2. Modern Languages 14131211109 87 6 54321 111098765 43 2 Row A Row B Row C Row D Row E Row F Row G Row H Row J Row K Row L Row M 212019181716 1514 13 12111098 212019181716 13 12111098 141312 table 7 6 54321 Row C Row D Row E Row F Row G Row H Row J Row K Row L Row M 321 21 1413 Projection Booth 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 7 6 5432 1 765 43 2 1 7 6 5432 1 765 43 2 1 7 6 54321 765 43 2 1 7 6 54321 765 43 2 1 7 6 54321 table Row C Row D Row E Row F Row G Row H Row J Row K Row L Row M 321 28 27 26252423 22 282726 2524 23 22 282726 2524 23 22 28 27 26252423 22 282726 2524 23 22 282726 2524 23 22 28 27 26252423 22 282726 2524 23 22 282726 2524 23 22 282726 2524 23 22 R/L handed broken desk Stage Lecturer’s desk Screen 1

3. MGMT 276: Statistical Inference in Management Spring 2015

5. Before our next exam (March 24 th ) Lind (5 – 11) Chapter 5: Survey of Probability Concepts Chapter 6: Discrete Probability Distributions Chapter 7: Continuous Probability Distributions Chapter 8: Sampling Methods and CLT Chapter 9: Estimation and Confidence Interval Chapter 10: One sample Tests of Hypothesis Chapter 11: Two sample Tests of Hypothesis 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 We’ll be jumping around some…we will start with chapter 7

6. Homework due – Tuesday (March 3 rd ) On class website: Please print and complete Homework Assignment 9 Chapter 5 Approaches to probabilities and Chapter 7 Interpreting probabilities using the normal curve Calculating z-score, raw scores and areas (probabilities) under normal curve

7. By the end of lecture today 2/26/15 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

8. 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

9. Always draw a picture! Homework worksheet

10. 1.6800 z =-1 z = 1

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

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

13. Homework worksheet 4.5000 z = 0

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

15. 6 z = 33-30 2 z = 1.5 Go to table.4332 Add area Lower half.4332 +.5000 =.9332 z = 1.5

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

17. 8 z = 29-30 2 = -.5 Go to table.1915 Add to upper Half of curve.5000 -.1915 =.6915 z = -.5

18. 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

19. 10 z = 27-30 2 = -1.5 Go to table.4332 Subtract From.5000.5000 -.4332 =.0668 z =-1.5

20. 11 z = 25-30 2 = -2.5 Go to table.4938 Add lower Half of curve.5000 +.4938 =.9938 z =-1.5

21. 12 z = 32-30 2 = 1.0 Go to table.3413 Subtract from.5000.5000 -.3413 =.1587 z =1

22. 13 50 th percentile = median 30 z =0

23. 14 28 32 z =-1 z = 1

24. 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

25. 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

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

27. 17.6800 z =-1 z = 1

28. .9500 18 z =-2 z = 2

29. .9970 19 z =-3 z = 3

30. 20 = 230-200 40 =.75 Go to table.2734 z =.75

31. 21 Go to table Subtract from.5000 z = 190-200 40 = -.25.0987.5000 -.0987 =.4013 z =-.5

32. 22 Go to table Add to upper Half of curve z = 180-200 40 = -.5.1915.5000 +.1915 =.6915 z =-.5

33. 23 z = 236-200 40 = 0.9 Go to table.3159 Subtract from.5000.5000 -.3159 =.1841 z =.9

34. 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

35. 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

36. 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

37. 27 z = 130-200 40 = -1.75.4599 Add to upper Half of curve Go to table.5000 +.4599 =.9599 z =-1.75

38. 28 40 z = 130-200 = -1.75.4599 Subtract from.5000.5000 -.4599 =.0401 Go to table z =-1.75

39. 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

40. 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

41. 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

42. 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

43. . 44 - 50 4 = -1.5 55 - 50 4 = +1.25 z of 1.5 = area of.4332.4332 +.3944 =.8276 z of 1.25 = area of.3944 55 - 50 4 = +1.25.5000 -.3944 =.1056 1.25 = area of.3944.3944 52 - 50 4 = +.5 55 - 50 4 = +1.25 z of.5 = area of.1915.3944 -.1915 =.2029 z of 1.25 = area of.3944.3944.1915.8276.1056.2029.4332.3944

44. 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

45. 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.”

46. 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

47. 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

48. 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

49. The probability of an event is the relative likelihood that the event will occur. The probability of event A [denoted P(A)], must lie within the interval from 0 to 1: 0 < P(A) < 1 If P(A) = 0, then the event cannot occur. If P(A) = 1, then the event is certain to occur.

50. The probabilities of all simple events must sum to 1 For example, if the following number of purchases were made by P(S) = P(E 1 ) + P(E 2 ) + … + P(E n ) = 1 credit card: 32% debit card: 20% cash: 35% check: 13% Sum =100% P(credit card) =.32 P(debit card) =.20 P(cash) =.35 P(check) =.13 Sum =1.0 Probability

51. What is the complement of the probability of an event The probability of event A = P(A). The probability of the complement of the event A’ = P(A’) A’ is called “A prime” Complement of A just means probability of “not A” P(A) + P(A’) = 100% P(A) = 100% - P(A’) P(A’) = 100% - P(A) Probability of getting into an educational program 66% chance of “admitted” 34% chance of “not admitted” 100% chance of admitted or not 5% chance of “rotten apple” Probability of getting a rotten apple 95% chance of “not rotten apple” 100% chance of rotten or not

52. Two mutually exclusive characteristics: if the occurrence of any one of them automatically implies the non-occurrence of the remaining characteristic Two mutually exclusive characteristics: if the occurrence of any one of them automatically implies the non-occurrence of the remaining characteristic Two events are mutually exclusive if they cannot occur at the same time (i.e. they have no outcomes in common). Two propositions that logically cannot both be true. http://www.thedailyshow.com/video/index.jhtml?videoId=188474&title=an-arab-family-manWarranty No Warranty For example, a car repair is either covered by the warranty (A) or not (B).

53. Events are collectively exhaustive if their union is the entire sample space S. Events are collectively exhaustive if their union is the entire sample space S. Two mutually exclusive, collectively exhaustive events are dichotomous (or binary) events. Two mutually exclusive, collectively exhaustive events are dichotomous (or binary) events. For example, a car repair is either covered by the warranty (A) or not (B). Warranty No Warranty Collectively Exhaustive Events

54. Satirical take on being “mutually exclusive” Recently a public figure in the heat of the moment inadvertently made a statement that reflected extreme stereotyping that many would find highly offensive. It is within this context that comical satirists have used the concept of being “mutually exclusive” to have fun with the statement. http://www.thedailyshow.com/video/index.jhtml?videoId=188474&title=an-arab-family-man Transcript: Speaker 1: “He’s an Arab” Speaker 2: “No ma’am, no ma’am. He’s a decent, family man, citizen…” Arab Decent, family man Warranty No Warranty

55. Union versus Intersection Union of two events means Event A or Event B will happen Intersection of two events means Event A and Event B will happen Also called a “joint probability” ∩ P(A B) P(A ∩ B)

56. 5A-55 union The union of two events: all outcomes in the sample space S that are contained either in event A or in event B or both (denoted A  B or “A or B”).  may be read as “or” since one or the other or both events may occur.

57. What is probability of drawing a red card or a queen? It is the possibility of drawing It is the possibility of drawing either a queen (4 ways) or a red card (26 ways) or both (2 ways). either a queen (4 ways) or a red card (26 ways) or both (2 ways). what is Q  R? union The union of two events: all outcomes contained either in event A or in event B or both (denoted A  B or “A or B”).

58. P(Q) = 4/52 (4 queens in a deck) = 4/52 + 26/52 – 2/52 P(Q  R) = P(Q) + P(R) – P(Q  R) = 28/52 =.5385 or 53.85% P(R) = 26/52 (26 red cards in a deck) P(Q  R) = 2/52 (2 red queens in a deck) Probability of picking a Queen Probability of picking a Red Probability of picking both R and Q 4/52 26/52 2/52 When you add the P(A) and P(B) together, you count the P(A and B) twice. So, you have to subtract P(A  B) to avoid over- stating the probability.

59. Union versus Intersection Union of two events means Event A or Event B will happen Intersection of two events means Event A and Event B will happen Also called a “joint probability” ∩ P(A B) P(A ∩ B)

60. what is Q  R? The intersection of two events: all outcomes contained in both event A and event B (denoted A  B or “A and B”) What is probability of drawing red queen? It is the possibility of drawing both a queen and a red card (2 ways).

61. If two events are mutually exclusive (or disjoint) their intersection is a null set (and we can use the “Special Law of Addition”) Intersection of two events means Event A and Event B will happen Examples: If A = Poodles If B = Labradors P(A ∩ B) = 0 Poodles and Labs: Mutually Exclusive (assuming purebred) mutually exclusive

62. Intersection of two events means Event A and Event B will happen P(A ∩ B) = 0 If two events are mutually exclusive (or disjoint) their intersection is a null set (and we can use the “Special Law of Addition”) Examples: If A = Poodles If B = Labradors ∩ P(A B) = P(A) +P(B) P(poodle or lab) = P(poodle) + P(lab) P(poodle or lab) = (.10) + (.15) = (.25) What’s the probability of picking a poodle or a lab at random from pound? Dog Pound (let’s say 10% of dogs are poodles) (let’s say 15% of dogs are labs) Poodles and Labs: Mutually Exclusive (assuming purebred )

63. Conditional Probabilities Probability that A has occurred given that B has occurred P(A | B) = P(A ∩ B) P(B) Denoted P(A | B): The vertical line “ | ” is read as “given.” The sample space is restricted to B, an event that has occurred. A  B is the part of B that is also in A. The ratio of the relative size of A  B to B is P(A | B).

64. Of the population aged 16 – 21 and not in college: Unemployed13.5% No high school diploma29.05% Unemployed with no high school diploma 5.32% What is the conditional probability that a member of this population is unemployed, given that the person has no diploma? Conditional Probabilities Probability that A has occurred given that B has occurred P(U) =.1350 P(ND) =.2905 P(U  ND) =.0532 P(A | B) = P(A ∩ B) P(B).0532.2905 = =.1831 or 18.31%

65. Of the population aged 16 – 21 and not in college: Unemployed13.5% No high school diploma29.05% Unemployed with no high school diploma 5.32% What is the conditional probability that a member of this population is unemployed, given that the person has no diploma? Conditional Probabilities Probability that A has occurred given that B has occurred P(U) =.1350 P(ND) =.2905 P(U  ND) =.0532 P(A | B) = P(A ∩ B) P(B).0532.2905 = =.1831 or 18.31%