APPENDIX B S OME B ASIC T ESTS IN S TATISTICS 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 Slides for Introduction to Stochastic Search and Optimization (ISSO) by J. C. Spall

B-2 The Standard One-Sample Test One set of data {X i } for testing on   E(X i ) 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 rejectionLarge values of | z | or | t | indicate rejection of null hypothesis that  is some chosen value (commonly  = 0)

B-3 P-Values at least as extremeP-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 based on null hypothesis that   0 –Fairly strong evidence that  > 0

B-4 Two-Sample Tests Two sets of data {X i } and {Y i } for testing  X =  Y –E.g., X i and Y i 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