screen Lecturers desk Row A 13 8 Row

screen Lecturer’s desk Row A 13 8 Row A 7 16 15 14 20 19 18 17 16 15 Row B 21 20 19 18 17 16 Row C 22 21 20 19 18 17 Row D 23 22 21 20 19 18 Row E 17 16 15 14 13 12 11 10 9 8 23 22 21 20 19 18 Row F 17 16 15 14 13 12 11 10 9 8 24 23 22 21 20 19 Row G 18 22 21 20 19 18 17 Row H 16 14 16 24 23 22 Row J 21 20 19 18 27 26 25 24 23 Row K 22 21 20 19 28 27 26 25 24 Row L 23 28 27 26 25 24 Row M 30 29 28 27 26 Row N 25 24 23 30 29 28 27 26 Row P 25 24 23 38 9 10 17 25 39 11 18 26 40 12 19 37 36 35 34 33 23 32 22 22 31 30 Physics- atmospheric Sciences (PAS) - Room 201 21 18 9 10 8 Row B 7 4 3 2 1 Row A 6 5 4 3 2 1 Row B 14 13 12 11 10 9 8 7 Row C 6 5 4 3 2 1 Row C 15 14 13 12 11 10 9 8 7 Row D 6 5 4 3 2 1 Row D 7 Row E 6 5 4 3 2 1 Row E 7 Row F 6 5 4 3 2 1 Row F Row G 6 5 4 3 2 1 Row G Row H 6 5 4 3 2 1 Row H 15 14 17 18 11 5 15 16 15 12 14 13 13 12 16 12 11 10 11 table 14 19 20 21 17 13 6 9 10 8 7 9 8 7 table 13 9 8 7 6 5 1 Row J 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row K 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row L 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row M 20 19 18 17 16 15 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row N 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Row P 5 4 3 2 1 Row Q 29 28 27 26 25 24 23 22 21 - 15 14 13 12 11 10 9 8 7 6

MGMT 276: Statistical Inference in Management Fall 2015

Just for Fun Assignments Go to D 2 L - 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

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Schedule of readings Before our next exam (October 20 th) Open. Stax 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

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

Use this as your study guide By the end of lecture today 10/6/15 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

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 400 66% chance of getting admitted Probability of getting a rotten apple Number of rotten apples Number of apples 5% chance of getting a rotten 100 apple 5

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

2. Classic probability: a priori probabilities based on logic rather than on data or experience. All options are equally likely (deductive rather than inductive). Likelihood get Chosen at Lottery question right random to be on multiple team captain choice test 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 1 = 6 16% chance of getting a two 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

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

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

Raw scores, z scores & probabilities Notice: 3 types of numbers z = • raw scores 2 • z scores • probabilities z= +2 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 convert Distance from the mean (z scores) Proportion of curve Raw Scores (actual data) We care about this! (area from mean) 68% z = -1 z=1 What is the actual number on this scale? “height” vs “weight” “pounds” vs “test score” Proportion of curve Raw Scores convert We care about this! “percentiles” “percent of people” “proportion of curve” “relative position” 68% (actual data) convert z = -1 z=1 Distance from the mean (z scores) (area from mean) convert

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

Scores, standard deviations, and probabilities Actually 95. 44 To be exactly 95% we will use z = 1. 96 Actually 68. 26

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

Mean = 50 Standard deviation = 10 Let’s do some problems ? 60 Find the area under the curve that falls below 60 means the same thing as Find the percentile rank for score of 60 Pro 1 ble m vie e r w

Mean = 50 Standard deviation = 10 Let’s do some problems ? Find the percentile rank for score of 60 . 5000 . 3413 60 Distance from the mean ( from raw to z scores) Raw Scores (actual data) 1) Find z score = 60 - 50 =1 10 2) Go to z table - find area under correct column (. 3413) z-table (from z to area) Proportion of curve (area from mean) 3) Look at your picture - add. 5000 to. 3413 =. 8413 4) Percentile rank or score of 60 = 84. 13% Pro 1 ble m Hint always draw a picture! vie e r w

Mean = 50 Standard deviation = 10 ? Find the percentile rank for score of 75 . 4938 75 1) Find z score = 75 - 50 10 z score = 25 = 2. 5 10 2) Go to z table Pr ob lem 2 Hint always draw a picture! vie e r w

Mean = 50 Standard deviation = 10 ? Find the percentile rank for score of 75 . 5000. 4938 75 1) Find z score = 75 - 50 10 z score = 25 = 2. 5 10 2) Go to z table 3) Look at your picture - add. 5000 to. 4938 =. 9938 4) Percentile rank or score of 75 = 99. 38% lem Hint always draw a picture! vie e r w Pro b 2

Mean = 50 Standard deviation = 10 Find the percentile rank for score of 45 ? Distance from the mean ( from raw to z scores) 45 1) Find z score = 45 - 50 10 z score = - 5 = -0. 5 10 Raw Scores (actual data) z-table (from z to area) Proportion of curve (area from mean) 2) Go to z table Pro b 3 lem vie e r w

Mean = 50 Standard deviation = 10 Find the percentile rank for score of 45 ? . 1915 ? 45 1) Find z score = 45 - 50 10 z score = - 5 = -0. 5 10 2) Go to z table Pro b 3 lem vie e r w

Mean = 50 Standard deviation = 10 ? Find the percentile rank for score of 45. 1915 . 3085 45 1) Find z score = 45 - 50 10 z score = - 5 = -0. 5 10 z-table (from z to area) Distance from the mean ( from raw to z scores) Raw Scores (actual data) Proportion of curve (area from mean) 2) Go to z table 3) Look at your picture - subtract. 5000 -. 1915 =. 3085 Pro b 3 lem 4) Percentile rank or score of 45 = 30. 85% vie e r w

Mean = 50 Standard deviation = 10 ? Find the percentile rank for score of 55 Distance from the mean ( from raw to z scores) 55 1) Find z score = 55 - 50 10 z score = 5 = 0. 5 10 2) Go to z table Pro b 4 lem Raw Scores (actual data) z-table (from z to area) Proportion of curve (area from mean)

Mean = 50 Standard deviation = 10 Find the percentile rank for score of 55 . 1915 55 1) Find z score = 55 - 50 10 z score = 5 = 0. 5 10 2) Go to z table Pro b 4 lem ?

Mean = 50 Standard deviation = 10 Find the percentile rank for score of 55 . 1915. 5 55 1) Find z score = 55 - 50 10 z score = 5 = 0. 5 10 Distance from the mean ( from raw to z scores) Raw Scores (actual data) 2) Go to z table 3) Look at your picture - add. 5000 +. 1915 =. 6915 Pro b 4 lem 4) Percentile rank or score of 55 = 69. 15% ? z-table (from z to area) Proportion of curve (area from mean)

Mean = 50 Standard deviation = 10 Find the score for z = -2 Find the score that is associated with a z score of -2 3? 0 Hint always draw a picture! 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 ztable ‘backwards’. We find the closest z to match our proportion Distance from the mean ( from raw to z scores) Raw Scores (actual data) z-table (from z to area) Proportion of curve (area from mean)

Find the score for percentile rank of 77%ile Mean = 50 Standard deviation = 10 ? Distance from the mean ( from raw to z scores) . 7700 ? Pro b 5 lem Please note: When we are looking for the score from proportion we use the ztable ‘backwards’. We find the closest z to match our proportion Raw Scores (actual data) z-table (from z to area) Proportion of curve (area from mean)

Find the score for percentile rank of 77%ile. 5 Mean = 50 Standard deviation = 10 . 27. 5 ? . 5 +. 27 =. 77. 27 . 7700 ? Pro b 5 1) Go to z table - find z score for area. 2700 (. 7700 -. 5000) =. 27 area =. 2704 (closest I could find to. 2700) z = 0. 74 lem 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

Find the score for percentile rank of 77%ile. 5 Mean = 50 Standard deviation = 10 . 27. 5 x = ? 57. 4 . 27 . 7700 ? 2) x = mean + (z)(standard deviation) x = 50 + (0. 74)(10) x = 57. 4 Pro b 5 lem 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

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

Find the score for percentile rank of 55%ile. 5 Mean = 50 Standard deviation = 10 . 05. 5 ? . 5 +. 05 =. 55. 05 . 5500 ? Pro b 7 1) Go to z table - find z score for area. 0500 (. 5500 -. 5000) =. 05 area =. 0517 (closest I could find to. 0500) z = 0. 13 lem 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

Find the score for percentile rank of 55%ile. 5 Mean = 50 Standard deviation = 10 . 05. 5 ? . 05 . 5500 ? Pro b 7 1) Go to z table - find z score for area. 0500 (. 5500 -. 5000) =. 05 area =. 0517 (closest I could find to. 0500) z = 0. 13 lem 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

Find the score for percentile rank of 55%ile. 5 Mean = 50 Standard deviation = 10 . 05. 5 ? x = 51. 3 . 05 . 5500 ? 1) Go to z table - find z score for area. 0500 (. 5500 -. 5000) =. 0500 area =. 0517 (closest I could find to. 0500) z = 0. 13 Pro b 2) x = mean + (z)(standard deviation) x = 50 + (0. 13)(10) x = 51. 3 7 lem 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

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. 4500 Go to table nearest z = 1. 64 x = mean + z σ = 50 + (1. 64)(4) = 56. 56 . 9500. 4500 . 5000 38 42 46 50 e ? 54 56. 6058 62 io lp a n A it dd lem b o Pr ra c cti 8

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 Go to. 4700 table nearest z = - 1. 88 x = mean + z σ = 2100 + (-1. 88)(250) = 1, 630 . 0300 ? 1, 630 . 4700 e 2100 io lp a n A it dd lem b o Pr ra c cti 9

Normal Distribution has a mean of 195 and standard deviation of 8. 5. Determine value for top 1% of hours listened. . 4900 Go to table nearest z = 2. 33 x = mean + z σ = 195 + (2. 33)(8. 5) = 214. 805 . 5000 . 4900 195 . 0100 ? 214. 8 e lp a n A o Pr ra m e l b io it dd c cti 10

. Find score associated with the 75 th percentile. 2500 Go to table nearest z =. 67 x = mean + z σ = 30 + (. 67)(2) = 31. 34 . 7500. 25 . 5000 24 26 28 30 z =. 67 ? 31. 34 34 e 36 lp a n A o Pr ra m e l b io it dd c cti 11

. Find the score associated with the 25 th percentile. 2500 Go to table nearest z = -. 67 x = mean + z σ = 30 + (-. 67)(2) = 28. 66 . 2500. 25 24 . 25 ? 28 26 28. 66 30 z = -. 67 34 e 36 lp a n A o Pr ra m e l b io it dd c cti 12

. 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. 4750 nearest z = 1. 96 table mean + z σ = 30 + (1. 96)(2) = 33. 92. 4750 Go to table nearest z = -1. 96 mean + z σ = 30 + (-1. 96)(2) = 26. 08 . 9500. 475 ? 24 26. 08 28 30 e ? 32 33. 92 36 lp a n A o Pr ra m e l b io it dd c cti 13

. 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. 4750 nearest z = 1. 96 table mean + z σ = 100 + (1. 96)(5) = 109. 80. 4750 Go to table nearest z = -1. 96 mean + z σ = 100 + (-1. 96)(5) = 90. 20 . 9500. 475 ? 85 90. 2 95 . 475 100 e ? 105 109. 8 115 lp a n A o Pr ra m e l b io it dd c cti 14

. 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. 4750 nearest z = 1. 96 table mean + z σ = 30 + (2. 58)(2) = 35. 16. 4750 Go to table nearest z = -1. 96 mean + z σ = 30 + (-2. 58)(2) = 24. 84 . 9900. 495 ? 24. 84 28 30 32 e ? 35. 16 lp a n A o Pr ra m e l b io it dd c cti 15

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

Always draw a picture! Homeworksheet

Homeworksheet. 6800 1 also fine: 68% also fine: . 6826 . 6800 1 sd -1 z = 28 1 sd 30 1 z = 32

Homeworksheet. 9500 2 also fine: 95. 00% also fine: . 9544 . 9500 2 sd -2 z = 26 28 2 sd 30 32 2 z = 34

Homeworksheet. 9970 3 also fine: 99. 70% also fine: . 9974 . 9970 3 sd -3 z = 24 26 28 3 sd 30 32 34 3 z = 36

Homeworksheet . 5000 4 also fine: 50% . 5000 24 26 28 0 z = 30 32 34 36

Homeworksheet 33 -30 z = 1. 5 z= 2 Go to table . 4332 5 also fine: 43. 32%. 4332 24 26 28 30 1. 5 = z 32 34 36

z= 33 -30 z = 1. 5 2 Go to table . 4332 Add area Lower half . 4332 +. 5000 =. 9332 6 also fine: 93. 32%. 9332. 5000 24 26 28 30 . 4332 1. 5 = z 32 34 36

Homeworksheet Go to 33 -30 z=. 4332 = 1. 5 table 2 Subtract from. 5000 -. 4332 =. 0668 7. 4332 also fine: 6. 68%. 0668 24 26 28 30 1. 5 = z 32 34 36

z= 29 -30 2 = -. 5 Go to. 1915 table Add to upper Half of curve. 5000 +. 1915 =. 6915 8 also fine: 69. 15%. 6915. 1915 24 26 -. 5 = z 28 30 . 5000 32 34 36

25 -30 2 31 -30 = 2 = = -2. 5 =. 5 Go to table . 4938. 1915 . 4938 +. 1915 =. 6853 9 also fine: 68. 53%. 6853. 4938 5 -2. = z 24 26 28 . 1915. 5 = 30 z 32 34 36

z= Go to 27 -30 = -1. 5 table 2 . 4332 Subtract From. 5000 -. 4332 =. 0668 10 also fine: 6. 68%. 5000. 0668. 5 24 =-1 z . 4332 26 28 30 32 34 36

z= 25 -30 2 = -2. 5 Go to table . 4938 Add lower Half of curve. 5000 +. 4938 =. 9938 11 also fine: 99. 38%. 9938. 5000 . 4938. 5 24 =-2 z 26 28 30 32 34 36

z= Go to 32 -30 = 1. 0 table 2 . 3413 Subtract from. 5000 -. 3413 =. 1587 12 also fine: 15. 87% . 3413 1587. . 5000 24 26 28 30 1 z = 32 34 36

50 th percentile = median 30 13 In a normal curve Median= Mean = Mode 24 26 28 0 z = 30 32 34 36

28 32 14. 6800 1 sd 24 26 -1 z = 28 1 sd 30 1 z = 32 34 36

77 th percentile Find area of interest . 7700 -. 5000 =. 2700 Find nearest z =. 74 15 x = mean + z σ = 30 + (. 74)(2) = 31. 48 z table provides area from mean to score . 5000. 2700. 7700 24 30 4 . 7 z = ? 31. 48 36

13 th percentile Find area of interest . 5000 -. 1300 =. 3700 Find nearest z = -1. 13 16 x = mean + z σ = 30 + (-1. 13)(2) = 27. 74 Note: . 13 +. 37 =. 50 z table provides area from mean to score . 1300 24 z ? . 13 1 = 27. 74 . 3700 30 36

Please use the following distribution with a mean of 200 and a standard deviation of 40. 80 120 160 200 240 280 320

. 6800 17 also fine: 68. 00% also fine: . 6826 . 6800 1 sd -1 z = 160 1 sd 1 z = 200 240

. 9500 18 also fine: 95. 00% also fine: . 9544 . 9500 2 sd -2 z = 120 2 sd 200 2 z = 280

. 9970 19 also fine: 99. 70% also fine: . 9974 . 9970 3 sd -3 z = 80 3 sd 200 3 z = 320

= 230 -200 =. 75 40 Go to table . 2734 20 also fine: 27. 34% 4 . 273 80 120 160 5 . 7 z = 200 240 280 320

190 -200 = -. 25 Go to z= table 40 . 0987 also fine: 40. 13 Subtract from. 5000 -. 0987 =. 4013 21 . 0987 . 4013. 5000 80 120 5 -. 2 = z 160 200 240 280 320

180 -200 = -. 5 Go to z= table 40 . 1915 Add to upper. 5000 +. 1915 =. 6915 Half of curve also fine: 22 69. 15%. 6915. 1915 80 120 -. 5 z = 160 200 . 5000 240 280 320

236 -200 = 0. 9 z= 40 Go to table . 3159 Subtract from. 5000 -. 3159 =. 1841 23. 3159 also fine: 18. 41%. 1841 80 120 160 =. 9 z 200 240 280 320

192 - 200 40 = 222 - 200 40 z= z = -. 2 =. 55 Go to table . 0793. 2088 also fine: 28. 81% . 0793 +. 2088 =. 2881 24 . 2881. 2088. 0793 80 120 2 =-. z 160 5 . 5 200 z = 240 280 320

z= 275 -200 = 1. 875 40 Go to table . 4693 or. 4699 Add area Lower half Please note: If z-score rounded to 1. 88, then percentile = 96. 99% . 4693 +. 5000 =. 9693. 4699 +. 5000 =. 9699 also fine: 25 96. 93%. 9693 . 5000 80 120 160 . 4693 5 . 87 200 z =1240 280 320

295 -200 z = 2. 375 z= 40 Go to table . 4911 or. 4913 . 5000 -. 4911 =. 0089 Add area Lower half. 5000 -. 4913 =. 0087 26 Please note: If z-score rounded to 2. 38, then area =. 0087 . 4911 also fine: 0. 89% . 0089 80 120 160 200 75 240 z =2. 3280 320

Add to upper 130 -200 = -1. 75 Go to. 5000 +. 4599 =. 9599 z=. 4599 table Half of curve 40 also fine: 27 95. 99%. 9599. 5000 . 4599 80 z . 75 1 120 = 160 200 240 280 320

Subtract 130 -200 = -1. 75 Go to z=. 4599 table from. 5000 40 . 5000 -. 4599 =. 0401 28 also fine: 4. 01% . 5000 . 4599 1 0 4. 0 80 z . 75 1 120 = 160 200 240 280 320

99 th percentile Find area of interest Find nearest z = 2. 33 . 9900 -. 5000 =. 4900 29 x = mean + z σ = 200 + (2. 33)(40) = 293. 2 . 5000 . 4900 . 9900 80 120 160 z table provides area from mean to score 200 240 33 2. z = ? 293. 2

33 rd percentile Find area of interest . 5000 -. 3300 =. 1700 Find nearest z = -. 44 30 x = mean + z σ = 200 + (-. 44)(40) = 182. 4 Note: . 33 +. 17 =. 50 z table provides area from mean to score . 3300 80 . 1700 ? 44 -. 182. 4 = z 200 240 280 320

40 th percentile Find area of interest . 5000 -. 4000 =. 1000 Find nearest z = -. 25 31 x = mean + z σ = 200 + (-. 25)(40) = 190 Note: . 40 +. 10 =. 50 z table provides area from mean to score . 4000 80 . 1000 ? 25 -. 182. 4 = z 200 240 280 320

67 th percentile Find area of interest . 6700 -. 5000 =. 1700 Find nearest z =. 44 32 x = mean + z σ = 200 + (. 44)(40) = 217. 6 z table provides area from mean to score . 1700 80 200 ? . 44 = z 217. 6 320