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t Test for Two Independent Samples t test for two independent samples Basic Assumptions Independent samples are not paired with other observations Null hypothesis states that there is no difference between the means of the groups Or H0: µ1 - µ2 ≤ 0 t test for two independent samples Basic Assumptions Alternate hypothesis H1: µ1 - µ2 > 0 Two other possible alternate hypotheses Directional less than H1: µ1 - µ2 < 0 Or Nondirectional H1: µ1 - µ2 ≠ 0 t ratio (X1 – X2) – (µ1 - µ2)hyp t= sx1 – x2 Calculation steps for t ratio for two independent means Phase I 1. Assign a value to n1 2. Sum all X1 scores 3. Find mean for X1 4. Square each X1 score 5. Sum all squared X1 scores 6. Solve for SS1 Repeat for X2 Calculation steps for t ratio for two independent means Phase II 7. Calculate pooled variance using formula p 290 SS1 + SS2 s2p = n1 + n2 – 2 8. Calculate standard error p 291 9. Substitute numbers to get t ratio Pooled variance estimate The pooled variance represents the mean of the variances for the two samples Estimated standard error uses calculated pooled variance p-value The p-value indicates the degree of rarity of the observed test result when combined with all potentially more deviant test results. Smaller p-values tend to discredit the null hypothesis and support the research hypothesis. Significance?? Statistical significance between pairs of sample means implies only that the null hypothesis is probably false, and not whether it’s false because of a large or small difference between the population means. Confidence intervals Confidence intervals for µ1 - µ2 specify ranges of values that, in the long run, include the unknown effect (difference between population means) a certain percent of the time. X1 – X2 ± (tconf )(sx1 – x2 ) But wait ……. there is more!! Significance?? Statistical significance between pairs of sample means implies only that the null hypothesis is probably false, and not whether it’s false because of a large or small difference between the population means. Effect size: Cohen’s d __mean difference_ d = standard deviation = X1 – X2 √ s2 p Effect size: Cohen’s d Interpreting d Effect size is small if d is less than 0.2 Effect size is medium if d is in the vicinity of 0.5 Effect size is large if d is more than 0.8 Assumptions when using t ratio Both underlying populations are normally distributed Both populations have equal variances If these are not met you might try: Increasing sample size Equate sample sizes Use a less sensitive (yet more complex) t test Use a less sensitive test such as Mann-Whitney U test