Hypothesis Tests: Two Independent Samples

1. Hypothesis Tests: Two Independent Samples

2. Violent Videos Again • Bushman (1998) Violent videos and aggressive behavior • Doing the study Bushman’s way--almost • Bushman had two independent groups • Violent video versus educational video • We want to compare mean number of aggressive associations between groups

3. The Data

4. Analysis • 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 want to test differences between sample means. • Not between a sample and a population mean Cont.

5. Analysis--cont. • How are sample means distributed if H0 is true? • Need sampling distribution of differences between means • Same idea as before, except statistic is (X1 - X2)

6. Sampling Distribution of Mean Differences • Mean of sampling distribution = m1 - m2 • Standard deviation of sampling distribution (standard error of mean differences) = Cont.

7. Sampling Distribution--cont. • Distribution approaches normal as n increases. • Later we will modify this to “pool” variances.

8. Analysis--cont. • Same basic formula as before, but with accommodation to 2 groups. • Note parallels with earlier t

9. Our Data

10. Degrees of Freedom • Each group has 100 subjects. • Each group has n - 1 = 100 - 1 = 99 df • Total df = n1- 1 + n2 - 1 = n1 + n2 - 2 100 + 100 - 2 = 198 df • t.025(198) = +1.97 (approx.)

11. Conclusions • Since 2.66 > 1.97, reject H0. • Conclude that those who watch violent videos produce more aggressive associates (M = 7.10) than those who watch nonviolent videos (M = 5.65), t(198) = 2.66, p < .05 (one-tailed).

12. IQ PillIndependent Samples

13. IQ PillIndependent Samples

14. Computing t

15. Conclusions Since -.88 < 1.97, fail to reject H0. Conclude that those who took the IQ pill did not score significantly higher (M = 102.4) on an IQ test than those who did not take the IQ pill (M = 99.4), t(8) = -.88, p > .10.

16. t value for related samples Larger t value for same data

17. Assumptions • Two major assumptions • Both groups are sampled from populations with the same variance • “homogeneity of variance” • Both groups are sampled from normal populations • Assumption of normality • Frequently violated with little harm.

18. Pooling Variances • If we assume both population variances are equal, then average of sample variances would be better estimate. Cont.

19. Pooling Variances--cont. • Substitute sp2 in place of separate variances in formula for t. • Will not change result if sample sizes equal • Do not pool if one variance more than 4 times the other. • Discuss

20. Heterogeneous Variances • Refers to case of unequal population variances. • We don’t pool the sample variances. • We adjust df and look t up in tables for adjusted df. • Minimum df = smaller n - 1. • Most software calculates optimal df.

21. Effect Size for Two Groups • Extension of what we already know. • 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. • Gives d • Which standard deviation? Cont.

22. Effect Size, cont. • Use either standard deviation or their pooled average. • We will pool because neither has any claim to priority. • Pooled st. dev. = √14.8 = 3.85 Cont.

23. Effect Size, cont. • This difference is approximately .38 standard deviations. • This is a medium effect. • Elaborate

24. Confidence Limits Cont.

25. Confidence Limits--cont. • 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. • Comment that reference is to “interval formed in this way,” not “this interval.”