2. Lecturer’s desk Physics- atmospheric Sciences (PAS) - Room 201 s c r e e n Row A Row B Row C Row D Row E Row F Row G Row H 131211109 87 Row A 14131211109 87 Row B 1514131211109 87 Row C 1514131211109 87 Row D 16 1514131211109 87 Row E 17 16 1514131211109 87 Row F 1716 1514131211109 87 Row G 1716 1514131211109 87 Row H 16 18 table Row A Row B Row C Row D Row E Row F Row G Row H 15141716 1819 16 15 18171920 17161918 2021 18172019 2122 19182120 2223 20192221 2324 18172019 2122 19182120 2223 2143 56 2143 56 2143 56 2143 56 2143 56 2143 56 2143 56 2143 56 Row J Row K Row L Row M Row N Row P 2143 5 2143 5 2143 5 2143 5 2143 5 1 5 Row J Row K Row L Row M Row N Row P 27262928 30 25242726 28 24232625 27 23222524 26 25242726 28 27262928 30 6 14 131211109 87 16151817 19 202122 614131211109 87 16 15 18 17 19 20212223 614131211109 87 16 15 18171920 2122 23 6 14 131211109 87 1624181719 20 2122 231525 6 14 131211109 87 1624181719 20 2122 231525 Row Q 2143 5 27262928 30 6 14 131211109 87 242223 21 - 15 25 37363938 40 34 3132 3335 69 87 13 table 14 18 192021

3. MGMT 276: Statistical Inference in Management Fall 2015

4. Just for Fun Assignments Go to D2L - Click on “Content” Click on “Interactive Online Just-for-fun Assignments” Please note: These are not worth any class points and are different from the required homeworks

5. Please re-register your clicker http://student.turningtechnologies.com/

6. Before our next exam (October 20 th ) OpenStax Chapters 1 – 11 Plous (10, 11, 12 & 14) Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability and Risk Chapter 14: The Perception of Randomness Schedule of readings

7. Homework due – Thursday (October 8 th ) On class website: Please print and complete homework worksheet #9 Approaches to probabilities Interpreting probabilities using the normal curve Due: Tuesday, October 8th

8. By the end of lecture today 10/6/15 Use this as your study guide Counting ‘standard deviationses’ – z scores Connecting raw scores, z scores and probability Connecting probability, proportion and area of curve Percentiles Approaches to probability: Empirical, Subjective and Classical

9. What is probability 1. Empirical probability: relative frequency approach Number of observed outcomes Number of observations Probability of getting into an educational program Number of people they let in Number of applicants Probability of getting a rotten apple Number of rotten apples Number of apples 5 100 5% chance of getting a rotten apple 400 600 66% chance of getting admitted

10. What is probability 1. Empirical probability: relative frequency approach Number of observed outcomes Number of observations Probability of hitting the corvette Number of carts that hit corvette Number of carts rolled 182 200 91% chance of hitting a corvette =.91 10% of people who buy a house with no pool build one. What is the likelihood that Bob will? “There is a 20% chance that a new stock offered in an initial public offering (IPO) will reach or exceed its target price on the first day.” “More than 30% of the results from major search engines for the keyword phrase “ring tone” are fake pages created by spammers.”

11. 2. Classic probability: a priori probabilities based on logic rather than on data or experience. All options are equally likely (deductive rather than inductive). Number of outcomes of specific event Number of all possible events In throwing a die what is the probability of getting a “2” Number of sides with a 2 Number of sides In tossing a coin what is probability of getting a tail Number of sides with a 1 Number of sides 1 2 50% chance of getting a tail 1 6 16% chance of getting a two = = Lottery Likelihood get question right on multiple choice test Chosen at random to be team captain

12. 3. Subjective probability: based on someone’s personal judgment (often an expert), and often used when empirical and classic approaches are not available. There is a 5% chance that Verizon will merge with Sprint Bob says he is 90% sure he could swim across the river Likelihood that company will invent new type of battery Likelihood get a ”B” in the class 60% chance that Patriots will play at Super Bowl

13. Approach Example Empirical There is a 2 percent chance of twins in a randomly-chosen birth Classical There is a 50 % probability of heads on a coin flip. Subjective There is a 5% chance that Verizon will merge with Sprint

14. Notice: 3 types of numbers raw scores z scores probabilities Mean = 50 Standard deviation = 10 If we go up two standard deviations z score = +2.0 and raw score = 70 If we go down two standard deviations z score = -2.0 and raw score = 30 Raw scores, z scores & probabilities z = -2 z = +2

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

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

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

18. Writing Assignment Let’s do some problems Mean = 50 Standard deviation = 10

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

20. Let’s do some problems Mean = 50 Standard deviation = 10 1) Find z score z score = 60 - 50 10 Hint always draw a picture! Find the percentile rank for score of 60 60 2) Go to z table - find area under correct column (.3413) 4) Percentile rank or score of 60 = 84.13% 3) Look at your picture - add.5000 to.3413 =.8413 ?.3413.5000 = 1 Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) Problem 1 review

21. Mean = 50 Standard deviation = 10 1) Find z score z score = 75 - 50 10 Hint always draw a picture! Find the percentile rank for score of 75 75 2) Go to z table ? z score = 25 10 = 2.5.4938 Problem 2 review

22. Mean = 50 Standard deviation = 10 1) Find z score z score = 75 - 50 10 Hint always draw a picture! Find the percentile rank for score of 75 75 2) Go to z table ? z score = 25 10 = 2.5.4938 4) Percentile rank or score of 75 = 99.38% 3) Look at your picture - add.5000 to.4938 =.9938.5000 Problem 2 review

23. Mean = 50 Standard deviation = 10 1) Find z score z score = 45 - 50 10 Find the percentile rank for score of 45 45 ? 2) Go to z table z score = - 5 10 = -0.5 Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) Problem 3 review

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

25. Mean = 50 Standard deviation = 10 1) Find z score z score = 45 - 50 10 Find the percentile rank for score of 45 45 2) Go to z table z score = - 5 10 = -0.5 4) Percentile rank or score of 45 = 30.85% 3) Look at your picture - subtract.5000 -.1915 =.3085.1915 ?.3085 Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) Problem 3 review

26. Mean = 50 Standard deviation = 10 1) Find z score z score = 55 - 50 10 Find the percentile rank for score of 55 ? 2) Go to z table z score = 5 10 = 0.5 55 Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) Problem 4

27. Mean = 50 Standard deviation = 10 1) Find z score z score = 55 - 50 10 Find the percentile rank for score of 55 2) Go to z table z score = 5 10 = 0.5 55.1915 ? Problem 4

28. Mean = 50 Standard deviation = 10 1) Find z score z score = 55 - 50 10 Find the percentile rank for score of 55 ? 2) Go to z table z score = 5 10 = 0.5 4) Percentile rank or score of 55 = 69.15% 3) Look at your picture - add.5000 +.1915 =.6915 55.1915.5 Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) Problem 4

29. Find the score for z = -2 Mean = 50 Standard deviation = 10 raw score = mean + (z score)(standard deviation) Raw score = 50 + (-2)(10) Raw score = 50 + (-20) = 30 Hint always draw a picture! Find the score that is associated with a z score of -2 ? 30 Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) 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

30. Mean = 50 Standard deviation = 10 Find the score for percentile rank of 77%ile.7700 ? ? Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) 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

31. Mean = 50 Standard deviation = 10 Find the score for percentile rank of 77%ile 1) Go to z table - find z score for for area.2700 (.7700 -.5000) =.27.7700 ? ?.5.27.5.27 area =.2704 (closest I could find to.2700) z = 0.74 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 5.5 +.27 =.77

32. Mean = 50 Standard deviation = 10 Find the score for percentile rank of 77%ile ?.5.27 2) x = mean + (z)(standard deviation) x = 50 + (0.74)(10) 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 x = 57.4.7700 ?.5.27 Problem 5

33. Mean = 50 Standard deviation = 10 Find the score for percentile rank of 55%ile.5500 ? ? Raw Scores (actual data) Distance from the mean ( from raw to z scores) Proportion of curve (area from mean) z-table (from z to area) Problem 6 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

34. Mean = 50 Standard deviation = 10 Find the score for percentile rank of 55%ile 1) Go to z table - find z score for for area.0500 (.5500 -.5000) =.05.5500 ? ?.5.05.5.05 area =.0517 (closest I could find to.0500) z = 0.13 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 7.5 +.05 =.55

35. Mean = 50 Standard deviation = 10 Find the score for percentile rank of 55%ile 1) Go to z table - find z score for for area.0500 (.5500 -.5000) =.05.5500 ? ?.5.05.5.05 area =.0517 (closest I could find to.0500) z = 0.13 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 7

36. Mean = 50 Standard deviation = 10 Find the score for percentile rank of 55%ile 1) Go to z table - find z score for for area.0500 (.5500 -.5000) =.0500 area =.0517 (closest I could find to.0500) z = 0.13.5500 ? ?.5.05.5.05 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

37. nearest z = 1.64 Go to table.4500 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 x = mean + z σ = 50 + (1.64)(4) = 56.56 Additional practice Problem 8

38. nearest z = - 1.88 Go to table.4700 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 3 rd percentile x = mean + z σ = 2100 + (-1.88)(250) = 1,630 Additional practice Problem 9

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

40. . 75 th percentile Go to table.2500 nearest z =.67 x = mean + z σ = 30 + (.67)(2) = 31.34 z =.67 Additional practice Problem 11

41. . 25 th percentile Go to table.2500 nearest z = -.67 x = mean + z σ = 30 + (-.67)(2) = 28.66 z = -.67 Additional practice Problem 12

42. . 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 Additional practice Problem 13

43. . 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 Additional practice Problem 14

44. . Try this one: Please find the (2) raw scores that border exactly the middle 99% of the curve Mean of 30 and standard deviation of 2 Go to table.4750 nearest z = 1.96 mean + z σ = 30 + (2.58)(2) = 35.16 Go to table.4750 nearest z = -1.96 mean + z σ = 30 + (-2.58)(2) = 24.84 Additional practice Problem 15

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

46. Always draw a picture! Homework worksheet

47. 1.6800 z =-1 z = 1

48. Homework worksheet 2.9500 z =-2 z = 2

49. Homework worksheet 3.9970 z =-3 z = 3

50. Homework worksheet 4.5000 z = 0

51. Homework worksheet 5 2 z = 33-30 z = 1.5 Go to table.4332 z = 1.5

52. 6 z = 33-30 2 z = 1.5 Go to table.4332 Add area Lower half.4332 +.5000 =.9332 z = 1.5

53. Homework worksheet 7 2 z = 33-30 = 1.5 Go to table.4332 Subtract from.5000.5000 -.4332 =.0668 z = 1.5

54. 8 z = 29-30 2 = -.5 Go to table.1915 Add to upper Half of curve.5000 +.1915 =.6915 z = -.5

55. 9.4938 +.1915 =.6853 = 25-30 2 = -2.5.4938 Go to table = 31-30 2 =.5.1915 Go to table z =.5 z =-2.5

56. 10 z = 27-30 2 = -1.5 Go to table.4332 Subtract From.5000.5000 -.4332 =.0668 z =-1.5

57. 11 z = 25-30 2 = -2.5 Go to table.4938 Add lower Half of curve.5000 +.4938 =.9938 z =-2.5

58. 12 z = 32-30 2 = 1.0 Go to table.3413 Subtract from.5000.5000 -.3413 =.1587 z =1

59. 13 50 th percentile = median 30 z =0

60. 14 28 32 z =-1 z = 1

61. 15 x = mean + z σ = 30 + (.74)(2) = 31.48 77 th percentile Find area of interest.7700 -.5000 =.2700 Find nearest z =.74 z =.74

62. 16 13 th percentile Find area of interest.5000 -.1300 =.3700 Find nearest z = -1.13 x = mean + z σ = 30 + (-1.13)(2) = 27.74 z =-1.13

63. Please use the following distribution with a mean of 200 and a standard deviation of 40.

64. 17.6800 z =-1 z = 1

65. .9500 18 z =-2 z = 2

66. .9970 19 z =-3 z = 3

67. 20 = 230-200 40 =.75 Go to table.2734 z =.75

68. 21 Go to table Subtract from.5000 z = 190-200 40 = -.25.0987.5000 -.0987 =.4013 z =-.25

69. 22 Go to table Add to upper Half of curve z = 180-200 40 = -.5.1915.5000 +.1915 =.6915 z =-.5

70. 23 z = 236-200 40 = 0.9 Go to table.3159 Subtract from.5000.5000 -.3159 =.1841 z =.9

71. 24.0793 +.2088 =.2881 z = 192 - 200 40 = -.2.0793 Go to table z = 222 - 200 40 =.55.2088 Go to table z =-.2 z =.55

72. 25 40 z = 275-200 = 1.875 Go to table.4693 or.4699 Add area Lower half.4693 +.5000 =.9693.4699 +.5000 =.9699 z =1.875

73. 26 z = 295-200 40 z = 2.375 Go to table.4911 or.4913 Add area Lower half.5000 -.4911 =.0089.5000 -.4913 =.0087 z =2.375

74. 27 z = 130-200 40 = -1.75.4599 Add to upper Half of curve Go to table.5000 +.4599 =.9599 z =-1.75

75. 28 40 z = 130-200 = -1.75.4599 Subtract from.5000.5000 -.4599 =.0401 Go to table z =-1.75

76. 29 x = mean + z σ = 200 + (2.33)(40) = 293.2 99 th percentile Find area of interest.9900 -.5000 =.4900 Find nearest z = 2.33 z =2.33

77. 30 33 rd percentile Find area of interest.5000 -.3300 =.1700 Find nearest z = -.44 x = mean + z σ = 200 + (-.44)(40) = 182.4 z =-.44

78. 31 40 th percentile Find area of interest.5000 -.4000 =.1000 Find nearest z = -.25 x = mean + z σ = 200 + (-.25)(40) = 190 z =-.25

79. 32 67 th percentile Find area of interest.6700 -.5000 =.1700 Find nearest z =.44 x = mean + z σ = 200 + (.44)(40) = 217.6 z =.44