Second Semester 2012 Page 1

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The Chi-Square Test

It is the test appropriate for inference of qualitative data (frequency/count). Characteristics of the Chi-Square Distribution:

1. It is not symmetric.

2. The shape of the chi-square distribution depends on the degrees of freedom. 3. The values 2 are nonnegative. That is, the values of 2 0. Basic Assumptions of the 2-test:

1. The data are randomly selected.

2. All expected frequencies are greater than 0.

3. No more than 20% of the expected frequencies are less than 5. Formula of the 2-test:

Where Oi represent the observed counts of category i (i.e. the data) and Ei represents the expected count of the category i, k represents the number of categories and the number of degrees of freedom =k – 1. Types of Chi-Square test:

1. Chi-Square test for Association or independence

2. Chi-Square test for Homogeneity of proportions

3. Chi-square test for Goodness of Fit

Hypotheses for Chi-Square Tests:

Chi-Square test for Association or independence:

Ho: There is no association between the 2 variables. (The two variables are independent) Ha: There is an association between the 2 variables. (The variables are dependent) Chi-Square test for Homogeneity of proportions:

Ho: The proportions are equal ()

Ha: At least one proportion is different.

Chi-square test for Goodness of Fit

Ho: The data follow a specific distribution.

Ha: The data do not follow a specific distribution.

BIOSTATISTICS

Second Semester 2012 Page 2

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Example 1. An obstetrician want to know whether the proportion of children born each day of the week is the same. She randomly selects 500 birth records and obtains the data shown below: Day

Frequency

Sunday

57

Monday

78

Tuesday

74

Wednesday

76

Thursday

71

Friday

81

Saturday

63

Is there reason to believe that the day of the week on which a child is born occurs with equal frequency at the =0.01 level of significance? Solution:

Ho: The...

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