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 2 as a test for goodness of fit

 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.  2 as a test for goodness of fit. But what if:

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 2 as a test for goodness of fit

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  1. 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

  2. 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?

  3. 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

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

  5. 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

  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 • 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?

  8. 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

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

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

  11. Step 3: Create the data table

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

  13. Step 4: Calculate the Expected Frequencies

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

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

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

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

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

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

  20. 2

  21. 2

  22. 2

  23. 2

  24. 2

  25. 2 15.52

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

  27. Step 6: Decision 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

  28. 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

  29. 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

  30. 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)?

  31. 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

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

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

  34. Step 3: Create the data table

  35. Step 4: Calculate the Expected Frequencies

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

  37. 2 19.42

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

  39. Step 6: Decision 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

  40. 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)

  41. 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.

  42. Results Side Dish Gender

  43. 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

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

  45. Results Side Dish Gender

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