Maths-AI----Goodness-of-Fit.pptx goodness

Goodness of fit

1. STARTER https://blogs.sas.com/content/iml/2017/02/20/proportion-of-colors-mandms.html

2. STARTER

3. Solution

4. Maths AI SL Chi Square Goodness of Fit Test

5. Video Icons Click on the following icons throughout the presentation to take you to a video

6. Chi-Square Distributions

7. Chi-Square Test

8. RECAP

9. RECAP

10. The X 2 Goodness of Fit Test

11. The X 2 Goodness of Fit Test

12. The X 2 Goodness of Fit Test The chi-square test for goodness-of-fit uses frequency data from a sample to test hypotheses about the shape or proportions of a population. Each individual in the sample is classified into one category on the scale of measurement. The data, called observed frequencies , simply count how many individuals from the sample are in each category.

13. The X 2 Goodness of Fit Test The null hypothesis specifies the proportion of the population that should be in each category. The proportions from the null hypothesis are used to compute expected frequencies that describe how the sample would appear if it were in perfect agreement with the null hypothesis.

14. The X 2 Goodness of Fit Test

15. The X 2 Goodness of Fit Test

16. The X 2 Goodness of Fit Test

17. Understanding the Test Rock-Paper-Scissors

18. Understanding the Test Rock-Paper-Scissors ROCK PAPER SCISSORS 66 39 14 How would we test whether all of these categories are equally likely?

19. Understanding the Test Hypothesis Testing State Hypotheses Calculate a statistic, based on your sample data Create a distribution of this statistic, as it would be observed if the null hypothesis were true Measure how extreme your test statistic from (2) is, as compared to the distribution generated in (3) test statistic

20. Hypotheses Let p i denote the proportion in the i th category. H : All p i s are the same H a : At least one p i differs from the others OR H : Every p i = 1/3 H a : At least one p i ≠ 1/3 Understanding the Test

21. Test Statistic Why can’t we use the familiar formula to get the test statistic? More than one sample statistic More than one null value We need something a bit more complicated… Understanding the Test

22. Observed Counts The observed counts are the actual counts observed in the study ROCK PAPER SCISSORS Observed 66 39 14 Example

23. Expected Counts The expected counts are the expected counts if the null hypothesis were true For each cell, the expected count is the sample size (n) times the null proportion, p i Example

24. Rock-Paper-Scissors ROCK PAPER SCISSORS Observed 66 39 14 Expected Example

25. Rock-Paper-Scissors ROCK PAPER SCISSORS Observed 66 39 14 Expected 39.7 39.7 39.7 Example

26. Chi-Square Statistic A test statistic is one number, computed from the data, which we can use to assess the null hypothesis The chi-square statistic is a test statistic for categorical variables: Example

27. What Next? We have a test statistic. What else do we need to perform the hypothesis test? A distribution of the test statistic assuming H is true How do we get this? Distributional Theory Example

28. To calculate the p-value for a chi-square test, we always look in the upper tail Why? Values of the χ 2 are always positive The higher the χ 2 statistic is, the farther the observed counts are from the expected counts, and the stronger the evidence against the null Upper-Tail P-Values

29. The X 2 Goodness of Fit Test

30. The X 2 Goodness of Fit Test

31. The X 2 Goodness of Fit Test

32. The X 2 Goodness of Fit Test

33. The X 2 Goodness of Fit Test

34. The X 2 Goodness of Fit Test

35. Summary

36. Example – Starter Copy and complete. Show all steps

37. GDC

38. Solution

39. Exercise

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41. Exercise

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43. Exercise

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45. Example

46. Stating the Null and Alternative Hypothesis

47. Solution

48. Solution

49. Limitations

50. Example

51. Solution

52. Test Your Understanding

53. Exercise

54. Solution

55. Exercise

56. Solution

57. The X 2 Goodness of Fit Test

58. Example

59. STARTER - Test Your Understanding

60. Solution

61. Example

62. Test Your Understanding

63. ANSWERS

64. IB Exam Questions

65. Solution

66. GOF: Normal Distribution

67. GOF: Normal Distribution

68. GOF: Normal Distribution

69. GOF: Normal Distribution

70. GOF: Normal Distribution

71. GDC

72. ANSWERS

73. Exercise

74. Exercise

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77. IB Exam Questions

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79. IB Exam Questions

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81. IB Exam Questions

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84. GOF: Binomial Distribution A  binomial distribution  can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times. The binomial is a type of distribution that has  two possible outcomes  (the prefix “ bi ” means two, or twice). For example, a coin toss has only two possible outcomes: heads or tails and taking a test could have two possible outcomes: pass or fail.

85. GOF: Binomial Distribution

86. Exercise

87. GOF: Binomial Distribution

88. GDC

89. GDC

90. Answers

91. Example

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94. Test Your Understanding

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98. Exercise

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101. IB Exam Questions

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103. IB Exam Questions

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105. IB Exam Questions page 135

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109. IB Exam Questions page 106

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