Expected frequencies refer to the anticipated or predicted frequencies of observations in each category or cell of a contingency table, based on the assumption that the null hypothesis is true. They are a crucial component in the calculation and interpretation of the chi-square statistic, which is used to assess the goodness of fit between observed and expected frequencies.
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Expected frequencies are calculated based on the assumption that the null hypothesis is true, which means that there is no significant difference between the observed and expected frequencies.
In a chi-square test for homogeneity, the expected frequencies are calculated based on the assumption that the distribution of observations across the categories is the same for all populations or groups.
For a chi-square test of independence, the expected frequencies are calculated based on the assumption that the two categorical variables are independent, meaning that the distribution of one variable is not influenced by the other.
The formula to calculate expected frequencies in a contingency table is: (row total × column total) / grand total.
Comparing the observed and expected frequencies is the key to interpreting the chi-square statistic and determining if the null hypothesis should be rejected or not.
Review Questions
Explain the role of expected frequencies in the context of the chi-square distribution and its associated tests.
Expected frequencies are a crucial component in the calculation and interpretation of the chi-square statistic, which is used to assess the goodness of fit between observed and expected frequencies. In the context of the chi-square distribution, expected frequencies represent the anticipated or predicted frequencies of observations in each category or cell of a contingency table, based on the assumption that the null hypothesis is true. These expected frequencies are then compared to the observed frequencies to determine if there is a significant difference, which is the basis for the chi-square test for homogeneity and the chi-square test of independence.
Describe how expected frequencies are calculated for a chi-square test of independence and explain their significance in the interpretation of the test results.
For a chi-square test of independence, the expected frequencies are calculated based on the assumption that the two categorical variables are independent, meaning that the distribution of one variable is not influenced by the other. The formula to calculate expected frequencies in a contingency table is: (row total × column total) / grand total. Comparing the observed and expected frequencies is the key to interpreting the chi-square statistic and determining if the null hypothesis, which states that the two variables are independent, should be rejected or not. If the observed frequencies differ significantly from the expected frequencies, it suggests that the two variables are not independent, and the null hypothesis would be rejected.
Analyze the importance of expected frequencies in the context of a chi-square test for homogeneity and explain how they are used to draw conclusions about the homogeneity of populations or groups.
In a chi-square test for homogeneity, the expected frequencies are calculated based on the assumption that the distribution of observations across the categories is the same for all populations or groups. By comparing the observed and expected frequencies, the chi-square test for homogeneity allows researchers to determine if there is a significant difference in the distribution of observations across the categories for the different populations or groups. If the observed frequencies differ significantly from the expected frequencies, it suggests that the populations or groups are not homogeneous, and the null hypothesis, which states that the distributions are the same, would be rejected. The analysis of expected frequencies is crucial in this context, as it provides the basis for evaluating the homogeneity of the populations or groups under investigation.
A tabular arrangement of data where the frequencies or counts of observations are displayed in rows and columns, typically used to analyze the relationship between two categorical variables.
A statistical test used to determine if there is a significant difference between the observed frequencies and the expected frequencies in one or more categories.
Null Hypothesis: A statement about the population parameter that the researcher believes is true and wants to test, typically stating that there is no significant difference or relationship between variables.