Independent Samples t-test

1. Independent Samples t-test

2. Review of Tests We�ve Learned So Far z-test and one-sample t-tests: Used to compare one sample mean to a population mean or some other known value. More common: Compare two (or more) sample means to each other Two general research strategies: Two completely separate (independent) samples Two related (dependent) samples

3. Did mental imagery raise scores or are the differences due to chance (sampling error)? Or, do the samples come from the same population? Introducing the Independent Samples t-test

4. A Quick Word on Notation Use subscripts to designate to which group statistics and parameters belong: n1, SS1, S21, M1, �1 n2, SS2, S22, M2, �2

5. Setting the Statistical Hypothesis for Ind. Samples For Two-Tailed (Directional) Tests: H0 H0: �1 � �2 = 0 Or H0: �1 = �2 H1 H1: �1 � �2 ? 0 Or H1: �1 ? �2

6. Independent Samples t Ratio

7. Two Sample t-test Example Research Question: Does mental imagery change memory scores? State Statistical Hypothesis: H0: � rote = � imagery H1: � rote ? � imagery

8. Two Sample t-test Example Group1: Rote Group2: Imagery 24 13 18 31 23 17 19 29 16 20 23 26 17 15 29 21 19 26 30 24 Sample Descriptive Statistics: Rote Group: n1=10, S21=17.78, SS1=160, M1=19 Imagery Group: n2=10, S22=22.22, SS2 =200, M2=25 Compute standard error of the mean difference:

9. Set Decision Criteria: df=(n1-1)+(n2-1)=(10-1)+(10-1)=18 If a=.05 and two-tailed test, from t-table, tcrit = + 2.101 Reject Ho if tobtained> tcrit Compute test statistic (tobtained): Make Decision: Reject HO Mental imagery memory scores are significantly higher compared to rote scores, t(18)=-3.00, p<.05. Two Sample t-test Example

10. Mental imagery improves memory scores. H0: �1>�2 or H0: �1-�2>0 H1: �1<�2 or H1: �1-�2<0 Where �1=Rote Group and �2=Imagery Group In this case, set decision criteria: Critical region is in the lower tail of the distribution. df=(n1-1)+(n2-1)=(10-1)+(10-1)=18 If a=.05 and one-tailed test, from t-table, tcrit = - 1.734 Reject Ho if tobtained falls within the critical region. Setting Decision Criteria for a Directional Test (one-tailed)

11. Mental imagery decreases memory scores. H0: �1<�2 or H0: �1-�2<0 H1: �1>�2 or H1: �1-�2>0 Where �1=Rote Group and �2=Imagery Group In this case, set decision criteria: Critical region is in the upper tail of the distribution. df=(n1-1)+(n2-1)=(10-1)+(10-1)=18 If a=.05 and one-tailed test, from t-table, tcrit = 1.734 Reject Ho if tobtained falls within the critical region. Setting Decision Criteria for a Directional Test (one-tailed)

12. Assumptions of the Independent Samples t-test Independent Observations Normality Homogeneity of Variance

13. How big is the effect? We can use Cohen�s d to estimate effect size: