Empirical Direction in Design and Analysis

By Norman H. Anderson | Go to book overview

NOTES
0.1.2a
A third answer to the proposal to select representative samples of subjects is that they are practically impossible to achieve. Individual behavior is highly unpredictable in novel situations.
0.1.3a
Although MSerror= s2is an unbiased estimate of σ 2, its square root, s, is not an unbiased estimate of σ. This bias is not ordinarily pertinent; in particular, it does not affect the t test or confidence intervals.
0.1.3b
The confidence interval will of course differ from sample to sample. However, 95% of the confidence intervals around the sample mean will contain the population mean. Hence you are entitled to 95% confidence that the confidence interval calculated from your one particular sample contains the population mean (see further The Concept of Confidence, page 94).
0.1.3c
The confidence interval also adds information about the size of the effect—measured jointly by the width of the confidence interval and the distance between the end of the interval and 0.
0.4.1a
Formula for Normal Distribution. Although the mathematical formula for the normal distribution is not needed to understand anything in this book, it is given here to show that it really does exist. For a normal distribution with mean μ and standard deviation, σ, this formula is

This formula gives the probability, Y, of each value of X, as illustrated in Figure 0.1 where X is women's heights. In this formula, it denotes the ratio of the circumference of a circle to its diameter, and e denotes the base for natural logarithms.

This formula shows that all normal distributions have the same overall shape; they differ only in their mean, μ, and standard deviation, σ. The multiplier, 1/σ √2π, makes the total area under the normal curve equal to 1, as it must to be a probability distribution.

For the curious, this formula contains three famous numbers. Two are π and e, workhorse numbers that pop up everywhere in mathematics. Both are transcendental numbers, so called because they are not the roots of any polynomial equation with rational coefficients. √2 is an irrational number—not expressible as the ratio of two whole numbers—and hence not really a number to the ancient Greeks. The discovery that √2 was irrational caused a crisis in the Pythagorean religion analogous to, although lesser than, the crisis in the Catholic church caused by Galileo's discoveries.

0.2.1a
Probability theory also includes the NOT rule: Prob(NOT A) = 1 − Prob(A).
0.3a
Runs of consecutive events present some interesting aspects of probability thinking. A couple has four girls in a row and are expecting their fifth child. Will it be another girl? Since height and other physical characteristics are correlated across siblings, it seems reasonable to expect the same for sex. On this plausible argument, the probability that the fifth child will also be a girl is greater than ½

-803-

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Empirical Direction in Design and Analysis
Table of contents

Table of contents

  • Title Page iii
  • Dedication v
  • Foreword vi
  • Contents vii
  • Preface xvi
  • Chapter 1 - Scientific Inference 1
  • Preface 30
  • Chapter 2 - Statistical Inference 31
  • How to Do Exercises 54
  • Exercises for Chapter 2 54
  • Preface 58
  • Chapter 3 - Elements of Analysis of Variance I 59
  • Notes 75
  • Appendix: How to Randomize 77
  • Exercises for Chapter 3 84
  • Preface 90
  • Chapter 4 - Elements of Analysis of Variance II 91
  • Notes 111
  • Exercises for Chapter 4 113
  • Preface 118
  • Chapter 5 - Factorial Design 119
  • Notes 145
  • Appendix: Hand Calculation for Factorial Design 148
  • Exercises for Chapter 5 151
  • Preface 158
  • Chapter 6 - Repeated Measures Design 159
  • Notes 177
  • Exercises for Chapter 6 181
  • Preface 188
  • Chapter 7 - Understanding Interactions 189
  • Notes 209
  • Exercises for Chapter 7 214
  • Preface 218
  • Chapter 8 - Confounding 219
  • Notes 250
  • Preface 258
  • Chapter 9 - Regression and Correlation 259
  • Notes 280
  • Exercises for Chapter 9 282
  • Preface 286
  • Chapter 10 - Frequency Data and Chi-Square 287
  • Notes 300
  • Exercises for Chapter 10 302
  • Preface 306
  • Chapter 11 - Single Subject Design 307
  • Notes 338
  • Exercises for Chapter 11 345
  • Preface 350
  • Chapter 12 - Nonnormal Data and Unequal Variance 351
  • Notes 373
  • Exercises for Chapter 12 378
  • Preface 382
  • Chapter 13 - Analysis of Covariance 383
  • Notes 395
  • Exercises for Chapter 13 397
  • Preface 400
  • Chapter 14 - Design Topics I 401
  • Notes 431
  • Exercises for Chapter 14 437
  • Preface 442
  • Chapter 15 - Design Topics II 443
  • Notes 475
  • Exercises for Chapter 15 481
  • Preface 484
  • Chapter 16 - Multiple Regression 485
  • Notes 514
  • Exercises for Chapter 16 520
  • Preface 524
  • Chapter 17 - Multiple Comparisons 525
  • Notes 546
  • Exercises for Chapter 17 548
  • Preface 550
  • Chapter 18 - Sundry Topics 551
  • Notes 589
  • Exercises for Chapter 18 596
  • Preface 602
  • Chapter 19 - Foundations of Statistics 603
  • Notes 637
  • Preface 646
  • Chapter 20 - Mathematical Models for Process Analysis 647
  • Notes 677
  • Exercises for Chapter 20 681
  • Preface 688
  • Chapter 21 - Toward Unified Theory 689
  • Notes 729
  • Exercises for Chapter 21 742
  • Preface 750
  • Chapter 22 - Principles and Tactics of Writing Papers 751
  • Notes 761
  • Preface 764
  • Chapter 23 - Lifelong Learning 765
  • Notes 780
  • Preface 782
  • Chapter 0 - Basic Statistical Concepts 783
  • Notes 803
  • Exercises for Chapter 0 805
  • Statistical Tables 808
  • References 820
  • Author Index 847
  • Subject Index 854
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