## Appendix A: Conditional Probability Calculations

In this section, we calculate several of the conditional probabilities mentioned in the foregoing discussion. For justification of these calculations, we refer the reader to any textbook on mathematical probability.

First, we compute

which was mentioned in our discussion of Healey [

8] in

Section 3.1. In the following, we leave off “in the given experiment” for brevity of expression. So we have

where we have denoted the (conditional) probability of Type II error by β according to the usual convention. In practice, it is impossible to determine β or Pr[

H0 is true], and these values could be anything from 0 to 1. Clearly this expression is not equal to α, as some writers claim.

Next we compute

which Hopkins and Glass [

5] claimed was equal to

q in a “statistically ideal” world. Again using abbreviated notation, we find

In general, both numerator and denominator in the expression $\left(\frac{\mathrm{Pr}\text{[}p-\text{value}=q\text{|}H0\text{isfalse}]}{\mathrm{Pr}\text{[}p-\text{value}=q\text{|}H0\text{istrue}]\text{}}\right)$ are zero, but the expression still corresponds to a well-defined ratio of probability densities which in practice are impossible to estimate (as is the probability that H0 is true). Certainly, the expression is not equal to q in general.

Finally we compute

which we mentioned in our discussion of Bakeman [

3] in

Section 4. The expression can be rewritten as (with our usual abbreviation)

The second term is zero, since it is impossible to wrongly reject

H0 in an experiment where

H0 is false. On the other hand, the first conditional probability is equal to α. In summary we have:

Since Pr[H0 is true in the experiment] is less than or equal to 1, one could say that the confidence level α for an experiment could be interpreted as a “maximum risk” of wrongly rejecting H0. We concur, but this means something completely different from what many writers appear to think. In particular, it has nothing whatsoever to do with the probability that rejection of H0 in an already-completed experiment constitutes a Type I error. In the district attorney example discussed in the text, this probability is equal to 1; and it is equally possible to construct a scenario where the probability is equal to 0.

## Appendix B: Textbooks Examined in the Study

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Bachman, Ronet, and Raymond Paternoster. Statistical Methods for Criminology and Criminal Justice. New York: Mcgraw-Hill, 1997.

Bakeman, Roger. Understanding Social Science Statistics: A Spreadsheet Approach. Hillsdale: Lawrence Erlbaum Associates, 1992.

Blalock, Hubert M. Social statistics, rev. 2nd ed. New York: McGraw-Hill, 1979.

Bland, Martin. An Introduction to Medical Statistics. Oxford: Oxford University Press, 1987.

Cotton, John W. Elementary Statistical Theory for Behavioral Scientists. Reading: Addison Wesley, 1967.

Couch, James V. Fundamentals of Statistics for the Behavioral Sciences. New York: St. Martin’s, 1982.

Daly, Leslie, and Geoffrey J. Bourke, Interpretation and Uses of Medical Statistics, 5th ed. Oxford: Blackwell Science, 2000.

Healey, Joseph F. Statistics: A Tool for Social Research, 9th ed. Stamford: Cengage Learning, 2011.

Hopkins, Kenneth D., and Gene V. Glass. Basic Statistics for the Behavioral Sciences. Englewood Cliffs: Prentice-Hall, 1978.

Jackson, Sherri. Statistics and Research Design (Custom edition for Psychology 418 at the University of Texas at Austin). Mason: Cengage Learning, 2008.

Jordan, Kelvin, Bie No Ong, and Peter Croft. Mastering Statistics: A Guide for Health Service Professionals and Researchers. Cheltenham: Stanley Thornes Ltd., 1998.

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Runyon, Richard, Kay Coleman, and David Pittenger. Fundamentals of Behavioral Statistics, 9th ed. Hawkins: McGraw-Hill Higher Education, 2000.

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Weisburd, David, and Chester Britt. Statistics in Criminal Justice, 3rd ed. New York: Springer Science & Business Media, LLC, 2007.

Welkowitz, Joan, Barry H. Cohen, and R. Brooke Lea. Introductory Statistics for the Behavioral Sciences. New York: Wiley, 2011.

Wilcox, Rand R. New Statistical Procedures for the Social Sciences: Modern Solutions to Basic Problems. Hillsdale: Lawrence Erlbaum Associates, 1987.

Willemsen, Eleanor W. Understanding Statistical Reasoning. San Francisco: W. H. Freeman, 1974.