3. Chi-square = 2.85
Chi-square crit = 5.99
Achievement is unrelated to whether or not a child attended preschool.

4. 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

5. 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?

6. 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

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

8. 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

9. 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

10. 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?

12. Step 1: State the Hypothesis
H0: The data do fit the model
i.e., the observed data does agree with M&M’s theory
H1: 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

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

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

15. Step 3: Create the data table

16. Step 3: Create the data table
Add the expected proportion of each category

17. Step 4: Calculate the Expected Frequencies

18. Step 4: Calculate the Expected Frequencies
Expected Frequency = (proportion)(N)

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

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

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

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

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

24. 2

25. 2

26. 2

27. 2

28. 2

29. 2
15.52

30. Step 6: Decision Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

31. Step 6: Decision Thus, if 2 > than 2critical
2 = 15.52
2 crit = 11.07
Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

32. Step 7: Put answer into words
H1: The data do not fit the model
M&M’s color “theory” did not significantly (.05) fit the data

34. 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

35. 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)?

37. Step 1: State the Hypothesis
H0: The data do fit the model
i.e., marital status is distributed the same way in the population of female executives as in the general population
H1: The data do not fit the model
i.e., marital status is not distributed the same way in the population of female executives as in the general population

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

39. Step 2: Find 2 critical df = number of categories - 1 df = 5 - 1 = 4
 = .05
2 critical = 9.49

40. Step 3: Create the data table

41. Step 4: Calculate the Expected Frequencies

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

43. 2
19.42

44. Step 6: Decision Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

45. Step 6: Decision Thus, if 2 > than 2critical
2 = 19.42
2 crit = 9.49
Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

46. Step 7: Put answer into words
H1: The data do not fit the model
Marital status is not distributed the same way in the population of female executives as in the general population ( = .05)

48. Practice
Is there a significant ( = .05) relationship between gender and a persons favorite Thanksgiving “side” dish?
Each participant reported his or her most favorite dish.

49. Results
Side Dish
Gender

50. Step 1: State the Hypothesis
H1: There is a relationship between gender and favorite side dish
Gender and favorite side dish are independent of each other

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

52. Results
Side Dish
Gender

53. Step 5: Calculate 2

54. Step 6: Decision Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

55. Step 6: Decision Thus, if 2 > than 2critical
2 = 13.15
2 crit = 5.99
Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

56. Step 7: Put answer into words
H1: There is a relationship between gender and favorite side dish
A person’s favorite Thanksgiving side dish is significantly (.05) related to their gender