2. Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Fall 2015 Room 150 Harvill Building 10:00 - 10:50 Mondays, Wednesdays & Fridays. http://courses.eller.arizona.edu/mgmt/delaney/d15s_database_weekone_screenshot.xlsx

4. By the end of lecture today 11/2/15 Logic of hypothesis testing Steps for hypothesis testing Levels of significance (Levels of alpha) what does p < 0.05 mean? what does p < 0.01 mean? Two-sample t-tests Preview of ANOVA

5. Before next exam (November 20 th ) Please read chapters 1 - 11 in OpenStax textbook Please read Chapters 2, 3, and 4 in Plous Chapter 2: Cognitive Dissonance Chapter 3: Memory and Hindsight Bias Chapter 4: Context Dependence

6. Homework Assignment Worksheet Assignment 17 More t-tests using Excel Due: Wednesday, November 4 th

7. Confidence Interval of 99% Has and alpha of 1% α =.01 Confidence Interval of 90% Has and alpha of 10% α =. 10 Confidence Interval of 95% Has and alpha of 5% α =.05 99%95%90% Area outside confidence interval is alpha Area in the tails is called alpha Area associated with most extreme scores is called alpha Critical statistic Critical Statistic Critical statistic Critical statistic Critical statistic Critical statistic

8. Rejecting the null hypothesis The result is “statistically significant” if: the observed statistic is larger than the critical statistic (which can be a ‘z” or “t” or “r” or “F” or x 2 ) observed stat > critical stat If we want to reject the null, we want our t (or z or r or F or x 2 ) to be big!! the p value is less than 0.05 (which is our alpha) p < 0.05 If we want to reject the null, we want our “p” to be small!! we reject the null hypothesis then we have support for our alternative hypothesis

9. Everyone will want to be enrolled in one of the lab sessions Labs continue this week, Project 3

10. Study Type 2: t-test Study Type 3: One-way Analysis of Variance (ANOVA) Comparing more than two means We are looking to compare two means Review

11. Comparing ANOVAs with t-tests Similarities still include: Using distributions to make decisions about common and rare events Using distributions to make inferences about whether to reject the null hypothesis or not The same 5 steps for testing an hypothesis The three primary differences between t-tests and ANOVAS are: 1. ANOVAs can test more than two means 2. We are comparing sample means indirectly by comparing sample variances 3. We now will have two types of degrees of freedom t(16) = 3.0; p < 0.05 F(2, 15) = 3.0; p < 0.05 Tells us generally about number of participants / observations Tells us generally about number of groups / levels of IV

12. A girl scout troop leader wondered whether providing an incentive to whomever sold the most girl scout cookies would have an effect on the number cookies sold. She provided a big incentive to one troop (trip to Hawaii), a lesser incentive to a second troop (bicycle), and no incentive to a third group, and then looked to see who sold more cookies. n = 5 x = 10 n = 5 x = 12 n = 5 x = 14 Troop 1 (nada) 10 8 12 7 13 Troop 2 (bicycle) 12 14 10 11 13 Troop 3 (Hawaii) 14 9 19 13 15 What is Independent Variable? How many groups? What is Dependent Variable? How many levels of the Independent Variable?

13. ANOVA: Using MS Excel A girlscout troop leader wondered whether providing an incentive to whomever sold the most girlscout cookies would have an effect on the number cookies sold. She provided a big incentive to one troop (trip to Hawaii), a lesser incentive to a second troop (bicycle), and no incentive to a third group, and then looked to see who sold more cookies. Troop 1 (Nada) 10 8 12 7 13 Troop 2 (bicycle) 12 14 10 11 13 Troop 3 (Hawaii) 14 9 19 13 15 n = 5 x = 10 n = 5 x = 12 n = 5 x = 14

14. Let ’ s do one Replication of study (new data)

15. Let ’ s do same problem Using MS Excel

17. SS within df within SS between df between 88 12 =7.33 40 2 =20 20 7.33 =2.73 40 2 88 12 MS between MS within # groups - 1 # scores - number of groups # scores - 1 3-1=2 15-3=12 15- 1=14

18. F critical (is observed F greater than critical F?) P-value (is it less than.05?) No, so it is not significant Do not reject null No, so it is not significant Do not reject null

19. Make decision whether or not to reject null hypothesis 2.7 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis Observed F = 2.73 Critical F (2,12) = 3.89 Also p-value is not smaller than 0.05 so we do not reject the null hypothesis Step 6: Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold

20. Make decision whether or not to reject null hypothesis 2.7 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis Observed F = 2.72727272 Critical F (2,12) = 3.88529 Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold F (2,12) = 2.73; n.s. The average number of cookies sold for three different incentives were compared. The mean number of cookie boxes sold for the “Hawaii” incentive was 14, the mean number of cookies boxes sold for the “Bicycle” incentive was 12, and the mean number of cookies sold for the “No” incentive was 10. An ANOVA was conducted and there appears to be no significant difference in the number of cookies sold as a result of the different levels of incentive F(2, 12) = 2.73; n.s.

25. Independent samples t-test Donald is a consultant and leads training sessions. As part of his training sessions, he provides the students with breakfast. He has noticed that when he provides a full breakfast people seem to learn better than when he provides just a small meal (donuts and muffins). So, he put his hunch to the test. He had two classes, both with three people enrolled. The one group was given a big meal and the other group was given only a small meal. He then compared their test performance at the end of the day. Please test with an alpha =.05 Big Meal 22 25 Small meal 19 23 21 Mean= 24 Mean= 21 Are the two means significantly different from each other, or is the difference just due to chance?

26. Mean= 24 Participant 1 2 3 Big Meal 22 25 Small meal 19 23 21 Mean= 21 Complete a t-test

27. Mean= 24 Participant 1 2 3 Big Meal 22 25 Small meal 19 23 21 Mean= 21 Complete a t-test

28. Mean= 24 Participant 1 2 3 Big Meal 22 25 Small meal 19 23 21 Mean= 21 Complete a t-test If checked you’ll want to include the labels in your variable range If checked, you’ll want to include the labels in your variable range

29. Finding Means

30. This is variance for each sample (Remember, variance is just standard deviation squared) Please note: “Pooled variance” is just like the average of the two sample variances, so notice that the average of 3 and 4 is 3.5

31. 3 4 Mean= 24 Squared Deviation 4 0 Σ = 8 Big Meal 22 25 Small meal 19 23 21 Big Meal Deviation From mean -2 1 Squared deviation 4 1 Mean= 21 Small Meal Deviation From mean -2 2 0 Σ = 6 6 2 1 8 2 1 2 2 Notice: s 2 = 3.0 Notice: s 2 = 4.0 Notice: Average = 3.5

32. This is “n” for each sample (Remember, “n” is just number of observations for each sample) df = “degrees of freedom” Remember, “degrees of freedom” is just (n-1) for each sample. So for sample 1: n-1 =3-1 = 2 And for sample 2: n-1=3-1 = 2 Then, df = 2+2=4 This is “n” for each sample (Remember, “n” is just number of observations for each sample)

33. Finding Observed t

34. Finding Critical t

36. Finding p value (Is it less than.05?)

38. We compared test scores for large and small meals. The mean test scores for the big meal was 24, and was 21 for the small meal. A t-test was calculated and there appears to be no significant difference in test scores between the two types of meals, t(4) = 1.964; n.s. Start summary with two means (based on DV) for two levels of the IV Describe type of test (t-test versus anova) with brief overview of results Finish with statistical summary t(4) = 1.96; ns Or if it *were* significant: t(9) = 3.93; p < 0.05 Type of test with degrees of freedom Value of observed statistic n.s. = “not significant” p<0.05 = “significant”

39. Hypothesis testing α =.05 Step 4: Make decision whether or not to reject null hypothesis Reject when: observed stat > critical stat 1.96396 is not bigger than 2.776 “p” is less than 0.05 (or whatever alpha is) p = 0.121 is not less than 0.05 Step 5: Conclusion - tie findings back in to research problem There was no significant difference, there is no evidence that size of meal affected test scores

40. The mean test score for participants who ate the big meal was 24, while the mean test score for participants who ate the small meal was 21. A t-test was completed and there appears to be no significant difference in the test scores as a function of the size of the meal, t(4) = 1.96; n.s. Start summary with two means (based on DV) for two levels of the IV Describe type of test (t-test versus anova) with brief overview of results Finish with statistical summary t(4) = 1.96; ns Type of test with degrees of freedom Value of observed statistic n.s. = “not significant” p<0.05 = “significant”

41. Graphing your t-test results

43. Graphing your t-test results Chart Layout

44. Graphing your t-test results Fill out titles

45. Independent samples t-test Donald is a consultant and leads training sessions. As part of his training sessions, he provides the students with breakfast. He has noticed that when he provides a full breakfast people seem to learn better than when he provides just a small meal (donuts and muffins). So, he put his hunch to the test. He had two classes, both with three people enrolled. The one group was given a big meal and the other group was given only a small meal. He then compared their test performance at the end of the day. Please test with an alpha =.05 What if we ran more subjects? Big Meal 22 25 22 25 22 25 Small meal 19 23 21 19 23 21 19 23 21 Mean= 24 Mean= 21

46. We compared test scores for large and small meals. The mean test score for the big meal was 24, and was 21 for the small meal. A t-test was calculated and there was a significant difference in test scores between the two types of meals t(16) = 3.928; p < 0.05 Let’s run more subjects using our excel!

47. What happened? We ran more subjects: Increased n So, we decreased variability Easier to find effect significant even though effect size didn’t change Big sampleSmall sample This is the sample size

48. What happened? We ran more subjects: Increased n So, we decreased variability Easier to find effect significant even though effect size didn’t change Big sampleSmall sample This is variance for each sample (Remember, variance is just standard deviation squared) This is variance for each sample (Remember, variance is just standard deviation squared)

49. Another format option Independent samples t-test Big Meal versus Small Meal Will use the sort function

50. Another format option Independent samples t-test Big Meal versus Small Meal Will use the sort function

51. Independent samples t-test Male versus Female Students Another format option Will use the sort function

52. Independent samples t-test Male versus Female Students Another format option Will use the sort function

53. The mean test score for female participants was 22.2, while the mean test score for male participants was 22.7. A t-test was completed and there appears to be no significant difference in the test scores as a function of gender, t(16) = -0.523; n.s. Type of test with degrees of freedom Value of observed statistic n.s. = “not significant” p<0.05 = “significant”

54. One paragraph summary of this study. Describe the IV & DV. Present the two means, which type of test was conducted, and the statistical results. We compared productivity for men and women. The mean productivity level for men was 3.65 and the mean productivity for women was 3.43. A t-test was calculated and there appears to be a significant difference in productivity between the two groups t(298) = 3.64; p < 0.05 Start summary with two means (based on DV) for two levels of the IV Describe type of test (t-test versus anova) with brief overview of results Type of test with degrees of freedom Value of observed statistic p<0.05 = “significant” Sample size150150

55. If this is less than.05 (or whatever alpha is) it is significant, and we the reject null df = (n 1 – 1) + (n 2 – 1) = (165 - 1) + (120 -1) = 283