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FOUNDATIONAL and APPLIED STATISTICS for BIOLOGISTS USING R This Page Intentionally Left Blank FOUNDATIONAL and APPLIED STATISTICS for BIOLOGISTS USING R FOUNDATIONAL and APPLIED STATISTICS for BIOLOGISTS USING R This page intentionally left blank FOUNDATIONAL and APPLIED STATISTICS for BIOLOGISTS USING R Ken A. Aho Idaho State University Pocatello, Idaho, USA Cover: View east from the Abiathar Peak in Northeastern Yellowstone National Park. Three mountain goats (Oreom- nos americanus) are visible at the edge of the summit plateau. Goat weights are approximately normally distributed with mean 90.5 kg and variance 225 kg2. This distribution is shown in the lower right hand corner of the cover. A ran- dom sample from this distribution is shown in the strip at the top. Photo credit: Scott Close. Goat illustration donated by Pearson Scott Foresman (an educational publisher) to Wikimedia Commons, http://commons.wikimedia.org/wiki/ Category:Pearson_Scott_Foresman_publisher; http://all-free-download.com. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2013 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20160203 International Standard Book Number-13: 978-1-4398-7339-7 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti- lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy- ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To JEF and OAA who traveled on this journey This page intentionally left blank Contents Preface ........................................................................................................................................... xix Acknowledgments ...................................................................................................................... xxi Section I Foundations . 1 Philosophical and Historical Foundations ........................................................................3 1.1 Introduction ...................................................................................................................3 1.2 Nature of Science ..........................................................................................................3 1.3 Scientic Principles .......................................................................................................4 1.3.1 Objectivity .........................................................................................................4 1.3.2 Realism ..............................................................................................................4 1.3.3 Communalism ..................................................................................................5 1.4 Scientic Method ..........................................................................................................5 1.4.1 A Terse History ................................................................................................6 1.4.1.1 Experimentation ..............................................................................8 1.4.1.2 Induction ...........................................................................................8 1.4.1.3 Probability ........................................................................................8 1.5 Scientic Hypotheses ...................................................................................................9 1.5.1 ​Falsiability ......................................................................................................9 1.6 Logic .............................................................................................................................. 10 1.6.1 Induction ......................................................................................................... 10 1.6.2 Deduction ........................................................................................................ 11 1.6.3 Induction versus Deduction ......................................................................... 11 1.6.4 Logic and Null Hypothesis Testing ............................................................12 1.6.4.1 Modus Tollens ................................................................................12 1.6.4.2 Reductio Ad Absurdum ............................................................... 13 1.7 Variability and Uncertainty in Investigations ........................................................ 14 1.8 Science and Statistics .................................................................................................. 16 1.9 Statistics and Biology .................................................................................................. 16 1.10 Summary ...................................................................................................................... 18 Exercises .................................................................................................................................. 18 2. Introduction to Probability ................................................................................................. 21 2.1 Introduction: Models for Random Variables........................................................... 21 2.1.1 Set Theory Terminology ...............................................................................23 2.1.2 Philosophical Conceptions of Probability .................................................. 24 2.2 Classical Probability ...................................................................................................26 2.2.1 Disjoint ............................................................................................................28 2.2.1.1 Boole’s Inequality ..........................................................................30 2.2.2 Independence ................................................................................................. 31 2.2.2.1 Bonferroni’s Inequality ................................................................. 31 2.3 Conditional Probability ..............................................................................................32 vii viii Contents 2.4 Odds ..............................................................................................................................34 2.4.1 Odds Ratio and Relative Risk ......................................................................34 2.5 Combinatorial Analysis .............................................................................................35 2.5.1 Multiplication Principle ................................................................................35 2.5.2 Permutations ...................................................................................................36 2.5.3 Combinations .................................................................................................37 2.6 Bayes Rule ....................................................................................................................38 2.7 Summary ......................................................................................................................43 Exercises ..................................................................................................................................43 3. Probability Density Functions ...........................................................................................49 3.1 Introduction .................................................................................................................49 3.1.1 How to Read This Chapter ...........................................................................50 3.1.2 So, What Is Density? ......................................................................................50 3.1.3 Cumulative Distribution Function ..............................................................52 3.2 Introductory Examples of pdfs .................................................................................54 3.2.1 Discrete pdfs ...................................................................................................55
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