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Normal Density Fitting: Melbourne Temperature and Excel Probs

Understand normal distribution, density curves, fitting data, computation of probabilities, and Z-score views for statistical analysis. Learn Excel functions like NORMDIST for calculations. Explore Z-score interpretations and the 68-95-99.7% rule.

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Normal Density Fitting: Melbourne Temperature and Excel Probs

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  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:

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