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Chi-square = 2.85 Chi-square crit = 5.99 Achievement is unrelated to whether or not a child attended preschool. 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.
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Chi-square = 2.85 • Chi-square crit = 5.99 • Achievement is unrelated to whether or not a child attended preschool.
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: • You have a theory or hypothesis that the frequencies should occur in a particular manner?
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
Example • Based on genetic theory you hypothesize that in the population: • 45% have brown eyes • 35% have blue eyes • 20% have another eye color
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
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
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?
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
Step 2: Find 2 critical • df = number of categories - 1
Step 2: Find 2 critical • df = number of categories - 1 • df = 6 - 1 = 5 • = .05 • 2 critical = 11.07
Step 3: Create the data table Add the expected proportion of each category
Step 4: Calculate the Expected Frequencies Expected Frequency = (proportion)(N)
Step 4: Calculate the Expected Frequencies Expected Frequency = (.30)(2081) = 624.30
Step 4: Calculate the Expected Frequencies Expected Frequency = (.20)(2081) = 416.20
Step 4: Calculate the Expected Frequencies Expected Frequency = (.20)(2081) = 416.20
Step 4: Calculate the Expected Frequencies Expected Frequency = (.10)(2081) = 208.10
Step 5: Calculate 2 O = observed frequency E = expected frequency
2 15.52
Step 6: Decision • Thus, if 2 > than 2critical • Reject H0, and accept H1 • If 2 < or = to 2critical • Fail to reject H0
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
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
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
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)?
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
Step 2: Find 2 critical • df = number of categories - 1
Step 2: Find 2 critical • df = number of categories - 1 • df = 5 - 1 = 4 • = .05 • 2 critical = 9.49
Step 5: Calculate 2 O = observed frequency E = expected frequency
2 19.42
Step 6: Decision • Thus, if 2 > than 2critical • Reject H0, and accept H1 • If 2 < or = to 2critical • Fail to reject H0
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
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)
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.
Results Side Dish Gender
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