Thursday, September 26, 2013

1. Thursday, September 26, 2013 Paired samples t-test

2. Last Time • Independent Samples t-test • Numerator: (MA-MB)-(μA- μB) • (μA- μB) always equals 0 • Denominator: (estimated standard error of the difference between two independent means) • Formula for denominator: • Formula for pooled variance: • Questions about these items before we move on? • Next topic: Paired/Related Samples t-tests

3. Practice problem You are conducting an experiment about methods for teaching reading. You have access to a sample of 10 3rd graders. You randomly assign half of the group to an experimental reading intervention, and the other half receives instruction as usual in their classrooms. After the intervention, you measure the number of words each child can read correctly in one minute, and obtain the following results. Group 1 (experimental) scores: 30, 35, 40, 20, 32 Group 2 scores: 25, 30, 20, 18, 18 Conduct a t-test to find out whether the groups are different in reading ability at the end of the study. Show all four steps of hypothesis testing. Use the t-table from the textbook or the one from “useful links” on the class website

4. Effect Size for the t Test for Independent Means Estimated effect size after a completed study Percentage of the variance in the DV explained by the IV

5. Power for the t Test for Independent Means (.05 significance level)

6. Approximate Sample Size Needed for 80% Power (.05 significance level)

7. Monitor Your Understanding http://teachertap.appspot.com/ Please click on the above link, and type (or paste) where it says “classroom”: 340.01_9/26/13 Then click buttons “got it” “unsure” or “lost me” to provide anonymous feedback about your understanding throughout the lecture

8. 1 sample • One score per subject • Population mean (μ) and standard deviation (s) are known • The one-sample z-test can be used when: Statistical analysis follows design

9. 1 sample • One score per subject • Population mean (μ) is known • but standard deviation (s) is NOT known • The one-sample t-test can be used when: Statistical analysis follows design

10. Statistical analysis follows design • 2 samples • Samples are independent • The independent samples t-test can be used when:

11. 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 pairs of individuals in the samples are related in some way (data points can be grouped into pairs) • Same procedure as t test for single sample, except • Use difference scores • Assume that the population mean is 0

12. 1 sample • Two scores per subject • The related-samples t-test can be used when: Statistical analysis follows design

13. 2 samples • Scores are related • The related-samples t-test can be used when: Statistical analysis follows design • 1 sample • Two scores per subject - OR -

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

15. Monitor Your Understanding http://teachertap.appspot.com/ Please click on the above link, and type (or paste) where it says “classroom”: 340.01_9/26/13 Then click buttons “got it” “unsure” or “lost me” to provide anonymous feedback about your understanding throughout the lecture

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

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

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

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

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

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

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

23. Pre-test Post-test 45 43 55 49 40 35 60 51 Performing your statistical test What are all of these “D’s” referring to? Difference scores Person D - MD (D - MD)2 1 2 -3.5 12.25 2 6 0.5 0.25 H0: Memory performance at the post-test are equal to memory performance at the pre-test. 3 5 -0.5 0.25 4 9 3.5 12.25 22 25 = SSD MD = 5.5 2.9 = sD

24. Pre-test Post-test 45 43 55 49 40 35 60 51 Performing your statistical test What are all of these “D’s” referring to? Difference scores Person D - MD (D - MD)2 1 2 -3.5 12.25 2 6 0.5 0.25 This is our tobs 3 5 -0.5 0.25 4 9 3.5 12.25 22 25 = SSD MD= 5.5 2.9 = sD

25. Pre-test Post-test 45 43 55 49 40 35 60 51 Performing your statistical test What are all of these “D’s” referring to? Difference scores Person D - MD (D - D)2 1 2 -3.5 12.25 2 6 0.5 0.25 tobs 3 5 -0.5 0.25 tcrit Two-tailed • = 0.05 4 9 3.5 12.25 22 25 = SSD MD= 5.5 2.9 = sD

26. Pre-test Post-test 45 43 55 49 40 35 tobs=3.8 60 51 +3.18 = tcrit Performing your statistical test What are all of these “D’s” referring to? Difference scores Person D - MD (D - MD)2 1 2 -3.5 12.25 2 6 0.5 0.25 tobs 3 5 -0.5 0.25 tcrit Two-tailed • = 0.05 4 9 3.5 12.25 22 25 = SSD MD = 5.5 2.9 = sD - Reject H0

27. Pre-test Post-test 45 43 55 49 40 35 60 51 Performing your statistical test What are all of these “D’s” referring to? Difference scores Person D - MD (D - MD)2 1 2 -3.5 12.25 2 6 0.5 0.25 tobs 3 5 -0.5 0.25 tcrit Two-tailed • = 0.05 4 9 3.5 12.25 22 25 = SSD Tobs > tcrit so we reject the H0 MD = 5.5 2.9 = sD

28. Monitor Your Understanding http://teachertap.appspot.com/ Please click on the above link, and type (or paste) where it says “classroom”: 340.01_9/26/13 Then click buttons “got it” “unsure” or “lost me” to provide anonymous feedback about your understanding throughout the lecture

29. Statistical Tests Summary Design Statistical test (Estimated) Standard error Degrees of freedom One sample, σ known One sample, σ unknown n– 1 Two independent samples, σ unknown (nA – 1) + (nB-1) Two related samples (or one sample with repeated measures), σ unknown

30. Effect Sizes & Power for t Test for Dependent Means Remember we don’t know these Estimated

31. Approximate Sample Size Needed for 80% Power (.05 significance level) • Using Power and effect sizes to determine how many participants you need

32. Monitor Your Understanding http://teachertap.appspot.com/ Please click on the above link, and type (or paste) where it says “classroom”: 340.01_9/26/13 Then click buttons “got it” “unsure” or “lost me” to provide anonymous feedback about your understanding throughout the lecture

33. Using spss to conduct t-tests One-sample t-test: Analyze =>Compare Means =>One sample t-test. Select the variable you want to analyze, and type in the expected mean based on your null hypothesis. Independent samples t-test: Analyze =>Compare Means =>Independent samples t-test. Specify test variable and grouping variable, and click on define groups to specify how grouping variable will identify groups. Paired or related samples t-test: Analyze =>Compare Means =>Paired samples t-test. Select the variables you want to compare and drag them into the “pair 1” boxes labeled “variable 1” and “variable 2”

34. Practice problem Use SPSS to conduct a paired samples t-test comparing opinions about two different kinds of chocolate chip cookies. Use SPSS to conduct an independent samples t-test comparing opinions about two different samples of one brand.

35. Using excel to compute t-tests =ttest(array1,array2,tails,type) Select the arrays that you want to compare, specify number of tails (1 or 2) and type of t-test (1=dependent, 2=independent w/equal variance assumed, 3=independent w/unequal variance assumed). Returns the p-value associated with the t-test.

36. Next Homework (due Monday, October 1) Chapter 10: 1-4, 6, 21, 22 Chapter 11: 1, 5, 9, 10, 18, 20