Homework Assignment

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

4. Sampling distribution of the mean

5. Sampling distribution of the mean

6. Sampling distribution of the mean 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 • 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.

14. 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. Mutually exclusive Venn diagram

25. Mutually exclusive Sample space Venn diagram

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

27. Mutually exclusive

28. Mutually exclusive 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. Probability distribution 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 Frequency Number rolled

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

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 Sum of areas Pr[1 or 2] = Pr[1]+Pr[2]

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. The addition rule What if they are not mutually exclusive?