Chapter 10. Discrete Data Analysis

1. Chapter 10. Discrete Data Analysis 10.1 Inferences on a Population Proportion 10.2 Comparing Two Population Proportions 10.3 Goodness of Fit Tests for One-Way Contingency Tables 10.4 Testing for Independence in Two-Way Contingency Tables 10.5 Supplementary Problems

2. Population Proportion p with characteristic Random sample of size n With characteristic Without characteristic Cell probability p Cell frequency x Cell probability 1-p Cell frequency n-x 10.1 Inferences on a Population Proportion

3. 10.1.1 Confidence Intervals for Population Proportions

4. 10.1.1 Confidence Intervals for Population Proportions • Example 55 : Building Tile Cracks Random sample n = 1250 of tiles in a certain group of downtown building for cracking. x = 98 are found to be cracked.

5. 10.1.1 Confidence Intervals for Population Proportions

6. 10.1.2 Hypothesis Tests on a Population Proportion

7. 10.1.2 Hypothesis Tests on a Population Proportion

8. 10.1.2 Hypothesis Tests on a Population Proportion

9. Example 55 : Building Tile Cracks 10% or more of the building tiles are cracked ? 10.1.2 Hypothesis Tests on a Population Proportion 0 z = -2.50

10. 10.1.3 Sample Size Calculations

11. 10.1.3 Sample Size Calculations • Example 59 : Political Polling To determine the proportion p of people who agree with the statement “The city mayor is doing a good job.” within 3% accuracy. (-3% ~ +3%), how many people do they need to poll?

12. 10.1.3 Sample Size Calculations • Example 55 : Building Tile Cracks

13. 10.2 Comparing Two Population Proportions

14. 10.2.1 Confidence Intervals for the Difference Between Two Population Proportions

15. 10.2.1 Confidence Intervals for the Difference Between Two Population Proportions • Example 55 : Building Tile Cracks Building A : 406 cracked tiles out of n = 6000. Building B : 83 cracked tiles out of m = 2000.

16. 10.2.2 Hypothesis Tests on the Difference Between Two Population Proportions

17. Population age 18-39 age >= 40 A B Random sample m=1043 Random sample n=952 “The city mayor is doing a good job.” Agree : x = 627 Disagree : n-x = 325 Agree : y = 421 Disagree : m-y = 622 10.2.2 Hypothesis Tests on the Difference Between Two Population Proportions • Example 59 : Political Polling

18. Summary problems • Why do we assume large sample sizes for statistical inferences concerning proportions? So that the Normal approximation is a reasonable approach. (2) Can you find an exact size test concerning proportions? No, in general.

19. 10.3 Goodness of Fit Tests for One-Way Contingency Tables10.3.1 One-Way Classifications

20. 10.3.1 One-Way Classifications • Example 1 : Machine Breakdowns n = 46 machine breakdowns. x1 = 9 : electrical problems x2 = 24 : mechanical problems x3 = 13 : operator misuse It is suggested that the probabilities of these three kinds are p*1 = 0.2, p*2 = 0.5, p*3 = 0.3.

21. 10.3.1 One-Way Classifications

22. 10.3.1 One-Way Classifications • Example 1 : Machine Breakdowns H0 : p1 = 0.2, p2 = 0.5, p3 = 0.3

23. 10.3.1 One-Way Classifications

24. 10.3.2 Testing Distributional Assumptions • Example 3 : Software Errors For some of expected values are smaller than 5, it is appropriate to group the cells. • Test if the data are from a Poisson distribution with mean=3.

25. 10.3.2 Testing Distributional Assumptions

26. 10.4 Testing for Independence in Two-Way Contingency Tables10.4.1 Two-Way Classifications • A two-way (r x c) contingency table. Second Categorization First Categorization Row marginal frequencies Column marginal frequencies

27. 10.4.1 Two-Way Classifications Example 55 : Building Tile Cracks Notice that the column marginal frequencies are fixed. ( x.1 = 6000, x.2 = 2000)

28. 10.4.2 Testing for Independence

29. 10.4.2 Testing for Independence • Example 55 : Building Tile Cracks

30. 10.4.2 Testing for Independence

31. Summary problems • Construct a goodness-of-fit test for testing a distributional assumption of a normal distribution by applying the one-way classification method.