APPENDIX B S OME B ASIC T ESTS IN S TATISTICS

1. Slides for Introduction to Stochastic Search and Optimization (ISSO)by J. C. Spall APPENDIX BSOME BASIC TESTSIN STATISTICS Organization of appendix in ISSO Standard one-sample test P-values Confidence intervals Basic two-sample tests Matched pairs t-test Unmatched pairs t-test with identical variances Unmatched pairs t-test with nonidentical variances Other approaches to testing One- and two-sample tests important in stochastic search, optimization, and Monte Carlo simulation

2. The Standard One-Sample Test • One set of data {Xi } for testing on   E(Xi) • Famous test statistics • z and t have a N(0, 1) and t-distribution, respectively • t-statistic useful in small samples; both z and t often used with non-normal samples • Large values of |z|or |t| indicate rejection of null hypothesis that  is some chosen value (commonly  = 0)

3. P-Values • P-value:Probability that future experiment would have value of test statistic at least as extreme as that observed in the current experiment • Provides info. beyond binary accept/reject null hypothesis • Useful as indicator of strength of rejection • Example: If z = 2.15, P-value is 0.016 based on null hypothesis that   0 • Fairly strong evidence that  > 0

4. Two-Sample Tests • Two sets of data {Xi } and {Yi } for testing X = Y • E.g.,Xi and Yi represent simulation outputs under two scenarios • Generic test statistic form where () denotes appropriate variance estimate • Three basic categories of tests affecting () • matched pairs • unmatched pairs; identical variances ( ) • unmatched pairs; non-identical variances ( ) • Large values of |t| indicate rejection of null hypothesis that X = Y