2. Seven Steps for Doing  2 1) State the hypothesis 2) Create data table 3) Find  2 critical 4) Calculate the expected frequencies 5) Calculate  2 6) Decision 7) Put answer into words

3. Example With whom do you find it easiest to make friends? Subjects were either male and female. Possible responses were: “opposite sex”, “same sex”, or “no difference” Is there a significant (.05) relationship between the gender of the subject and their response?

4. Results

5. Step 1: State the Hypothesis H 1: There is a relationship between gender and with whom a person finds it easiest to make friends H 0 :Gender and with whom a person finds it easiest to make friends are independent of each other

6. Step 2: Create the Data Table Add “total” columns and rows

7. Step 3: Find  2 critical df = (R - 1)(C - 1) df = (2 - 1)(3 - 1) = 2  =.05  2 critical = 5.99

8. Step 4: Calculate the Expected Frequencies

9. Step 5: Calculate  2 O = observed frequency E = expected frequency

10. 22

11. 22

12. 22

13. 22

14. 22 8.5

15. Step 6: Decision Thus, if  2 > than  2 critical –Reject H 0, and accept H 1 If  2 < or = to  2 critical –Fail to reject H 0

16. Step 6: Decision Thus, if  2 > than  2 criticalThus, if  2 > than  2 critical –Reject H 0, and accept H 1 If  2 < or = to  2 critical –Fail to reject H 0  2 = 8.5  2 crit = 5.99

17. Step 7: Put answer into words H 1: There is a relationship between gender and with whom a person finds it easiest to make friends A persons gender is significantly (.05) related with whom it is easiest to make friends.

18. Practice Is there a significant (  =.01) relationship between opinions about the death penalty and opinions about the legalization of marijuana? 933 Subjects responded yes or no to: “Do you favor the death penalty for persons convicted of murder?” “Do you think the use of marijuana should be made legal?”

19. Results Marijuana ? Death Penalty ?

20. Step 1: State the Hypothesis H 1: There is a relationship between opinions about the death penalty and the legalization of marijuana H 0 :Opinions about the death penalty and the legalization of marijuana are independent of each other

21. Step 2: Create the Data Table Marijuana ? Death Penalty ?

22. Step 3: Find  2 critical df = (R - 1)(C - 1) df = (2 - 1)(2 - 1) = 1  =.01  2 critical = 6.64

23. Step 4: Calculate the Expected Frequencies Marijuana ? Death Penalty ?

24. Step 5: Calculate  2

25. Step 6: Decision Thus, if  2 > than  2 critical –Reject H 0, and accept H 1 If  2 < or = to  2 critical –Fail to reject H 0

26. Step 6: Decision Thus, if  2 > than  2 critical –Reject H 0, and accept H 1 If  2 < or = to  2 criticalIf  2 < or = to  2 critical –Fail to reject H 0  2 = 3.91  2 crit = 6.64

27. Step 7: Put answer into words H 0 :Opinions about the death penalty and the legalization of marijuana are independent of each other A persons opinion about the death penalty is not significantly (.01) related with their opinion about the legalization of marijuana

29.  2 as a test for goodness of fit So far.... The expected frequencies that we have calculated come from the data They test rather or not two variables are related

30.  2 as a test for goodness of fit But what if: You have a theory or hypothesis that the frequencies should occur in a particular manner?

31. Example M&Ms claim that of their candies: 30% are brown 20% are red 20% are yellow 10% are blue 10% are orange 10% are green

32. Example Based on genetic theory you hypothesize that in the population: 45% have brown eyes 35% have blue eyes 20% have another eye color

33. To solve you use the same basic steps as before (slightly different order) 1) State the hypothesis 2) Find  2 critical 3) Create data table 4) Calculate the expected frequencies 5) Calculate  2 6) Decision 7) Put answer into words

34. Example M&Ms claim that of their candies: 30% are brown 20% are red 20% are yellow 10% are blue 10% are orange 10% are green

35. Example Four 1-pound bags of plain M&Ms are purchased Each M&Ms is counted and categorized according to its color Question: Is M&Ms “theory” about the colors of M&Ms correct?

37. Step 1: State the Hypothesis H 0 : The data do fit the model –i.e., the observed data does agree with M&M’s theory H 1: The data do not fit the model –i.e., the observed data does not agree with M&M’s theory –NOTE: These are backwards from what you have done before

38. Step 2: Find  2 critical df = number of categories - 1

39. Step 2: Find  2 critical df = number of categories - 1 df = 6 - 1 = 5  =.05  2 critical = 11.07

40. Step 3: Create the data table

41. Add the expected proportion of each category

42. Step 4: Calculate the Expected Frequencies

43. Expected Frequency = (proportion)(N)

44. Step 4: Calculate the Expected Frequencies Expected Frequency = (.30)(2081) = 624.30

45. Step 4: Calculate the Expected Frequencies Expected Frequency = (.20)(2081) = 416.20

46. Step 4: Calculate the Expected Frequencies Expected Frequency = (.20)(2081) = 416.20

47. Step 4: Calculate the Expected Frequencies Expected Frequency = (.10)(2081) = 208.10

48. Step 5: Calculate  2 O = observed frequency E = expected frequency

49. 22

50. 22

51. 22

52. 22

53. 22

54. 22 15.52

55. Step 6: Decision Thus, if  2 > than  2 critical –Reject H 0, and accept H 1 If  2 < or = to  2 critical –Fail to reject H 0

56. Step 6: Decision Thus, if  2 > than  2 criticalThus, if  2 > than  2 critical –Reject H 0, and accept H 1 If  2 < or = to  2 critical –Fail to reject H 0  2 = 15.52  2 crit = 11.07

57. Step 7: Put answer into words H 1: The data do not fit the model M&M’s color “theory” did not significantly (.05) fit the data

58. Practice Among women in the general population under the age of 40: 60% are married 23% are single 4% are separated 12% are divorced 1% are widowed

59. Practice You sample 200 female executives under the age of 40 Question: Is marital status distributed the same way in the population of female executives as in the general population (  =.05)?