Chapter Fourteen

Chapter Fourteen - Lecture Notes Chapter Fourteen:...

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Lecture Notes Chapter Fourteen: Nonparametric Statistics Randall Miller 1 | Page 1. Introduction: Distribution-Free Tests Definition 14.1 Distribution-free tests are statistical that do not rely on any underlying assumptions about the probability distribution of the sampled population. Definition 14.2 The branch of inferential statistics devoted to distribution-free test is call nonparametrics . Definition 14.3 Nonparametric statistics (or test) based on the ranks of measurements are called rank statistics (or rank test ). 2. Single-Population Inferences Sign Test for a Population Median η One-Tailed Test [ ] 00 : : or : aa H HH ηη = >< Test statistic : S = Number of sample measurements greater than 0 [or S = number of measurements less than 0 ] Two-Tailed Test 0 : : a H H = Test Statistic : S = Larger of 12 and SS where 1 S is the number of measurements less than 0 and 2 S is the number of measurements greater than 0 [ Note : Eliminate observation from the analysis that are exactly equal to the hypothesized median, 0 .] Observed significance level : p -value = ( ) Px S Observed significance level : p -value = ( ) 2 where x has a binomial distribution with parameter n and p = .5. (Use Table II, Appendix A.) Rejection region : zz α > Rejection region : /2 > Where tabulated z 0values can be found in Table IV, Appendix A.
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Lecture Notes Chapter Fourteen: Nonparametric Statistics Randall Miller 2 | Page Conditions Required for a Valid Application of this Sign Test The sample is selected randomly from a continuous probability distribution. [ Note : No assumptions need to be made about the shape of the probability distribution.] Large Sample Sign Test for a Population Median η One-Tailed Test [ ] 00 : : or : aa H HH ηη = >< Two-Tailed Test 0 : : a H H = Test statistic : ( ) .5 .5 .5 Sn z n −− = [ Note : S is calculated as shown above. We subtract .5 from S as the “correction for continuity.” The null-hypothesized mean value is .5 np n = , and the standard deviation is ( )( ) .5 .5 .5 npq n n = (See Chapter 5 for details on the normal approximation to the binomial distribution.)] Rejection region : zz α > Rejection region : /2 > where tabulated z -values can be found in Table IV, Appendix A. 3. Comparing Two Populations: Independent Samples Wilcoxon Rank Sum Test: Independent Samples Let 12 and DD represent the probability distribution for populations 1 and 2, respectfully. One-Tailed Test [ ] 01 2 : and are identical : is shifted to the right of or : is shifted to the left of a a HD D D D Test statistic : 1 1 22 2 1 if : if T n nT n n << (Either rank sum can be used if nn = .) Rejection region : [ ] [ ] 11 1 22 2 : or : or UL LU TTT TT ≥≤ ≤≥ Two-Tailed Test 2 1 2 : and are identical : is shifted either to the left or to the right of a D D Test statistic : (Either rank sum can be used if = .) We will denote this rank sum as T .
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This note was uploaded on 07/23/2011 for the course STA 3123 taught by Professor Staff during the Fall '08 term at FIU.

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Chapter Fourteen - Lecture Notes Chapter Fourteen:...

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