Two Way Tables Chi Square Test

1. Two Way Tables Chi Square Test

2. 2 categorical variables

3. Data collected in a table with counts.

4. Can test whether there is association between the 2 variables.

5. Market researchers think that background music can influence the mood and purchasing behavior of customers. One study compared 3 treatments: no music, French accordion music, and Italian string music. Under each condition, the researchers recorded the number of bottles of French, Italian, and other wine purchased.

6. Market researchers think that background music can influence the mood and purchasing behavior of customers. One study compared 3 treatments: no music, French accordion music, and Italian string music. Under each condition, the researchers recorded the number of bottles of French, Italian, and other wine purchased.

7. Music and wine data in counts 243 bottles total

8. Marginal Distribution

9. What percent of wine bought was Italian?

10. What percent of wine bought was Italian?

11. What percent of wine was purchased from a store with no music playing

12. What percent of wine was purchased from a store with no music playing

13. Marginal Distributions Only ask about 1 variable Use values in “margins”

14. What percent of all wine bought was Italian with French music playing in the store?

15. What percent of all wine bought was Italian with French music playing in the store?

16. Joint distribution Ask about both variables Use values inside table and total

17. What percent of all wine bought was Italian with French music playing in the store?

18. Of the Italian wine purchased, what percent was from a store playing French music?

19. What percent of the wine purchased from French music-playing stores was French?

20. What percent of the wine purchased from French music-playing stores was French?

21. Conditional distribution Asks about both variables Given one variable Use values inside table and margins

22. Hypothesis Testing

23. H0 Null Hypothesis • Ha Alternative Hypothesis

25. H0 Half of the deck is black • Ha Half of the deck is notblack

26. Possible Outcomes • Reject H0 • Fail to reject H0 • Never • Prove • Accept

27. Standard: α = 0.05

28. Chi-square hypothesis tests H0: There is NO association between music and wine in the population. Ha: There is an association between music and wine in the population.

29. Observed Data vs. Expected Data

30. Have: Observed Counts Want: Expected Counts if Music and Wine were Independent

33. Expected counts

34. Expected Counts

35. Expected Counts

36. Expected Counts vs Observed Counts

37. Chi-square test statistic

38. Partial Chi-Square values

39. Chi-Square Value χ2 = 0.52 + 2.31 + 0.52 + 0.0084 + 7.70 + 6.44 + 0.39 + 0.00029 + 0.43 χ2 = 18.31

40. Chi-Square Table

41. Degrees of Freedom DF = (r-1)(c-1) r = 3 c = 3 DF= (3-1)*(3-1)= 2x2 =4

42. Chi-Square Table Test statistic = 9.49

43. Test Statistic If table value < χ2 then reject H0 If table value > χ2 then fail to reject H0

44. Chi-square hypothesis tests X2 = 18.31 > 9.49 Reject H0. We have evidence there is an association between music and wine in the population.

45. Is table value < χ2 Yes No reject H0 Fail to reject H0 We have evidence Ha We do not have evidence Ha

46. When is chi-square test appropriate?

47. Tables larger than 2x2: • the smallest expected count ≥ 1 • <20% of cells have expected counts < 5. 2x2 tables: • all 4 expected cell counts ≥ 5

48. Psychological and social factors can influence the survival of patients with serious diseases. One study examined the relationship between survival of patients with coronary heart disease and pet ownership. Each of 92 patients was classified as having a pet or not and by whether they survived for one year. The researchers suspect that having a pet might be connected to the patient status.

49. Pet ownership and heart disease data (counts)

50. Assuming a patient is still alive, what is the probability he owns a pet?