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Hypothesis tests for two samples Test F test on ratio of two variances Null hypothesis Test statistic The ratio of the variances of 2 the two populations is 1 2 . 2 F s12 / 12 s2 2 / 2 2 s 2 1 s22 Distribution Fn1 1, n2 1 D* n1n2 D where KolmogorovSmirnov 2-sample test The two samples are drawn D = largest difference between from the same underlying cumulative probabilities based on As in statistical tables population. the two samples and n1 , n2 are the sample sizes. Rank all the data values from both samples together; for sample sizes m, n : m n , calculate T1 and T2 1 S 2 n(n 1) where The two samples are drawn R is the sum of ranks of from the same underlying population. sample size m, S is the sum of [H1: the two samples come from populations with different ranks of sample size n. The test statistic is the smaller of medians] T1 , T2 . The test statistic can also be found by counting the number of values in sample 2 which exceed each of the values in sample 1, repeat for all the values in sample 2. There are online calculators such as: Mann Whitney U test 1 R 2 m(m 1) As in statistical tables. Approx Normal for large samples with Mean 12 mn , Variance 121 mn m n 1 http://socr.stat.ucla.edu/Applets.dir/U_Test.html Normal test for paired samples with known variance The difference in the population means has value k. z Normal test for paired samples with unknown variance The difference in the population means has value k. z x1 x2 k d k n n x1 x2 k d k s s n Normal test for x y 1 2 unpaired samples The difference in the means of z 1 1 the two populations is 1 2 . with common n1 n2 known variance © MEI 2008 N(0, 1) N(0, 1) for large samples n N(0, 1) x y 1 2 where z Normal test for 1 1 unpaired samples The difference in the means of s n1 n2 with common the two populations is 1 2 . unknown n 1 s12 n2 1 s22 variance s 1 N(0, 1) for large samples x y 1 2 Normal test for unpaired samples The difference in the means of z 12 2 2 the two populations is 1 2 . with different n1 n2 known variances N(0, 1) Normal test for x y 1 2 unpaired samples The difference in the means of z with different s12 s2 2 the two populations is 1 2 . unknown n1 n2 variances N(0, 1) for large samples n1 n2 2 Sign test Population of differences has median = 0. r = number of values of di 0 t test for paired samples The difference in the population means has value k. t where di xi yi x1 x2 k d k s s n B(n, ½) where n = number of values ≠ 0 tn 1 n x y 1 2 t where t test for 1 1 unpaired samples The difference in the means of s n1 n2 with common the two populations is 1 2 . unknown n1 1 s12 n2 1 s2 2 variance s tn n 2 1 2 n1 n2 2 Wilcoxon paired sample test Test statistic: T = min [P, Q] P, Q are the sums of the ranks corresponding to positive and negative deviations d i M Population of differences has median = M. As in statistical tables where di xi yi Wilcoxon rank sum 2-sample test The two samples are drawn from the same underlying population. [H1: the two samples come from populations with different medians] Statistical tables give critical values for the lower tail; Rank all the data values from both critical value for the upper tail is m(m n 1) WC samples together; W sum of ranks of sample size where WC is the critical m. value from tables. where sample sizes are Approx Normal for large m, n : m n . samples with Mean 12 mn 12 m m 1 , Variance 121 mn m n 1 © MEI 2008