Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world

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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 chapter problem Reaching new heights Ergonomics is the study of people fitting into their environments, and heights are extremely important in many applications. Section 4.2 of the Americans with Disabilities Act relates to height clearances with this statement: Walks, halls, corridors, passageways, aisles, or other circulation spaces shall have 80 in. (2030 mm) minimum clear head room. A CBS News report identified many low-hanging signs in New York City subway walkways that violated that requirement with height clearances less than 80 in. Even when that 80 in. minimum height clearance is maintained, not all people can walk through without bending. Due to aircraft cabin designs and other considerations, British Airways and many other carriers have a cabin crew height requirement between 5 ft 2 in. and 6 ft 1 in. For aesthetic reasons, Rockette dancers at New York s Radio City Music Hall must be between 66.5 in. and 71.5 in. tall. For practical reasons, the U.S. Army requires that women must be between 58 in. and 80 in. tall. For social reasons, Tall Clubs International requires that male members must be 1 Review and Preview 2 The Standard Normal Distribution 3 Applications of Normal Distributions 4 Sampling Distributions and Estimators at least 6 ft 2 in. tall, and women members must be at least 5 ft 10 in. tall. Given that heights are so important in so many different circumstances, what do we know about heights? We know that an investigation of heights should involve much more than simply finding a mean. We should consider the CVDOT elements of center, variation, distribution, outliers, and changes over time. We might use the mean as a measure of center, the standard deviation as a measure of variation, and the histogram as a tool for visualizing the distribution of the data, and we should determine whether outliers are present. We should also consider whether we are dealing with a static population or one that is changing over time. For heights of adults, we might refer to Data Set 1 in Appendix: Data Sets to estimate that heights of adult males have a mean of 69.5 in. and a standard deviation of 2.4 in., while heights of adult females have a mean of 63.8 in. and a standard deviation of 2.6 in. (These values are very close to the values that would be obtained by using a much larger sample.) For distributions of heights, we might examine histograms, such as those shown here (based on Data Set 1 in Appendix: Data Sets). Note that the histograms appear to be roughly bell-shaped, suggesting that the heights are from populations having normal distributions (as described in Section 3). For outliers, we might examine the histograms and note that there is one male with a height that is somewhat, but not dramatically, different from the others. Also, we know that heights are changing over time, so our studies will focus on current heights, not heights from past or future centuries. 5 Central Limit Theorem 6 Assessing Normality 7 Normal as Approximation to Binomial 247

4 This chapter introduces the statistical tools that are basic to good ergonomic design. After completing this chapter, we will be able to solve problems in a wide variety of different disciplines as well. We will be able to answer questions such as these: What percentages of men and women can easily navigate in an area with the height clearance of 80 in. that is stipulated in the Americans with Disabilities Act? What percentages of men and women satisfy the flight cabin crew requirement of having a height between 5 ft 2 in. and 6 ft 1 in.? What percentage of women are eligible for membership in Tall Clubs International because they are at least 5 ft 10 in. tall? Current doorways are typically 6 ft 8 in. tall, but if we were to redesign doorways to accommodate 99% of the population, what should the height be? 1 Review and Preview In this chapter we introduce continuous probability distributions. To illustrate the correspondence between area and probability, we begin with a uniform distribution, but most of this chapter focuses on normal distributions. Normal distributions occur often in real applications, and they play an important role in methods of inferential statistics. Here we present concepts of normal distributions. Several of the statistical methods are based on concepts related to the central limit theorem discussed in Section 5. Many other sections require normally distributed populations, and Section 6 presents methods for analyzing sample data to determine whether the sample appears to be from a normally distributed population. DEFINITION If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped, as in Figure 1, and it can be described by the equation given as Formula 1, we say that it has a normal distribution. Curve is bell-shaped and symmetric Formula 1 y = e x - m s 22 s22p m Value Figure 1 The Normal Distribution We won t actually use Formula 1, and we include it only to illustrate that any particular normal distribution is determined by two parameters: the mean, m, and standard deviation, s. In that formula, the letter p represents the constant value and e represents the constant value The symbols m and s represent 248

5 fixed values for the mean and standard deviation, respectively. Once specific values are selected for m and s, we can graph Formula 1 and the result will look like Figure 1. From Formula 1 we see that a normal distribution is determined by the fixed values of the mean m and standard deviation s. Fortunately, that s all we need to know about that formula. 2 The Standard Normal Distribution Key Concept In this section we present the standard normal distribution, which has these three properties: 1. The graph of the standard normal distribution is bell-shaped (as in Figure 1). 2. The standard normal distribution has a mean equal to 0 (that is, m = 0). 3. The standard normal distribution has a standard deviation equal to 1 (that is, s = 1). In this section we develop the skill to find areas (or probabilities or relative frequencies) corresponding to various regions under the graph of the standard normal distribution. In addition, we find z scores that correspond to areas under the graph. These skills become important as we study nonstandard normal distributions and all of the real and important applications that they involve. Uniform Distributions The focus of this chapter is the concept of a normal probability distribution, but we begin with a uniform distribution. The uniform distribution allows us to see the following two very important properties: 1. The area under the graph of a probability distribution is equal to There is a correspondence between area and probability (or relative frequency), so some probabilities can be found by identifying the corresponding areas in the graph. We consider continuous probability distributions, beginning with the uniform distribution. DEFINITION A continuous random variable has a uniform distribution if its values are spread evenly over the range of possibilities. The graph of a uniform distribution results in a rectangular shape. Example 1 Subway to Mets Game For New York City weekday late-afternoon subway travel from Times Square to the Mets stadium, you can take the #7 train that leaves Times Square every 5 minutes. Given the subway departure schedule and the arrival of a passenger, the waiting time x is between 0 min and 5 min, as described by the uniform distribution depicted in Figure 2. Note that in Figure 2, waiting times can be any value between 0 min and 5 min, so it is possible to have a waiting time of min. Note also that all of the different possible waiting times are equally likely. The graph of a continuous probability distribution, such as in Figure 2, is called a density curve. A density curve must satisfy the following two requirements. 249

6 0.2 P(x) Area = x (waiting time in minutes) Figure 2 Uniform Distribution of Waiting Time Requirements for a Density Curve 1. The total area under the curve must equal Every point on the curve must have a vertical height that is 0 or greater. (That is, the curve cannot fall below the x-axis.) By setting the height of the rectangle in Figure 2 to be 0.2, we force the enclosed area to be 0.2 * 5 = 1, as required. (In general, the area of the rectangle becomes 1 when we make its height equal to the value of 1>range.) The requirement that the area must equal 1 simplifies probability problems, so the following statement is important: Because the total area under the density curve is equal to 1, there is a correspondence between area and probability. Example 2 Subway Waiting Time Given the uniform distribution illustrated in Figure 2, find the probability that a randomly selected passenger has a waiting time greater than 2 minutes. Solution The shaded area in Figure 3 represents waiting times greater than 2 minutes. Because the total area under the density curve is equal to 1, there is a correspondence between area and probability. We can find the desired probability by using areas as follows: P(wait time greater than 2 min) = area of shaded region in Figure 3 = 0.2 * 3 = Area = = 0.6 P(x) x (waiting time in minutes) Figure 3 Using Area to Find Probability 250

7 Interpretation The probability of randomly selecting a passenger with a waiting time greater than 2 minutes is 0.6. Standard Normal Distribution The density curve of a uniform distribution is a horizontal straight line, so we can find the area of any rectangular region by applying this formula: Area = height * width. Because the density curve of a normal distribution has a complicated bell shape as shown in Figure 1, it is more difficult to find areas. However, the basic principle is the same: There is a correspondence between area and probability. In Figure 4 we show that for a standard normal distribution, the area under the density curve is equal to 1. DEFINITION The standard normal distribution is a normal distribution with the parameters of m = 0 and s = 1. The total area under its density curve is equal to 1 (as in Figure 4). Area z Score Figure 4 Standard Normal Distribution It is not easy to manually find areas in Figure 4, but we have two other relatively easy ways of finding those areas: (1) Use technology; (2) use Table 2 in Appendix: Tables (the Standard Normal Distribution table in the Appendix). Finding Probabilities when Given z Scores We can find areas (or probabilities) for many different regions in Figure 4 by using a TI-83>84 Plus calculator or computer software such as STATDISK, Minitab, Excel, or StatCrunch, or we can also use Table 2 (in Appendix: Tables and the Formulas and Tables insert card) the Normal Distribution table. Key features of the different methods are summarized in Table 1 that follows. Because calculators or computer software generally give more accurate results than Table 2, we strongly recommend using technology. If using Table 2 the table, it is essential to understand these points: 1. Table 2 is designed only for the standard normal distribution, which has a mean of 0 and a standard deviation of