Expected frequencies refer to the anticipated or predicted values of frequencies in a statistical analysis, particularly in the context of hypothesis testing and the chi-square test. They represent the frequencies that would be expected to occur under the null hypothesis, assuming there is no significant difference or association between the variables being studied.
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Expected frequencies are calculated based on the null hypothesis and the total number of observations in the dataset.
In a chi-square test for homogeneity, expected frequencies represent the frequencies that would be expected if the null hypothesis of homogeneity is true.
For a chi-square test of independence, expected frequencies represent the frequencies that would be expected if the null hypothesis of independence is true.
The difference between the observed and expected frequencies is used to calculate the chi-square test statistic, which is then compared to a critical value to determine the statistical significance.
Expected frequencies are essential for evaluating the goodness of fit between the observed data and the expected distribution or model.
Review Questions
Explain the role of expected frequencies in the chi-square test for homogeneity.
In a chi-square test for homogeneity, expected frequencies represent the frequencies that would be expected if the null hypothesis of homogeneity is true. The null hypothesis states that the proportions or distributions of a categorical variable are the same across different populations or groups. The expected frequencies are calculated based on the total number of observations and the proportions under the null hypothesis. The difference between the observed and expected frequencies is then used to calculate the chi-square test statistic, which is compared to a critical value to determine if the null hypothesis can be rejected, indicating a significant difference in the distributions across the groups.
Describe how expected frequencies are used in the chi-square test of independence.
In a chi-square test of independence, expected frequencies represent the frequencies that would be expected if the null hypothesis of independence is true. The null hypothesis states that there is no association or relationship between two categorical variables. The expected frequencies are calculated based on the row and column totals of the contingency table, assuming the variables are independent. The difference between the observed and expected frequencies is then used to calculate the chi-square test statistic, which is compared to a critical value to determine if the null hypothesis can be rejected, indicating a significant association between the variables.
Evaluate the importance of expected frequencies in assessing the goodness of fit between observed data and a theoretical model or distribution.
Expected frequencies are essential for evaluating the goodness of fit between the observed data and the expected distribution or model. The chi-square goodness-of-fit test compares the observed frequencies in each category to the expected frequencies under the null hypothesis, which represents the theoretical or expected distribution. The difference between the observed and expected frequencies is used to calculate the chi-square test statistic, and the resulting p-value indicates the probability of observing the given data if the null hypothesis (the theoretical model or distribution) is true. A statistically significant p-value suggests that the observed data does not fit the expected distribution, and the null hypothesis should be rejected, indicating a poor goodness of fit.
The chi-square test is a statistical hypothesis test used to determine whether there is a significant difference between the observed and expected frequencies in one or more categories.
Goodness of fit refers to the degree to which the observed data fits the expected or theoretical distribution, which is assessed using a chi-square test.