Adler

Mathematics for Life Scientists Frederick R. Adler

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c

Frederick R. Adler, 1994

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Department of Mathematics and Department of Biology, University of Utah, Salt Lake City, Utah 84112

Preface Mathematics for Life Scientists teaches calculus, probability, and statistics as a way to introduce freshman and sophomore life science majors to the insights mathematics can provide into many aspects of biology. Why should there be a special book for this audience? Although the importance of quantitative skills in the life sciences is much discussed, current realities tend to conceal their vital role. Too often, biology is the natural science of last resort for students who believe “they aren’t cut out for math.” Most colleges and universities require little calculus for their biology majors, and those that do require a full calculus course doubt its worth when students emerge unable to apply even pre-calculus mathematics in new contexts. Students are left with similar doubts when the techniques they learned for tests vanish as swiftly from the curriculum as from their memories. Students, biology faculty, and administrators see that biology is burgeoning as a science and as a major, apparently unhindered by pervasive mathematical illiteracy. In fact, mathematics has played an important if under-appreciated role in biology, providing the impetus for breakthroughs in epidemiology, genetics, statistics, physiology, and many other areas. As a theoretical biologist who uses mathematics to make sense of complex biological systems, I see this role expanding, not contracting. Although a great deal of biology can be done without any mathematics, the powerful new technologies that are transforming fields of biology from genetics and physiology to ecology are increasingly quantitative, as are many of the questions at the frontiers of knowledge. Mathematics is the language, the “technology of thought,” with which these developments are created and controlled. Students who speak this language will be the leaders of the next generation of biologists. As biology becomes more important in society, mathematical literacy becomes as necessary for doctors, business people, lawyers, and art historians as for researchers. My goal in this book is simple: to teach biology majors the mathematical ideas I use every day in my own research and in collaborations with my more empirical colleagues. These ideas are not specific techniques like “differentiation,” but concepts of modeling. The skills include describing a system, translating appropriate aspects into equations, and interpreting results in terms of the original problem. In this process, the science is central and “solving” the equations is in some ways the least important step. Because a few dynamical principles underlie a remarkable diversity of biological processes, this book follows three themes throughout: growth, diffusion and selection. Each theme is studied in turn with the three kinds of model that structure the course: discrete-time dynamical systems, differential equations, and stochastic processes. Techniques and insights build on each other throughout the course. Along the way, students learn and apply the standard material of a calculus course (differentiation, integration, and their applications). In addition, the course introduces matrices, vectors, and some basic calculus in two dimensions, all in a dynamical context. Most significantly, the final section of the book teaches probability 1

2 and statistics from the same perspective, using discrete-time dynamical systems and differential equations to describe simple stochastic processes. This section shows that correct and flexible application of statistics requires understanding the processes that generate data, and introduces the fundamental statistical notions of likelihood, parameter estimation, and hypothesis testing. In many ways, students go farther than in a traditional calculus (or probability) class. Time is saved by skipping methods made obsolete by computers. Learning more concepts and fewer techniques is definitely more challenging. As a sweetener, students are given the keys to the powerful techniques professionals use when equations cannot be solved: graphical methods (cobwebbing and phase space), approximation (leading behavior), and computers (labs using a computer algebra/graphics program). These techniques emphasize reasoning and visualization, and show that applied mathematics has less to do with algebraic wizardry than with the clear formulation of ideas. Working with computers has proven to be particularly successful in this context. What are the benefits of this approach? All instructors know that students will not remember every technique they have learned. This course emphasizes understanding what a model is, and recognizing what models say. To be able to recognize a differential equation, interpret the terms, and use the solution is far more important than knowing how to find the solution. These reasoning skills, in addition to familiarity with models in general, are what stay with the motivated student, and are what matter most in the end. The book is designed to mesh in a logical way with a general biology curriculum. The dynamical themes are distilled from the material covered in standard introductory courses: genetics, cell biology, physiology, and ecology. When instructors of these courses find themselves freed from reviewing basic quantitative methods, they can begin to use quantitative reasoning as an integral part of each course. Students forge the connections that make learning stick when they see ideas from their math course pay off in biology, and vice versa, or develop the confidence to play with the numbers with algebraic, graphical, or computer tools. Most importantly, the course is fun to teach. Leading students through an integrated course for a full year removes the pressure for instant instructor gratification. (“All of my students could take the derivatives of polynomials.”) Instead, one can allow understanding to develop as concepts return for the second or third time. Students find this unsettling and yearn for instant gratification too. But with time, they accept the challenge of thinking. When they begin to apply their new powers to their own problems, when they solve a problem on the computer without being told to, or when they teach me something about biology in the context of a mathematical idea, delayed gratification starts to feel like the best possible kind.

Acknowledgments This book would never have been written without the support of a Hughes Foundation Grant to the University of Utah which included as part of its mission an attempt to more effectively teach mathematics to biology majors. That grant brought together a committee of faculty to guide creation of this book and course consisting of Aaron Fogelson, David Goldenberg, Jim Keener, Mark Lewis, David Mason, Larry Okun, Hans Othmer, Jon Seger and Ryk Ward. Each, in his own way, added much to this work. Particular thanks to Jon Seger and Mark Lewis for discussion and ideas. Frank Wattenberg, Lou Gross, and Simon Levin for looked over the book and delivered much-needed advice on the whole. Alan Rogers kindly let me use his exercise style and Nelson Beebe helped smooth over many technical problems. Thanks to my editor Gary Ostedt for agreeing to

3 support a preliminary edition, for providing bird watching opportunities, and for making me feel important. This draft of the book benefited greatly from the comments and complaints of the students who survived the rocky first run of the course: Jennifer Aiman, Ty “Captain Flail” Corbridge, Brett Doxey, Ambur Economou, Brad Hasna, Robert Kane, Laura Krause, Jennifer Layman, Eric Mortensen, Scott Nord, Kevin Rapp, Chris Reilly, Mindi Robinson, Rachael Rosenfeld, Stephanie Spindler, Mark Stevens, Sheri Williams, Richard Wood and Gentry Yost. Ranging from those with an excess of comments to those who suffered in eloquent silence, they helped give this book whatever value it might have as a teaching tool. The veterans of the second run offered the same range of honest and helpful advice: Elissa Ashby, William Bleazard, Aaron Campbell, Timothy Christensen, Christina Davenport, Elizabeth Gloyn, Marc Hammerlund, Stacey Hansen, Catherine Hatt, Christopher Horne, Katina Lessard, Michelle Madsen, Stacy Meola, Jennifer Mercier, Wendy Pendry, Karin Rattlingourd, Alison Schick, Helene Segal, Sara Sharpsteen, Christen Sowards, Samuel Webb, and Luann Witt. They corrected many errors and cheerfully pointed out pedagogical shortcomings. My teaching assistants, Stephen Proulx, Peter Spiro, Vicky Solomon, Colonel Tim Lewis and Kristina Bogar, were always ready to stand behind me and tell me what I was doing wrong. In addition to her extraordinary sartorial advice and culinary support, I thank Anne Collopy for her inspirational example of writing with clear transitions, extended metaphors and elegant sentence structure. And for filling the work-free interstices of life with the same.

4

Contents I

Introduction to Discrete Dynamical Systems

1 Biology and Dynamics 1.1 Growth: Models of malaria . . . 1.2 Maintenance: Models of neurons 1.3 Replication: Models of genetics . 1.4 Types of dynamical systems . . .

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2 Updating Functions: Describing Growth 2.1 A model population: bacterial growth . . 2.2 A model organism: a growing tree . . . . 2.3 Functions: terminology and graphs . . . . 2.4 Exercises . . . . . . . . . . . . . . . . . .

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11 11 13 14 18

3 Units and Dimensions 3.1 Converting between units . . . . . . 3.2 Translating between dimensions . . . 3.3 Checking: dimensions and estimation 3.4 Exercises . . . . . . . . . . . . . . .

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21 21 23 26 29

4 Linear Functions and Their Graphs 4.1 Proportional relations . . . . . . . . 4.2 The equation of a line . . . . . . . . 4.3 Finding equations and graphing lines 4.4 Exercises . . . . . . . . . . . . . . .

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31 31 33 36 39

5 Finding Solutions: Describing the Dynamics 5.1 Bacterial population growth . . . . . . . . . . 5.2 Tree height . . . . . . . . . . . . . . . . . . . 5.3 Finding solutions in more complicated cases . 5.4 Exercises . . . . . . . . . . . . . . . . . . . .

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6 Combining and manipulating functions 53 6.1 Adding functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.2 Composition of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.3 Inverse functions: looking backwards . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5

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CONTENTS 6.4

Exercises

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7 Solutions and Exponential Functions 7.1 Bacterial population growth in general 7.2 Laws of exponents and logs . . . . . . 7.3 Expressing results with exponentials . 7.4 Exercises . . . . . . . . . . . . . . . .

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65 65 67 70 74

8 Power Functions and Allometry 8.1 Power relations and exponential growth . . 8.2 Power relations and lines . . . . . . . . . . . 8.3 Power relations in biology: shape and flight 8.4 Exercises . . . . . . . . . . . . . . . . . . .

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77 77 80 83 85

9 Oscillations and Trigonometry 9.1 Sine and cosine: a review . . . . . . . 9.2 Describing oscillations with the cosine 9.3 More complicated shapes . . . . . . . 9.4 Exercises . . . . . . . . . . . . . . . .

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89 89 91 94 96

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10 Modeling and Cobwebbing 10.1 A model of the lungs . . . . . . . . . . . . . 10.2 The lung updating function in general . . . 10.3 Cobwebbing: a graphical solution technique 10.4 Exercises . . . . . . . . . . . . . . . . . . .

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101 . 101 . 104 . 105 . 107

11 Equilibria 11.1 Graphical approach . . . . . . 11.2 Algebraic approach . . . . . . 11.3 Algebra involving parameters 11.4 Exercises . . . . . . . . . . .

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111 111 113 115 117

12 An Example of Nonlinear Dynamics 12.1 A model of selection . . . . . . . . . 12.2 Equilibria in the general case . . . . 12.3 Stable and unstable equilibria . . . . 12.4 Exercises . . . . . . . . . . . . . . .

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121 . 121 . 125 . 127 . 129

13 Excitable Systems I: The Heart 13.1 A simple heart . . . . . . . . . 13.2 Second degree block . . . . . . 13.3 The Wenckebach phenomenon . 13.4 Exercises . . . . . . . . . . . .

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135 135 138 140 141

CONTENTS

II

7

Limits and Derivatives

153

14 Differential Equations 14.1 Bacterial growth measured continuously 14.2 Rates of change . . . . . . . . . . . . . . 14.3 The limit . . . . . . . . . . . . . . . . . 14.4 Exercises . . . . . . . . . . . . . . . . .

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157 157 161 167 168

15 Slopes of Curves 15.1 The tangent line . . . . . . . . . 15.2 The equation for the tangent line 15.3 Estimating slopes from data . . . 15.4 Exercises . . . . . . . . . . . . .

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173 . 173 . 178 . 180 . 181

16 Stability and the Derivative 16.1 Motivation . . . . . . . . . . . 16.2 Stability and the slope . . . . . 16.3 Estimating slopes at equilibria 16.4 Exercises . . . . . . . . . . . .

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185 185 188 190 192

17 More Complex Dynamics 17.1 The logistic dynamical system . . 17.2 Qualitative dynamical systems . 17.3 Analysis of the logistic dynamical 17.4 Exercises . . . . . . . . . . . . .

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197 . 197 . 199 . 202 . 204

18 Limits 18.1 Limits of functions 18.2 Properties of limits 18.3 Infinite limits . . . 18.4 Exercises . . . . .

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19 Continuity 19.1 Continuous functions . . . . 19.2 Input and output tolerances 19.3 Hysteresis . . . . . . . . . . 19.4 Exercises . . . . . . . . . .

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207 207 211 213 215

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219 . 219 . 223 . 224 . 227

20 Building blocks for derivatives 20.1 Linear functions . . . . . . . 20.2 A quadratic function . . . . . 20.3 The sum rule . . . . . . . . . 20.4 Exercises . . . . . . . . . . .

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231 231 233 235 239

8

CONTENTS

21 Derivatives of Powers and Polynomials 21.1 Derivatives of basic power functions . . . . . . 21.2 Derivatives of polynomials . . . . . . . . . . . . 21.3 The power rule: negative and fractional powers 21.4 Exercises . . . . . . . . . . . . . . . . . . . . . 22 Derivatives of products and quotients 22.1 The product rule . . . . . . . . . . . . 22.2 Special cases and examples . . . . . . 22.3 The quotient rule . . . . . . . . . . . . 22.4 Exercises . . . . . . . . . . . . . . . .

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23 Derivatives of Exponential and Logarithmic Functions 23.1 The exponential function . . . . . . . . . . . . . . . . . . 23.2 The natural logarithm . . . . . . . . . . . . . . . . . . . . 23.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Derivatives of Trigonometric Functions 24.1 Sine and cosine . . . . . . . . . . . . . . 24.2 Other trigonometric functions . . . . . . 24.3 Applications . . . . . . . . . . . . . . . . 24.4 Exercises . . . . . . . . . . . . . . . . . 25 The 25.1 25.2 25.3 25.4

III

Chain Rule The derivative of a composite function Derivatives of inverse functions . . . . Applications . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . .

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243 . 243 . 245 . 250 . 253

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287 . 287 . 291 . 293 . 295

Applications of Derivatives and Dynamical Systems

26 Maximization 26.1 Minima and maxima . . . . 26.2 Maximizing food intake rate 26.3 Maximizing fish harvest . . 26.4 Exercises . . . . . . . . . .

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27 Reasoning about functions 27.1 The Intermediate Value Theorem . . . . . . . . 27.2 Maximization: The Extreme Value Theorem . . 27.3 Rolle’s Theorem and the Mean Value Theorem 27.4 Exercises . . . . . . . . . . . . . . . . . . . . .

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319 . 319 . 326 . 328 . 330

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335 . 335 . 338 . 340 . 343

CONTENTS

9

28 Limits at Infinity 28.1 The behavior of functions at infinity 28.2 Application to absorption functions . 28.3 Limits of sequences . . . . . . . . . . 28.4 Exercises . . . . . . . . . . . . . . .

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29 Leading behavior and L’Hopital’s Rule 29.1 Leading behavior of functions at infinity 29.2 Matched leading behaviors . . . . . . . . 29.3 L’Hopital’s Rule . . . . . . . . . . . . . 29.4 Exercises . . . . . . . . . . . . . . . . . 30 Approximating functions 30.1 The tangent and secant lines 30.2 Quadratic approximation . . 30.3 Taylor polynomials . . . . . . 30.4 Exercises . . . . . . . . . . . 31 Newton’s method 31.1 Finding equilibria . . . 31.2 Newton’s method . . . 31.3 Why Newton’s method 31.4 Exercises . . . . . . .

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32 Panting and Deep Breathing 32.1 Breathing at different rates 32.2 Deep breathing . . . . . . . 32.3 Panting . . . . . . . . . . . 32.4 Exercises . . . . . . . . . .

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347 . 347 . 353 . 354 . 356

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359 . 359 . 363 . 367 . 369

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397 . 397 . 398 . 399 . 401

Differential Equations, Integrals, and Their Applications

33 Differential Equations 33.1 Differential equations: examples and terminology 33.2 Euler’s method: pure-time . . . . . . . . . . . . . 33.3 Euler’s method: autonomous . . . . . . . . . . . 33.4 Exercises . . . . . . . . . . . . . . . . . . . . . . 34 Basic differential equations 34.1 Newton’s Law of Cooling . . . . . . 34.2 Diffusion across a membrane . . . . 34.3 A continuous time model of selection 34.4 Exercises . . . . . . . . . . . . . . .

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417 . 418 . 422 . 426 . 428

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431 . 431 . 434 . 434 . 438

10

CONTENTS

35 The 35.1 35.2 35.3 35.4

Antiderivative Pure-time differential equations . . . . . Rules for antiderivatives . . . . . . . . . Solving polynomial differential equations Exercises . . . . . . . . . . . . . . . . .

36 Special functions and substitution 36.1 Integrals of special functions . . . . 36.2 The chain rule and integration . . 36.3 Getting rid of excess constants . . 36.4 Exercises . . . . . . . . . . . . . . 37 Integrals and sums 37.1 Approximating integrals with sums 37.2 Approximating integrals in general 37.3 The definite integral . . . . . . . . 37.4 Exercises . . . . . . . . . . . . . .

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38 Definite and indefinite integrals 38.1 The Fundamental Theorem of Calculus . . . . 38.2 The summation property of definite integrals 38.3 General solution . . . . . . . . . . . . . . . . 38.4 Exercises . . . . . . . . . . . . . . . . . . . . 39 Applications of integrals 39.1 Integrals and areas . . 39.2 Integrals and averages 39.3 Integrals and mass . . 39.4 Exercises . . . . . . .

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40 Improper integrals 40.1 Infinite limits of integration . 40.2 Improper integrals: examples 40.3 Infinite integrands . . . . . . 40.4 Exercises . . . . . . . . . . .

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453 . 453 . 454 . 457 . 461

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465 . 465 . 467 . 471 . 472

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Analysis of Differential Equations

39 Autonomous Differential Equations 39.1 Review of autonomous differential equations 39.2 Equilibria . . . . . . . . . . . . . . . . . . . 39.3 Display of differential equations . . . . . . . 39.4 Exercises . . . . . . . . . . . . . . . . . . .

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479 . 479 . 481 . 483 . 486

CONTENTS

11

40 Stable and unstable equilibria 40.1 Recognizing stable and unstable equilibria 40.2 Applications of the stability theorem . . . 40.3 A model of a disease . . . . . . . . . . . . 40.4 Exercises . . . . . . . . . . . . . . . . . . 41 Solving autonomous equations 41.1 Separation of variables . . . . . . . . . 41.2 Pure-time equations revisited . . . . . 41.3 Applications of separation of variables 41.4 Exercises . . . . . . . . . . . . . . . . 42 Two dimensional equations 42.1 Predator-prey dynamics . 42.2 Newton’s law of cooling . 42.3 Euler’s method . . . . . . 42.4 Exercises . . . . . . . . . 43 The 43.1 43.2 43.3 43.4

Phase-Plane Equilibria and nullclines: Equilibria and nullclines: Equilibria and nullclines: Exercises . . . . . . . .

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predator-prey equations . selection equations . . . Newton’s law of cooling . . . . . . . . . . . . . . . .

44 Solutions in the phase-plane 44.1 Euler’s method in the phase-plane . . . . 44.2 Direction arrows: predator-prey equations 44.3 More direction arrows . . . . . . . . . . . 44.4 Exercises . . . . . . . . . . . . . . . . . . 45 The 45.1 45.2 45.3 45.4

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dynamics of a neuron A mathematician’s view of a neuron The mathematics of sodium channels The FitzHugh-Nagumo equations . . Exercises . . . . . . . . . . . . . . .

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545 . 545 . 547 . 549 . 553

Probability Theory and Descriptive Statistics

46 Probabilistic Models 46.1 Probability and statistics . . 46.2 Stochastic population growth 46.3 Markov chains . . . . . . . . 46.4 Exercises . . . . . . . . . . .

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569 . 569 . 570 . 573 . 575

12

CONTENTS

47 Stochastic models of diffusion 47.1 Stochastic diffusion . . . . . . 47.2 Stochastic diffusion . . . . . . 47.3 Stochastic diffusion . . . . . . 47.4 Exercises . . . . . . . . . . .

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48 Stochastic models of genetics 48.1 The genetics of inbreeding . 48.2 The dynamics of height . . 48.3 Blending inheritance . . . . 48.4 Exercises . . . . . . . . . .

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49 Probability Theory 49.1 Sample spaces and events . . . . 49.2 Set theory . . . . . . . . . . . . . 49.3 Assigning probabilities to events 49.4 Exercises . . . . . . . . . . . . .

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51 Independence and Markov Chains 51.1 Independence . . . . . . . . . . . . . . . . . . . 51.2 The multiplication rule for independent events 51.3 Markov chains and conditional probability . . . 51.4 Exercises . . . . . . . . . . . . . . . . . . . . .

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52 Displaying Probabilities 52.1 Probability and cumulative distributions 52.2 The probability density function . . . . 52.3 The cumulative distribution function . . 52.4 Exercises . . . . . . . . . . . . . . . . .

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53 Random Variables 53.1 Types of random variable . . 53.2 Expectation: discrete case . . 53.3 Expectation: continuous case 53.4 Exercises . . . . . . . . . . .

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50 Conditional Probability 50.1 Conditional probability . . 50.2 The law of total probability 50.3 Bayes’ theorem and the rare 50.4 Exercises . . . . . . . . . .

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54 Descriptive Statistics 643 54.1 The median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 54.2 The mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647 54.3 The geometric mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648

CONTENTS 54.4 Exercises

13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

55 Descriptive statistics for spread 55.1 Range and percentiles . . . . . 55.2 Mean absolution deviation . . . 55.3 Variance . . . . . . . . . . . . . 55.4 Exercises . . . . . . . . . . . .

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Probability Models

677

56 Joint distributions 56.1 Joint distributions . . . . . . . . . . . . . . . . . 56.2 Marginal probability distributions . . . . . . . . 56.3 Joint distributions and conditional distributions . 56.4 Exercises . . . . . . . . . . . . . . . . . . . . . . 57 Covariance and Correlation 57.1 Covariance . . . . . . . . 57.2 Correlation . . . . . . . . 57.3 Perfect correlation . . . . 57.4 Exercises . . . . . . . . .

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13 Matrices and Markov Chains 13.1 Matrices and conditional distributions 13.2 Equilibria of Markov chains . . . . . . 13.3 Generalized Markov chains . . . . . . 13.4 Exercises . . . . . . . . . . . . . . . .

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58 Sums and products of random variables 58.1 Expectation of a sum . . . . . . . . . . . 58.2 Expectation of a product . . . . . . . . 58.3 Variance of a sum . . . . . . . . . . . . 58.4 Exercises . . . . . . . . . . . . . . . . .

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713 713 715 718 723

. . . . . . . distribution . . . . . . . . . . . . . .

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725 725 728 730 732

Binomial Distribution The binomial distribution defined . . . Computing the binomial . . . . . . . . Binomial distribution: the general case Exercises . . . . . . . . . . . . . . . .

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60 Applications of the Binomial Distribution 60.1 Application to genetics and calculation of mode . . . . 60.2 Application to Markov chains: definition of cumulative 60.3 Applications to diffusion . . . . . . . . . . . . . . . . . 60.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

14

CONTENTS

61 Exponential distributions 61.1 The geometric distribution . . 61.2 The exponential distribution 61.3 The memoryless property . . 61.4 Exercises . . . . . . . . . . .

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62 The 62.1 62.2 62.3 62.4

Poisson Distribution The Poisson process . . . . . . . The Poisson distribution in space The Poisson and the binomial . . Exercises . . . . . . . . . . . . .

63 The 63.1 63.2 63.3 63.4

Normal Distribution The normal distribution: an example . . . The Central Limit Theorem for Sums . . The Central Limit Theorem for Averages Exercises . . . . . . . . . . . . . . . . . .

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64 Applying the Normal Approximation 64.1 The standard normal distribution . . . 64.2 Normal approximation of the binomial 64.3 Normal approximation of the Poisson . 64.4 Exercises . . . . . . . . . . . . . . . .

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735 . 735 . 739 . 742 . 743

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757 . 757 . 760 . 762 . 765

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769 . 769 . 775 . 777 . 779

Introduction to Statistical Reasoning

65 Statistics: Estimating Parameters 65.1 Estimating the binomial proportion . 65.2 Maximum likelihood . . . . . . . . . 65.3 Estimating a rate . . . . . . . . . . . 65.4 Exercises . . . . . . . . . . . . . . . 66 Confidence limits 66.1 Exact confidence limits . 66.2 Monte Carlo method . . 66.3 Likelihood, support, and 66.4 Exercises . . . . . . . .

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67 Estimating the Mean 67.1 Estimating the mean . . . . . 67.2 Confidence limits . . . . . . . 67.3 Sample variance and standard 67.4 Exercises . . . . . . . . . . .

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797 . 797 . 799 . 803 . 805

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807 . 807 . 811 . 813 . 816

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819 . 819 . 821 . 825 . 828

CONTENTS

15

68 Hypothesis Testing 68.1 Hypothesis testing: an example . . . . 68.2 Power and confidence limits . . . . . . 68.3 Likelihood and the method of support 68.4 Exercises . . . . . . . . . . . . . . . .

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831 831 835 836 838

69 Hypothesis Testing: Normal Theory 69.1 Computing p-values with the normal approximation 69.2 The power of normal tests . . . . . . . . . . . . . . . 69.3 Likelihood and the normal distribution . . . . . . . . 69.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . .

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841 841 844 847 849

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70 Comparing experiments 70.1 Unpaired normal distributions . . . 70.2 Comparing population proportions 70.3 Likelihood . . . . . . . . . . . . . . 70.4 Exercises . . . . . . . . . . . . . .

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851 . 851 . 853 . 855 . 858

71 Regression 71.1 Linear regression . . . . . . . . 71.2 Using linear regression . . . . . 71.3 The theory of linear regression 71.4 Exercises . . . . . . . . . . . .

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72 Answers

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861 861 865 867 870 883