1. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter Twelve The Two-Sample t-Test

2. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 3 Understanding the Two-Sample Experiment

4. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 6 Relationship in the Population in a Two-sample Experiment

7. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 7 The Independent Samples t-Test

8. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 8 Independent Samples t-Test The independent samples t-test is the parametric procedure used for testing two sample means from independent samples Two samples are independent when we randomly select participants for a sample, without regard to who else has been selected for either sample

9. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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 at least roughly normal distributions. 3.The populations have homogeneous variance. Homogeneity of variance means the variance of all populations being represented are equal. 4.While n s may be different, they should not be massively unequal.

10. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 11 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.

12. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 12 Computing the Independent Samples t-Test 1.Calculate the estimated population variance for each condition

13. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 13 Computing the Independent Samples t-Test 2.Compute the pooled variance

14. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 14 Computing the Independent Samples t-Test 3.Compute the standard error of the difference

15. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 15 Computing the Independent Samples t-Test 4.Compute t obt for two independent samples

16. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 16 Computing the Independent Samples t-Test These steps can be combined into the following computational formula for the independent samples t-test

17. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 17 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

18. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 18 Confidence Interval When the t-test for independent samples is significant, a confidence interval for the difference between two  s should be computed

19. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 20 Related Samples t-Test

21. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 21 Related Samples The related samples t-test is the parametric inferential procedure used with 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 23 Repeated-Measures Design In a repeated-measures design, each participant is tested under all conditions of the independent variable.

24. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 24 Assumptions of the Related Samples t-Test 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 26 Statistical Hypotheses Two-tailed test One-tailed test −If we expect the−If we expect the difference to be larger than 0less than 0

27. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 27 Estimated Population Variance of the Difference Scores The formula for the estimated population variance of the difference scores is

28. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 28 Standard Error of the Mean Difference The formula for the standard error of the mean difference is

29. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 29 Computing the Related Samples t-Test The computational formula for the related samples t-test is

30. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 30 Critical Values The critical value (t crit ) is determined based on degrees of freedom df = N – 1 where N equals the number of difference scores The selected , and whether a one-tailed or two-tailed test is used

31. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 31 Confidence Interval When the t-test for related samples is significant, a confidence interval for  D should be computed

32. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 33 Describing the Relationship in a Two-Sample Experiment

34. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. 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. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 35 Sample Line Graphs Describing a Significant Relationship

36. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 36 Effect size indicates the amount of influence changing the conditions of the independent variable has on dependent scores The larger the effect size, the more scientifically important the independent variable is There are several ways to measure effect size in a two-sample experiment Effect Size

37. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Cohen’s d Cohen’s d measures effect size as the magnitude of the difference between the conditions, relative to the population standard deviation Chapter 12 - 37 Cohen’s d for Independent Samples Cohen’s d for Related Samples

38. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Cohen’s d Values of dInterpretation of Effect Size d = 0.2Small effect d = 0.5Medium effect d = 0.8Large effect Chapter 12 - 38

39. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 39 Proportion of Variance Accounted For The squared point-biserial correlation coefficient indicates the proportion of variance accounted for in a two-sample experiment

40. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 40 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

41. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 41 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.

42. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 42 Example 1

43. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 43 Example 1

44. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 44 Example 1 Because t obt does not lie within the rejection region, we fail to reject H 0

45. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 45 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.

46. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 46 Sample 1 Sample 2 Differences 14 131511153-2 1310121314130-4 14151714 15012 Example 2 First, we find the differences between the matched scores

47. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 47 Example 2

48. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 12 - 48 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

49. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Key Terms Cohen’s d confidence interval for  D confidence interval for the difference between two  s effect size homogeneity of variance independent samples Chapter 12 - 49 independent-samples t-test matched-samples design point-biserial correlation coefficient pooled variance related samples related-samples t-test

50. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Key Terms Chapter 12 - 50 repeated-measures design sampling distribution of differences between the means sampling distribution of mean differences standard error of the difference standard error of the mean difference