Goodness-of-Fit Test

The chi-square test is a statistical test commonly used to determine whether there is a significant difference between the expected and observed frequencies in a categorical data set. It is a goodness-of-fit test, which means that it is used to assess how well the observed data fit a particular theoretical distribution.

To perform a Chi-square test, the following steps are typically followed:

1. Specify the null and alternative hypotheses. The null hypothesis is usually that the observed and expected frequencies are the same, while the alternative hypothesis is that they are different.
2. Collect and summarize the data. Calculate the observed frequencies and the expected frequencies for each category.
3. Calculate the Chi-square statistic using the formula: $$\LARGE{\chi^2 = \sum_{i=1}^{n}\frac{(O_i - E_i)^2}{E_i}} $$  Where: • \(O_i\) is the observed frequency in category i • \(E_i\) is the expected frequency in category i • \(n\) is the number of categories
4. Determine the critical value of the Chi-square statistic based on the significance level (alpha) of the test and the degrees of freedom. The degrees of freedom are calculated as the number of categories minus 1 (df = n - 1).
5. Compare the calculated Chi-square statistic to the critical value to determine whether to reject or fail to reject the null hypothesis. If the calculated Chi-square statistic exceeds the critical value, the null hypothesis is rejected, and the alternative hypothesis is accepted.

The assumptions of the Chi-square test are that the data is randomly sampled from a population, that the expected frequencies are greater than 5 in each category, and that all observations are independent.

In addition to testing the goodness-of-fit of a categorical data set, the Chi-square test can also be used to test the independence of two categorical variables. The steps to perform a Chi-square independence test (Contingency Table) are similar to those outlined above, except for using a two-dimensional contingency table to organize the data and calculate the expected frequencies.