1. Chapter 14 Hypothesis Tests: Two Independent Samples Fundamental Statistics for the Behavioral Sciences, 5th edition David C. Howell © 2003 Brooks/Cole Publishing Company/ITP

2. 2Chapter 14 Hypothesis Tests: Two Independent Samples Major Points An exampleAn example Distribution of differences between meansDistribution of differences between means Heterogeneity of VarianceHeterogeneity of Variance NonnormalityNonnormality Effect sizeEffect size Cont.

3. 3Chapter 14 Hypothesis Tests: Two Independent Samples Major Points--cont Confidence limitsConfidence limits SPSS printoutSPSS printout Review questionsReview questions

4. 4Chapter 14 Hypothesis Tests: Two Independent Samples Violent Videos Again Bushman (1998) Violent videos and aggressive behaviorBushman (1998) Violent videos and aggressive behavior XDoing the study Bushman’s way--almost Bushman had two independent groupsBushman had two independent groups XViolent video versus educational video XWe want to compare mean number of aggressive associations between groups

5. 5Chapter 14 Hypothesis Tests: Two Independent Samples The Data

6. 6Chapter 14 Hypothesis Tests: Two Independent SamplesAnalysis These are Bushman’s data, though he had more dimensions.These are Bushman’s data, though he had more dimensions. We still have means of 7.10 and 5.65, but both of these are sample means.We still have means of 7.10 and 5.65, but both of these are sample means. We want to test differences between sample means.We want to test differences between sample means. XNot between a sample and a population mean Cont.

7. 7Chapter 14 Hypothesis Tests: Two Independent SamplesAnalysis--cont. How are sample means distributed if H 0 is true?How are sample means distributed if H 0 is true? Need sampling distribution of differences between meansNeed sampling distribution of differences between means XSame idea as before, except statistic is (X 1 - X 2 )

8. 8Chapter 14 Hypothesis Tests: Two Independent Samples Sampling Distribution of Mean Differences Mean of sampling distribution =  1 -  2Mean of sampling distribution =  1 -  2 Standard deviation of sampling distribution (standard error of mean differences) =Standard deviation of sampling distribution (standard error of mean differences) = Cont.

9. 9Chapter 14 Hypothesis Tests: Two Independent Samples Sampling Distribution--cont. Distribution approaches normal as n increases.Distribution approaches normal as n increases. Later we will modify this to “pool” variances.Later we will modify this to “pool” variances.

10. 10Chapter 14 Hypothesis Tests: Two Independent SamplesAnalysis--cont. Same basic formula as before, but with accommodation to 2 groups.Same basic formula as before, but with accommodation to 2 groups. Note parallels with earlier tNote parallels with earlier t

11. 11Chapter 14 Hypothesis Tests: Two Independent Samples Our Data

12. 12Chapter 14 Hypothesis Tests: Two Independent Samples Degrees of Freedom Each group has 100 subjects.Each group has 100 subjects. Each group has n - 1 = 100 - 1 = 99 dfEach group has n - 1 = 100 - 1 = 99 df Total df = n 1 - 1 + n 2 - 1 = n 1 + n 2 - 2 100 + 100 - 2 = 198 dfTotal df = n 1 - 1 + n 2 - 1 = n 1 + n 2 - 2 100 + 100 - 2 = 198 df t.025 (198) = +1.97 (approx.)t.025 (198) = +1.97 (approx.)

13. 13Chapter 14 Hypothesis Tests: Two Independent SamplesConclusions Since 2.66 > 1.97, reject H 0.Since 2.66 > 1.97, reject H 0. Conclude that those who watch violent videos produce more aggressive associates than those who watch nonviolent videos.Conclude that those who watch violent videos produce more aggressive associates than those who watch nonviolent videos.

14. 14Chapter 14 Hypothesis Tests: Two Independent SamplesAssumptions Two major assumptionsTwo major assumptions XBoth groups are sampled from populations with the same variance “homogeneity of variance”“homogeneity of variance” XBoth groups are sampled from normal populations Assumption of normalityAssumption of normality XFrequently violated with little harm.

15. 15Chapter 14 Hypothesis Tests: Two Independent Samples Pooling Variances If we assume both population variances are equal, then average of sample variances would be better estimate.If we assume both population variances are equal, then average of sample variances would be better estimate. Cont.

16. 16Chapter 14 Hypothesis Tests: Two Independent Samples Pooling Variances--cont. Substitute s p 2 in place of separate variances in formula for t.Substitute s p 2 in place of separate variances in formula for t. Will not change result if sample sizes equalWill not change result if sample sizes equal Do not pool if one variance more than 4 times the other.Do not pool if one variance more than 4 times the other. XDiscuss

17. 17Chapter 14 Hypothesis Tests: Two Independent Samples Heterogeneous Variances Refers to case of unequal population variances.Refers to case of unequal population variances. We don’t pool the sample variances.We don’t pool the sample variances. We adjust df and look t up in tables for adjusted df.We adjust df and look t up in tables for adjusted df. Minimum df = smaller n - 1.Minimum df = smaller n - 1. XMost software calculates optimal df.

18. 18Chapter 14 Hypothesis Tests: Two Independent Samples Effect Size for Two Groups Extension of what we already know.Extension of what we already know. We can often simply express the effect as the difference between means (=1.45)We can often simply express the effect as the difference between means (=1.45) We can scale the difference by the size of the standard deviation.We can scale the difference by the size of the standard deviation. XGives d XWhich standard deviation? Cont.

19. 19Chapter 14 Hypothesis Tests: Two Independent Samples Effect Size, cont. Use either standard deviation or their pooled average.Use either standard deviation or their pooled average. XWe will pool because neither has any claim to priority. XPooled st. dev. = √14.8 = 3.85 Cont.

20. 20Chapter 14 Hypothesis Tests: Two Independent Samples Effect Size, cont. This difference is approximately.38 standard deviations.This difference is approximately.38 standard deviations. This is a medium effect.This is a medium effect. ElaborateElaborate

21. 21Chapter 14 Hypothesis Tests: Two Independent Samples Confidence Limits Cont.

22. 22Chapter 14 Hypothesis Tests: Two Independent Samples Confidence Limits--cont. p =.95 that interval formed in this way includes the true value of  1 -  2.p =.95 that interval formed in this way includes the true value of  1 -  2. Not probability that  1 -  2 falls in the interval, but probability that interval includes  1 -  2.Not probability that  1 -  2 falls in the interval, but probability that interval includes  1 -  2. XComment that reference is to “interval formed in this way,” not “this interval.”

23. 23Chapter 14 Hypothesis Tests: Two Independent Samples Computer Example SPSS reproduces the t value and the confidence limits.SPSS reproduces the t value and the confidence limits. All other statistics are the same as well.All other statistics are the same as well. Note different df depending on homogeneity of variance.Note different df depending on homogeneity of variance. Don’t pool if heterogeneityDon’t pool if heterogeneity and modify degrees of freedomand modify degrees of freedom

24. 24Chapter 14 Hypothesis Tests: Two Independent Samples

25. 25Chapter 14 Hypothesis Tests: Two Independent Samples Review Questions Why do you suppose it makes a difference whether we compare a sample and a population mean or two sample means?Why do you suppose it makes a difference whether we compare a sample and a population mean or two sample means? What do we know about the sampling distribution of differences between means?What do we know about the sampling distribution of differences between means? Cont.

26. 26Chapter 14 Hypothesis Tests: Two Independent Samples Review Questions--cont. What do we mean by “the standard error of differences between means?”What do we mean by “the standard error of differences between means?” How do we calculate the degrees of freedom with two groups?How do we calculate the degrees of freedom with two groups? What are the underlying assumptions?What are the underlying assumptions? What does it mean to “pool” the variances?What does it mean to “pool” the variances? Cont.

27. 27Chapter 14 Hypothesis Tests: Two Independent Samples Review Questions--cont. When would we not pool the variances?When would we not pool the variances? What is the difference between how we calculate confidence limits with one or with two means?What is the difference between how we calculate confidence limits with one or with two means? What does the text mean by modifying the df when we have heterogeneity of variance?What does the text mean by modifying the df when we have heterogeneity of variance?