2. SPSS Problem and slides

3. Is this quarter fair? How could you determine this? You assume that flipping the coin a large number of times would result in heads half the time (i.e., it has a.50 probability)

4. Is this quarter fair? Say you flip it 100 times 52 times it is a head Not exactly 50, but its close –probably due to random error

5. Is this quarter fair? What if you got 65 heads? 70? 95? At what point is the discrepancy from the expected becoming too great to attribute to chance?

6. Basic logic of research

7. Start with two equivalent groups of subjects

8. Treat them alike except for one thing

9. See if both groups are different at the end

10. Or – Single Group Subjects Give Treatment -- Prozac Dependent Variable Happiness

11. Do something Subjects Give Treatment -- Prozac Dependent Variable Happiness

12. Measure DV Subjects Give Treatment -- Prozac Dependent Variable Happiness

13. Compare Group to Population Subjects Give Treatment -- Prozac Dependent Variable Happiness Population Happiness Score

14. Example You randomly select 100 college students living in a dorm They complete a happiness measure –(1 = unhappy; 4 = neutral; 7 = happy) You wonder if the mean score of students living in a dorm is different than the population happiness score (M = 4)

15. The Theory of Hypothesis Testing Data are ambiguous Is a difference due to chance? –Sampling error

16. Population You are interested in the average self- esteem in a population of 40 people Self-esteem test scores range from 1 to 10.

17. Population Scores 1,1,1,1 2,2,2,2 3,3,3,3 4,4,4,4 5,5,5,5 6,6,6,6 7,7,7,7 8,8,8,8 9,9,9,9 10,10,10,10

18. Histogram

19. What is the average self-esteem score of this population? Population mean = 5.5 Population SD = 2.87 What if you wanted to estimate this population mean from a sample?

20. What if.... Randomly select 5 people and find the average score

21. Group Activity Why isn’t the average score the same as the population score? When you use a sample there is always some degree of uncertainty! We can measure this uncertainty with a sampling distribution of the mean

22. EXCEL

23. INTERNET EXAMPLE http://www.ruf.rice.edu/~lane/stat_sim/sam pling_dist/index.html

24. Sampling Distribution of the Mean Notice: The sampling distribution is centered around the population mean! Notice: The sampling distribution of the mean looks like a normal curve! –This is true even though the distribution of scores was NOT a normal distribution

25. Central Limit Theorem For any population of scores, regardless of form, the sampling distribution of the mean will approach a normal distribution a N (sample size) get larger. Furthermore, the sampling distribution of the mean will have a mean equal to  and a standard deviation equal to  / N

26. Sampling Distribution Tells you the probability of a particular sample mean occurring for a specific population

27. Sampling Distribution You are interested in if your new Self- esteem training course worked. The 5 people in your course had a mean self-esteem score of 5.5

28. Sampling Distribution Did it work? –How many times would we expect a sample mean to be 5.5 or greater? Theoretical vs. empirical –5,000 random samples yielded 2,501 with means of 5.5 or greater –Thus p =.5002 of this happening

29. Sampling Distribution 5.5 2,499 2,501 P =.4998 P =.5002

30. Sampling Distribution You are interested in if your new Self- esteem training course worked. The 5 people in your course had a mean self-esteem score of 5.8

31. Sampling Distribution Did it work? –How many times would we expect a sample mean to be 5.8 or greater? –5,000 random samples yielded 2,050 with means of 5.8 or greater –Thus p =.41 of this happening

32. Sampling Distribution 5.8 2,700 2,300 P =.59 P =.41

33. Sampling Distribution The 5 people in your course had a mean self-esteem score of 9.8. Did it work? –5,000 random samples yielded 4 with means of 9.8 or greater –Thus p =.0008 of this happening

34. Sampling Distribution 9.8 4,996 4 P =.9992 P =.0008

35. Logic 1) Research hypothesis –H 1 –Training increased self-esteem –The sample mean is greater than general population mean 2) Collect data 3) Set up the null hypothesis –H 0 –Training did not increase self-esteem –The sample is no different than general population mean

36. Logic 4) Obtain a sampling distribution of the mean under the assumption that H 0 is true 5) Given the distribution obtain a probability of a mean at least as large as our actual sample mean 6) Make a decision –Either reject H 0 or fail to reject H 0

37. Hypothesis Test – Single Subject You think your IQ is “freakishly” high that you do not come from the population of normal IQ adults. Population IQ = 100 ; SD = 15 Your IQ = 125

38. Step 1 and 3 H 1 : 125 > μ H o : 125 < or = μ

39. Step 4: Appendix Z shows distribution of Z scores under null -3  -2  -1   1  2  3 

40. Step 5: Obtain probability -3  -2  -1   1  2  3  125

41. Step 5: Obtain probability -3  -2  -1   1  2  3  125 (125 - 100) / 15 = 1.66

42. Step 5: Obtain probability -3  -2  -1   1  2  3  125 Z = 1.66.0485

43. Step 6: Decision Probability that this score is from the same population as normal IQ adults is.0485 In psychology –Most common cut-off point is p <.05 –Thus, your IQ is significantly HIGHER than the average IQ

44. One vs. Two Tailed Tests Previously wanted to see if your IQ was HIGHER than population mean –Called a “one-tailed” test –Only looking at one side of the distribution What if we want to simply determine if it is different?

45. One-Tailed -3  -2  -1   1  2  3  p =.05 Did you score HIGHER than population mean? Want to see if score falls in top.05 μ H 1 : IQ > μ H o : IQ < or = μ

46. Two-Tailed -3  -2  -1   1  2  3  p =.05 Did you score DIFFERNTLY than population mean? μ p =.05 H 1 : IQ = μ H o : IQ = μ

47. Two-Tailed -3  -2  -1   1  2  3  p =.05 Did you score DIFFERNTLY than population mean? PROBLEM: Above you have a p =.10, but you want to test at a p =.05 μ p =.05 H 1 : IQ = μ H o : IQ = μ

48. Two-Tailed -3  -2  -1   1  2  3  p =.025 Did you score DIFFERNTLY than population mean? μ p =.025 H 1 : IQ = μ H o : IQ = μ

49. Step 6: Decision Probability that this score is from the same population as normal IQ adults is.0485 In psychology –Most common cut-off point is p <.05 –Note that on the 2-tailed test the point of significance is.025 (not.05) –Thus, your IQ is not significantly DIFFERENT than the average IQ

51. Problems Problems with Null hypothesis testing Logic is backwards: Most think we are testing the probability of the hypothesis given the data Really testing the probability of the data given the null hypothesis!