2. BNAD 276: Statistical Inference in Management Winter, 2015 http://www.youtube.com/watch?v=Ahg6qcgoay4&watch_response Green sheet Seating Chart

3. Daily group portfolios Beginning of each lecture (first 5 minutes) Meet in groups of 3 or 4 Meet in groups of 3 or 4 Quiz one another on class material Quiz one another on class material Discuss the questions and determine the correct answer for each question Discuss the questions and determine the correct answer for each question Five copies (one for each group member – and typed) 3 multiple choice questions based on lecture Five copies (one for each group member – and typed) 3 multiple choice questions based on lecture Include 4 options (a, b, c, and d) Include 4 options (a, b, c, and d) Include a name and describe a person in a certain situation Include a name and describe a person in a certain situation Margaret was interested in taking a Statistics course. It is likely she was interested in studying which of the following? a. economic theories of communism b. theological perspectives of life after death c. musical compositions of the 12th century d. statistical techniques and inference They can be funny or serious, and must be clear and have only one correct answer.

5. . Homework Assignment

6. Hand out z tables

7. Mean = 100 Standard deviation = 5 If we go up one standard deviation z score = +1.0 and raw score = 105 If we go down one standard deviation z score = -1.0 and raw score = 95 If we go up two standard deviations z score = +2.0 and raw score = 110 If we go down two standard deviations z score = -2.0 and raw score = 90 If we go up three standard deviations z score = +3.0 and raw score = 115 If we go down three standard deviations z score = -3.0 and raw score = 85 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 85 90 95 100 105 110 115 z = -1 z = +1 z = -2 z = +2 z = -3 z = +3 68% 95% 99.7%

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

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

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

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

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

13. 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%

14. 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 Mean = 50 Standard deviation = 10 60 50 10 z = 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) Find z score for raw score of 60

15. 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 30 50 10 z = - 2 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) Find z score for raw score of 30 Mean = 50 Standard deviation = 10

16. 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 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) Raw scores, z scores & probabilities Find z score for raw score of 70 Mean = 50 Standard deviation = 10

17. 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 80 50 10 z = 3 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) Find z score for raw score of 80 Mean = 50 Standard deviation = 10

18. 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 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) Raw scores, z scores & probabilities Find z score for raw score of 20 Mean = 50 Standard deviation = 10

19. 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

20. Hand out z tables

21. Ties together z score with Draw picture of what you are looking for... Find z score (using formula)... Look up proportions on table probability proportion percent area under the curve 68% 34%

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

23. z score = raw score - mean standard deviation z score = 60 - 50 10 z score = 10 = 1.0 10 z table z score of 1 = area of.3413 Hint always draw a picture! Find the area under the curve that falls between 40 and 60 Mean = 50 Standard deviation = 10 z score = 40 - 50 10 z score = 10 = -1.0 10 z table z score of 1 = area of.3413.3413 +.3413 =.6826 68.26% 34.13%

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

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

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

27. Let’s do some problems.4772 +.4772 =.9544 Hint always draw a picture! Find the area under the curve that falls between 30 and 70 Mean = 50 Standard deviation = 10 z score of 2 = area of.4772.4772 95.44% 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)

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

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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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 6.5 +.05 =.55

48. 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 6

49. 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 6

50. 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 7

51. 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 8

52. 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 9

53. . 75 th percentile Go to table.2500 nearest z =.67 x = mean + z σ = 30 + (.67)(2) = 31.34 z =.67 Additional practice Problem 10

54. . 25 th percentile Go to table.2500 nearest z = -.67 x = mean + z σ = 30 + (-.67)(2) = 28.66 z = -.67 Additional practice Problem 11

55. . 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 12

56. . 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 13

57. . 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 14

58. 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

59. 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