Week 4 Lecture: Paired-Sample Hypothesis Tests (Chapter 9)

Transcription

1 Week 4 Lecture: Pare-Sample Hypothess Tests (Chapter 9) The two-sample proceures escrbe last week only apply when the two samples are nepenent. However, you may want to perform a hypothess tests to ata that are relate. Some examples of relate (non-nepenent) ata nclue: before an after, observe vs. precte, rght vs. left, entcal twns, etc. Parng s a goo ea when you expect greater varaton between the pars when compare to varaton wthn a par. Testng Mean Dfference In a smlar fashon to the two-tale hypotheses from last week, we can efne a mean populaton fference, µ, as µ 1 µ 2, a test the null hypothess: Ho: µ = 0 Ha: µ 0 The test statstc for Ho s a t-statstc: t =, s where = mean fference an s = stanar error of the mean fference. Thus, the pare- sample t-test s essentally a one-sample t-test. Note, however, that each observaton n one sample must be correlate to one an only one observaton n the secon sample. The paresample t-test oes not have the normalty an homogenety of varances assumptons as wth the two-sample t-test, but t oes assume that the fferences are normally strbute. 1

2 Example: We want to compare groun versus ar-base temperature sensors to etermne the earth s temperature, whch s mportant for agrcultural moelng, etc. Groun-base sensors are expensve, an ar-base (from satelltes or ar-planes) of nfrare wavelengths may be base. We collecte temperature ata from groun an ar-base sensors at ten locatons, an we want to test f they are fferent. We wll test the followng hypothess: Ho: µ = 0 Ha: µ 0 α = 0.05 Locaton Groun ( o C) Ar ( o C) Dfference ( ) = 15.5 = 1.55 o s C s = = = n 10 o C 2

3 1.55 t-statstc: t = = = s 0.24 Crtcal Value: ( ) = t ( ) t α 2, ν= n , 9 = Decson Rule: If t , then reject Ho; otherwse, o not reject Ho. Concluson: Snce > (P < 0.001), reject Ho an conclue that the mean fference between groun an ar-base sensors at these ten locatons s sgnfcantly fferent. We can also test one-tale hypotheses for the mean fference n a manner analogous to that of the two-sample t-test (see Zar p. 181) for an example. Confence Intervals for the Populaton Mean Dfference Just as we calculate confence ntervals n preceng chapters, we can also obtan confence ntervals for the mean populaton fference: ± t α( 2), νs. Example: Contnung wth our earler example of temperature sensors, we can obtan a 95% confence nterval for the populaton mean fference: 1.55 ± o ( 2.262)( 0.24) 1.55 ± 0.54 C We can also etermne power of the test an necessary sample sze for a specfe level of precson as we for a one-sample t-test (chapter 7). Smply substtute for X, an 2 s for s 2. 3

4 Wlcoxon Pare-Sample Test Ths s the non-parametrc analog of the pare t-test, also known as the Wlcoxon Pare- Sample Test. It s use for pare-sample testng wth ornal ata. The proceure for ths test s: 1. Compute for each par 2. Rank s the absolute values of the fferences (assgn te ranks as before) 3. Calculate the sum of the ranks for: a) postve fferences, an b) negatve fferences 4. Apply approprate ecson rule (crtcal values from Table B.12): When Ho: Populaton 1 = Populaton 2 an Ha: Populaton 1 Populaton 2, an f T+ or T Tα, then reject Ho ( 2), n When Ho: Populaton 1 Populaton 2 an Ha: Populaton 1 > Populaton 2, an f T T α ()n 1,, then reject Ho. When Ho: Populaton 1 Populaton 2 an Ha: Populaton 1 < Populaton 2, an f T+ Tα, then reject Ho. ()n 1, 4

5 Example: Let s reo our prevous example usng the non-parametrc test: Ho: Ha: Groun base sensors = Ar base sensors Groun base sensors Ar base sensors α = 0.05 Locaton Groun ( o C) Ar ( o C) Dfference ( ) Rank of Sgne Rank of Sum of Ranks: T + = 0 T = 55. Decson Rule: If T+ or T T0.05( 2), 10 = 8, then reject Ho; otherwse, o not reject Ho. Concluson: Snce T + = 0 < 8 (P < 0.005), reject Ho an conclue that the mean fference between groun an ar-base sensors at these fve locatons s sgnfcantly fferent. 5