1. Chapter Twelve The Two-Sample t-Test

2. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 2 is the mean of the first sample is the mean of the second sample is the estimated population standard deviation of the first sample is the estimated population standard deviation of the second sample is the number of scores in the first sample is the number of scores in the second sample New Statistical Notation

3. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 3 Understanding the Two-Sample Experiment

4. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 4 Participants’ scores are measured under two conditions of the independent variable Condition 1 produces sample mean that represents Condition 2 produces sample mean that represents Two-Sample Experiment

5. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 5 Two-Sample t-Test The parametric statistical procedure for determining whether the results of a two-sample experiment are significant is the two-sample t-test There are two versions of the two- sample t-test –The independent samples t-test –The related samples t-test.

6. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 6 Relationship in the Population in a Two-sample Experiment

7. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 7 The Independent Samples t-Test

8. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 8 Independent Samples t-Test The independent samples t-test is the parametric procedure used for significance testing of two sample means from independent samples Two samples are independent when we randomly select and assign a participant to a sample

9. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 9 Assumptions of the Independent Samples t-Test 1.The dependent scores measure an interval or ratio variable 2.The populations of raw scores form normal distributions 3.The populations have homogeneous variance. Homogeneity of variance means that the variance of all populations being represented are equal. 4.While N s may be different, they should not be massively unequal.

10. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 10 Two-tailed test One-tailed test –If  1 is expected to If  2 is expected to be larger than  2 be larger than  1 Statistical Hypotheses

11. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 11 Critical Values Critical values for the independent samples t-test (t crit ) are determined based on degrees of freedom df = ( n 1 - 1) + ( n 2 - 1), the selected , and whether a one-tailed or two-tailed test is used

12. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 12 Sampling Distribution The sampling distribution of differences between means is the distribution of all possible differences between two means when they are drawn from the raw score population described by H 0

13. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 13 Computing the Independent Samples t-Test 1.Calculate the estimated population variance for each condition

14. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 14 Computing the Independent Samples t-Test 2.Compute the pooled variance

15. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 15 Computing the Independent Samples t-Test 3.Compute the standard error of the difference

16. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 16 Computing the Independent Samples t-Test 4.Compute t obt for two independent samples

17. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 17 Computing the Independent Samples t-Test These steps can be combined into the following computational formula for the independent samples t-test

18. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 18 Confidence Interval When the t-test for independent samples is significant, a confidence interval for the difference between two ms should be computed

19. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 19 Power To maximize power in the independent samples t-test, you should maximize the difference produced by the two conditions Minimize the variability of the raw scores Maximize the sample n s

20. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 20 Related Samples t-Test

21. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 21 Related Samples The related samples t-test is the parametric inferential procedure used when we have two sample means from two related samples Related samples occur when we pair each score in one sample with a particular score in the other sample Two types of research designs that produce related samples are matched samples design and repeated measures design

22. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 22 Matched Samples Design In a matched samples design, the researcher matches each participant in one condition with a participant in the other condition We do this so that we have more comparable participants in the conditions

23. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 23 Repeated Measures Design In a repeated measures design, each participant is tested under all conditions of the independent variable.

24. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 24 Assumptions of the Related Samples t-Test The assumptions of the related samples t-test are when the dependent variable involves an interval or ratio scale The raw score populations are at least approximately normally distributed The populations being represented have homogeneous variance Because related samples form pairs of scores, the n in the two samples must be equal

25. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 25 Transforming the Raw Scores In a related samples t-test, the raw scores are transformed by finding each difference score The difference score is the difference between the two raw scores in a pair The symbol for a difference score is D

26. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 26 Statistical Hypotheses Two-tailed test One-tailed testIf we expect the difference to be larger than 0less than 0

27. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 27 Estimated Population Variance of the Difference Scores The formula for the estimated population variance of the difference scores is

28. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 28 Standard Error of the Mean Difference The formula for the standard error of the mean difference is

29. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 29 Computing the Related Samples t-Test The computational formula for the related samples t-test is

30. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 30 Critical Values The critical value (t crit ) is determined based on degrees of freedom df = N - 1 The selected , and whether a one-tailed or two-tailed test is used

31. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 31 Confidence Interval When the t-test for related samples is significant, a confidence interval for  D should be computed

32. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 32 Power The related samples t-test is intrinsically more powerful than an independent samples t-test To maximize the power you should –Maximize the differences in scores between the conditions. –Minimize the variability of the scores within each condition. –Maximize the size of N.

33. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 33 Describing the Relationship in a Two-Sample Experiment

34. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 34 Describing the Relationship Once a t-test has been shown to be significant, the next step is to describe the relationship In order to describe the relationship, you should –Compute a confidence interval –Graph the relationship –Compute the effect size –Compute the appropriate correlation coefficient to determine the strength of the relationship

35. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 35 Sample Line Graphs Describing a Significant Relationship

36. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 36 Using r pb Because a two-sample t-test involves one dichotomous X variable (the two conditions of the independent variable) and one continuous interval or ratio Y variable, the point-biserial correlation coefficient is the appropriate coefficient to use

37. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 37 Degrees of Freedom for the r pb For independent samples, df = ( n 1 - 1) + ( n 2 - 1), where n is the number of scores in a sample For related samples, df = N - 1, where N is the number of difference scores

38. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 38 In a two-sample experiment, equals the proportion of the variance accounted for This proportion of variance accounted for also is called the effect size in an experiment Effect size indicates how big a role changing the conditions of the independent variable plays in determining differences in dependent scores Effect Size

39. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 39 Sample 1 Sample 2 14 13151115 131012131413 14151714 15 Example 1 Using the following data set, conduct an independent samples t-test. Use  = 0.05 and a two-tailed test.

40. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 40 Example 1

41. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 41 Example 1

42. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 42 Example 1 Because t obt does not lie within the rejection region, we fail to reject H 0

43. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 43 Sample 1 Sample 2 14 13151115 131012131413 14151714 15 Example 2 Using the following data set, conduct a related samples t-test. Use  = 0.05 and a two-tailed test.

44. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 44 Sample 1 Sample 2 Differences 14 131511153-2 1310121314130-4 14151714 15012 Example 2 First, we find the differences between the matched scores

45. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 45 Example 2

46. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 46 Example 2 Using  = 0.05 and df = 8, t crit = 2.306 Because t obt does not lie within the rejection region, we fail to reject H 0