Last Time

1. Last Time • Normal Distribution • Density Curve (Mound Shaped) • Family Indexed by mean and s. d. • Fit to data, using sample mean and s.d. • Computation of Normal Probabilities • Using Excel function, NORMDIST • And Big Rules of Probability

2. Reading In Textbook Approximate Reading for Today’s Material: Pages 61-62, 66-70, 59-61, 322-326 Approximate Reading for Next Class: Pages 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. Computation of Normal Probs EXCEL Computation: probs given by “lower areas” E.g. for X ~ N(1,0.5) P{X ≤ 1.3} = 0.726

6. Computation of Normal Probs Computation of upper areas: (use “1 –”, i.e. “not” formula) = 1 -

7. Computation of Normal Probs Computation of areas over intervals: (use subtraction) = -

8. Z-score view of populations Idea: Reproducible view of “where data point lies in population”

9. Z-score view of populations Idea: Reproducible view of “where data point lies in population” Context 1: List of Numbers Context 2: Probability distribution

10. Z-score view of Lists of #s Idea: Reproducible view of “where data point lies in population”

11. Z-score view of Lists of #s Idea: Reproducible view of “where data point lies in population” • Thought model: population is Normal

12. Z-score view of Lists of #s Idea: Reproducible view of “where data point lies in population” • Thought model: population is Normal • Population mean: μ

13. Z-score view of Lists of #s Idea: Reproducible view of “where data point lies in population” • Thought model: population is Normal • Population mean: μ • Population standard deviation: σ

14. Z-score view of Lists of #s Idea: Reproducible view of “where data point lies in population” • Thought model: population is Normal • Population mean: μ • Population standard deviation: σ Interpret data as “s.d.s away from mean”

15. Z-score view of Lists of #s Approach: • Transform data

16. Z-score view of Lists of #s Approach: • Transform data • By subtracting mean & dividing by s.d

17. Z-score view of Lists of #s Approach: • Transform data • By subtracting mean & dividing by s.d. • To get

18. Z-score view of Lists of #s Approach: • Transform data • By subtracting mean & dividing by s.d. • To get (gives mean 0, s.d. 1)

19. Z-score view of Lists of #s Approach: • Transform data • By subtracting mean & dividing by s.d. • To get (gives mean 0, s.d. 1) • Interpret as

20. Z-score view of Lists of #s Approach: • Transform data • By subtracting mean & dividing by s.d. • To get (gives mean 0, s.d. 1) • Interpret as • I.e. “ is sd’s above the mean”

21. Z-score view of Normal Dist. Approach: • For

22. Z-score view of Normal Dist. Approach: • For • Subtract mean & divide by s.d

23. Z-score view of Normal Dist. Approach: • For • Subtract mean & divide by s.d. • To get

24. Z-score view of Normal Dist. Approach: • For • Subtract mean & divide by s.d. • To get (gives mean 0, s.d. 1, i.e. Standard Normal)

25. Z-score view of Normal Dist. Approach: • For • Subtract mean & divide by s.d. • To get (gives mean 0, s.d. 1, i.e. Standard Normal) • Interpret as

26. Z-score view of Normal Dist. Approach: • For • Subtract mean & divide by s.d. • To get (gives mean 0, s.d. 1, i.e. Standard Normal) • Interpret as • I.e. “ is sd’s above the mean”

27. Z-score view of Normal Dist. HW: 1.117

28. Interpretation of Z-scores Z-scores

29. Interpretation of Z-scores Z-scores are on N(0,1) scale,

30. Interpretation of Z-scores Z-scores are on N(0,1) scale,

31. Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them

32. Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them Important Areas:

33. Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them Important Areas: • Within 1 sd of mean

34. Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them Important Areas: • Within 1 sd of mean

35. Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them Important Areas: • Within 1 sd of mean “the majority”

36. Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them Important Areas: • Within 1 sd of mean “the majority” ≈ 68%

37. Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them Important Areas: 2. Within 2 sd of mean “really most” ≈ 95%

38. Interpretation of Z-scores Z-scores are on N(0,1) scale, so use areas to interpret them Important Areas: 3. Within 3 sd of mean “almost all” ≈ 99.7%

39. Interpretation of Z-scores Summary: these are called the “68 - 95 - 99.7 % Rule”

40. Interpretation of Z-scores Summary: these are called the “68 - 95 - 99.7 % Rule” Mean +- 1 - 2 – 3 sd’s

41. Interpretation of Z-scores Summary: “68 - 95 - 99.7 % Rule” Excel Calculation From Class Example 9: http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg9.xls

42. Interpretation of Z-scores Summary: “68 - 95 - 99.7 % Rule” Excel Calculation

43. Interpretation of Z-scores HW: 1.115, 1.116 (50%, 2.5%, 0.18-0.22) 1.119

44. Inverse Normal Probs Idea, for a given cutoff value, x

45. Inverse Normal Probs Idea, for a given cutoff value, x Calculated P{X < x}

46. Inverse Normal Probs Idea, for a given cutoff value, x Calculated P{X < x} as Area under normal density

47. Inverse Normal Probs Idea, for a given cutoff value, x Calculated P{X < x} as Area under normal density Using Excel function: NORMDIST

48. Inverse Normal Probs Now for a given P{X < x}, i.e. Area

49. Inverse Normal Probs Now for a given P{X < x}, i.e. Area Find corresponding cutoff x

50. Inverse Normal Probs Now for a given P{X < x}, i.e. Area Find corresponding cutoff x Terminology: