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

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

Step 4: Calculate the Expected Frequencies

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

2

2

2

2

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 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

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 3: Create the data table

Step 4: Calculate the Expected Frequencies

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 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

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

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

Results Side Dish Gender

Step 5: Calculate 2

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 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

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