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Lecturer’s desk

Screen. Cabinet. Cabinet. Lecturer’s desk. Table. Computer Storage Cabinet. Row A. 3. 4. 5. 19. 6. 18. 7. 17. 16. 8. 15. 9. 10. 11. 14. 13. 12. Row B. 1. 2. 3. 4. 23. 5. 6. 22. 21. 7. 20. 8. 9. 10. 19. 11. 18. 16. 15. 13. 12. 17. 14. Row C. 1. 2.

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Lecturer’s desk

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  1. Screen Cabinet Cabinet Lecturer’s desk Table Computer Storage Cabinet Row A 3 4 5 19 6 18 7 17 16 8 15 9 10 11 14 13 12 Row B 1 2 3 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 Row C 1 2 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 Row D 1 2 25 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 Row E 1 26 2 25 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 Row F 27 1 26 2 25 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 28 Row G 27 1 26 2 25 3 24 4 23 5 6 22 21 7 20 8 9 29 10 19 11 18 16 15 13 12 17 14 28 Row H 27 1 26 2 25 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 Row I 1 26 2 25 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 1 Row J 26 2 25 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 28 27 1 Row K 26 2 25 3 24 4 23 5 6 22 21 7 20 8 9 10 19 11 18 16 15 13 12 17 14 Row L 20 1 19 2 18 3 17 4 16 5 15 6 7 14 13 INTEGRATED LEARNING CENTER ILC 120 9 8 10 12 11 broken desk

  2. BNAD 276: Statistical Inference in ManagementSpring 2016 Welcome Green sheets

  3. By the end of lecture today 2/11/16 Characteristics of a distribution Central Tendency Dispersion The Normal Curve Raw scores, probability and z scores

  4. Schedule of readings • Before next exam: February 18th • Please read • Chapters 1 - 4 in OpenStax • Supplemental reading (Appendix D) • Supplemental reading (Appendix E) • Supplemental reading (Appendix F) • Please read Chapters 1, 5, 6 and 13 in Plous • Chapter 1: Selective Perception • Chapter 5: Plasticity • Chapter 6: Effects of Question Wording and Framing • Chapter 13: Anchoring and Adjustment

  5. Scores, standard deviations, and probabilities The normal curve always has the same shape. They differ only by having different means and standard deviation

  6. Scores, standard deviations, and probabilities Actually 68.26 Actually 95.44 To be exactly 95% we will use z = 1.96

  7. Scores, standard deviations, and probabilities What is total percent under curve? What proportion of curve is above the mean? .50 100% The normal curve always has the same shape. They differ only by having different means and standard deviation

  8. Scores, standard deviations, and probabilities What score is associated with 50th percentile? What percent of curve is below a score of 50? 50% median Mean = 50 Standard deviation = 10

  9. Scores, standard deviations, and probabilities What score is associated with 50th percentile? What percent of curve is below a score of 100? 50% median Mean = 100 Standard deviation = 5

  10. Raw scores, z scores & probabilities Distance from the mean (z scores) convert convert Proportion of curve (area from mean) Raw Scores (actual data) 68% We care about this! What is the actual number on this scale?“height” vs “weight” “pounds” vs “test score” We care about this! “percentiles” “percent of people” “proportion of curve” “relative position” z = -1 z = -1 z = 1 z = 1 68% Proportion of curve (area from mean) Raw Scores (actual data) Distance from the mean (z scores) convert convert

  11. Normal distribution Raw scores z-scores probabilities Z Scores Have z Find raw score Have z Find area z table Formula Have area Find z Area & Probability Raw Scores Have raw score Find z

  12. Find z score for raw score of 60 z score: A score that indicates how many standard deviations an observation is above or below the mean of the distribution z score = raw score - mean standard deviation 50 60 z = 1 10 Mean = 50 Standard deviation = 10

  13. 50 60 Find the area under the curve that falls between 50 and 60 34.13% 1) Draw the picture 2) Find z score 3) Go to z table - find area under correct column 4) Report the area z = 1 50 60 10 Review

  14. Find z score for raw score of 30 z score: A score that indicates how many standard deviations an observation is above or below the mean of the distribution z score = raw score - mean standard deviation 50 30 z = - 2 10 Mean = 50 Standard deviation = 10

  15. Find z score for raw score of 70 Raw scores, z scores & probabilities If we go up to score of 70 we are going up 2.0 standard deviations Then, z score = +2.0 z score = raw score - mean standard deviation z score = 70 – 50 . 10 = 20. 10 = 2 Mean = 50 Standard deviation = 10

  16. Find z score for raw score of 80 z score: A score that indicates how many standard deviations an observation is above or below the mean of the distribution z score = raw score - mean standard deviation 50 80 z = 3 10 Mean = 50 Standard deviation = 10

  17. Find z score for raw score of 20 Raw scores, z scores & probabilities If we go down to score of 20 we are going down 3.0 standard deviations Then, z score = -3.0 z score = raw score - mean standard deviation z score = 20 – 50 10 = - 30 . 10 = - 3 Mean = 50 Standard deviation = 10

  18. z table z table Mean = 50 Standard deviation = 10 68.26% Find the area under the curve that falls between 40 and 60 34.13% 34.13% z score = raw score - mean standard deviation Hint always draw a picture! z score = 60 - 50 10 z score = 40 - 50 10 z score = 10 = 1.0 10 z score = 10 = -1.0 10 z score of 1 = area of .3413 z score of 1 = area of .3413 .3413 + .3413 = .6826

  19. Mean = 50 Standard deviation = 10 1) Draw the picture 2) Find z score 3) Go to z table - find area under correct column 4) Report the area Find the area under the curve that falls between 30 and 50 z score = raw score - mean standard deviation z score = 30 - 50 10 z score = - 20 = - 2.0 10 Hint always draw a picture!

  20. z table Mean = 50 Standard deviation = 10 1) Draw the picture 2) Find z score 3) Go to z table - find area under correct column 4) Report the area 47.72% Find the area under the curve that falls between 30 and 50 z score = raw score - mean standard deviation z score = 30 - 50 10 z score = - 20 = - 2.0 10 z score of - 2 = area of .4772 Hint always draw a picture! Hint always draw a picture!

  21. Let’s do some problems z table Mean = 50 Standard deviation = 10 47.72% Find the area under the curve that falls between 70 and 50 z score = raw score - mean standard deviation z score = 70 - 50 10 z score = 20 = +2.0 10 z score of 2 = area of .4772 Hint always draw a picture!

  22. Let’s do some problems Mean = 50 Standard deviation = 10 .4772 .4772 95.44% z score of 2 = area of .4772 Find the area under the curve that falls between 30 and 70 .4772 + .4772 = .9544 Hint always draw a picture!

  23. Scores, standard deviations, and probabilities Actually 68.26 Actually 95.44 To be exactly 95% we will use z = 1.96

  24. Writing AssignmentLet’s do some problems Mean = 50Standard deviation = 10

  25. Let’s do some problems ? Mean = 50Standard deviation = 10 60 Find the area under the curve that falls below 60 means the same thing as Find the percentile rank for score of 60 Problem 1

  26. Let’s do some problems ? 60 Mean = 50Standard deviation = 10 Find the percentile rank for score of 60 .3413 .5000 1) Find z score z score = 60 - 50 10 = 1 2) Go to z table - find area under correct column (.3413) 3) Look at your picture - add .5000 to .3413 = .8413 4) Percentile rank or score of 60 = 84.13% Problem 1 Hint always draw a picture!

  27. ? 75 Mean = 50Standard deviation = 10 Find the percentile rank for score of 75 .4938 1) Find z score z score = 75 - 50 10 z score = 25 10 = 2.5 2) Go to z table Problem 2 Hint always draw a picture!

  28. ? 75 Mean = 50Standard deviation = 10 Find the percentile rank for score of 75 .4938 .5000 1) Find z score z score = 75 - 50 10 z score = 25 10 = 2.5 2) Go to z table 3) Look at your picture - add .5000 to .4938 = .9938 4) Percentile rank or score of 75 = 99.38% Problem 2 Hint always draw a picture!

  29. ? 45 Mean = 50Standard deviation = 10 Find the percentile rank for score of 45 1) Find z score z score = 45 - 50 10 z score = - 5 10 = -0.5 2) Go to z table Problem 3

  30. ? 45 Mean = 50Standard deviation = 10 Find the percentile rank for score of 45 .1915 ? 1) Find z score z score = 45 - 50 10 z score = - 5 10 = -0.5 2) Go to z table Problem 3

  31. ? 45 Mean = 50Standard deviation = 10 Find the percentile rank for score of 45 .1915 .3085 1) Find z score z score = 45 - 50 10 z score = - 5 10 = -0.5 2) Go to z table 3) Look at your picture - subtract .5000 -.1915 = .3085 Problem 3 4) Percentile rank or score of 45 = 30.85%

  32. ? Mean = 50Standard deviation = 10 Find the percentile rank for score of 55 55 1) Find z score z score = 55 - 50 10 z score = 5 10 = 0.5 2) Go to z table Problem 4

  33. ? Mean = 50Standard deviation = 10 Find the percentile rank for score of 55 .1915 55 1) Find z score z score = 55 - 50 10 z score = 5 10 = 0.5 2) Go to z table Problem 4

  34. ? Mean = 50Standard deviation = 10 Find the percentile rank for score of 55 .1915 .5 55 1) Find z score z score = 55 - 50 10 z score = 5 10 = 0.5 2) Go to z table 3) Look at your picture - add .5000 +.1915 = .6915 4) Percentile rank or score of 55 = 69.15% Problem 4

  35. Find the score for z = -2 ? Mean = 50Standard deviation = 10 30 Hint always draw a picture! Find the score that is associated with a z score of -2 raw score = mean + (z score)(standard deviation) Raw score = 50 + (-2)(10) Raw score = 50 + (-20) = 30 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion

  36. ? .7700 ? Mean = 50Standard deviation = 10 Find the score for percentile rank of 77%ile Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion Problem 5

  37. ? .7700 ? Mean = 50Standard deviation = 10 .27 Find the score for percentile rank of 77%ile .5 .5 + .27 = .77 .5 .27 1) Go to z table - find z score for for area .2700 (.7700 - .5000) = .27 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion area = .2704 (closest I could find to .2700) z = 0.74 Problem 5

  38. ? .7700 ? Mean = 50Standard deviation = 10 .27 Find the score for percentile rank of 77%ile .5 x = 57.4 .5 .27 2) x = mean + (z)(standard deviation) x = 50 + (0.74)(10) x = 57.4 x = 57.4 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion Problem 5

  39. ? .5500 ? Mean = 50Standard deviation = 10 Find the score for percentile rank of 55%ile Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion Problem 6

  40. ? .5500 ? Mean = 50Standard deviation = 10 .05 Find the score for percentile rank of 55%ile .5 .5 + .05 = .55 .5 .05 1) Go to z table - find z score for for area .0500 (.5500 - .5000) = .05 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion area = .0517 (closest I could find to .0500) z = 0.13 Problem 7

  41. ? .5500 ? Mean = 50Standard deviation = 10 .05 Find the score for percentile rank of 55%ile .5 .5 .05 1) Go to z table - find z score for for area .0500 (.5500 - .5000) = .05 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion area = .0517 (closest I could find to .0500) z = 0.13 Problem 7

  42. ? .5500 ? Mean = 50Standard deviation = 10 .05 Find the score for percentile rank of 55%ile .5 x = 51.3 .5 .05 1) Go to z table - find z score for for area .0500 (.5500 - .5000) = .0500 area = .0517 (closest I could find to .0500) z = 0.13 2) x = mean + (z)(standard deviation) x = 50 + (0.13)(10) x = 51.3 Please note: When we are looking for the score from proportion we use the z-table ‘backwards’. We find the closest z to match our proportion x = 51.3 Problem 7

  43. Normal Distribution has a mean of 50 and standard deviation of 4. Determine value below which 95% of observations will occur.Note: sounds like a percentile rank problem Go to table .4500 nearest z = 1.64 x = mean + z σ = 50 + (1.64)(4) = 56.56 .9500 .4500 .5000 Additional practice Problem 8 38 62 54 46 58 ? 42 50 56.60

  44. Normal Distribution has a mean of $2,100 and s.d. of $250. What is the operating cost for the lowest 3% of airplanes.Note: sounds like a percentile rank problem = find score for 3rd percentile Go to table .4700 nearest z = - 1.88 x = mean + z σ = 2100 + (-1.88)(250) = 1,630 .0300 .4700 Additional practice Problem 9 ? 2100 1,630

  45. Normal Distribution has a mean of 195 and standard deviation of 8.5. Determine value for top 1% of hours listened. Go to table .4900 nearest z = 2.33 x = mean + z σ = 195 + (2.33)(8.5) = 214.805 .4900 .0100 .5000 Additional practice Problem 10 195 ? 214.8

  46. . Find score associated with the 75th percentile 75th percentile Go to table nearest z = .67 .2500 x = mean + z σ = 30 + (.67)(2) = 31.34 .7500 .25 .5000 24 36 ? 28 34 26 30 31.34 Additional practice Problem 11 z = .67

  47. . Find the score associated with the 25th percentile 25th percentile Go to table nearest z = -.67 .2500 x = mean + z σ = 30 + (-.67)(2) = 28.66 .2500 .25 .25 28.66 24 ? 36 28 34 26 30 Additional practice Problem 12 z = -.67

  48. . Try this one: Please find the (2) raw scores that border exactly the middle 95% of the curve Mean of 30 and standard deviation of 2 Go to table .4750 nearest z = 1.96 mean + z σ = 30 + (1.96)(2) = 33.92 Go to table .4750 nearest z = -1.96 mean + z σ = 30 + (-1.96)(2) = 26.08 .9500 .475 .475 Additional practice Problem 13 26.08 33.92 ? ? 24 32 36 28 30

  49. . Try this one: Please find the (2) raw scores that border exactly the middle 95% of the curve Mean of 100 and standard deviation of 5 Go to table .4750 nearest z = 1.96 mean + z σ = 100 + (1.96)(5) = 109.80 Go to table .4750 nearest z = -1.96 mean + z σ = 100 + (-1.96)(5) = 90.20 .9500 .475 .475 Additional practice Problem 14 90.2 109.8 ? ? 85 105 115 95 100

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