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Analyzing Two Samples for Differences in Meristic/Mensural Means I BIOL457 8 February 2016 The t-Test for Paired Data, Equal Variances Paired t-Test DV meristic/mensural Two DV records per subject Ex: H0: There is no difference in stem diameter between hemlocks on tip-up mounds and on adjacent flat areas. Paired t-Test Compares mean difference between __ observations (x) to a hypothesized mean of zero* __ __ t = x/(SD/√N) SD is standard deviation of differences N is number of pairs df = N1 _ *Or, another hypothesized mean difference may be substituted via subtraction from x in the formula for t HANDOUT Tellería et al., 1995 Spanish Juniper Woodland Birds HANDOUT Manter et al., 2000 Firs, Fungi, and CO2 The t-Test for Unpaired Data, Equal Variances t-Test—Equal Variances DV meristic/mensural One DV record per subject, two independent sets of subjects to compare Data pass test for equality of variance using Ftest Ex: H0: There is no difference in carapace length between Oklahoma and Arkansas razorback musk turtle females The F-test for Equality of Variances Divide larger variance by smaller variance Compare to F-test tabular values (Table B) Calculated F < Ftabular: Variances not shown to differ Calculated F > Ftabular: Variances differ significantly—don’t use t-test that assumes equal variances! The F-test for Equality of Variances Exs: Var1 = 8.31, N1 = 21 Var2 = 7.29, N2 = 21 F=1.14, dfnum=20, dfden=20, Ftabular[0.05]=2.12 Data pass test (p0.40) Var1 = 14.46, N1 = 26 Var2 = 5.52, N2 = 25 F= 2.62, dfnum=25, dfden=24, Ftabular[0.05]=1.96 Data fail test (p=0.01) Var1 = 31.6, N1 = 29 Var2 = 18.4, N2 = 31 F=1.72, dfnum=28, dfden=30, Ftabular[0.05]=1.87 Data pass test (p0.075) t-Test for Equal Variances __ __ (x1x2) _________________________ /√[Vpooled(1/N1+1/N2)], where t= Vpooled = [(N11)V1+(N21)V2]/(N1+N22) df = N1+N22 The t-Test for Unpaired Data, Unequal Variances t-Test—Unequal Variances DV must be meristic or mensural One DV record per subject, two independent sets of subjects to compare Data fail test for equality of variance using Ftest Less powerful than t-test for equal variance t-Test for Unequal Variances __ __ (x1x2) __________________ /√(V1/N1+V2/N2) t= df = (V1/N1+V2/N2)2 [(V1/N1)2/(N11)+ (V2/N2)2/(N21)] / (Not necessarily a whole number—round to nearest whole number to look up critical value of t in Table C, or integrate between values) HANDOUT Negro et al., 1998 Kestrel Carotenoids One Tail or Two? Two-tailed test No a priori assumption which group will have larger mean Ex: Female crickets preferring or avoiding male groups One-tailed test Good a priori reason to believe A may have larger mean than B /2 Ex: Bees learning to associate shape with food One Tail or Two? Two-tailed tests have twice the p values of one-tailed tests HANDOUT Ferguson and Fox, 1984 Juvenile Side-Blotched Lizards Standard Error ___ SE = SD/√N SE decreases as N increases Shows up as tic marks on bar graphs—overlap relates to p-value in test for differences Fig. 5-10 p. 128 Confidence Interval on the Mean Range of probable population mean under a set level of confidence 95% CL = Sample mean ± 1.96(SE) 95% of samples with mean and variance like the one drawn will be drawn from populations with their true mean within this range 98% CL = Sample mean ± 2.33(SE) 99% CL = Sample mean ± 2.58(SE)