Hypothesis Testing

1. Hypothesis Testing

2. Hypothesis • An educated opinion • What you think will happen, based on • previous research • anecdotal evidence • reading the literature

3. Body fat level of 8th graders • National Norm: • Mean = 23%, SD = 7% • postulated parameter ( and ) • Your 8th grade PE program (N=200) How does my program compare??

4. Your gut feeling • You expect to find, you want to find, your instincts tell you that your students are better.

5. Your gut feeling • You expect to find, you want to find, your instincts tell you that your students are better. But are they??

6. Question • Is any observed difference between your sample mean (representative of your 8th grade population mean) and the National Norm (population of all 8th graders) attributable to random sampling errors, or is there a real difference?

7. Question • Is any observed difference between your sample mean (representative of your 8th grade population mean) and the National Norm (population of all 8th graders) attributable to random sampling errors, or is there a real difference? • Is the mean of your class REALLY the same as the National Norm?

8. How to determine this • Research Question • is my POPULATION mean really 23% • Statistical Question • = 23% • set the Null Hypothesis that the mean of YOUR group is 23% (equal to the National Norm) • assume that your group is NOT REALLY different

9. Null Hypothesis • Ho:  = 23% • The true difference between your sample and the population mean is 0. • There is NO real difference between your sample mean and the population mean. • The performance of your students is not really different from the national norm.

10. Null Hypothesis • In inferential statistics, we usually want to reject the Null hypothesis • to say that the differences are more than what would be expected by random sampling error • this was our initial gut feeling • our program is better

11. 3 Possible Outcomes • No difference between groups • do not reject the null hypothesis

12. 3 Possible Outcomes • No difference between groups • One specific group is higher than the other • directional hypothesis • What you EXPECT to happen when planning the experiment/measurement

13. 3 Possible Outcomes • No difference between groups • One specific group is higher than the other • Either group mean is higher • non-directional hypothesis • The possible outcome of the experiment/measurement

14. Alternative Hypothesis • Our research hypothesis (what we expect to see) • HA:  23% • non-directional hypothesis • interested to see if my grade body composition is better than or worse than the national norm

15. Alternative Hypothesis • Our research hypothesis (what we expect to see) • HA:  < 23% (HA:  > 23%) • directional hypothesis • expect to see my grade mean less than (better than) the national norm • expect to see my grade mean greater than (worse than) that of the national norm

16. Comparing My Class to the National Norm • My 8th grade PE program (N = 200) • National Norm = 23% • postulated parameter • At the end of the semester, calculate the mean % body fat • Using a random sample ( n = 25) • mean % body fat of 20 % Is my sample mean different from the National Norm?

17. Need to Test Ho • Determine whether the observed difference is means is attributable to random sampling error rather than a true difference between the groups (my class and the national norm) • treatment effect Hypothesis Testing

18. Null Hypothesis • No true difference between two means (sample mean and national norm) • Infers: my sample is drawn from the identified population • Nothing more than random sampling errors accounts for any observed difference between the means. An element of uncertainty is inherent in any act of observation (Menard’s Philosophy)

19. Alternative Hypothesis • A true difference does exist between two means • Infers: my sample is notdrawn from the identified population • Observed difference between the means is larger than what we are willing to attribute to random sampling error

20. Testing Ho • Test the probability that the observed difference between means is attributable to random sampling error alone • Evaluate the probability that Ho is not to be rejected • reject or do not reject Ho What amount of risk are you willing to take?

21. Weatherman Example • 85% chance of rain • put up the sunroof • 5% chance of rain • it may happen, but the chance is slight • not very likely to rain • willing to risk being wrong to avoid the inconvenience of having to put up the sunroof.

22. If we do not put up the sunroof: We reject the hypothesis that it will rain

23. If we do not put up the sunroof: We could be right or We could be wrong

24. Wait for certainty means to wait forever

25. What risk are YOU willing to take 1%?? 5%?? 10%%

26. Applied Research = 0.10 = 0.05 = 0.01

27. = 0.05 • With these observed conditions • 5 times in 100 it will rain • 5 times in 100 it will rain when we have kept the sunroof down • 95 times in 100 it will not rain • 95 times in 100 it will not rain when we have kept the sunroof down

28. = 0.05 • Reject Ho if the observed mean difference is greater than what we would expect to occur by chance (random sampling error) less than 5 times in 100 instances • reported in research as a statistically significant difference

29. Testing Ho at = 0.05 • If p > 0.05 : do not reject Ho • difference is attributable to random sampling error (expected variability in mean drawn from a population) • If p 0.05 : reject Ho • difference is attributable to something other than random sampling error

30. Decision Table DECISION

31. Decision Table DECISION R E A L I T Y

32. Decision Table: Correct DECISION R E A L I T Y

33. Decision Table: Incorrect (RT1) DECISION R E A L I T Y

34. Decision Table: Incorrect (AFII) DECISION R E A L I T Y

36. Belief in God as Decision Table Ho: God does not exist DECISION R E A L I T Y Life no hope Lived life of hope Eternal life Lost out on Eternal life

37. To this juncture • Sampling involves error • Expect differences between samples

38. To this juncture • Sampling involves error • Expect differences between samples • If we expect a difference between treatments/conditions, BUT we also expect a difference because of random sampling error

39. To this juncture • Sampling involves error • Expect differences between samples • If we expect a difference between treatments/conditions, BUT we also expect a difference because of random sampling error • HOW do we determine if difference is statistically significant (> than RSE)?

40. Testing Ho requires • Mean value • measure of typical performance level • Standard deviation • measure of the variability • n of cases • known to affect • variability expected with the estimate of the population mean

41. z test for one sample • Our beginning point • National Norm BF = 23% (SD = 7%) • Our sample performance • n = 25 • Mean = 20% • SD = 6% Do my students differ from the National Norm??

42. Our hypotheses • Research Hypothesis • Do my students differ from the national norm • want to know if better OR worse • Ho • There is no real difference in the BF% of my students and the national norm • = 0.05

43. Recall • z-score of > 1.96 or < -1.96 occurs less than 5% of the time • see table of the Normal Curve • That is, the probability of obtaining a z-score value this extreme purely by chance is 5% (only 5 times in 100) (explain).

44. Relevance to Hypothesis Testing • Use the same general idea to evaluate the probability of obtaining a sample mean score of 20% with n = 25 if the true population mean is 23% • Recall the concept of the distribution of sampling means

45. Recall: Z score equation X - X Z = SD

46. Introduce: Z test equation X - Z = SEm

47. Standard Error of the Mean

48. Z test equation X -  Mean difference Z = SEm

49. Z test equation X -  Z = SEm Expected variability in sample means

50. Our given & required data • X = 20% • SD = 6% • n = 25 • = 23% •  = 7% • SEm = ??? • X -  = ??? • Z = ??? X -  Z = SEm