1. NIPRL 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. NIPRL 2 10.1 Inferences on a Population Proportion 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

3. NIPRL 3 10.1.1 Confidence Intervals for Population Proportions

4. NIPRL 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. NIPRL 5 10.1.1 Confidence Intervals for Population Proportions

6. NIPRL 6 10.1.2 Hypothesis Tests on a Population Proportion

7. NIPRL 7 10.1.2 Hypothesis Tests on a Population Proportion

8. NIPRL 8 10.1.2 Hypothesis Tests on a Population Proportion

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

10. NIPRL 10 10.1.3 Sample Size Calculations

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

13. NIPRL 13 10.2 Comparing Two Population Proportions

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

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

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

18. NIPRL 18 Summary problems (1) 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. NIPRL 19 10.3 Goodness of Fit Tests for One-Way Contingency Tables 10.3.1 One-Way Classifications

20. NIPRL 20 10.3.1 One-Way Classifications Example 1 : Machine Breakdowns n = 46 machine breakdowns. x 1 = 9 : electrical problems x 2 = 24 : mechanical problems x 3 = 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. NIPRL 21 10.3.1 One-Way Classifications

22. NIPRL 22 10.3.1 One-Way Classifications Example 1 : Machine Breakdowns H 0 : p 1 = 0.2, p 2 = 0.5, p 3 = 0.3 ElectricalMechanical Operator misuse Observed cell freq. x 1 = 9x 2 = 24x 3 = 13n = 46 Expected cell freq. e 1 = 46*0.2 = 9.2 e 2 = 46*0.5 =23.0 e 3 = 46*0.3 =13.8 n = 46

23. NIPRL 23 10.3.1 One-Way Classifications

24. NIPRL 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. NIPRL 25 10.3.2 Testing Distributional Assumptions

26. NIPRL 26 10.4 Testing for Independence in Two-Way Contingency Tables 10.4.1 Two-Way Classifications Level 1Level 2Level jLevel c Level 1x 11 x 12 x 1c x1.x1. Level 2x 21 x 22 x 2c x2.x2. Level ix ij xi.xi. Level rx r1 x r2 x rc xr.xr. x. 1 x. 2 x. j x. c n = x.. A two-way (r x c) contingency table. Second Categorization First Categorization Row marginal frequencies Column marginal frequencies

27. NIPRL 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) Location Tile Condition Building ABuilding B Undamagedx 11 = 5594x 12 = 1917x 1. = 7511 Crackedx 21 = 406x 22 = 83x 2. = 489 x. 1 = 6000x. 2 = 2000n = x.. = 8000

28. NIPRL 28 10.4.2 Testing for Independence

29. NIPRL 29 10.4.2 Testing for Independence Example 55 : Building Tile Cracks Building ABuilding B Undamaged x 11 = 5594 e 11 = 5633.25 x 12 = 1917 e 12 = 1877.75 x 1. = 7511 Cracked x 21 = 406 e 21 = 366.75 x 22 = 83 e 22 = 122.25 x 2. = 489 x. 1 = 6000x. 2 = 2000n = x.. = 8000

30. NIPRL 30 10.4.2 Testing for Independence

31. NIPRL 31 Summary problems 1.Construct a goodness-of-fit test for testing a distributional assumption of a normal distribution by applying the one-way classification method. 2.