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2 Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6 10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. ISBN 10: ISBN 10: ISBN 13: ISBN 13: British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America

3 188 C HAPTER 4 Discrete Probability Distributions Cooperative Group Activities 2. Determining the Effectiveness of an HIV Training Program The New York State Health Department reports a 10% rate of the HIV virus for the at-risk population. In one region, an intensive education program is used in an attempt to lower that 10% rate. After running the program, a follow-up study of 150 at-risk individuals is conducted. a. Assuming that the program has no effect, find the mean and standard deviation for the number of HIV cases in groups of 150 at-risk people. b. Among the 150 people in the follow-up study, 8% (or 12 people) tested positive for the HIV virus. If the program has no effect, is that rate unusually low? Does this result suggest that the program is effective? 1. In-class activity: Suppose we want to identify the probability distribution for the number of children born to randomly selected couples. For each student in the class, find the number of brothers and sisters and record the total number of children (including the student) in each family. Construct the relative frequency table for the result obtained. (The values of the random variable x will be 1, 2, 3, cd What is wrong with using this relative frequency table as an estimate of the probability distribution for the number of children born to randomly selected couples? 2. Out-of-class activity: See Cumulative Review Exercise 1, which suggests that an analysis of the last digits of data can sometimes reveal whether the data have been collected through actual measurements or reported by the subjects. Refer to an almanac or the Internet and find a collection of data (such as lengths of rivers in the world), then analyze the distribution of last digits to determine whether the values were obtained through actual measurements. Technology Project In mornings and afternoons, the Newport Hospital emergency room can accommodate a maximum of 12 patients in one hour, and the admissions procedure is designed to accommodate a maximum of five patients in one hour. Past experience suggests that for this hospital, an average of 28% of emergency room patients are admitted to the hospital for further care. We can analyze the probability distribution of admissions by considering a binomial distribution with n 5 12 and p Find the probability that when 12 patients show up in the emergency room in one hour, the admissions procedure is overloaded. That is, find the probability of at least 6 patients requiring admission, assuming that 12 patients show up in the emergency room in one hour. Because of the value of p , Table A-1 cannot be used, and calculations with the binomial probability formula would be tedious. The best approach is to use statistics software or a TI-83> 84 Plus calculator. Use technology to find the relevant probability. Is the probability of an overloaded admissions procedure small enough so that it does not happen very often, or does it seem too high so that changes must be made to make it lower? 188

From Data to Decision 189 from DATA to DECISION Critical Thinking: Determining criteria for concluding that a genderselection method is effective You are responsible for analyzing results from a

4 From Data to Decision 189 from DATA to DECISION Critical Thinking: Determining criteria for concluding that a genderselection method is effective You are responsible for analyzing results from a clinical trial of the effectiveness of a new method of gender selection. Assume that the sample size of n 5 50 couples has already been established, and each couple will have one child. Further assume that each of the couples will be subjected to a treatment that supposedly increases the likelihood that the child will be a girl. There is a danger in obtaining results first, then making conclusions about the results. If the results are close to showing the effectiveness of a treatment, it might be tempting to conclude that there is an effect when, in reality, there is no effect. It is better to establish criteria before obtaining results. Using the methods of this chapter, identify the criteria that should be used for concluding that the treatment is effective in increasing the likelihood of a girl. Among the 50 births, how many girls would you require in order to conclude that the gender-selection procedure is effective? Explain how you arrived at this result. 189

Normal Probability Distributions 5-1 Overview 5-2 The Standard Normal Distribution 5-3 Applications of Normal Distributions 5-4 Sampling Distributions and Estimators 5-5 The Central Limit Theorem 5-6

5 Normal Probability Distributions 5-1 Overview 5-2 The Standard Normal Distribution 5-3 Applications of Normal Distributions 5-4 Sampling Distributions and Estimators 5-5 The Central Limit Theorem 5-6 Normal as Approximation to Binomial 5-7 Assessing Normality From Chapter 5 of Biostatistics for the Biological and Health Sciences with Statdisk, First Edition. Marc M. Triola, Mario F. Triola. Copyright 2006 by Pearson Education, Inc. All rights reserved. 190

C HAPTER P R OBLEM Ergonomics: Engineering with biology A relatively new discipline is ergonomics, the study of people fitting into their environments.

Ergonomics is used in applications including the design of ambulance stretchers, car dashboards, cycle helmets, screw bottle tops, door handles, manhole covers, keyboards, air traffic control

6 C HAPTER P R OBLEM Ergonomics: Engineering with biology A relatively new discipline is ergonomics, the study of people fitting into their environments. Good ergonomic design results in an environment that is safe, functional, efficient, and comfortable. Ergonomics is used in applications including the design of ambulance stretchers, car dashboards, cycle helmets, screw bottle tops, door handles, manhole covers, keyboards, air traffic control centers, and computer assembly lines. For example, in Vail, Colorado, a gondola carries skiers to the top of a mountain. It bears a plaque stating that the maximum capacity is 12 people or 2004 pounds. Reading that plaque causes many passengers to look around wondering whether they are in danger because there are too many fellow passengers or because there are 12 or fewer passengers who, by chance random variation, are all exceptionally heavy. How likely is it that 12 randomly selected people will have a total weight greater than 2004 pounds? The ability to comfortably endure a long transcontinental flight is affected by the width of the seat we must occupy. Most U.S. commercial aircraft have seat widths of 17 in. to 18 in., which just barely accommodates the 16 in. required by the average passenger. Business and first-class seats usually have widths between 19 in. and 21 in., so the extra room allows a greater degree of comfort. If American Airlines wants to gain business by increasing passenger comfort, what width should be used in redesigned seats? When visiting preserved living quarters that are hundreds of years old, many people are struck by the fact that the doorways have openings that are too low for most people living today. When walking through a modern doorway, most of us fit comfortably under the top threshold, which is typically 80 in. high. However, some people are exceptionally tall and must bend to avoid a bump on the head. What percent of people are too tall for the current doorway design standards? In recent years, the United States Air Force recognized that women make perfectly good pilots of fighter jets. Cockpits of fighter jets were originally designed for men only, so various changes were required to better accommodate the new women pilots. One such change involved a redesign of the ACES-II ejection seats. Because they were originally designed for men who weighed between 140 and 211 1b, the ejection seats provided a greater chance of injury for any women pilots weighing less than 140 1b or more than 211 1b. What weights should be used for the new cockpit design? In this chapter we address questions such as those posed above. 191

192 C HAPTER 5 Normal Probability Distributions 5-1 Overview In Chapter 2 we considered important measures of data sets, including measures of center and variation, as well as the distribution of

7 192 C HAPTER 5 Normal Probability Distributions 5-1 Overview In Chapter 2 we considered important measures of data sets, including measures of center and variation, as well as the distribution of data. In Chapter 3 we discussed basic principles of probability, and in Chapter 4 we presented the following concepts: Changing Populations Included among the five important data set characteristics listed in Chapter 2 is the changing pattern of data over time. Some populations change, and their important statistics change as well. Car seat belt standards haven t changed in 40 years, even though the weights of Americans have increased considerably since then. In 1960, 12.8% of adult Americans were considered obese, compared with 22.6% in According to the National Highway Traffic Safety Administration, seat belts must fit a standard crash dummy (designed according to 1960 data) placed in the most forward position, with 4 in. to spare. In theory, 95% of men and 99% of women should fit into seat belts, but those percentages are now lower because of the increases in weight over the last half-century. Some car companies provide seat belt extenders, but some do not. A random variable is a variable having a single numerical value, determined by chance, for each outcome of some procedure. A probability distribution for a discrete random variable describes the probability for each value of the random variable. A discrete random variable has either a finite number of values or a countable number of values. That is, the number of possible values that x can assume is 0, or 1, or 2, and so on. A continuous random variable has infinitely many values, and those values are often associated with measurements on a continuous scale with no gaps or interruptions. In Chapter 4 we considered only discrete probability distributions, but in this chapter we present continuous probability distributions. Although we begin with a uniform distribution, most of the chapter will focus on normal distributions. Normal distributions are extremely important because they occur so often in real applications and they play such an important role in methods of inferential statistics. Normal distributions will be used often throughout the remainder of this text. Definition If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped, as in Figure 5-1, and it can be described by the equation given as Formula 5-1, we say that it has a normal distribution. 1 Formula 5-1 y 5 e2 2 Qx2m s R 2 s 22p The complexity of Formula 5-1 can be intimidating, but it isn t really necessary for us to actually use Formula 5-1. That formula does show, however, that any particular normal distribution is determined by two parameters: the mean, m, FIGURE 5-1 The Normal Distribution Curve is bell-shaped and symmetric m Value 192

Research in Education. Tenth Edition

Research in Education. Tenth Edition Research in Education John W. Best James V. Kahn Tenth Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World