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## Forensic Analysis Weighing Bullet Lead Evidence (2004) Board on Chemical Sciences and Technology (BCST)

### Citation Manager

. "Appendix F: Simulating False Match Probabilities Based on Normal Theory." Forensic Analysis Weighing Bullet Lead Evidence . Washington, DC: The National Academies Press, 2004.

 Page 143

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Forensic Analysis Weighing Bullet Lead Evidence

…+ s2B)/B. The ratio of to is the same as the distribution of namely a Student’s tv (v degrees of freedom), so the two-sample t statistic is distributed as a (central) Student’s t on v degrees of freedom:

The FBI criterion for a match on this one element can be written

Because E(sx) = E(sy) = 0.8812σ, and E(sp) ≈ σ if v > 60, this reduces very roughly to

The approximation is very rough because E(P{t < S}) ≠ P{t < E(S)}, where t stands for the two-sample t statistic and S stands for But it does show that if δ is very large, this probability is virtually zero (very small false match probability because the probability that the sample means would, by chance, end up very close together is very small). However, if δ is small, the probability is quite close to 1.

The equivalence t test proceeds as follows. Assume

where H0 is the null hypothesis that the true population means differ by at least δ, and the alternative hypothesis is that they are within δ of each other. The two-sample t test would reject H0 in favor of H1 if the sample means are too close, that is, if where Kα(n,δ) is chosen so that does not exceed a preset per-element risk level of α (in Chapter 3, we used α = 0.30). Rewriting that equation, and writing Kα for Kα(n,δ),

When v is large sp ≈ 0, and therefore the quantity

 Page 143
 Front Matter (R1-R12) Executive Summary (1-7) 1. Introduction (8-11) 2. Compositional Analysis (12-25) 3. Statistical Analysis of Bullet Lead Data (26-70) 4. Interpretation (71-108) 5. Major Findings and Recommendations (109-114) Appendix A: Statement of Task (115-117) Appendix B: Committee Membership (118-123) Appendix C: Committee Meeting Agendas (124-129) Appendix D: Glossary (130-132) Appendix E: Basic Principles of Statistics (133-141) Appendix F: Simulating False Match Probabilities Based on Normal Theory (142-150) Appendix G: Data Analysis of Table 1, Randich et al. (151-156) Appendix H: Principal Components Analysis: How Many Elements Should Be Measured? (157-162) Appendix I: The Birthday Problem Analogy (163-164) Appendix J: Understanding the Significance of the Results of Trace Elemental Analysis of Bullet Lead (165-168) Appendix K: Statistical Analysis of Bullet Lead Data by Karen Kafadar and Clifford Spiegelman (169-214)