Introductory Probability and Statistics: Applications for Forestry and Natural Sciences

424 Pages·2008·5.085 MB·English

by  Antal Kozak, Robert A. Kozak, Christina L. Staudhammer, Susan B. Watts

Introductory Probability and Statistics
Applications for Forestry and Natural Sciences
Dedicated to the Hungarian freedom fighters of 1956, without whom this book
would never have materialized.
Introductory Probability and Statistics
Applications for Forestry and Natural Sciences
Antal Kozak
Professor Emeritus, Faculty of Forestry
The University of British Columbia, Vancouver, BC, Canada
Robert A. Kozak
Associate Professor, Faculty of Forestry
The University of British Columbia, Vancouver, BC, Canada
Christina L. Staudhammer
Assistant Professor, School of Forest Resources and Conservation
University of Florida, Gainesville, Florida, USA
Susan B. Watts
Lecturer, Faculty of Forestry
The University of British Columbia, Vancouver, BC, Canada
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Library of Congress Cataloging-in-Publication Data
Introductory probability and statistics : applications for forestry and natural sciences
/ Antal Kozak ... [et al.].
p. cm. -- (Modular texts)
Includes bibliographical references and index.
ISBN 978-1-84593-275-6 (alk. paper)
1. Forests and forestry--Statistical methods. 2. Probabilities. I. Kozak, Antal. II.
Title: Applications for forestry and natural sciences. III. Series.
SD387.S73I585 2007
634.901(cid:1)5192--dc22
2007005704
ISBN: 978 1 84593 275 6
Typeset by Columns Design, Reading, UK.
Printed and bound in the UK by Cambridge University Press, Cambridge.
Contents
List of Figures ix
List of Tables xiii
Preface xv
1 Statistics and Data: What do Numbers have to do with Trees? 1
1.1 What is Statistics? 1
1.2 Data 2
1.3 Measurement Scales 3
1.4 Data Collection 4
Exercises 6
2 Descriptive Statistics: Making Sense of Data 9
2.1 Tables 9
2.2 Graphical Tools 14
2.3 Measures of Central Location 19
2.4 Measures of Variation 23
2.5 Measures of Position 29
2.6 Computers and Statistical Software 30
Exercises 31
3 Probability: the Foundation of Statistics 35
3.1 Sample Space and Events 35
3.2 Counting Techniques 39
3.3 Probability 44
3.4 Rules for Probabilities 46
3.5 Bayes’ Theorem 53
Exercises 55
4 Random Variables and Probability Distributions: Outcomes of Random
Experiments 61
4.1 Random Variables 61
4.2 Probability Distributions 62
4.3 Mean of a Random Variable 66
4.4 Variance of a Random Variable 70
4.5 Rules of Mathematical Expectations Related to the Mean
and Variance 72
Exercises 75
Contents v
5 Some Discrete Probability Distributions: Describing Data that are Counted 79
5.1 Uniform Distribution 79
5.2 Binomial and Multinomial Distributions 80
5.3 Hypergeometric and Multivariate Hypergeometric Distributions 84
5.4 Geometric and Negative Binomial Distributions 87
5.5 Poisson Distribution 88
Exercises 89
6 Continuous Distributions and the Normal Distribution: Describing Data
that are Measured 93
6.1 Uniform Distribution 93
6.2 Exponential Distribution 95
6.3 Normal Distribution 96
6.4 Normal Approximation to the Binomial Distribution 104
Exercises 107
7 Sampling Distributions: The Foundation of Inference 111
7.1 Sampling and Sampling Distributions 111
7.2 Sampling Distribution of the Mean 115
7.3 Sampling Distribution of the Sample Proportion 122
7.4 Sampling Distribution of the Differences between Two Means 125
Independent populations 125
Dependent populations 132
7.5 Sampling Distribution of the Differences between Two Proportions 134
7.6 Sampling Distribution of the Variance 136
7.7 Sampling Distribution of the Ratios of Two Variances 138
7.8 Some Concluding Remarks about Sampling Distributions 140
Exercises 141
8 Estimation: Determining the Value of Population Parameters 147
8.1 Point Estimation 147
8.2 Interval Estimation 148
8.3 Estimating the Mean 149
8.4 Estimating Proportions 155
8.5 Estimating the Difference between Two Means 157
Independent samples 157
Dependent samples 161
8.6 Estimating the Difference of Two Proportions 162
8.7 Estimating the Variance 163
8.8 Estimating the Ratio of Two Variances 165
Exercises 167
9 Tests of Hypotheses: Making Claims about Population Parameters 173
9.1 Statistical Hypothesis and Test Procedures 173
9.2 Tests Concerning Means 179
9.3 Tests Concerning Proportions 182
vi Contents
9.4 Tests Concerning Variances 183
9.5 Tests Concerning the Difference between Two Means 185
Independent populations 185
Dependent populations 189
9.6 Tests Concerning the Difference between Two Proportions 190
9.7 Tests Concerning the Ratio of Two Variances 192
9.8 p-Values 196
Exercises 196
10 Goodness-of-fit and Test for Independence: Testing Distributions 201
10.1 Goodness-of-fit Test 201
10.2 Test for Independence 206
Exercises 213
11 Regression and Correlation: Relationships between Variables 217
11.1 Simple Linear Regression 218
Determination of the regression equation 218
Regression analysis 224
Sampling distributions and tests concerning the regression
coefficients and predictions 230
Lack of fit 236
11.2 Correlation Analysis 239
11.3 Multiple Regression 240
11.4 Non-linear Models 244
Exercises 246
12 Analysis of Variance: Testing Differences between Several Means 251
12.1 One-way Analysis of Variance 252
12.2 Multiple Comparisons 261
Bonferroni’s Procedure 262
Scheffé’s Method 263
12.3 Test for Equality of Variances 265
12.4 Two-way Analysis of Variance 266
Exercises 274
13 Sampling Methods and Design of Experiments: Collecting Data 277
13.1 Sampling Methods 277
Simple random sampling 278
Stratified random sampling 278
Two-stage sampling 278
Systematic sampling 279
Survey design 280
13.2 Experimental Designs 280
Completely randomized design 281
Randomized complete block design 282
Latin square design 283
Factorial experiments 285
Contents vii
14 Non-parametric Tests: Testing when Distributions are Unknown 287
14.1 Sign Test 288
14.2 Wilcoxon Signed Rank Test 291
14.3 Wilcoxon Rank Sum Test 293
14.4 Kruskal–Wallis Test 296
14.5 Runs Test 297
14.6 Spearman’s Rank Correlation Test 300
Exercises 301
15 Quality Control: Statistics for Production and Processing 305
15.1 Variable Charts 308
15.2 Attribute Charts 312
Exercises 313
Bibliography 317
Solutions to Odd-numbered Questions 321
Appendix A 349
A.1 Binomial Probabilities 351
A.2 Poisson Probabilities 358
A.3 Areas Under the Normal Curve 363
A.4 Random Numbers 364
A.5 Critical Values for the tDistribution 365
A.6 Critical Values for the (cid:2)2Distribution 366
A.7 Critical Values for the FDistribution 367
A.8 Critical Values for the rDistribution 373
A.9 Critical Values for the Bonferroni tStatistic 377
A.10 Critical Values for the Wilcoxon Signed Rank Test 378
A.11 Critical Values for the Wilcoxon Rank Sum Test 379
A.12 Critical Values for the Runs Test 381
A.13 Critical Values for Spearman’s Rank Correlation Coefficient Test 383
Appendix B 385
Summation Notation 385
Glossary 387
Index 401
viii Contents
List of Figures
Fig. 2.1. Bar graph for crown class data. 15
Fig. 2.2. Stick graph for number of neighbouring trees. 15
Fig. 2.3. Histogram for dbh data. 16
Fig. 2.4. Histogram for dbh data (relative frequencies). 16
Fig. 2.5. Pie chart for crown class data. 17
Fig. 2.6. Frequency polygon for dbh data. 17
Fig. 2.7. Cumulative frequency graph (ogive) for dbh data. 18
Fig. 2.8. Inverse cumulative frequency graph (ogive) for dbh data. 18
Fig. 2.9. Symmetric distributions. 19
Fig. 2.10. Skewed distributions. 19
Fig. 2.11. Deviations from sample mean. 24
Fig. 2.12. Empirical Rule. 27
Fig. 3.1. Venn diagram ofsample space and events. 37
Fig. 3.2. Complement of event B. 37
Fig. 3.3. Union of two events, AandB. 37
Fig. 3.4. Intersection of two events, AandB. 38
Fig. 3.5. Mutually exclusive events. 38
Fig. 3.6. Tree diagram for selecting 3 boards. 39
Fig. 3.7. Tree diagram for flipping a coin and rolling a die. 40
Fig. 3.8. Common elements (intersection) of events AandB. 47
Fig. 3.9. Tree diagram for selecting US and Canadian dimes. 52
Fig. 3.10. Venn diagram for events A ,A ,A andB. 54
1 2 3
Fig. 3.11. Tree diagram for Bayes’ Theorem (F, Douglas-fir; H,
western hemlock). 54
Fig. 4.1. Stick graph of the probability distribution for stunted and
normally growing seedlings (from Example 4.1). 63
Fig. 4.2. Examples of probability densities. 64
Fig. 4.3. Probability density for the time it takes a tree planter to plant a
seedling. 64
Fig. 5.1. Probability distribution for Example 5.1 (randomly selecting from
8 wood pieces). 79
Fig. 5.2. Tree diagram for drawing 2 cards from a deck of 52 cards
(without replacement; S, spades; N, other card). 85
Fig. 6.1. Probability density function for the uniform distribution. 94
Fig. 6.2. Probability density functions for some exponential distributions. 95
Fig. 6.3. Probability density function for a normal distribution. 97
Fig. 6.4. Normal curves for µ ≠µ andσ =σ. 97
1 2 1 2
Fig. 6.5. Normal curves for µ =µ andσ ≠σ. 97
1 2 1 2
Fig. 6.6. Normal curves for µ ≠µ andσ ≠σ. 97
1 2 1 2
Fig. 6.7. P(x <X<x ) from a normal curve (shaded area). 98
1 2
Fig. 6.8. Original and transformed normal distribution. 99
List of Figures ix
Fig. 6.9. P(Z< 1.58) and P(Z> 1.58). 100
Fig. 6.10. (a) P(X< 6). 101
Fig. 6.11. (b) P(X> 12). 101
Fig. 6.12. (c) P(6 < X< 12). 101
Fig. 6.13. (d) P(12 < X< 14). 101
Fig. 6.14. (a) Replace motor with guarantee 2% of the time.
(b) Replace motor with guarantee 4% of the time. 102
Fig. 6.15. Binomial probabilities for n= 2, 10 and 20, p= 0.5. 106
Fig. 7.1. Probability distribution of the population 0, 2, 4 and 6. 115
Fig. 7.2. Probability distribution of all possible means from sampling with
replacement (n= 2). 116
Fig. 7.3. Probability distribution of all possible means from sampling with
replacement (n= 3). 117
Fig. 7.4. Probability distribution of all possible means from sampling
without replacement (n= 2). 117
Fig. 7.5. Areas for Example 7.3. 119
Fig. 7.6. Areas for Example 7.5. 120
Fig. 7.7. Distribution of tfor 5, 10 and ∞degrees of freedom. 121
Fig. 7.8. Areas for Example 7.6. 122
Fig. 7.9. Areas for Example 7.7. 124
Fig. 7.10. Areas for Example 7.8. 125
Fig. 7.11. Sampling distribution of the differences between two means. 127
Fig. 7.12. Areas for Example 7.9. 128
Fig. 7.13. Areas for Example 7.10. 130
Fig. 7.14. Areas for Example 7.11. 131
Fig. 7.15. Areas for Example 7.12. 131
Fig. 7.16. Areas for Example 7.13. 133
Fig. 7.17. Areas for Example 7.14. 134
Fig. 7.18. Areas for Example 7.15a. 135
Fig. 7.19. Probability distribution of (a) sample variances and (b) some (cid:2)2curves. 137
Fig. 7.20. Areas for Example 7.16. 138
Fig. 7.21. Sampling distribution of (a) the ratio of variances and
(b) the Fdistribution. 138
Fig. 7.22. Typical Fcurves. 139
Fig. 7.23. The two tails of the Fdistribution. 139
Fig. 7.24. Areas for Example 7.17. 140
Fig. 8.1. TheZdistribution. 149
Fig. 8.2. 95% confidence intervals calculated from 20 sample means. 151
Fig. 8.3. Thetdistribution. 152
Fig. 8.4. The(cid:2)2distribution. 164
Fig. 8.5. TheFdistribution. 165
Fig. 9.1. Sampling distribution of the means of battery lives based on
16 observations. 175
Fig. 9.2. Acceptance and rejection regions for α= 0.05 one-tailed test. 175
Fig. 9.3. Type I and Type II errors. 176
Fig. 9.4. The nature of Type I and Type II errors. 177
Fig. 9.5. Critical regions for: (a) and (b) one-tailed and (c) two-tailed tests. 178
x List of Figures