1. Homework Assignment Chapter 1, Problems 6, 15 Chapter 2, Problems 6, 8, 9, 12 Chapter 3, Problems 4, 6, 15 Chapter 4, Problem 16 Due a week from Friday: Sept. 22, 12 noon. Your TA will tell you where to hand these in

2. Random Sampling - what did we learn? It’s difficult to do properly Why not just point? Computers and random numbers Can you tell if your numbers were random?

3. Sampling distribution of the mean

6. How confident can we be about this one estimate of the mean?

7. Estimating error of the mean Hard method: take a few MORE random samples, and get more estimates for the mean Easy method: use the formula:

8. Confidence interval –a range of values surrounding the sample estimate that is likely to contain the population parameter We are 95% confident that the true mean lies in this interval

11.  = 5.14 Y = 5.26

12. What if we calculate 95% confidence intervals? Approximately ± 2 S.E. Expect that 95% of the intervals from the class will contain the true population mean, 5.14 70 invervals * 5% = 3.5 Expect that 3 or 4 will not contain the mean, and the rest will

13. Mean ± 95% C.I.

15. What if we took larger samples? Say, n=20 instead of n=10?

18. Probability

19. The Birthday Challenge

20. Probability The proportion of times the event occurs if we repeat a random trial over and over again under the same conditions Pr[A] –The probability of event A

22. (cannot both occur simultaneously)

23. Mutually exclusive

24. Venn diagram

25. Mutually exclusive Venn diagram Sample space

26. Mutually exclusive Venn diagram Sample space Possible outcome Pr[B] proportional to area

27. Mutually exclusive

28. Pr(A and B) = 0

29. Mutually exclusive Visual definition - areas do not overlap in Venn diagram

30. Not mutually exclusive Pr(A and B)  0 Pr(purple AND square)  0

31. For example

32. Probability distribution

33. Random variable - a measurement that changes from one observation to the next because of chance

34. Probability distribution for the outcome of a roll of a die Number rolled Frequency

35. Probability distribution for the sum of a roll of two dice Sum of two dice Frequency

36. The addition rule

37. Addition Rule Pr[1 or 2] = ?

38. Addition Rule Pr[1 or 2] = Pr[1]+Pr[2]

39. Addition Rule Pr[1 or 2] = Pr[1]+Pr[2]

40. Addition Rule Pr[1 or 2] = Pr[1]+Pr[2] Sum of areas

41. The probability of a range For families of 8 children, Pr[Number of boys ≥ 6] = ?

42. The probability of a range For families of 8 children, Pr[Number of boys ≥ 6] = Pr[6 or 7 or 8] = Pr[6]+Pr[7]+Pr[8]

43. The probabilities of all possibilities add to 1.

44. Addition Rule Pr[1 or 2 or 3 or 4 or 5 or 6] = ?

45. Addition Rule Pr[1 or 2 or 3 or 4 or 5 or 6] = 1

46. Probability of Not Pr[NOT rolling a 2] = ?

47. Probability of Not Pr[NOT rolling a 2] = 1 - Pr[2] = 5/6

48. Probability of Not Pr[NOT rolling a 2] = 1 - Pr[2] = 5/6 Pr[not A] = 1-Pr[A]

49. The addition rule

50. What if they are not mutually exclusive?

51. General Addition Rule A B Pr[A or B] = ?

52. General Addition Rule A B Pr[A or B] = ? A B

53. General Addition Rule A B Pr[A or B] = ? A B

54. General Addition Rule A B Pr[A or B] = ? A B

55. General Addition Rule A B A B

57. Pr[Walks or flies] = ?

58. General Addition Rule Pr[Walks or flies] = ?

59. General Addition Rule Pr[Walks or Flies] = Pr[Walks] + Pr[Flies] - Pr[Walks and Flies]

60. General Addition Rule

61. Independence

62. Equivalent definition: The occurrence of one does not change the probability that the second will occur

63. Multiplication rule If two events A and B are independent, then Pr[A and B] = Pr[A] x Pr[B]

64. Pr[boy]=0.512

66. General Addition Rule

67. Multiplication rule If two events A and B are independent, then Pr[A and B] = Pr[A] x Pr[B]

68. OR versus AND OR statements: –Involve addition –It matters if the events are mutually exclusive AND statements: –Involve multiplication –It matters if the events are independent

69. Probability trees

70. Sex of two children family

72. Dependent events Variables are not always independent; in fact they are often not

73. Fig wasps

75. Testing independence Are the previous state of the fig and the sex of an egg laid independent? Test the multiplication rule: Pr[A and B] ?=? Pr[A] x Pr[B] Pr[fig already has eggs and male] ?=? P[fig already has eggs] x Pr[male]

80. ≠

83. Conditional probability Pr[X|Y]

85. Law of total probability:

87. The general multiplication rule

88. Does not require independence between A and B

89. Bayes' theorem

90. In class exercise

91. Answer

96. Homework Assignment Chapter 1, Problems 6, 15 Chapter 2, Problems 6, 8, 9, 12 Chapter 3, Problems 4, 6, 15 Chapter 4, Problem 16 Due a week from Friday: Sept. 22, 12 noon. Your TA will tell you where to hand these in