have one of the intervals that actually contains m. Another appropriate statement is that at the c condence level, our interval contains m. Please see Using Technology at the end of this chapter for a computer demonstration of this discussion about confidence intervals. COMMENT FIGURE 7-4 0.90 Condence Intervals for Samples of the Same Size GUIDED EXERCISE 1 Confidence interval for M with S known Walter usually meets Julia at the track. He prefers to jog 3 miles. From long experience, he knows that s 2.40 minutes for his jogging times. For a random sample of 90 jogging sessions, the mean time was x 22.50 minutes. Let m be the mean jogging time for the entire distribution of Walter's 3-mile running times over the past several years. Find a 0.99 condence interval for m. (a) Check Requirements Is the x distribution approximately normal? Do we know s? Yes; we know this from the central limit theorem. Yes, s 2.40 minutes. (b) What is the value of z0.99? (See Table 7-2.) z0.99 2.58 (c) What is the value of E? E zc (d) What are the endpoints for a 0.99 condence interval for m? The endpoints are given by s 2.40 2.58 a b 0.65 1n 190 x E 22.50 0.65 21.85 x E 22.50 0.65 23.15 (e) Interpretation Explain what the condence interval tells us. We are 99% certain that the interval from 21.85 to 23.15 is an interval that contains the population mean time m. When we use samples to estimate the mean of a population, we generate a small error. However, samples are useful even when it is possible to survey the entire population, because the use of a sample may yield savings of time or effort in collecting data. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 7.1 Estimating m When s Is Known 341 LO O K I N G F O R WA R D The basic structure for most condence intervals for population parameters is sample statistic E 6 population parameter 6 sample statistic E where E is the maximal margin of error based on the sample statistic distribution and the level of condence c. We will see this same format used for condence intervals of the mean when s is unknown (Section 7.2), for proportions (Section 7.3), for differences of means from independent samples (Section 7.4), for differences of proportions (Section 7.4), and for parameters of linear regression (Chapter 9). This structure for condence intervals is so basic that some software packages, such as Excel simply give the value of E for a condence interval and expect the user to nish the computation. TE C H N OTE S The TI-84Plus/TI-83Plus/TI-nspire calculators, Excel 2007, and Minitab all support condence intervals for m from large samples. The level of support varies according to the technology. When a condence interval is given, the standard mathematical notation (lower value, upper value) is used. For instance, the notation (15.23, 15.97) means the interval from 15.23 to 15.97. TI-84Plus/TI-83Plus/TI-nspire (with TI-84Plus keypad) These calculators give the most exten- sive support. The user can opt to enter raw data or just summary statistics. In each case, the value of s must be specied. Press the STAT key, then select TESTS, and use 7:ZInterval. The TI-84Plus/TI-83Plus/TI-nspire output shows the results for Example 2. Excel 2007 Excel gives only the value of the maximal error of estimate E. On the Home screen click the Insert Function fx . In the dialogue box, select Statistical for the category, and then select Condence. In the resulting dialogue box, the value of alpha is 1 - condence level. For example, alpha is 0.05 for a 95% condence interval. The values of s and n are also required. The Excel output shows the value of E for Example 2. fx An alternate approach incorporating raw data (using the Student's t distribution presented in the next section) uses a selection from the Data Analysis package. Click the Data tab on the home ribbon. From the Analysis group, select Data Analysis. In the dialogue box, select Descriptive Statistics. Check the box by Condence Level for Mean, and enter the condence level. Again, the value of E for the interval is given. Minitab Raw data are required. Use the menu choices Stat Basic Statistics 1-SampleZ. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 342 Chapter 7 ESTIMATION Sample Size for Estimating the Mean M In the design stages of statistical research projects, it is a good idea to decide in advance on the condence level you wish to use and to select the maximal margin of error E you want for your project. How you choose to make these decisions depends on the requirements of the project and the practical nature of the problem. Whatever specications you make, the next step is to determine the sample size. Solving the formula that gives the maximal margin of error E for n enables us to determine the minimal sample size. P ROCEDU R E HOW TO FIND THE s is known SAMPLE SIZE n for estimating m when Requirements The distribution of sample means x is approximately normal. Formula for sample size na zc s 2 b E (11) where E specified maximal margin of error s population standard deviation zc critical value from the normal distribution for the desired condence level c. Commonly used values of zc can be found in Table 5(b) of Appendix II. If n is not a whole number, increase n to the next higher whole number. Note that n is the minimal sample size for a specied condence level and maximal error of estimate E. COMMENT If you have a preliminary study involving a sample size of 30 or larger, then for most practical purposes it is safe to approximate s with the sample standard deviation s in the formula for sample size. EX AM P LE 3 Sample size for estimating m A wildlife study is designed to nd the mean weight of salmon caught by an Alaskan fishing company. A preliminary study of a random sample of 50 salmon showed s 2.15 pounds. How large a sample should be taken to be 99% condent that the sample mean x is within 0.20 pound of the true mean weight m? SOLUTION: In this problem, z0.99 2.58 (see Table 7-2) and E 0.20. The preliminary study of 50 sh is large enough to permit a good approximation of s by s 2.15. Therefore, Equation (6) becomes Photo Researchers na Salmon moving upstream (2.58)(2.15) 2 zc s 2 b a b 769.2 E 0.20 Note: In determining sample size, any fractional value of n is always rounded to the next higher whole number. We conclude that a sample size of 770 will be large enough to satisfy the specications. Of course, a sample size larger than 770 also works. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 7.1 VI EWPOI NT Estimating m When s Is Known 343 Music and Techno Theft Performing rights organizations ASCAP (American Society of Composers, Authors, and Publishers) and BMI (Broadcast Music, Inc.) collect royalties for songwriters and music publishers. Radio, television, cable, nightclubs, restaurants, elevators, and even beauty salons play music that is copyrighted by a composer or publisher. The royalty payment for this music turns out to be more than a billion dollars a year (Source: Wall Street Journal). How do ASCAP and BMI know who is playing what music? The answer is, they don't! Instead of tracking exactly what gets played, they use random sampling and condence intervals. For example, each radio station (there are more than 10,000 in the United States) has randomly chosen days of programming analyzed every year. The results are used to assess royalty fees. In fact, Deloitte & Touche (a nancial services company) administers the sampling process. Although the system is not perfect, it helps bring order into an otherwise chaotic accounting system. Such methods of \"copyright policing\" help prevent techno theft, ensuring that many songwriters and recording artists get a reasonable return for their creative work. SECTION 7.1 P ROB LEM S In Problems 1-8, answer true or false. Explain your answer. 1. Statistical Literacy The value zc is a value from the standard normal distribution such that P(zc 6 x 6 zc) c. 2. Statistical Literacy The point estimate for the population mean m of an x distribution is x, computed from a random sample of the x distribution. 3. Statistical Literacy Consider a random sample of size n from an x distribution. For such a sample, the margin of error for estimating m is the magnitude of the difference between x and m. 4. Statistical Literacy Every random sample of the same size from a given population will produce exactly the same condence interval for m. 5. Statistical Literacy A larger sample size produces a longer condence interval for m. 6. Statistical Literacy If the original x distribution has a relatively small standard deviation, the condence interval for m will be relatively short. 7. Statistical Literacy If the sample mean x of a random sample from an x distribution is relatively small, then the condence interval for m will be relatively short. 8. Statistical Literacy For the same random sample, when the condence level c is reduced, the condence interval for m becomes shorter. 9. Critical Thinking Sam computed a 95% condence interval for m from a specic random sample. His condence interval was 10.1 6 m 6 12.2. He claims that the probability that m is in this interval is 0.95. What is wrong with his claim? 10. Critical Thinking Sam computed a 90% condence interval for m from a specic random sample of size n. He claims that at the 90% condence level, his condence interval contains m. Is his claim correct? Explain. Answers may vary slightly due to rounding. 11. Basic Computation: Condence Interval Suppose x has a normal distribution with s 6. A random sample of size 16 has sample mean 50. (a) Check Requirements Is it appropriate to use a normal distribution to compute a condence interval for the population mean m? Explain. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 344 Chapter 7 ESTIMATION (b) Find a 90% condence interval for m. (c) Interpretation Explain the meaning of the condence interval you computed. 12. Basic Computation: Condence Interval Suppose x has a mound-shaped distribution with s 9. A random sample of size 36 has sample mean 20. (a) Check Requirements Is it appropriate to use a normal distribution to compute a condence interval for the population mean m? Explain. (b) Find a 95% condence interval for m. (c) Interpretation Explain the meaning of the condence interval you computed. 13. Basic Computation: Sample Size Suppose x has a mound-shaped distribution with s 3. (a) Find the minimal sample size required so that for a 95% condence interval, the maximal margin of error is E 0.4. (b) Check Requirements Based on this sample size, can we assume that the x distribution is approximately normal? Explain. 14. Basic Computation: Sample Size Suppose x has a normal distribution with s 1.2. (a) Find the minimal sample size required so that for a 90% condence interval, the maximal margin of error is E 0.5. (b) Check Requirements Based on this sample size and the x distribution, can we assume that the x distribution is approximately normal? Explain. 15. Zoology: Hummingbirds Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther (Reference: Hummingbirds by K. Long and W. Alther). A small group of 15 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with s 0.33 gram. (a) Find an 80% condence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (b) What conditions are necessary for your calculations? (c) Interpret Compare your results in the context of this problem. (d) Sample Size Find the sample size necessary for an 80% condence level with a maximal margin of error E 0.08 for the mean weights of the hummingbirds. 16. Diagnostic Tests: Uric Acid Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advance indication of illness such as gout, leukemia, or lymphoma (Reference: Manual of Laboratory and Diagnostic Tests by F. Fischbach). Over a period of months, an adult male patient has taken eight blood tests for uric acid. The mean concentration was x 5.35 mg/dl. The distribution of uric acid in healthy adult males can be assumed to be normal, with s 1.85 mg/dl. (a) Find a 95% condence interval for the population mean concentration of uric acid in this patient's blood. What is the margin of error? (b) What conditions are necessary for your calculations? (c) Interpret Compare your results in the context of this problem. (d) Sample Size Find the sample size necessary for a 95% condence level with maximal margin of error E 1.10 for the mean concentration of uric acid in this patient's blood. 17. Diagnostic Tests: Plasma Volume Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is inuenced by the overall health and physical activity of an individual. (Reference: See Problem 16.) Suppose that a random sample of 45 male reghters are tested and that they have a plasma Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 7.1 Estimating m When s Is Known 345 volume sample mean of x 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that s 7.50 ml/kg for the distribution of blood plasma. (a) Find a 99% condence interval for the population mean blood plasma volume in male reghters. What is the margin of error? (b) What conditions are necessary for your calculations? (c) Interpret Compare your results in the context of this problem. (d) Sample Size Find the sample size necessary for a 99% condence level with maximal margin of error E 2.50 for the mean plasma volume in male reghters. 18. Agriculture: Watermelon What price do farmers get for their watermelon crops? In the third week of July, a random sample of 40 farming regions gave a sample mean of x $6.88 per 100 pounds of watermelon. Assume that s is known to be $1.92 per 100 pounds (Reference: Agricultural Statistics, U.S. Department of Agriculture). (a) Find a 90% condence interval for the population mean price (per 100 pounds) that farmers in this region get for their watermelon crop. What is the margin of error? (b) Sample Size Find the sample size necessary for a 90% condence level with maximal margin of error E 0.3 for the mean price per 100 pounds of watermelon. (c) A farm brings 15 tons of watermelon to market. Find a 90% condence interval for the population mean cash value of this crop. What is the margin of error? Hint: 1 ton is 2000 pounds. 19. FBI Report: Larceny Thirty small communities in Connecticut (population near 10,000 each) gave an average of x 138.5 reported cases of larceny per year. Assume that s is known to be 42.6 cases per year (Reference: Crime in the United States, Federal Bureau of Investigation). (a) Find a 90% condence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (b) Find a 95% condence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (c) Find a 99% condence interval for the population mean annual number of reported larceny cases in such communities. What is the margin of error? (d) Compare the margins of error for parts (a) through (c). As the condence levels increase, do the margins of error increase? (e) Critical Thinking: Compare the lengths of the condence intervals for parts (a) through (c). As the condence levels increase, do the condence intervals increase in length? 20. Condence Intetvals: Values of s A random sample of size 36 is drawn from an x distribution. The sample mean is 100. (a) Suppose the x distribution has s 30. Compute a 90% condence interval for m. What is the value of the margin of error? (b) Suppose the x distribution has s 20. Compute a 90% condence interval for m. What is the value of the margin of error? (c) Suppose the x distribution has s 10. Compute a 90% condence interval for m. What is the value of the margin of error? (d) Compare the margins of error for parts (a) through (c). As the standard deviation decreases, does the margin of error decrease? (e) Critical Thinking Compare the lengths of the condence intervals for parts (a) through (c). As the standard deviation decreases, does the length of a 90% condence interval decrease? 21. Condence Intervals: Sample Size A random sample is drawn from a population with s 12. The sample mean is 30. (a) Compute a 95% condence interval for m based on a sample of size 49. What is the value of the margin of error? Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 346 Chapter 7 ESTIMATION (b) Compute a 95% condence interval for m based on a sample of size 100. What is the value of the margin of error? (c) Compute a 95% condence interval for m based on a sample of size 225. What is the value of the margin of error? (d) Compare the margins of error for parts (a) through (c). As the sample size increases, does the margin of error decrease? (e) Critical Thinking Compare the lengths of the condence intervals for parts (a) through (c). As the sample size increases, does the length of a 90% condence interval decrease? 22. Ecology: Sand Dunes At wind speeds above 1000 centimeters per second (cm/sec), signicant sand-moving events begin to occur. Wind speeds below 1000 cm/sec deposit sand, and wind speeds above 1000 cm/sec move sand to new locations. The cyclic nature of wind and moving sand determines the shape and location of large dunes (Reference: Hydraulic, Geologic, and Biologic Research at Great Sand Dunes National Monument and Vicinity, Colorado, Proceedings of the National Park Service Research Symposium). At a test site, the prevailing direction of the wind did not change noticeably. However, the velocity did change. Sixty wind speed readings gave an average velocity of x 1075 cm/sec. Based on long-term experience, s can be assumed to be 265 cm/sec. (a) Find a 95% condence interval for the population mean wind speed at this site. (b) Interpretation Does the condence interval indicate that the population mean wind speed is such that the sand is always moving at this site? Explain. 23. Prots: Banks Jobs and productivity! How do banks rate? One way to answer this question is to examine annual profits per employee. Forbes Top Companies, edited by J. T. Davis (John Wiley & Sons), gave the following data about annual profits per employee (in units of one thousand dollars per employee) for representative companies in nancial services. Companies such as Wells Fargo, First Bank System, and Key Banks were included. Assume s 10.2 thousand dollars. 42.9 43.8 48.2 60.6 54.9 55.1 52.9 54.9 42.5 33.0 33.6 36.9 27.0 47.1 33.8 28.1 28.5 29.1 36.5 36.1 26.9 27.8 28.8 29.3 31.5 31.7 31.1 38.0 32.0 31.7 32.9 23.1 54.9 43.8 36.9 31.9 25.5 23.2 29.8 22.3 26.5 26.7 (a) Use a calculator or appropriate computer software to verify that, for the preceding data, x 36.0. (b) Let us say that the preceding data are representative of the entire sector of (successful) nancial services corporations. Find a 75% condence interval for m, the average annual prot per employee for all successful banks. (c) Interpretation Let us say that you are the manager of a local bank with a large number of employees. Suppose the annual prots per employee are less than 30 thousand dollars per employee. Do you think this might be somewhat low compared with other successful nancial institutions? Explain by referring to the condence interval you computed in part (b). (d) Interpretation Suppose the annual prots are more than 40 thousand dollars per employee. As manager of the bank, would you feel somewhat better? Explain by referring to the condence interval you computed in part (b). (e) Repeat parts (b), (c), and (d) for a 90% condence level. 24. Prots: Retail Jobs and productivity! How do retail stores rate? One way to answer this question is to examine annual prots per employee. The following data give annual prots per employee (in units of one thousand dollars per employee) for companies in retail sales. (See reference in Problem 23.) Companies such as Gap, Nordstrom, Dillards, JCPenney, Sears, Wal-Mart, Ofce Depot, and Toys \" R \" Us are included. Assume s 3.8 thousand dollars. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 7.2 347 Estimating m When s Is Unknown 4.4 11.9 1.7 6.5 8.2 9.4 4.2 6.4 5.5 8.9 4.7 5.8 8.7 5.5 4.7 8.1 4.8 6.2 6.1 3.0 15.0 6.0 4.3 4.1 2.6 6.0 3.7 2.9 1.5 5.1 8.1 2.9 4.2 1.9 4.8 Images Etc Ltd /Getty images (a) Use a calculator or appropriate computer software to verify that, for the preceding data, x 5.1. (b) Let us say that the preceding data are representative of the entire sector of retail sales companies. Find an 80% condence interval for m, the average annual prot per employee for retail sales. (c) Interpretation Let us say that you are the manager of a retail store with a large number of employees. Suppose the annual prots per employee are less than 3 thousand dollars per employee. Do you think this might be low compared with other retail stores? Explain by referring to the condence interval you computed in part (b). (d) Interpretation Suppose the annual prots are more than 6.5 thousand dollars per employee. As store manager, would you feel somewhat better? Explain by referring to the condence interval you computed in part (b). (e) Repeat parts (b), (c), and (d) for a 95% condence interval. S E C T I O N 7. 2 25. Ballooning: Air Temperature How hot is the air in the top (crown) of a hot air balloon? Information from Ballooning: The Complete Guide to Riding the Winds by Wirth and Young (Random House) claims that the air in the crown should be an average of 100C for a balloon to be in a state of equilibrium. However, the temperature does not need to be exactly 100C. What is a reasonable and safe range of temperatures? This range may vary with the size and (decorative) shape of the balloon. All balloons have a temperature gauge in the crown. Suppose that 56 readings (for a balloon in equilibrium) gave a mean temperature of x 97C. For this balloon, s 17C. (a) Compute a 95% condence interval for the average temperature at which this balloon will be in a steady-state equilibrium. (b) Interpretation If the average temperature in the crown of the balloon goes above the high end of your condence interval, do you expect that the balloon will go up or down? Explain. Estimating M When S Is Unknown FOCUS POINTS Learn about degrees of freedom and Student's t distributions. Find critical values using degrees of freedom and condence levels. Compute condence intervals for m when s is unknown. What does this information tell you? Student's t distribution In order to use the normal distribution to nd condence intervals for a population mean m, we need to know the value of s, the population standard deviation. However, much of the time, when m is unknown, s is unknown as well. In such cases, we use the sample standard deviation s to approximate s. When we use s to approximate s, the sampling distribution for x follows a new distribution called a Student's t distribution. Student's t Distributions Student's t distributions were discovered in 1908 by W. S. Gosset. He was employed as a statistician by Guinness brewing company, a company that discouraged publication of research by its employees. As a result, Gosset published Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 348 Chapter 7 ESTIMATION his research under the pseudonym Student. Gosset was the rst to recognize the importance of developing statistical methods for obtaining reliable information from samples of populations with unknown s. Gosset used the variable t when he introduced the distribution in 1908. To this day and in his honor, it is still called a Student's t distribution. It might be more tting to call this distribution Gosset's t distribution; however, in the literature of mathematical statistics, it is known as a Student's t distribution. The variable t is dened as follows. A Student's t distribution depends on sample size n. Assume that x has a normal distribution with mean m. For samples of size n with sample mean x and sample standard deviation s, the t variable t Degrees of freedom, d.f. xm s 1n (12) has a Student's t distribution with degrees of freedom d.f. n 1. If many random samples of size n are drawn, then we get many t values from Equation (12). These t values can be organized into a frequency table, and a histogram can be drawn, thereby giving us an idea of the shape of the t distribution (for a given n). Fortunately, all this work is not necessary because mathematical theorems can be used to obtain a formula for the t distribution. However, it is important to observe that these theorems say that the shape of the t distribution depends only on n, provided the basic variable x has a normal distribution. So, when we use a t distribution, we will assume that the x distribution is normal. Table 6 of Appendix II gives values of the variable t corresponding to what we call the number of degrees of freedom, abbreviated d.f. For the methods used in this section, the number of degrees of freedom is given by the formula d.f. n 1 (13) where d.f. stands for the degrees of freedom and n is the sample size. Each choice for d.f. gives a different t distribution. The graph of a t distribution is always symmetrical about its mean, which (as for the z distribution) is 0. The main observable difference between a t distribution and the standard normal z distribution is that a t distribution has somewhat thicker tails. Figure 7-5 shows a standard normal z distribution and Student's t distribution with d.f. 3 and d.f. 5. FIGURE 7-5 A Standard Normal Distribution and Student's t Distribution with d.f. 3 and d.f. 5 Properties of a Student's t distribution 1. The distribution is symmetric about the mean 0. 2. The distribution depends on the degrees of freedom, d.f. (d.f. n 1 for m condence intervals). 3. The distribution is bell-shaped, but has thicker tails than the standard normal distribution. 4. As the degrees of freedom increase, the t distribution approaches the standard normal distribution. 5. The area under the entire curve is 1. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 7.2 349 Estimating m When s Is Unknown Using Table 6 to Find Critical Values for Condence Intervals Critical values tc FIGURE 7-6 Area Under the t Curve Between tc and tc Area = c tc 0 tc Table 6 of Appendix II gives various t values for different degrees of freedom d.f. We will use this table to nd critical values tc for a c condence level. In other words, we want to nd tc such that an area equal to c under the t distribution for a given number of degrees of freedom falls between tc and tc. In the language of probability, we want to nd tc such that P(tc 6 t 6 tc) c This probability corresponds to the shaded area in Figure 7-6. Table 6 of Appendix II has been arranged so that c is one of the column headings, and the degrees of freedom d.f. are the row headings. To nd tc for any specic c, we nd the column headed by that c value and read down until we reach the row headed by the appropriate number of degrees of freedom d.f. (You will notice two other column headings: one-tail area and two-tail area. We will use these later, but for the time being, just ignore them.) Convention for using a Student's t distribution table If the degrees of freedom d.f. you need are not in the table, use the closest d.f. in the table that is smaller. This procedure results in a critical value tc that is more conservative, in the sense that it is larger. The resulting condence interval will be longer and have a probability that is slightly higher than c. EX AM P LE 4 Student's t distribution Use Table 7-3 (an excerpt from Table 6 of Appendix II) to nd the critical value tc for a 0.99 condence level for a t distribution with sample size n 5. SOLUTION: (a) First, we nd the column with c heading 0.990. (b) Next, we compute the number of degrees of freedom: d.f. n 1 5 1 4. (c) We read down the column under the heading c 0.99 until we reach the row headed by 4 (under d.f.). The entry is 4.604. Therefore, t0.99 4.604. TABLE 7-3 Student's t Distribution Critical Values (Excerpt from Table 6, Appendix II) one-tail area two-tail area . . . 0.900 0.950 0.980 0.990 . . . 3 . . . 2.353 3.182 4.541 5.841 . . . 4 . . . 2.132 2.776 3.747 4.604 . . . 7 . . . 1.895 2.365 2.998 3.449 . . . 8 . . . 1.860 2.306 2.896 3.355 . . . d.f. c o o GUIDED EXERCISE 2 Student's t distribution table Use Table 6 of Appendix II (or Table 7-3, showing an excerpt from the table) to nd tc for a 0.90 condence level for a t distribution with sample size n 9. Continued Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 350 Chapter 7 GUIDED EXERCISE 2 ESTIMATION continued (a) We nd the column headed by c __________. c 0.900. (b) The degrees of freedom are given by d.f. n 1 __________. d.f. n 1 9 1 8. (c) Read down the column found in part (a) until you reach the entry in the row headed by d.f. 8. The value of t0.90 is __________ for a sample of size 9. t0.90 1.860 for a sample of size n 9. (d) Find tc for a 0.95 condence level for a t distribution with sample size n 9. t0.95 2.306 for a sample of size n 9. LO O K I N G F O R WA R D Student's t distributions will be used again in Chapter 8 when testing m and when testing differences of means. The distributions are also used for condence intervals and testing of parameters of linear regression (Sections 9.3 and 9.4). Condence Intervals for M When S Is Unknown Maximal margin of error, E s 1n In Section 7.1, we found bounds E on the margin of error for a c condence level. Using the same basic approach, we arrive at the conclusion that E tc is the maximal margin of error for a c confidence level when s is unknown (i.e., 0x m 0 6 E with probability c). The analogue of Equation (1) in Section 7.1 is P atc s s 6 x m 6 tc bc 1n 1n (14) Comparing Equation (14) with Equation (1) in Section 7.1, it becomes evident that we are using the same basic method on the t distribution that we used on the z distribution. COMMENT where E tc(s/ 1n). Let us organize what we have been doing in a convenient summary. Likewise, for samples from normal populations with unknown s, Equation (9) of Section 7.1 becomes P(x E 6 m 6 x E) c P ROCEDU R E Condence interval for m with s unknown HOW TO FIND A unknown CONFIDENCE INTERVAL FOR (15) m when s is Requirements Let x be a random variable appropriate to your application. Obtain a simple random sample (of size n) of x values from which you compute the sample mean x and the sample standard deviation s. Continued