Last Time

1. Last Time • Z-score interpretation of populations • Normal probability distributions • Lists of #s • Inverse Normal Probabilities • Quantiles, i.e. Percentiles • Excel computation: NORMINV • Quality control • Q-Q plots • Visual diagnostic for normality

2. Reading In Textbook Approximate Reading for Today’s Material: Pages 61-62, 66-70, 59-61, 335-346 Approximate Reading for Next Class: Pages 322-326, 337-344, 488-498

3. Normal Density Fitting Idea: Choose μ and σ to fit normal density to histogram of data, Approach: IF the distribution is “mound shaped” & outliers are negligible THEN a “good” choice of normal model is:

4. Normal Density Fitting Melbourne Average Temperature Data

5. Checking Normality Idea: For which data sets, will the normal distribution be a good model?

6. Checking Normality Q-Q plot, e.g. Buffalo Snowfalls • Approximately linear • Suggests normal • But some wiggles? • Due to natural sampling variation? Study with smaller simulation

7. Checking Normality Q-Q plot, e.g. n = 100 from N(0,1) • Approximately linear • Some wiggliness • Suggests Buffalo variation is usual • Make this more precise?

8. Research Corner Melbourne Average Temperature Data • Mound shaped

9. Research Corner Melbourne Average Temperature Data • Mound shaped • But really Normal?

10. Research Corner Melbourne Average Temperature Data • Mound shaped • But really Normal? • Check with QQ – plot

11. Research Corner Melbourne Average Temperature Data • QQ-plot

12. Research Corner Melbourne Average Temperature Data • QQ-plot • For N(xbar,s)

13. Research Corner Melbourne Average Temperature Data • QQ-plot • For N(xbar,s) • Overlay 45o line

14. Research Corner Melbourne Average Temperature Data • QQ-plot • For N(xbar,s) • Overlay 45o line • Helps assess how linear

15. Research Corner Melbourne Average Temperature Data • Is this curved? (suggests non-Normal)

16. Research Corner Melbourne Average Temperature Data • Is this curved? • Or just result of natural sampling variation?

17. Research Corner Melbourne Average Temperature Data • Is this curved? • Or just result of natural sampling variation? Approach: simulate from Normal

18. Research Corner Melbourne Average Temperature Data • Is this curved? • Or just result of natural sampling variation? Approach: simulate from Normal

19. Research Corner Melbourne Average Temperature Data • Is this curved? • Or just result of natural sampling variation? Approach: simulate from Normal

20. Research Corner Melbourne Average Temperature Data • Is this curved? • Or just result of natural sampling variation? Approach: simulate from Normal

21. Research Corner Melbourne Average Temperature Data • Is this curved? • Or just result of natural sampling variation? Approach: simulate from Normal

22. Research Corner Melbourne Average Temperature Data • Is this curved? • Or just result of natural sampling variation? Approach: simulate from Normal (repeat 100 times)

23. Research Corner Melbourne Average Temperature Data QQ Envelope Plot

24. Research Corner Melbourne Average Temperature Data QQ Envelope Plot • Curvature not far from natural sampling variation

25. Research Corner Melbourne Average Temperature Data QQ Envelope Plot • Curvature not far from natural sampling variation • But does stick out

26. Research Corner Melbourne Average Temperature Data QQ Envelope Plot • Curvature not far from natural sampling variation • But does stick out • Conclude not Normal

27. Research Corner Recall Buffalo Snowfall Data

28. Research Corner Recall Buffalo Snowfall Data QQ Envelope Plot

29. Research Corner Recall Buffalo Snowfall Data QQ Envelope Plot • Always within band

30. Research Corner Recall Buffalo Snowfall Data QQ Envelope Plot • Always within band • Conclude: Normal distribution fits this data set

31. Research Corner Recall British Suicide Data

32. Research Corner Recall British Suicide Data QQ Envelope Plot

33. Research Corner Recall British Suicide Data QQ Envelope Plot • Way outside band

34. Research Corner Recall British Suicide Data QQ Envelope Plot • Way outside band • Normal does not fit this data set

35. Research Corner Recall British Suicide Data QQ Envelope Plot • Way outside band • Normal does not fit this data set (so transformation was a good approach)

36. Research Corner Recall log10 British Suicide Data Idea: log10 transforms to normality

37. Research Corner Recall log10 British Suicide Data QQ Envelope Plot

38. Research Corner Recall log10 British Suicide Data QQ Envelope Plot • Now much closer to normal

39. Research Corner Recall log10 British Suicide Data QQ Envelope Plot • Now much closer to normal • But three 0s, so not quite normal

40. Research Corner Recall log10 British Suicide Data QQ Envelope Plot • Now much closer to normal • But three 0s, so not quite normal Approach: Shifted log transformation

41. Research Corner log10 (10 + British Suicide Data) QQ Envelope Plot

42. Research Corner log10 (10 + British Suicide Data) QQ Envelope Plot • Shift reduces impact of 0s

43. Research Corner log10 (10 + British Suicide Data) QQ Envelope Plot • Shift reduces impact of 0s • Now conclude data are Normal

44. Research Corner log10 (10 + British Suicide Data) QQ Envelope Plot • Shift reduces impact of 0s • Now conclude data are Normal • Shift is good to know about

45. Research Corner QQ Envelope Plot Summary • Useful in close cases • Makes interpretation over different sample sizes much easier • Generally more quantitative approach

46. Applications of Normal Dist’n • Population Modeling

47. Applications of Normal Dist’n • Population Modeling Examples: • Heights of people • Scores on SAT ...

48. Applications of Normal Dist’n • Population Modeling Examples: • Heights of people • Scores on SAT ... (for almost anything measured)

49. Applications of Normal Dist’n • Population Modeling Often want to make statements about: • The population mean, μ • The population standard deviation, σ

50. Applications of Normal Dist’n • Population Modeling Often want to make statements about: • The population mean, μ • The population standard deviation, σ Based on a sample from the population