1. Statistics for the Social Sciences Psychology 340 Spring 2005 Using t-tests

2. Statistics for the Social Sciences Outline (for 2 weeks) Review t-tests –One sample, related samples, independent samples –Additional assumptions Levene’s test

3. Statistics for the Social Sciences Statistical analysis follows design The one-sample z-test can be used when: –1 sample –One score per subject –Population mean (  ) and standard deviation (  )are known

4. Statistics for the Social Sciences Statistical analysis follows design The one-sample t-test can be used when: –1 sample –One score per subject –Population mean (  ) is known –but standard deviation (  ) is NOT known

5. Statistics for the Social Sciences Testing Hypotheses –Step 1: State your hypotheses –Step 2: Set your decision criteria –Step 3: Collect your data –Step 4: Compute your test statistics Compute your estimated standard error Compute your t-statistic Compute your degrees of freedom –Step 5: Make a decision about your null hypothesis Hypothesis testing: a five step program

6. Statistics for the Social Sciences Performing your statistical test What are we doing when we test the hypotheses? –Consider a variation of our memory experiment example Population of memory patients MemoryTest  is known Memory treatment Memory patients Memory Test X Compare these two means Conclusions: the memory treatment sample are the same as those in the population of memory patients. they aren’t the same as those in the population of memory patients H0:H0: HA:HA:

7. Statistics for the Social Sciences Performing your statistical test What are we doing when we test the hypotheses? Real world (‘truth’) H 0 : is false (is a treatment effect) Two populations XAXA they aren’t the same as those in the population of memory patients H 0 : is true (no treatment effect) One population XAXA the memory treatment sample are the same as those in the population of memory patients.

8. Statistics for the Social Sciences Performing your statistical test What are we doing when we test the hypotheses? –Computing a test statistic: Generic test Could be difference between a sample and a population, or between different samples Based on standard error or an estimate of the standard error

9. Statistics for the Social Sciences Performing your statistical test Test statistic One sample z One sample t identical

10. Statistics for the Social Sciences Performing your statistical test Test statistic Diff. Expected by chance Standard error One sample z One sample t different don’t know this, so need to estimate it

11. Statistics for the Social Sciences Performing your statistical test Test statistic Diff. Expected by chance Standard error Estimated standard error One sample z One sample t different don’t know this, so need to estimate it Degrees of freedom

12. Statistics for the Social Sciences One sample t-test The t-statistic distribution (a transformation of the distribution of sample means transformed) –Varies in shape according to the degrees of freedom New table: the t-tablet-table

13. Statistics for the Social Sciences One sample t-test If test statistic is here Fail to reject H 0 Distribution of the t-statistic If test statistic is here Reject H 0 –To reject the H 0, you want a computed test statistics that is large The alpha level gives us the decision criterion New table: the t-tablet-table The t-statistic distribution (a transformation of the distribution of sample means transformed)

14. Statistics for the Social Sciences One sample t-test New table: the t-table One tailed - or - Two-tailed  levels Critical values of t t crit Degrees of freedom df

15. Statistics for the Social Sciences One sample t-test What is the t crit for a two-tailed hypothesis test with a sample size of n = 6 and an  -level of 0.05? Distribution of the t-statistic  = 0.05 Two-tailed n = 6 df = n - 1 = 5 t crit = + 2.571

16. Statistics for the Social Sciences One sample t-test Distribution of the t-statistic  = 0.05 One-tailed n = 6 df = n - 1 = 5 t crit = +2.015 What is the t crit for a one-tailed hypothesis test with a sample size of n = 6 and an  -level of 0.05?

17. Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal,  = 60? After the treatment they have an average score of = 55, s = 8 memory errors. Step 1: State your hypotheses H0:H0: the memory treatment sample are the same as those in the population of memory patients. HA:HA: they aren’t the same as those in the population of memory patients  Treatment >  pop = 60  Treatment <  pop = 60

18. Statistics for the Social Sciences One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. Step 2: Set your decision criteria H 0 :  Treatment >  pop = 60 H A :  Treatment <  pop = 60  = 0.05 One -tailed How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal,  = 60? After the treatment they have an average score of = 55, s = 8 memory errors. An example: One sample t-test

19. Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment. Step 2: Set your decision criteria H 0 :  Treatment >  pop = 60 H A :  Treatment <  pop = 60  = 0.05 One -tailed How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal,  = 60? After the treatment they have an average score of = 55, s = 8 memory errors.

20. Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment.  = 0.05 One -tailed Step 3: Collect your data H 0 :  Treatment >  pop = 60 H A :  Treatment <  pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal,  = 60? After the treatment they have an average score of = 55, s = 8 memory errors.

21. Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment.  = 0.05 One -tailed Step 4: Compute your test statistics = -2.5 H 0 :  Treatment >  pop = 60 H A :  Treatment <  pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal,  = 60? After the treatment they have an average score of = 55, s = 8 memory errors.

22. Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment.  = 0.05 One -tailed Step 4: Compute your test statistics t = -2.5 H 0 :  Treatment >  pop = 60 H A :  Treatment <  pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal,  = 60? After the treatment they have an average score of = 55, s = 8 memory errors.

23. Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment.  = 0.05 One -tailed Step 5: Make a decision about your null hypothesis H 0 :  Treatment >  pop = 60 H A :  Treatment <  pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal,  = 60? After the treatment they have an average score of = 55, s = 8 memory errors. t crit = -1.753

24. Statistics for the Social Sciences One sample t-test An example: One sample t-test Memory experiment example: We give a n = 16 memory patients a memory improvement treatment.  = 0.05 One -tailed Step 5: Make a decision about your null hypothesis H 0 :  Treatment >  pop = 60 H A :  Treatment <  pop = 60 How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal,  = 60? After the treatment they have an average score of = 55, s = 8 memory errors. -1.753 = t crit t obs =-2.5 - Reject H 0

25. Statistics for the Social Sciences t Test for Dependent Means Unknown population mean and variance Two situations –One sample, two scores for each person Repeated measures design –Two samples, but individuals in the samples are related Same procedure as t test for single sample, except –Use difference scores –Assume that the population mean is 0

26. Statistics for the Social Sciences Statistical analysis follows design The related-samples t- test can be used when: –1 sample –Two scores per subject

27. Statistics for the Social Sciences Statistical analysis follows design The related-samples t- test can be used when: –1 sample –Two scores per subject –2 samples –Scores are related - OR -

28. Statistics for the Social Sciences Performing your statistical test Diff. Expected by chance Estimated standard error of the differences Test statistic What are all of these “D’s” referring to? Mean of the differences Number of difference scores Difference scores –For each person, subtract one score from the other –Carry out hypothesis testing with the difference scores Population of difference scores with a mean of 0 –Population 2 has a mean of 0

29. Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Difference scores 2 6 5 9 22 Person Pre-test Post-test 1 2 3 4 45 55 40 60 43 49 35 51 (Pre-test) - (Post-test) H0:H0: There is no difference between pre-test and post- test HA:HA: There is a difference between pre-test and post- test  D = 0  D ≠ 0

30. Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Difference scores 2 6 5 9 22 = 5.5 Person Pre-test Post-test 1 2 3 4 45 55 40 60 43 49 35 51 (Pre-test) - (Post-test)

31. Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person Pre-test Post-test Difference scores 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5

32. Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person Difference scores 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 -3.5- 5.5 = 0.5- 5.5 = -0.5- 5.5 = 3.5- 5.5 = 12.25 0.25 12.25 25 = SS D D - D(D - D) 2

33. Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 -3.5 0.5 -0.5 3.5 D - D Difference scores 12.25 0.25 12.25

34. Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 2.9 = s D -3.5 0.5 -0.5 3.5 D - D Difference scores 12.25 0.25 12.25

35. Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 2.9 = s D 1.45 = s D ? Think back to the null hypotheses -3.5 0.5 -0.5 3.5 D - D Difference scores 12.25 0.25 12.25

36. Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 2.9 = s D 1.45 = s D H 0 : Memory performance at the post-test are equal to memory performance at the pre-test. -3.5 0.5 -0.5 3.5 D - D Difference scores 12.25 0.25 12.25

37. Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 2.9 = s D 1.45 = s D This is our t obs -3.5 0.5 -0.5 3.5 D - D Difference scores 12.25 0.25 12.25

38. Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 2.9 = s D 1.45 = s D t obs  = 0.05 Two-tailed t crit -3.5 0.5 -0.5 3.5 D - D Difference scores 12.25 0.25 12.25

39. Statistics for the Social Sciences +3.18 = t crit - Reject H 0 Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 2.9 = s D 1.45 = s D t obs  = 0.05 Two-tailed t crit -3.5 0.5 -0.5 3.5 D - D Difference scores 12.25 0.25 12.25 t obs =3.8

40. Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person 1 2 3 4 Pre-test Post-test 45 55 40 60 43 49 35 51 2 6 5 9 22 D = 5.5 25 = SS D (D - D) 2 2.9 = s D 1.45 = s D t obs  = 0.05 Two-tailed t crit T obs > t crit so we reject the H 0 -3.5 0.5 -0.5 3.5 D - D Difference scores 12.25 0.25 12.25

41. Statistics for the Social Sciences Next time Finishing up related samples t-tests Independent samples t-tests Using SPSS with t-tests (maybe next Monday)